# Evaluation of Loading Bay Restrictions for the Installation of Offshore Wind Farms Using a Combination of Mixed-Integer Linear Programming and Model Predictive Control

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## Abstract

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## Featured Application

**This article demonstrates a combination of Mixed-Integer Linear Programming with methods usually applied for short-term control, namely the Model Predictive Control scheme, to achieve decision support for the scheduling of installation activities for offshore wind farms. The general approach applies to several areas of application, where time-dependent uncertainties complicate mid- to long-term planning.**

## Abstract

## 1. Introduction

#### 1.1. Installation Process for Offshore Wind Farms

#### 1.2. Planning Approaches for the Installation of Offshore Wind Farms

#### 1.3. General Classes of Scheduling Problems

**Basic JSSP:**- The Basic JSSP constitutes the simplest class of job-shop scheduling problems. The Basic JSSP assumes that each operation can only be conducted by a single machine, thus resulting in a sequencing problem.
**Flexible JSSP:**- The Flexible JSSP extends the basic formulation by allowing a specified set of alternative machines to perform operations. This flexibility increases the problem’s complexity: In addition to the already present sequencing problem (order of operations per machine), the Flexible JSSP also has to cope with an additional routing problem, deciding which machine to choose for which operation [26].
**Multi-Resource JSSP:**- The Multi-Resource JSSP extends the formulation of the Flexible JSSP by another dimension. In addition to the machine, each operation additionally requires other limited resources. These could, for example, be workers, tools, or dies.
**Multi-Plant Multi-Resource JSSP:**- This type of model extends the Multi-Resource JSSP by regarding multiple plants or production facilities. Therefore, it is also necessary to include the transport of orders between those facilities within the scheduling.

**Time-Indexed formulation:**- These types of models index the generated schedules over discrete time instances, e.g., minutes or hours. Therefore, the schedules denote for each machine and each time instance, which operation of which job is currently processed. The drawback of these formulations is the high number of decision variables, which is at least quadratic over the number of machines and time instances. According to Demir and İs̨leyen [31], a significant drawback of these formulations is that the planning horizon must be estimated in advance so that the optimizer can schedule all operations.
**Sequence-Position formulations:**- This type of formulation uses discrete slots instead of time instances, which refer to semantic concepts, e.g., to machining operations. Therefore, it represents each machine by a sequence of slots, assigning operations to them. This type of formulation closely resembles a time-indexed formulation but reduces the number of decision variables by summarizing these into discrete slots of equal length in time. For example, if a machine performs three operations of lengths 2 h, 2.5 h, and 3 h, a formulation of this class would most probably use slots of 3 h length. As a trade-off, this abstraction reduces the resolution of the model (for the example above by factor 3 compared to an hourly time-indexed model) but may lead to weaker solutions, e.g., if operations finish in less time than assigned to a sequencing slot. These formulations require an a-priori estimation of the number of slots to allow the optimizer to schedule all operations. Consequently, this class summarizes all indexed formulations, which do not focus on single time instances but higher-level concepts. This reduces the complexity and simplifies the formulation of semantic constraints, e.g., like not assigning a morning shift directly after a night shift.
**Precedence-based formulations:**- In contrast to the formulations above, precedence-based models use a continuous representation of time. Therefore, the generated solutions are indexed over the number of operations to be performed, usually resulting in fewer decision variables. The schedules denote start or end times, and the assigned machine for each operation.

#### 1.4. Dynamic Scheduling Approaches

**Reactive and Continuous Approaches:**- Reactive approaches generally do not incorporate uncertainties into the initial planning but rely on so-called recourse actions in cases of plan failures, i.e., a rescheduling. Therefore, some authors distinguish between approaches that only react to disturbances and between approaches that perform a continuous rescheduling. The first class of reactive approaches mostly focuses on a reduction of rescheduling times, e.g., by only selecting affected operations or by applying dispatching rules. The second class often uses receding horizon techniques (e.g., variations of the Model Predictive Control) and combines these with state-space descriptions to avoid expensive optimizations.
**Preventive (Robust and Stochastic) Approaches:**- Preventive approaches try to provide schedules that can cope with the dynamics of the system in general. At this, the plans are generally designed to either focus on the worst possible case (robust plans) or to provide alternatives for possible scenarios using stochastic programming techniques. In general, robust plans tend to be too conservative as they usually assume the worst case. In contrast, stochastic plans come at very high computational costs, as the solver needs to compute alternative plans for different probable cases in advance.
**Predictive-Reactive Approaches:**- This class of approaches tries to assume the extent of disturbances during the initial planning stage and employ reactive recourse actions if additional disturbances occur. Dias and Ierapetritou provide the example of fuzzy-based approaches, which translate uncertainties into a fuzzy description of the resulting variables. These descriptions are then used to derive schedules [34]. This type of reformulation enables a reduction of the models’ complexity as done for robust scheduling approaches, but does not limit the results to the worst case. Ouelhadj and Petrovic provide an overview of approaches, which use a bi-objective formulation, minimizing the makespan, and the impact of disturbances on the overall plan [33].

#### 1.5. Summary of the Research Aim and the Assumptions

- Each scenario only considers the actual installation part of the given supply chain. Consequently, this article assumes that components are always available at the base port.
- As in practice, this article only considers the so-called season from April to October, as it provides suitable weather conditions to perform offshore operations.
- If a scenario applies several installation vessels, all installation vessels are considered to be equal.
- The number of installation vessels is fixed for each scenario.
- The number of loading bays is fixed for each scenario.
- This article assumes that vessels host several crews to assure the availability of personnel seven days a week in three shifts (24 h).
- This article does not include staff planning and thus neglects differences in labor costs for night shifts or weekend shifts.

#### 1.6. Notation

## 2. Materials and Methods

#### 2.1. Overall Methodology

#### 2.2. First Step: Representation of the State and Simulation of Weather Forecasts

#### 2.3. Second Step: Duration Estimation by Sliding Windows and Markov-Based Approaches

#### 2.3.1. Discretization by Sliding Windows

#### 2.3.2. Discretization by Markov Chains

#### 2.4. Third Step: MILP Formulation for the Incremental Scheduling of Offshore Operations

#### 2.5. Fourth Step: Control Extraction and Plan Synchronization

#### 2.6. Fifth Step: Apply Short-Term Plan and Simulation of the Real-World System

## 3. Results

#### 3.1. Comparison of Approaches for the Estimation of Operation Durations

#### 3.2. Evaluation of the Influence of Limited Resources Without Forecast Uncertainties

#### 3.3. Evaluation of the Influence of the Number of Planning Periods on the Incremental Approach

#### 3.4. Evaluation of Restricted Optimization Times

## 4. Discussion

#### 4.1. Conclusions Regarding Loading Bay Restrictions

#### 4.2. Conclusions Regarding the General Approach

#### 4.3. Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The conventional installation process for top structures of offshore wind farms (following [16]).

**Figure 2.**Schematic depiction of the Model Predictive Control scheme (following [9]).

**Figure 5.**Example transition matrix ${P}^{n}$ with three operations of duration 2, 3 and 1, which results in 7 states: The first operation with one helper state, the second operation with two helper states, the third operation with no helper state and the final state.

**Figure 6.**Estimation for the Markov-Chain and Sliding-Window approaches compared to the real duration.

**Figure 7.**Absolute number of finished turbines per vessel (

**a**) and normalized bay use (

**b**) for different ratios of bays and vessels.

Operation | Base Duration (h) | Max. Wind (m/s) | Max. Wave (m) |
---|---|---|---|

Traveling | 4 | 21 | 2.5 |

Positioning | 1 | 14 | 2 |

Jack-up/-down | 2 | 14 | 1.8 |

Load Tower | 3 | 12 ^{⋆} | 5 ^{⋆} |

Load Nacelle | 2 | 12 ^{⋆} | 5 ^{⋆} |

Load Blade | 2 | 10 ^{⋆} | 5 ^{⋆} |

Load Hub | 1 | 12 ^{⋆} | 5 ^{⋆} |

Install Tower | 3 | 12 | 2.5 ^{†} |

Install Nacelle | 3 | 12 | 2.5 ^{†} |

Install Blade | 2 | 10 | 2.5 ^{†} |

Install Hub | 2 | 12 | 2.5 ^{†} |

Indices: | ||
---|---|---|

k, v, o | Indices | Indices for Time Instances (k), Vessels (v), and Operations (o) with $\mathit{k},\text{}\mathit{v}\in {\mathbb{N}}^{+}$ and $\mathit{o}\in \{1,2,3,4\}$. |

Parameters determined a-priori: | ||

N | Integer | Length of the prediction horizon as $N=P*T$ |

T | Integer | Sampling step size |

P | Integer | Number of planning periods considered by the MILP |

V | Integer | Number of installation vessels |

$CAP$ | Integer | Maximum loading capacity for vessels |

$BAYS$ | Integer | Number of loading bays |

${D}_{o,k}$ | Integer | Estimated duration of operation o at time instance k |

${START}_{v}$ | Integer | Index of time instance when a vessel v is available |

${INUSE}_{k}^{bays}$ | Integer | Denotes the number of busy loading bays for each time instance k |

${C}_{v}^{o}$ | Float | Cost for being offshore per hour by vessel v |

${C}_{v}^{m}$ | Float | Cost for moving between port and site by vessel v |

${C}_{v}^{p}$ | Float | Cost for port operations per hour by vessel v |

${B}^{owt}$ | Float | Benefit for installing a turbine |

${B}^{early}$ | Float | Benefit for finishing an operation early |

Decision variables (start of operation events): | ||

${Y}_{v,k}^{owt}$ | Binary | Denotes if a vessel v starts an installation operation at time instance k |

${Y}_{v,k}^{load}$ | Binary | Denotes if a vessel v starts a loading operation at time instance k |

${Y}_{v,k}^{toPort}$ | Binary | Denotes if a vessel v starts to move to the port at time instance k |

${Y}_{v,k}^{toSite}$ | Binary | Denotes if a vessel v starts to move to the construction site at time instance k |

Support variables (ongoing operations and states): | ||

${X}_{v,k}^{cap}$ | Integer | Amount of currently loaded components |

${Y}_{v,k}^{loc}$ | Binary | Denotes if vessel v is in port (0) or offshore (1) |

${Y}_{v,k}^{busy}$ | Binary | Denotes if vessel v is currently performing an operation (1) or is idle (0) |

${Y}_{v,k}^{atBay}$ | Binary | Denotes if vessel v is currently using a loading bay (1) or not (0) |

${X}_{v,k}^{fOwt}$ | Integer | Denotes at which time instance an installation operation conducted by vessel v and started at instance k will finish |

${X}_{v,k}^{fOp}$ | Integer | Denotes at which time instance a non-installation operation conducted by vessel v and started at instance k will finish |

Operation | Base Duration (h) | Max. Wind (m/s) | Max. Wave (m) |
---|---|---|---|

Travel to Base Port | 4 | 21 | 2.5 |

Travel to Construction Site | 4 | 21 | 2.5 |

Load OWT | 12 | - | - |

Install OWT | 19 | Depending on progress |

2000 | Apr. | May | Jun. | Jul. | Aug. | Sep. | Oct. | Mean |
---|---|---|---|---|---|---|---|---|

Act. dur. Min | 19 | 19 | 19 | 19 | 19 | 19 | 19 | 19.00 |

Act. dur. Mean | 23 | 23 | 28 | 30 | 20 | 26 | 31 | 25.86 |

Act. dur. Max | 69 | 81 | 85 | 101 | 41 | 74 | 130 | 83.00 |

Standard Deviation: Sliding Window in hours | ||||||||

1 Week | 0.22 | 1.77 | 1.41 | 2.21 | 0.00 | 0.57 | 0.86 | 1.01 |

2 Weeks | 2.16 | 1.25 | 3.29 | 3.67 | 0.00 | 3.11 | 2.10 | 2.23 |

3 Weeks | 2.10 | 3.07 | 20.81 | 20.15 | 0.55 | 10.18 | 15.93 | 10.40 |

4 Weeks | 35.82 | 77.29 | 31.79 | 17.45 | 1.79 | 94.42 | 335.87 | 84.92 |

Standard Deviation: Markov in hours | ||||||||

1 Week | 0.26 | 0.67 | 2.56 | 0.58 | 0.00 | 1.47 | 0.77 | 0.90 |

2 Weeks | 1.07 | 0.53 | 1.99 | 2.28 | 0.08 | 2.86 | 6.30 | 2.16 |

3 Weeks | 1.14 | 0.95 | 4.06 | 12.73 | 0.29 | 5.02 | 5.59 | 4.25 |

4 Weeks | 1.08 | 3.07 | 6.78 | 11.02 | 1.26 | 5.26 | 77.85 | 15.19 |

Scenario | June | August | October | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Bays | Vessels | Bays/Vessels | OWTs Finished | Per Vessel | Bay Use | OWTs Finished | Per Vessel | Bay Use | OWTs Finished | Per Vessel | Bay Use |

1 | 1 | 1.00 | 14 | 14.00 | 0.31 | 14 | 14.00 | 0.31 | 12 | 12.00 | 0.28 |

2 | 2 | 1.00 | 28 | 14.00 | 0.31 | 28 | 14.00 | 0.31 | 24 | 12.00 | 0.28 |

3 | 3 | 1.00 | 42 | 14.00 | 0.31 | 42 | 14.00 | 0.31 | 36 | 12.00 | 0.28 |

4 | 4 | 1.00 | 56 | 14.00 | 0.31 | 56 | 14.00 | 0.31 | 48 | 12.00 | 0.28 |

3 | 4 | 0.75 | 51 | 12.75 | 0.37 | 56 | 14.00 | 0.41 | 41 | 10.25 | 0.32 |

1 | 2 | 0.50 | 24 | 12.00 | 0.52 | 28 | 14.00 | 0.61 | 20 | 10.00 | 0.46 |

2 | 4 | 0.50 | 48 | 12.00 | 0.52 | 56 | 14.00 | 0.61 | 40 | 10.00 | 0.46 |

1 | 3 | 0.33 | 33 | 11.00 | 0.72 | 35 | 11.67 | 0.76 | 26 | 8.67 | 0.60 |

1 | 4 | 0.25 | 33 | 8.25 | 0.72 | 35 | 8.75 | 0.76 | 26 | 6.50 | 0.60 |

1 | 5 | 0.20 | 33 | 6.60 | 0.72 | 35 | 7.00 | 0.76 | 26 | 5.20 | 0.60 |

1 | 10 | 0.10 | 33 | 3.30 | 0.72 | 35 | 3.50 | 0.76 | 26 | 2.60 | 0.60 |

OWT | Plans | OWT | Plans | OWT | Plans | OWT | Plans | OWT | Plans | OWT | Plans | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

T/P | 84/1 | 84/2 | 84/3 | 168/1 | 168/2 | 168/3 | ||||||

Apr. | 16 | 10 | 17 | 8 | 19 | 8 | 16 | 5 | 17 | 4 | 18 | 4 |

May | 17 | 10 | 18 | 8 | 18 | 8 | 17 | 5 | 17 | 5 | 18 | 4 |

Jun. | 13 | 11 | 17 | 8 | 17 | 9 | 16 | 5 | 16 | 5 | 16 | 5 |

Jul. | 14 | 10 | 18 | 8 | 17 | 8 | 16 | 6 | 17 | 4 | 17 | 5 |

Aug. | 18 | 10 | 19 | 8 | 18 | 8 | 18 | 5 | 18 | 4 | 18 | 4 |

Sep. | 12 | 10 | 18 | 8 | 18 | 8 | 18 | 5 | 17 | 5 | 16 | 5 |

Oct. | 12 | 9 | 13 | 9 | 14 | 9 | 13 | 6 | 13 | 6 | 13 | 6 |

Sum | 102 | 70 | 120 | 57 | 121 | 58 | 114 | 37 | 115 | 33 | 116 | 33 |

April | June | August | ||||
---|---|---|---|---|---|---|

Duration [h] | O/T [h] | Duration [h] | O/T [h] | Duration [h] | O/T [h] | |

2 | 1776 | 22.36 | 1807 | 21.90 | 1814 | 21.44 |

5 | 1786 | 22.10 | 1774 | 21.90 | 1772 | 21.30 |

15 | 1786 | 22.10 | 1774 | 21.90 | 1772 | 21.30 |

30 | 1786 | 22.10 | 1774 | 21.90 | 1772 | 21.30 |

60 | 1786 | 22.10 | 1774 | 21.90 | 1772 | 21.30 |

Mean | 1784.00 | 22.15 | 1780.60 | 21.90 | 1780.40 | 21.33 |

Std.Dev | 4.00 | 0.10 | 13.20 | 0.00 | 16.80 | 0.06 |

Mean (5+) | 1786.00 | 22.10 | 1774.00 | 21.90 | 1772.00 | 21.30 |

Std.Dev (5+) | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

April | June | August | ||||
---|---|---|---|---|---|---|

Duration [h] | O/T [h] | Duration [h] | O/T [h] | Duration [h] | O/T [h] | |

2 | 993 | 21.54 | 1097 | 22.12 | 1003 | 21.64 |

5 | 902 | 21.48 | 963 | 21.84 | 932 | 21.30 |

15 | 972 | 21.26 | 963 | 21.64 | 912 | 21.24 |

30 | 907 | 21.24 | 963 | 21.64 | 907 | 21.12 |

60 | 888 | 21.70 | 963 | 22.04 | 885 | 21.26 |

Mean | 932.40 | 21.44 | 989.80 | 21.86 | 927.80 | 21.31 |

Std.Dev | 41.91 | 0.17 | 53.60 | 0.20 | 40.46 | 0.17 |

Mean (5+) | 917.25 | 21.42 | 963.00 | 21.79 | 909.00 | 21.23 |

Std.Dev (5+) | 37.38 | 0.22 | 0.00 | 0.19 | 19.30 | 0.08 |

April | June | August | ||||
---|---|---|---|---|---|---|

Duration [h] | O/T [h] | Duration [h] | O/T [h] | Duration [h] | O/T [h] | |

2 | 731 | 21.76 | 769 | 22.52 | 729 | 22.28 |

5 | 717 | 21.90 | 760 | 22.58 | 696 | 21.78 |

15 | 735 | 21.10 | 766 | 21.48 | 689 | 21.54 |

30 | 717 | 21.80 | 767 | 22.60 | 707 | 22.00 |

60 | 697 | 21.50 | 736 | 22.02 | 689 | 21.16 |

Mean | 719.40 | 21.61 | 759.60 | 22.24 | 702.00 | 21.75 |

Std.Dev | 13.35 | 0.29 | 12.18 | 0.44 | 15.02 | 0.38 |

Mean (5+) | 716.50 | 21.58 | 757.25 | 22.17 | 695.25 | 21.62 |

Std.Dev (5+) | 15.52 | 0.36 | 14.50 | 0.53 | 8.50 | 0.36 |

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## Share and Cite

**MDPI and ACS Style**

Rippel, D.; Jathe, N.; Lütjen, M.; Freitag, M. Evaluation of Loading Bay Restrictions for the Installation of Offshore Wind Farms Using a Combination of Mixed-Integer Linear Programming and Model Predictive Control. *Appl. Sci.* **2019**, *9*, 5030.
https://doi.org/10.3390/app9235030

**AMA Style**

Rippel D, Jathe N, Lütjen M, Freitag M. Evaluation of Loading Bay Restrictions for the Installation of Offshore Wind Farms Using a Combination of Mixed-Integer Linear Programming and Model Predictive Control. *Applied Sciences*. 2019; 9(23):5030.
https://doi.org/10.3390/app9235030

**Chicago/Turabian Style**

Rippel, Daniel, Nicolas Jathe, Michael Lütjen, and Michael Freitag. 2019. "Evaluation of Loading Bay Restrictions for the Installation of Offshore Wind Farms Using a Combination of Mixed-Integer Linear Programming and Model Predictive Control" *Applied Sciences* 9, no. 23: 5030.
https://doi.org/10.3390/app9235030