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University of Groningen
Faculty of Economics and Business
“Offshore Wind Turbine Installation
Under Uncertainty”
Master Thesis, Msc, Supply Chain Management
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Abstract
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Table of Content
1. Introduction ... 4
2. Problem Description... 5
2.1 OWT installation process ... 5
2.2 OWT installation vessel ... 6
2.3 Uncertain weather factor ... 8
3. Literature review ... 8
4. The Wind Turbine Installation Model ... 9
4.1 Assumptions ... 10
4.2 Deterministic model ... 10
5. Solution methodology ... 12
5.1 Benders decomposition algorithm ... 13
5.2 Deterministic equivalent of stochastic model ... 16
6. Computational experiments ... 16
6.1 Computational results on small-scale problems ... 16
6.2 Computational results on large-scale problems... 18
7. Conclusion and further research ... 20
References ... 22
Appendix I ... 24
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1. Introduction
Driven by world’s ever-increasing concern for sustainability, renewable energy is catching great attention over the last decades. Limited fossil energy leads people to explore all the possible renewable energy such as wind and solar. Wind power is one of the greenest ways to produce vast amounts of electricity with low CO2 emissions (Diamond, 2010). Wind turbines are the most prevalent tools to convert wind energy to electricity. Offshore wind turbine (OWT) outperforms onshore ones on three aspects (Hibbert, 2011). Firstly the noise pollution can be avoided from being too close to citizens. Second, the limitation of machine size does not need to be taken into account because of the larger space at sea. Third, OWT can generate more electricity than the onshore one and the disturbance for wind caused by architectures nearby can also be decreased. The potential development of OWT is promising, but the problems are also critical. The investment cost for the installation and maintenance of OWT is 50 % higher than the onshore one (Montilla-DJesus et al., 2010). In a lot of countries that plan for OWT, the projects come to a standstill because of the extremely high investment. Offshore foundations, turbines and installations are the most costly parts in the investments (Bilgili et al., 2011). Since the cost for the components and machines cannot be decreased under current material and technology situation, an effective and efficient way to lower cost is reducing installation time. With reduced time, the labor cost and rental cost for installation vessel can be lowered. Labor cost is directly in proportion to the installation time, and the rental cost which is always costly can also drop with less rental time (Dijk, 2013). Scheduling is a helpful tool to plan the process in order to reduce the total service time. An improved scheduling method might reduce the installation time from supply chain perspective. This study will provide solutions to schedule the OWT installation so as to minimize the service duration.
Most studies on OWT in the supply chain area concern production, installation and maintenance. The studies on installation scheduling of OWT are quite limited. Scholz-Reiter et al. (2011) firstly proposed a mathematical model to get optimal schedule for installation. The model considers the weather condition and operational time that includes travelling time of installation vessels and building time for the turbines. However, the values of traveling time and building time are chosen to be deterministic which will not change under different weather conditions. In the practical situation, the operational time is affected by weather variation. Stochastic modeling studies the problem in different scenarios under uncertainty and provides a robust solution which can deal with various scenarios. Some stochastic scheduling models are already developed for the maintenance and operation of OWT (Kovácsa, A., 2011; Zhang, J., 2012; Besnard and Bertling, 2010; Pappalla et al., 2009; Meibom et al., 2011).
Few scholars have studied the stochastic model in the installation phase. Most existing stochastic models regard demands or cost as the uncertain parameters. Sherali et al. (2013), Liu and Nagurney (2013), Keller and Bayraksan (2009) developed stochastic models for scheduling under uncertain demand or cost conditions. They applied Benders decomposition (Benders, 1961) approach to solve the stochastic model. Benders decomposition is a quite popular method to settle stochastic programming problems (You and Grossmann, 2011). However, to our best knowledge, few studies specially consider weather condition as uncertain parameters in the stochastic model and solve the model with Benders decomposition method.
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method to solve the problem. The research question is how to schedule the OWT installation considering uncertain weather conditions and operational time with the aim of minimizing total service time? The initial contribution of this study is to schedule OWT installation under fluctuating weather conditions. Next, the contribution is then to schedule other offshore projects such as offshore plant form construction which may also be weather-dependent and help to minimize the service time. Contribution to scientific knowledge is to extend previous literatures and include more complex weather and operational time variation. The proposed solution will be tested with real data from OWT installation project “BARD1” in the North Sea. Based on experiment results, conclusion and suggestion for future study are offered.
The reminder of this paper is organized as follows. The second chapter will introduce the installation processes and related problems. The third chapter will provide a brief review of relevant literatures. The OWT installation model will be presented in chapter 4. Chapter 5 will illustrate the solution algorithms applied to solve this model under uncertainty. Some numerical and computational results will be given in Chapter 6. The final chapter will conclude this paper and provide some suggestion for future research.
2. Problem Description
In the installation phase of OWT, site condition and the operation of installation vessels are the major factors that bring challenges (Uraz, 2011). The site condition includes seabed properties and water depth. The whole offshore construction procedure is highly weather-dependent. Some installation processes can only be proceeded under suitable weather conditions, and some can still carry on in less ideal situations. This chapter will introduce the processes of OWT installation and the relevance of installation vessel in detail. We will also illustrate the interference of uncertain weather conditions during the process.
2.1 OWT installation process
The installation process of OWT is mainly separated into two stages (Dijk, 2013). First is the pre-assembly stage. This stage is always done onshore. Some components (e.g. nacelle, hub and blades) are assembled together before the construction at sea. The next stage is the installation of the whole turbine in the offshore wind farm. This stage is generally separated into two parts. One is the installation for the sub-structure, and the other is for the top part (Scholz-Reiter et al., 2011). As shown in Figure 1, the foundation under water is the sub-structure. The turbine above the water is the top-structure. It is easy to understand that only with the sub-structure installed can the top part be built upon. Besides these two major parts, the piles and cables are also important and difficult to install. The piles are used to fasten the foundation to the seabed (Ait-Alla et al., 2013). The cables connect the turbines to the electricity grid. Due to technical requirements, cables must be installed before the installation of top-structure. To conclude, the installation sequence is as follows:
1. Build sub-structure; 2. Install piles and cables; 3. Build top-structure.
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major sections of construction and weather is not that influential for the installation of piles and cables. Figure 2 shows the operational processes of installation. For each vessel tour, the installation vessel is first loaded at the harbor with some separate sub and top parts. After that, it travels to the wind farm, and the construction process begins. There are multiple processes for each vessel tour. For each building process, either one top-structure or one subpart can be built. When all the processes have been finished, the vessel returns back and the new vessel tour begins.
Figure 1: Top- and sub-structure of offshore wind turbine, Redwave
Estimated loading time at harbor plus travelling time from harbor to wind farm of vessel tour Estimated travelling time from wind farm to harbor of vessel tour
Estimated building time for a sub-structure of vessel tour and building process j Estimated building time for a top-structure of vessel tour and building process j Start time of vessel tour process j
Start time of vessel tour Return time of vessel tour
Figure 2: Operational processes of OWT installation
2.2 OWT installation vessel
The installation vessel is vital during an installation process, because it transports the pre-assembled parts from harbor to the construction site at sea (Uraz, 2011). The crane on the vessel also assists the installation. As shown in Figure 3, for the construction of top- and sub-structures, a Jack-up installation vessel is required. For the installation of cables, another type of cable vessel is used. Various OWTs have different structures and sizes which require different vessel types (Ait-Alla et al., 2013). The high-standard requirements for the cranes and infrastructure on the vessel make it extremely expensive to rent one. According to the research of Dijk (2013) on the “BARD1” OWT project in the North Sea, the daily rental fee for a Jack-up
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vessel is around €140.000. The labor cost is €110 per person per hour. If the installation time can be lowered, the relative costs can also be reduced in proportion.
It is comprehensible that more installation vessels can decrease the installation time to a great extent. Another factor that influences the service time is the distance between the port and wind farm. Since the location of wind farm is decided by the wind conditions and other technical factors, we will not discuss this in this paper. However, it is interesting to see the influence of the number of turbines on both the installation cost and service time. According to the data from Uraz (2011) and Dijk (2013), take a wind farm that builds 100 turbines as an example. The distance between the wind farm and the port is 110 miles. This distance is similar to the one in the “BRAD1” project. Figure 4 illustrates the total rental costs and the total service time in relation to the number of installation vessels. Although the total service time drops dramatically with increased vessel number, the costs also swell apparently. This cost does not consider the additional labor costs for multiple vessels. Because multiple vessels will lead to another routing problems for these vessels, this scheduling research will use one vessel to simplify the problem.
Figure 3: Jack-up installation vessels, Blue Ocean Ships
Figure 4: Total service time and vessel rental cost in relation to the number of vessels Uraz (2011) Distance between wind farm and port is 110 miles. Vessel speed is 8 knots. Wind farm has 100
0 5000000 10000000 15000000 20000000 25000000 0 20 40 60 80 100 120 140 1 2 3 4 5 6 7 8 9 10 11 12 Number of vessels Total service time(day)
8 turbines.
2.3 Uncertain weather factor
From both theoretical and practical aspects, weather is regarded as the most influential factor during the installation. Bilgili et al. (2011) claimed that offshore costs are greatly dependent on weather conditions. From the experience of installation of Middelgrunden 40 MW OWT, Sørensen et al. (2011) concluded that unexpected weather change is the major reason for project delay. Sun et al. (2012) also state that harsh weather is a big challenge for the safe operation during installation and maintenance. The top-structures take a lot of wind. When the wind is too strong, it is dangerous to build the top part, because the crane on the vessel will operate under unstable condition which cannot guarantee the safety (Munich Re, 2013).
Figure 2 has already shown the ideal operation process without consideration of the interference of bad weather. The practical processes that incorporate various weather conditions are shown in Figure 5. When the weather condition is bad, the wind is so strong that no construction can be done. During medium weather period, the sub-structures can still be installed; however, the practical construction time will be longer than the time that is estimated under good weather condition. The inaccurate weather forecast makes the case more complicated. Take the first bad weather block in Figure 5 as an example. When the vessel leaves the port for the first vessel tour, this is the first day of construction. The bad weather is forecasted to begin from the sixth day and end at the seventh day. The practical situation can be different. It may happen that the bad weather actually begins from the eighth day of construction and ends at the tenth day. If these uncertain weather factors are not considered when making the schedule, there will be added waiting time during installation. As a result, this paper focuses on OWT installation under uncertainty.
Figure 5: Operational processes of OWT installation under uncertainty
3. Literature review
Current studies related with OWT installation consider weather as a vital factor. In this chapter, we will first review the current studies about OWT installation and similar stochastic models for other phases of OWT such as maintenance. The current Benders-based stochastic modeling researches are presented afterwards.
Typian et al. (2011) compared two mathematical methods to deal with weather downtime and operational time during OWT installation. The number of OWT is limited to one and scheduling is not considered this study. Scholz-Reiter et al. (2011) proposed a mixed-integer liner programming model for the installation of OWT that takes the weather condition and operational time into
Start time of first bad weather condition:
Estimated: the 6th day
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consideration. This method aims at minimizing the installation time. However, weather condition and operational time are assumed to be constant values. Only one weather scenario is considered in the model. Ait-Alla et al. (2013) provided an aggregate planning method for minimizing the installation cost. They figure out a mathematical model which takes the capacity and weather factors into consideration. But capacity and weather condition are still considered as deterministic parameters. Lütjen (2012) presented a simulation approach for the inventory control during the installation process of OWT. In addition, the reactive scheduling heuristics are proposed for the installation vessels under different weather conditions. Nonetheless, this concentrates on the inventory phase instead of the construction phase. So far, studies about OWT installation are quite limited.
Compared with installation, there are more intensive studies about maintenance and operation of OWT. Kovácsa, A. (2011), Zhang, J. (2012) and Besnard and Bertling (2010) researched scheduling models for the maintenance of OWT. Weather is also included in these models. For the daily operation of constructed OWT, more stochastic scheduling models are studied in order to minimize cost (Pappalla et al., 2009; Meibom et al. 2011). Compared with simple modeling, stochastic modeling takes different scenarios into account in order to handle more realistic uncertainties and up-to-date information. In the case of OWT installation, weather is the uncertain factor because actual weather condition always deviates from the forecasted one. Meanwhile, operational time also changes under different weather conditions. As a result, stochastic modeling is also necessary for planning OWT installation.
Stochastic modeling is popular in the supply chain management area. It provides more robust solution than the deterministic one, because it computes different input value with different probabilities and provides a solution that is on average optimal (Velarde and Laguna, 2004). Demand and cost are the most common stochastic parameters. Sherali et al. (2013) propose a stochastic model for airline schedule design which regards flight time and customer demands as the stochastic input. Similarly, Cerisola et al. (2009) provide a stochastic model for power generation company which incorporates demands as uncertain factors. Liu et al. (2013) and
Velarde et al. (2004) present the stochastic models for the international outsourcing problems which consider various demands and exchange rate. Few stochastic models take weather as the stochastic parameter. Stochastic modeling for OWT installation is an uncovered part. This work will provide a solution to tackle different scenarios with varied weather conditions and operational time through stochastic modeling.
4. The Wind Turbine Installation Model
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4.1 Assumptions
1. One installation vessel for the installation of one wind turbine farm. As already mentioned before, multiple vessels may cause additional routing problems for each vessel. As a result, we only consider one installation vessel in this model. In the practical case, the capacity of the vessel may limit the number of building processes for each tour. Some small vessels only have the capacity for one turbine for each vessel tour. Some other vessels may have bigger capacity to install three or four turbines in one vessel tour (Dijk 2013). In this model we assume the vessel has bigger capacity;
2. Installation vessel is the vessel with jack-ups (see Figure 3). This is the vessel used in the real project “BRAD1”. The travelling and loading data we gathered are also from the jack-up vessel. Using this vessel in the model will make the computational results more in line with the practical situation;
3. Either one sub-structure or one top-structure can be built within each building process;
4. Weather can change in every other weather period t. If we consider the weather changes every other hour, the sample size will become too big even for a small-scale problem. So we assume the weather changes in days. In addition, the variation within one day is already considered when we process and collect the data in Appendix II.
4.2 Deterministic model
The related indices, parameters, decision variables and the mixed-integer linear programming model for OWT installation are defined as follows:
Notation:
Indices
= (1,…,V) Set of all vessel tours
= (1,…,J) Set of all processes at each vessel tour = (1,…,T) Set of weather periods
Parameters
Estimated travelling time from wind farm to harbor of vessel tour
Estimated loading time at harbor plus travelling time from harbor to wind farm of
vessel tour
Estimated construction time for a sub-structure of vessel tour Estimated construction time for a top-structure of vessel tour
Number of sub-structures in loading set
Number of top-structures in loading set
Number of sub-structures in loading set
Number of top-structures in loading set
Total number of turbines to be installed
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Length of bad weather in weather period
Forecasted start time of medium weather in weather period
Length of medium weather in weather period
M A big number
Decision variables
Start time of vessel tour building process
Start time of vessel tour
Return time of vessel tour
1 if sub-structure is built at vessel tour and building process
1 if top-structure is built at vessel tour and building process j
Number of available sub-structures on which the top part can be built at vessel tour building process
1 if vessel loading a is selected on tour
1 if vessel loading b is selected on tour
1 if vessel tour building process is performed before bad weather condition
1 if vessel tour building process is performed before medium weather condition
, , , and are mixed-integer variables. The objective of
optimization of this mixed-integer liner model is to minimize the time of installation. This model aims to minimize the total return time of all the vessel tours and the total building time, which means the building phase utilize minimum time and the vessel comes back as early as possible. Further, in order to make sure that all process starts quicker, the sum of building start time also needs to be minimized, because there might be some idle time between every process. The objective function is:
12 ( ) (12) , (13) = + - , (14) = 0 (15) (16) + + , ,t (17) ( ) (18) (19) ( ) (20)
Constraints (2) to (5) introduce the basic operational processes of installation. After the vessel has arrived from the previous tour, it can leave the harbor for next tour. After the vessel has arrived at the wind farm, the first building process can start. Start time of every vessel tour and process must be after the start time of vessel tour and process plus the building time of either the sub-structure or the top-structure. When all the processes have been finished, the vessel can return. Constraints (6) to (8) are related with the loading set. Either loading set or loading set needs to be selected for each vessel tour. The number of sub-structure that can be built is limited by the maximum number of loaded sub–structures. This is also true for top-structures. Constraints (9) to (12) claim that the sum of built sub-structures must be equal to the number of total turbines, and this is also true for top-structures. Either a sub-structure or a top-structure can be built at each process . For the first vessel tour, the first process must build sub structure; otherwise there is no sub-structure where the top part can be built. Constraints (13) to (16) are added to guarantee the availability of built sub-structure. The available number of sub-structure is 0 at the beginning and must be bigger than or equal to zero. When = 0, this process
must build sub-structure. Constraints (17) to (20) are made to deal with the interference of weather. For bad weather period, all the building processes are either finished before the bad weather or begin after the bad condition is over. For medium weather period, only the construction of sub-structure is allowed, but it is not allowed to build the top part because the wind is too strong. All the processes can either be finished before the medium weather or begin after the medium conditions is over.
In order to get planning solutions for OWT installation, we need to know the value of decision variables and to decide which loading set to choose before the start of each vessel tour. More important decisions are and . For each building process, whether to build top-structure or sub-structure must be decided with the aim of minimizing service time. The developed model is adapted into a stochastic version and solved with the Benders decomposition approach in the next chapter.
5. Solution methodology
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stochastic programming problems (Birge and Louveaux, 1997; Infanger, 1994; Shapiro, 2008; You and Grossmann, 2011). In the two-stage stochastic programming method, the decision variables are decomposed into two parts (Ahmed et al., 2009). The first stage decisions are made “here and now” before the entry of uncertainty, while the second-stage decisions are resolved in the “wait and see” way after the uncertainties are revealed (You and Grossmann, 2011).The first stage variables are the variables that need to be decided before the input of uncertain parameters. When the stochastic values enter the second stage, the second stage variables are calculated together with the provided values of first stage variables. The objective is then to get the first stage variables so that the sum of first stage variables and expected values of second stage variables are minimized. Cerisola et al. (2009) pointed that for a practical representation of uncertainty, decomposition must be used. In order to decompose the decision variables, two-stage or multi-stage methods are developed to separate the variables that have different properties clearly. This principle is similar to separating the classrooms of senior students and junior students so that the teachers can teach them different knowledge. Two-stage stochastic programming is the starting point of most stochastic models (Ahmed et al., 2009). We will also use this method in our model. Benders decomposition is the mostly used algorithms to solve the stochastic models. The advantage of this method is that it is not limited by the number of scenarios which is quite helpful for practical implementation. In comparison, another approach, deterministic equivalent approach of stochastic model, is limited by the sample size. We will also introduce this method in short.
5.1 Benders decomposition algorithm
Several computational methods have been developed to solve a two-stage stochastic programing problem. Benders decomposition (Benders, 1962) is one of the mostly used approaches because it is easier to implement (You and Grossmann, 2011) when compared with other approaches such as disjunctive decomposition (Ntaimo, 2010). Benders decomposition is a way to separate complex programming models into two, and simplifying the solution by solving one master problem and one sub problem afterwards. First stage variables that are independent of stochastic parameter are calculated in the master problem. The sub problems only include the second stage variables which are directly related with the stochastic input values and generate duality cuts based on the sub gradient iteratively (You and Grossmann, 2011). In this model, constraints involving variables and are included in the master problem. The baseline schedule is provided after calculation. This baseline schedule is originally not influenced by the stochastic weather conditions. Remaining constraints are left in the sub problem, and decision variables , are calculated in the sub problems. They are directly related with
uncertain parameter about weather. The decomposition is as follows: (Master problem)
MP: Minimize
∑ + (21)
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, (23)
(24)
Constraints (22) and (23) are adapted from constraints (2) (3) (5). Since variables , are put in the sub problem, adjustment is needed in the Master problem. is introduced as the Benders cut that iterates between master problem and sub problem. The sub problem for scenario is represented by: (Sub problem) SP( ̅̅̅ ̅̅̅̅) Minimize ∑ ( ) (∑ ∑ ( ) ∑ ∑ ( ) ∑ ∑ ( )) (25) ( ) ̅̅̅̅ , (26) ( ) ( ) ( ) ( ) , (27) ( ) ( ) ( ) ̅̅̅ , (28) ( ) ( ) (29) ∑ ( ) ( ) + ( ) (30) ∑ ( ) ( ) + ( ) (31) ∑ ∑ ( ) (32) ∑ ∑ ( ) (33) ( ) ( ) , (34) ( )( ) (35) ( ) ( ) ( ) ( ) 0, (36) ( ) ( ) ( )+ ( ) , (37) ( )= 0 (38) ( ) (39) ( ) ( )- ( ) ( ) ( ) ( ) , ,t (40) ( ) ( ( )) ( ) ( ) (41) ( ) ( ) ( ) ( ) ( ) ( ) (42) ( ) ( ( )) ( ) ( ) (43)
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were already calculate in the master problem, theirs value can directly be used and they are represented as ̅̅̅ ̅̅̅̅ in the sub problems. The corresponding dual to this sub problem can be stated as:
(Dual sub problem)
DSP( ̅̅̅ ̅̅̅̅): Maximize ∑ ( ) (∑ ( ̅̅̅̅̅̅ ) ∑ ( ̅̅̅) + ∑ ∑ ∑ ( ( ) ( ) ( ) ) + ∑ ∑ ∑ ( ( ) ( ) ) + ∑ ∑ ∑ ( ( ) ( ) ( ) ) + ∑ ∑ ∑ ( ( ) ( ) ) (44) (45) (46) (47)
denote the dual variables associated with constraints (26) to (43). These solved dual variables in the sub problem are used to create the benders cut and (24) is updated to:
∑ ( ) (∑ ( ) ∑ ( ) + ∑ ∑ ∑ ( ( ) ( ) ( ) ) + ∑ ∑ ∑ ( ( ) ( ) )
+ ∑ ∑ ∑ ( ( ) ( ) ( ) )
+ ∑ ∑ ∑ ( ( ) ( ) ) ) (48)
The major procedures for Benders decomposition is shown in Figure 6:
Figure 6: Algorithm for Benders cut, adopted from (You and Grossmann, 2011)
No Yes
Solve Master problem equation (21) and get lower bond (LB)
Solve Sub problem equation (25) and get upper bond (UB)
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5.2 Deterministic equivalent of stochastic model
The deterministic equivalent of stochastic model is the most original way to solve the stochastic model. At first stage, deterministic model is solved as the core model (1) subject to constraints (2) to (20) with estimated weather condition (Kalveleagen, 2003). The value of is calculated in the first stage. The second stage is the deterministic equivalent of this stochastic model. The objective function value is presented as follows:
Minimize
∑ + ∑ ( ) (∑ ∑ ( ) ∑ ∑
( ) ∑ ∑ ( ))
(49)
As shown in (49), the second stage variables , and change into the form with
additional ( ) in order to make difference with the form in the first stage. However, the first stage variable still keeps its original notation as in the first stage. This is because the calculated value of from the first stage needs to be used in the second stage. The constraints are similar to constraints (2) to (20). Variables , and in constraints (3), (4), (5), (7),
(8), (9), (10), (11), (13), (14), (17), (18), (19) and (20) are replaced with ( ), ( )
and ( ). And the weather-related parameters with single set of input in constraints (17)-(20)
are replaced with multiple scenarios in company with their joint probabilities. The deterministic equivalent approach stores all the possible solutions for all the scenarios during computation and provides the optimal one. Next chapter provides computational results of both methods, and compare the solutions of these two methods with the one from deterministic model in order to see the difference.
6. Computational experiments
Two sets of computational experiments are performed in this chapter. We use the real operational data collected from the offshore wind project “BARD1” in the North Sea as input for the computation. The real weather data was also collected. This will make the model more in line with practical situation. Firstly, the computational results and the comparison between solutions from the deterministic model and stochastic models with two different algorithms will be presented. Weather is the only stochastic input in the small-scale case. In the large-scale problems, additional stochastic input parameter such as travelling time and operational time are added in order to see its influence on the results. CPU time of the two algorithms is also compared in the larger problem. The proposed models and algorithms were implemented in Gams 22.5 with CPLEX solver for solving the mixed-integer linear problems. All the computations were carried on an personal computer with AMD Phenom™ II P960 1.8GHz PC running Windows 7.
6.1 Computational results on small-scale problems
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is directly related to the number of vessel tours (V) and number of processes (P), we set = 16, V = 4, P = 8. In the large-scale problems, we will only increase the values of and V. Figure 7 illustrates the solution from the deterministic model. Figure 8 and 9 provide the solutions from the stochastic model calculated with Benders-based approach and deterministic equivalent approach. Only one weather condition ( ) is considered in the deterministic model, while three different scenarios ( , , ) are calculated in the two stochastic models. The values of the parameters applied in the model can be found in Appendix I. Appendix II lists all the weather data input ( , , ) for the small-scare problems. The probabilities of relative scenarios are all shown in the appendix. The interference of weather is not shown in these three figures because there are different weather conditions. This is mainly used to illustrate the variations between these solutions. The colored blocks are the decision variables for the solutions. Before the start of a vessel tour, decision variables and are decided in order to choose one loading set. The block is black if we choose loading set and white for loading set . As already introduced in chapter 4, denotes the estimated travelling time from wind farm to harbor of vessel tour and denotes the estimated loading time at harbor plus travelling time from harbor to wind farm of vessel tour . During the eight building processes, decisions of either building sub-structure or top-structure in every process are made. The black block ‘sub’ means the decision for building sub-structure. The white one with ‘top’ means the decision for building top part. Because the building time for top-structure is longer than the sub-structure as already listed in Appendix I, the white block is longer than the black one. With three different solutions presented, we want to know the difference in between.
Process number j Vessel Tour i
j1 j2 j3 j4 j5 j6 j7 j8
Vessel tour 1
va sub sub sub top sub top top top
Vessel tour 2
vb sub sub sub top top sub top top
Vessel tour 3
va sub sub sub sub top top top top
Vessel tour 4
va sub sub sub sub top top top top
Figure 7:OWT planning solutions from deterministic model
Process number j Vessel Tour i
j1 j2 j3 j4 j5 j6 j7 j8
Vessel tour 1
vb sub sub top sub sub top top top
Vessel tour 2
vb sub sub sub top sub top top top
Vessel tour 3
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Vessel tour 4
va sub sub sub sub top top top top
Figure 8:OWT planning solutions from Benders-based stochastic model
Process number j Vessel Tour i
j1 j2 j3 j4 j5 j6 j7 j8
Vessel tour 1
va sub sub sub top top top sub top
Vessel tour 2
va sub sub sub sub top top top top
Vessel tour 3
va sub sub sub sub top top top top
Vessel tour 4
va sub sub sub sub top top top top
Figure 9:OWT planning solutions from deterministic equivalent stochastic model
Since the quality of the solution is the most important indicator to measure the solution, we compare the three solutions with their quality. In order to know the quality, we compare the objective function value ( ) calculated in (1) of the stochastic model to that of the deterministic one. Table1 shows the results on the small-scale case. denotes the objective function value calculated in the deterministic model. and are the results obtained from the deterministic equivalent of stochastic model and Benders decomposition approach respectively. The estimated optimality gap and the standard deviation of the gap for the three solutions are listed in the table. The Benders-based solutions have zero optimality gaps which precisely show the preponderance of the stochastic model solved with this method. Further, the average ( ) value from this solution is also the lowest one which indicates lowest average service time. Lowest service time means this solution outperforms other two methods. From the small-scale experiments, we can preliminarily conclude that a Benders-based stochastic model with multiple scenarios presents solutions of better quality to tackle the uncertain weather variations. In the next section, we want to test whether this conclusion still holds in large-scale situation. ( ) Average 1412.73 1410.42 1372.33 Min 1243.75 1237.75 1231.00 Max 1513.23 1514.29 1526.42 Gap(%) 1.77 4.95 0 Std. Dev. 1.78 2.49 0
Table 1: Comparison of deterministic and stochastic solutions
6.2 Computational results on large-scale problems
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CPU time and estimated optimality gap of computing different values in the two stochastic models are listed in Table 2. In the large-scale case, the CPU time of Benders-based stochastic models increases in proportion to the number of turbines, while the optimality gap does not vary a lot as shown in Figure 10. Deterministic equivalent approach for the stochastic model requires less CPU time, however, when is higher than 80, the calculation cannot proceed because there is not enough storage space and the computer is running out of memory. We can now confirm that the Benders-based algorithm provides better solutions for the stochastic OWT installation model than the deterministic equivalent approach even under larger-scale case. In addition, the proposed solution can minimize the total service time to a great extent. With similar data input from “BARD1” project, the total service time (return time of last vessel tour plus the traveling time from the wind farm back to the harbor) is 405 days which is less than the simulation result from Dijk(2013) that is 500days. Although part of the variation may be due to different usage of data, it still indicates the potential of decreasing the total service time through this Benders-based stochastic model.
Stochastic input: weather data Stochastic input: weather data
Deterministic equivalent of stochastic model
Stochastic model calculated with Benders Deterministic equivalent of stochastic model Stochastic model calculated with Benders CPU time(seconds) Gap(%) CPU time(seconds) Gap(%) 16 0.234 4.95 5.476 0 153.188 76.185 48 1.685 0.53 41.169 0.41 385.917 181.872 80 Out of memory 274.546 0.34 - 279.521 120 Out of memory 390.362 0.04 - 405.1655 144 Out of memory 503.820 0.21 - 468.853
Table 2: CPU time of two algorithms with varied values
Figure 10: CPU time and optimality gap from Benders-based solution in large-scale problem
Figure 11 and 12 illustrate the influence of travelling time and operational time variation on the total service time. The number of turbines is 48. In the two figures, the traveling and building time vary from 0.5 to 8. The total service time in both figures goes up with the variation growing.
0 100 200 300 400 500 600 16 48 80 120 144
Number of turbines to be built
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Further, the variation of building time has a greater impact on the service time than the travelling time. This is reasonable because there are only two traveling performances in every vessel tour while there are eight building processes in a single tour. In summary, the variation of weather conditions, travelling time and operational time all affect the total service time. If these variations are not considered when making the schedule, actual service time will increase when the model is applied to practical project.
Figure11: The total service time under different variation of travelling time
Figure 12: The total service time under different variation of building time
7. Conclusion and further research
This paper has developed a two-stage scheduling model for the OWT installation problem under uncertain meteorological and operational conditions. The stochastic model is solved with the Benders-based algorithms. The deterministic equivalent approach for the stochastic model cannot
0 100 200 300 400 500 600 0.5 1 2 3 4 5 6 7 8 to to al s erv ic e ti m e
Variaton of travelling time
0 200 400 600 800 1000 1200 1400 1600 1800 2000 0.5 1 2 3 4 5 6 7 8 to tl al s e rv ic e ti m e
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handle the problems with large sample size because of the storage space limitation. We propose the Benders-based algorithms that can incorporate various stochastic inputs and larger sample size. Variation of traveling time and construction time is proved to have influence on the total service time.
The major theoretical contribution is extending the existing deterministic model from Scholz-Reiter et al. (2011) to a stochastic one which can deal with more complex situations. This enables the model to be applied to real life practice to build large number of turbines. The long-term practical contribution of this study is that this stochastic model can also help to deal with other offshore projects such as offshore platform construction that are also weather-dependent. The structure of the offshore platform can also be separated into two parts. The jackup unit is the sub-structure. The pile and tower are the top-structures. For installation of multiple offshore platforms, the proposed solution can also be adapted to schedule the process.
For practical application of the solution, the relative distance between each OWT needs to be considered. If the distance is too long, there might be additional traveling time which may cause inaccuracy of the modeling results. Further, the data collection for the weather condition is another important part. We suggest combining the weather forecast together with the historical records for the same period together as the stochastic input for the model. This may increase the reliability of the solution, because most of the possible scenarios have been incorporated in the model. The limitation of this work is that it does not link the operational time with the weather conditions.
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Appendix I
Input data collected from “BRAD1” project, these are the initial value in the deterministic model. The input values for stochastic model are adjusted based on these data.
Parameter Description Value
las Numebr of sub-s in loading set a 6
lat Number of top-s in loading set a 4
lbs Number of sub-s in loading set b 4
lbt Number of top-s in loading set b 6
th(i) Traveling time from wind farm to harbor of vessel tour i in day
0.3333 thl(i) Traveling time plus loading time from harbor to wind farm
of vessel tour i in day
0.1875
bts(i) Building time for sub-structure of vessel tour I in day 1
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Appendix II
Weather data input for small-scare problems
Start time of bad weather in weather period
Start time of medium weather in weather period
Since the weather condition in the original data is recorded every two hours, we calculate the average wind speed every day. When the average wind speed is bigger than or equal to 16 Km/h, this means more than half of the day is bad weather. Then we regard this day as ‘bad weather day’. Similarly, when the average wind speed is between 10 and 16 Km/h, we regard this day as ‘medium weather day’.
