Executing Offshore Maintenance ActivitiesLivia Stark1 Toby Kingsman (Supervisor)1

1STOR-i, Lancaster University

Introduction

To increase its renewable energy capacity, the UK isinstalling an increasing number of wind farms off-shore.New technologies enable the construction of largerturbines that can be placed further from the shore,however these need to be subsidised to be a competitivealternative to fossil fuels. Reducing the cost ofmaintenance activities by carrying them out moreefficiently would reduce the reliance on this subsidy.

We considered two approaches to finding an efficientsolution:

Mathematical programming

Heuristics

Figure 1: Off-shore wind turbines

The Capacitated Multi-Vehicle Routing Problem

An important subproblem to the offshore maintenance scheduling problem is the CapacitatedMulti-Vehicle Routing Problem (CMVRP).It is a generalisation of the travelling salesman problem, where each location does not only needto be visited, but a certain amount of product needs to be delivered, while the salesmen havelimited carrying capacities. This applies to transferring the required personnel to the windturbines where the tasks are located, using the vessels available.It was modelled as an integer programming problem and was compared to a heuristic approach.

Formulation of CMVRP using Integer Programming

The n locations of the tasks and the port is represented by a set of vertices V = {v0, v1, ..., vn},while a set of arcs A = {aij = ( vi , vj) | vi , vj ∈ V ; i 6= j} represents all possible connectionsbetween them. A cost matrix C stores the distance between any two vertices. The demand atthe vertices is stored in vector q. There are m vessels available, each with capacity Q.The model uses 2 variables:

xij : xij = 1 if arc aij is used, xij = 0 if it is not.

ui : The amount delivered so far on the route that includes vertex vi .

The objective is to minimise the distance travelled by the vessels:

min∑

i∈V∑

j∈V cijxijUnder the constraints:∑

i∈V

xi0 = m(1) ∑j∈V

x0j = m(2) ∑i∈V

xij = 1; ∀j ∈ V \ {0}(3) ∑j∈V

xij = 1; ∀i ∈ V \ {0}(4)

xij ∈ {0, 1}; ∀i , j ∈ V(5)

uj ≤ Q + ( qj − Q) x0j ; ∀j ∈ V(6)

ui − uj + Qxij ≤ Q − qj ; ∀i , j ∈ V(7)

qi ≤ ui ≤ Q; ∀i ∈ V(8)

Constraints 1 and 2 ensure that the number of arcs leaving and entering the vertice representingthe port equal the number of vessels respectively, and constraints 3 and 4 ensures that all othervertices are only visited once. To impose connectivity and capacity requirements the MTZconstraints are used, displayed as constraint 6, 7 and 8.

Heuristics

Heuristics are used to find a feasible solution to problems where finding an optimal solution isimpossible or impractical. Heuristic approaches are generally faster than models aiming to find theideal solution.In this case the routes taken by the vessels is assembled according to the following logic:

Starting from the port, the routes of the different vessels are assembled in parallel.

The next stop is determined by proximity, if there are enough personnel on the vessel to carryout the task that is closest, the vessel is routed to it.

If the closest task requires more personnel then available, the second closest stop is considered.If it can be carried out the vessel is routed to it, otherwise the vessel returns to the port.

Once a vessel does not have enough personnel on board to carry out any more tasks or all tasksare done, it is routed to go back to the port.

The above logic aims to find a simple but short route where no vessel transfers more personnelthan its capacity.

Comparision of solutions to the CMVRP

A set of data containing 50 turbines with 20 tasks to be carried out was constructed, withdistances measured in an arbitrary unit. There were multiple vessels available, assumed to have auniform capacity of 15 personnel.The comparision focused on the time taken to produce a solution, and the overall distancetravelled by the vessels. The performance of the two approaches can be seen in the following table.

Number of tasks5 10 15 17 18 19 20

OverallDistanceTraveled

Heuristics 59.54 96.71 141.20 144.45 146.85 149.65 155.82Linear P. 44.86 75.51 104.02 110.18 114.54 118.60 118.83

Gap 32.7 % 28.1 % 35.7 % 31.1 % 28.2 % 26.2 % 31.1 %

TimeHeuristics 0.11 s 0.14 s 0.19 s 0.22 s 0.22 s 0.28 s 0.25 sLinear P. 0.09 s 0.24 s 1.10 s 8.87 s 116.80 s 512.50 s 3107.78 s

Table 1: Comparision of performance of the heuristics and the integer programming model for CMVRP

The heuristic method has produced routes longer than the optimal solution, with a gap of around30%, however, this gap showed large variation for different task locations and vessel capacities.

Figure 2: Routes suggested by the mathematical model and the heuristic method.

For example the case shown in Figure 2 only has a gap of 5.1%, with the optimal route covering adistance of 40.15 and heuristic route covering 42.20.

For the cases where only a small number of tasks needed to be carried out, both models producedresults quickly. When the number of tasks were increased, finding the optimal solution becameincreasingly time consuming making it impractical for larger sets of tasks. The same problem isexpected to arise if further constraints were to be added and therefore the heuristic method waschosen to be developed further.

Further development of the heuristic method

The capacitated multi-vehicle routing problem is is only a sub-problem to the offshore maintenancescheduling problem.The heuristic approach was modified to include project and vessel selection.

The tasks to carry out are selected in a way to maximise the value added back when the tasksare done. Their locations are added to a list for the vessels to visit.

There are expected to be multiple types of vessels used to transport personnel. The previouslydescribed routing logic is carried out for all possible number and permutations of the availablevessels. The selected route is the one that completes all the chosen tasks, or most in the caseswhere the available resources do not allow for full completion, in the shortest distance.

The maintenance personnel do not only need to be transported to the location of a task, but alsoback to the port once their task is complete.

Figure 3: Flowchart illustrating how pick-up of personnel is incorporated in the heuristic approach

A discretised time with one unit being 0.05 hours was incorporated. The start times andexpected finish time of all tasks were recorded.

Once the maintenance was completed, to pick the personnel up their location is added to the listof locations to visit.

The time maintenance activities take varies, and vessels often need to wait out at sea for them tofinish to be able to return all personnel to the port.

Before a vessel is allowed to return to port checks are carried out to see if all tasks are completed.

The vessels are only allowed to return to the port once no more personnel need picking up, orthey can not carry any more personnel.

Further work

Improve project selection

Comparison of different routing strategies

Develop variable that incorporates both the value of tasks completed and the associated costs.This could be used to select best solution.

References

Paolo Toth, Daniele Vigo. Discrete Applied Mathematics, Volume 123, Issue 1, 2002, Pages 487-512.

Models, relaxations and exact approaches for the capacitated vehicle routing problem

http://www.stor-i.lancs.ac.uk/intern/interns/2017 l.stark@lancaster.ac.uk