As stated in Sections 1 and 2, the tidal cable routing problem may be formulated via the TSP approach. The TSP approach is adopted in this paper and extended to the multiple hub problem. Prior to presenting the TSP formulations for the single and multiple hub problem variants employed in this paper, a concise problem statement is presented in the following subsection.
Single hub problem statement
Given the following pieces of information:
-
1)
A set of identical tidal current turbines, with each of which a spatial location and a nominal power rating are associated.
-
2)
A set of identical electrical power cables, with each of which a unit cost per length and a maximum electrical power capacity are associated.
-
3)
A single collection point with each of which a spatial position and a distance from shore are associated.
-
4)
Each turbine possesses at most one input cable segment and at most one output cable segment.
-
5)
For each turbine pair and each inter-turbine cable segment, a cable segment length (distance between a turbine pair) and a cable power flow are associated.
It is necessary to find the least cost cable routing configuration, which will be the configuration using the shortest cable length.
In the single hub problem, the multiple traveling salesmen problem (mTSP) model is employed [18], which may be stated as follows.
$$\hbox{min} \, \sum\limits_{(i,j) \in T} {c_{ij} } x_{ij}$$
(1)
s.t.
$$\sum\limits_{j = 2}^{n} {x_{1j} } = \, C$$
(2)
$$\sum\limits_{j = 2}^{n} {x_{j1} } = \, C$$
(3)
$$\sum\limits_{i = 1}^{n} {x_{ij} } = \, 1 \qquad j = 2,3, \ldots ,n$$
(4)
$$\sum\limits_{j = 1}^{n} {x_{ij} } = \, 1 \qquad i = 2,3, \ldots ,n$$
(5)
$$u_{i} + \left( {L - 2} \right)x_{1i} - x_{i1} \le L \qquad i = 2,3, \ldots ,n$$
(6)
$$u_{i} + x_{i1} \ge 2\qquad i = 2,3, \ldots ,n$$
(7)
$$u_{i} - u_{j} + Lx_{ij} + \left( {L - 2} \right)x_{ji} \,\le\, L - 1 \qquad 2 \le i \ne j \le n$$
(8)
$$\sum\limits_{j = 2}^{n} { {c_{i1} } } = 0$$
(9)
$$x_{ij} \in \left\{ {0,1} \right\} \qquad\forall \left( {i,j} \right) \in V$$
(10)
where i, j denote index pair for cable segment connecting turbines i and j; C denotes set of cables; L denotes maximum allowed number of turbines connected to one cable (electrical power flow capacity); T denotes set of turbines; cij denotes cost of cable segment connecting turbines i and j; xij denotes binary variable, 1 if turbines i and j are connected, and 0 otherwise; ui denotes number of connected turbines by cable C and passing through turbine i.
Objective function (1) depicts the total cost of electrical power cable segments employed. Constraints (2) and (3) ensure that exactly C cables leave from and return to the hub. Constraints (4) and (5) are the degree constraints. The inequality given in (6) serves as an upper bound for the number of turbines connected with one cable. Inequality (7) serves to initialize the value of ui to 1 if and only if i is the first connected turbine for a given cable. The inequalities given in (8) ensure that uj = ui + 1 if and only if xij = 1. Thus, they prohibit the formation of any subtour between nodes in T; i.e., they are the subtour elimination constraints (SECs) of the mTSP model. Constraint (9) ensures that the cost of a cable segment returning to the hub is equal to 0. Constraint (10) defines the domain of the decision variables.
The hub is numbered as node 1 in this model. There is no lower limit to the number of turbines connected to one cable in this model, but this can also be implemented by adding a constraint to set a lower limit, say K. The number of separate cables C connecting hub and turbines must be predefined by the user as a parameter, which is varied as part of the solution strategy of the model.
Multiple hub problem statement
Given the following pieces of information:
-
1)
A set of identical tidal current turbines, with each of which a spatial location and a nominal power rating are associated.
-
2)
A set of identical electrical power cables, with each of which a unit cost per length and a maximum electrical power capacity are associated.
-
3)
A set of cable collection points, with each of which a spatial position and a distance from shore are associated.
-
4)
Each tidal turbine possesses at most one input electrical cable segment and at most one output electrical cable segment.
-
5)
For each tidal turbine pair and each inter-turbine electrical cable segment, an electrical cable segment length and an electrical cable power flow are associated.
It is necessary to find the least cost electrical cable routing configuration, which will be the configuration using the shortest cable length.
The same notation is employed in the multiple hub optimization model as that in the single hub model, albeit with some modification. In the multiple hub problem, several hubs are connected to several cables. In order to include this possibility in the multiple hub optimization problem model, the set of nodes is divided into two subsets; i.e. the set of nodes V = D ∪ V’, where D represents the set of hub nodes, D = {1, 2, …, d} and V’ represents the set of turbine nodes, V’ = {d + 1, d + 2, …, d + n}. At each hub node, there are mi cable segments as input and as output.
In the multiple hub problem, the mTSP model is employed [18], which may be stated as follows.
$$\hbox{min} \, \sum\limits_{(i,j) \in A} {c_{ij} x_{ij} }$$
(11)
s.t.
$$\sum\limits_{j \in V} {x_{ij} } = m_{i} \qquad i \in D$$
(12)
$$\sum\limits_{i \in V} {x_{ij} } = m_{j} \qquad j \in D$$
(13)
$$\sum\limits_{j \in V} {x_{ij} } = 1 \qquad i \in V$$
(14)
$$\sum\limits_{i \in V} {x_{ij} } = 1 \qquad j \in V$$
(15)
$$u_{i} + (L - 2)\sum\limits_{k \in D} {x_{ki} } - \sum\limits_{k \in D} {x_{ik} } \le L \qquad i \in V$$
(16)
$$- u_{i} - \sum\limits_{k \in \,D} {x_{ki} } \le 2 \qquad i \in V$$
(17)
$$u_{i} - u_{j} + Lx_{ij} + \left( {L - 2} \right)x_{ij} \le L - 1 \qquad i \ne j; \, i,j \in V$$
(18)
$$\sum\limits_{i \in V} {{c_{ik} } } = \, 0 \qquad k \in D$$
(19)
$$x_{ij} \in \left\{ {0,1} \right\} \qquad \forall \left( {i,j} \right) \in V$$
(20)
Objective function (11) depicts the total cost of the electrical power cable segments employed. For each i ∈ D, mi outward and mi inward arcs are guaranteed by constraints (12) and (13). Equations (14) and (15) are the degree constraints for the customer (turbine) nodes. Constraints (16) and (17) impose bounds on the number of nodes a salesman (cable) visits together when initializing the value of the ui as 1 if i is the first node visited on the tour. Constraint (18) is a sub-elimination constraint (SEC) in that it breaks all subtours between the customer (turbine) nodes. Constraint (19) ensures that there is no cost related to the return from the last visited turbine to the hub. Constraint (20) defines the domain of the decision variables.