EXTENDING THE SOFTWARE CAPABILITIES

Sound results have been obtained with the application of the deterministic model of the HSR planning optimization problem to the synthetic case-study, solved with the implementation of the SAA. The synthetic case-study is intentionally simple to allow testing the tool performance. However, aiming at solving real-world HSR planning problems requires extended software capabilities.

Besides the fact that real HSR problems involve dealing with larger datasets, other complexities need to be addressed. This research identified two important aspects: dealing

with “islands” of forbidden land-use and establishing a minimum length of the HSR linear sections.

Existing “islands” of forbidden land-use have implications for the generation of new HSR candidate configurations while the lack of a minimum length of linear sections may affect the performance of the SAA. The consideration of these issues is presented in sections 4.4.1 and 4.4.2.

4.4.1 INFLUENCE OF THE FORBIDDEN LAND-USE AREAS ON THE GENERATION OF HSR CANDIDATE CONFIGURATIONS

As discussed in section 4.1.2 the constraints define the feasible search space and can impose difficulties to the search that should be overcome by the algorithm implementation. An example is the fact that a variety of forbidden land-use areas, differing in size and shape (concave and convex) may exist, which are not represented in the synthetic case-study used for the tool development.

If “islands” of forbidden land-use exist an initial solution is always located on one side of these areas and, depending on the size of the areas and the refinement of the discretization mesh ΩN, so will the new HSR candidate configurations. Such a case is depicted in the land-use layer of Figure 4-17, in which the yellow area covering most of the search space is exempt from restrictions, the black area is outside the feasibility space (ocean) and the remaining colored areas represent forbidden land-use. Starting from the initial HSR configuration of Figure 4-17 a), the SAA is not able to search the entire feasible space unless mechanisms are adopted in the generation of new HSR candidate configurations which allow leaping over these “islands”. Figure 4-17 b) shows an optimized solution, for which the SAA search was able to leap over the smaller forbidden land-use areas but not the larger ones.

To address this issue, mechanisms which allow leaping over forbidden land-use areas of variable sizes and shapes were incorporated in the generation of the HSR candidate configurations. Additionally, these have to take into account the existing constraints to the horizontal angle at intermediate nodes of the HSR configurations.

a) b)

Figure 4-17 HSR configurations overlaying the protected land-use layer map (scale 1:2 000 000): a) initial configuration and b) best configuration found by the SAA.

Consider the example of Figure 4-18, for which the node N of the current HSR configuration is randomly chosen to be horizontally repositioned to the right (N’). If a small “island” of forbidden land-use exists, such as depicted in Figure 4-18 a), the respective move to N’ can successfully leap over the forbidden land-use area. However, for a larger area such as shown in Figure 4-18 b), this repositioning is not sufficient.

The procedures adopted consist in incrementing the repositioning distance in the original direction by the spacing of the discretization mesh δ. One observes in Figure 4-18 b) that

HSR HSR

Unable to leap over

Able to leap over

Exempt Exempt

repositioning for both ∆1 = 1δ and ∆2 = 2δ still overlays the forbidden land-use but for ∆3 = 3δ it does not. However, this overstretching effect leads to increasingly smaller horizontal angles at nodes N’, N-1 and N+1.

A new feasible candidate configuration may not be obtained with the incremental displacement of the node, either by not complying with the land-use constraint or the horizontal angle constraint. In this case, a new tentative generation procedure is considered, which also repositions the anterior and posterior nodes. Figure 4-19 shows how the area from Figure 4-18 b) can be successfully dealt with by the repositioning of nodes N, (N-1) and (N+1).

a) b)

Figure 4-18 Leaping over “islands” of forbidden land-use by moving only one node of the current HSR configuration: a) successfully and b) unsuccessfully.

Figure 4-19 Leaping over the forbidden land-use area of Figure 4-18 (b) with the repositioning of node N, the anterior node (N-1) and the posterior node (N+1).

Given that the procedures to implement may have to deal with various shapes and sizes of forbidden land-use areas, a general procedure was developed and implemented. It consists in the tentative generation of feasible HSR configurations in 5 sequential steps and stopped when a feasible configuration is obtained. Each step relates with a number of anterior and posterior nodes to be displaced, besides the randomly chosen node N. Each step is detailed below.

STEP 0

This step considers only the displacement of the node N, which is sequentially incremented until a feasible configuration is found. It is illustrated by Figure 4-18. If the displacement causes the non-compliance with the horizontal angle constraint, Step 0 is abandoned and Step 1 is adopted.

STEP 1

This step considers the displacement of the nodes N, (N-1) and (N+1). It is illustrated by Figure 4-19. (N-1) and (N+1) are repositioned at 1/2 of the N displacement, in the same

direction. The displacement of N is incremented until a feasible configuration is found but if it causes the non-compliance with the horizontal angle constraint, Step 1 is abandoned and Step 2 is adopted.

STEP 2

This step considers the displacement of the nodes N, (N-1), (N-2), (N+1) and (N+2). It is illustrated by Figure 4-20 (a). (N-1) and (N+1) are repositioned at 2/3 of the N displacement and (N-2) and (N+2) are repositioned at 1/3, all in the same direction as N. The displacement of N is incremented until a feasible configuration is found but if it causes the non-compliance with the horizontal angle constraint, Step 2 is abandoned and Step 3 is adopted.

STEP 3

This step considers the displacement of the nodes N, (N-1), (N-2), (N-3), (N+1), (N+2) and (N+3). It is illustrated by Figure 4-20 (b). (N-1) and (N+1) are repositioned at 3/4 of the N displacement, (N-2) and (N+2) are repositioned at 1/2 of the N displacement and (N-3) and (N+3) are repositioned at 1/4 of the N displacement, all in the same direction as N. The displacement of N is incremented until a feasible configuration is found but if it causes the non-compliance with the horizontal angle constraint, Step 3 is abandoned and Step 4 is adopted.

STEP 4

This step considers the displacement of the nodes N, (N-1), (N-2), (N-3), (N-4), (N+1), (N+2), (N+3) and (N+4). It is illustrated by Figure 4-21. (N-1) and (N+1) are repositioned at 8/9 of the N displacement, 2) and (N+2) are repositioned at 7/9 of the N displacement, (N-3) and (N+(N-3) are repositioned at 2/9 of the N displacement and (N-4) and (N+4) are repositioned at 2/9 of the N displacement, all in the same direction as N. The displacement of N is incremented until a feasible configuration is found but if it causes the non-compliance with the horizontal angle constraint, Step 4 is abandoned. At this point, the generation of HSR candidate configurations is restarted and a new node N is randomly chosen to be displaced in the neighborhood.

a) b)

Figure 4-20 Leaping over “islands” of forbidden land-use: a) displacement of 2 anterior and 2 posterior nodes and b) displacement of 3 anterior and 3 posterior nodes.

Figure 4-21 Leaping over “islands” of forbidden land-use with displacement of 4 anterior and 4 posterior nodes.

4.4.2 MINIMUM LENGTH OF LINEAR SECTIONS

Besides the capabilities of the SAA to deal with forbidden land-use areas, one additional requisite was observed when transitioning from an intentionally simple and synthetic case-study (section 4.3) to a real-world case-case-study: the establishment of a minimum length for the HSR linear sections.

Initial applications to a real world case-study (detailed in section Chapter 5) have shown the formation of node clusters such as shown in Figure 4-22.

Figure 4-22 Plan view a HSR configuration showing evidence of node clustering, overlaying a land-use layer.

Node Cluster

Exempt

It is important to note that such a cluster would hardly represent an optimal or near-optimal solution of a real-world high-speed intercity connection such as in the fundamental principles of HSR. However, escaping from node clusters, can be a cumbersome task, mainly due to the fact that for closely positioned nodes, moves in the neighborhood are very conditioned by the horizontal angle feasibility. Furthermore, for SAA low temperature stages, in which the probability of accepting worsening configurations is also low, the HSR configurations required for the escape may yield unaccepted objective function values. In fact, if a cluster forms and propagates to low temperature stages of the SAA implementation most of the moves that would reverse the cluster are either forbidden because of the horizontal angle feasibility requirements or will not be accepted because the probability of accepting worse solutions is very low. It is not impossible for the clusters to be reversed, but it is very unlikely.

A minimum length for the HSR linear sections is to be defined to avoid the formation of the clusters and its propagation to the optimized HSR configurations. This minimum length aims at disallowing candidate configurations in which the nodes are positioned very close to each other and thus avoiding the cluster formation.

However, the value of the linear sections minimum length is conditioned by the case-study and should be defined according to the normal and limit values for the horizontal angles at intermediate nodes. This is discussed for a specific real-world case-study in section 6.1.2.