Maximising the Worth of Nascent Networks

Abstract

The roll-out of new infrastructural networks in space-constrained areas requires the careful consideration of limited paths. This design task is aggravated if the number and/or location of connectors is unknown. The novel combination of graph theory and concepts of exploratory modelling in this contribution allow for an analysis of most likely paths that maximise the value for the planners. We apply this approach to two proposed energy networks in the Netherlands: a biogas network of farmers in the province of Overijssel and an LNG pipeline connecting industries in the Port of Rotterdam. The examples demonstrate the ease of use and simplicity of this approach that transparently deals with unknowns.

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Acknowledgments

This research was partly supported by the Next Generation Infrastructures Foundation.

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Correspondence to P. W. Heijnen or A. Ligtvoet.

Appendix A: Pseudo-Code of the Main Procedures

Appendix A: Pseudo-Code of the Main Procedures

This section shows pseudo-code of the main procedures in the heuristic algorithm complementary to the flowchart in Fig. 8.

A.1 [REROUTING] – Procedure to Reroute the Network to Allowed Region

The procedure needs as input the network, defined as a graph with nodes and edges, and the trangulation of the allowed region, defined as a simple polygon. The output of the procedure is a network that is completely located within the allowed region.

  1. 1.

    FOR all edges in network DO

  2. 2.

    Replace edge by shortest path in polygon

  3. 3.

    END FOR

  4. 4.

    Remove cycles from the network

  5. 5.

    Remove obsolete Steiner and corner points

  6. 6.

    RETURN rerouted network

A.2 [MINIMUM ANGLE] – Procedure to Find the Minimum Angle in the Network

The procedure needs as input the network, defined as a graph with nodes and edges, where the nodes are on specified locations and the capacities are given as weights to the edges. The output of the procedure is the minimum angle between two edges of the network.

  1. 1.

    minimum angle ←

  2. 2.

    FOR all nodes V in the network DO

  3. 3.

         FOR all pair of vertices (A, B) adjacent to node V DO

  4. 4.

              Angle ← calculate angle between edge A, V and V, B using cosinus rule

  5. 5.

              IF angle < minimum angle THEN

  6. 6.

                  angle nodes ← A, V, B

  7. 7.

                  minimum angle ← angle

  8. 8.

              END IF

  9. 9.

         END FOR

  10. 10.

    END FOR

  11. 11.

    RETURN angle nodes

A.3 [EWSM-ST] – Procedure to find the Edge Weighted Steiner Minimal Subtree

The procedure needs as input a set T of terminals and the net flow entering or leaving these terminals. The output of the procedure is the edge-weighted Steiner minimal subtree that connects these terminals directly by a minimal cost spanning tree or by using | T | − 2 Steiner points.

  1. 1.

    Determine minimal cost spanning tree for set T of terminals

  2. 2.

    min costs ← cost of minimal cost spanning tree

  3. 3.

    best tree ← minimal cost spanning tree

  4. 4.

    FOR all possible full Steiner topologies ST of terminals in T DO

  5. 5.

         Find optimal location of the Steiner points in ST

  6. 6.

         Add capacity to the edges in ST

  7. 7.

         IF costs of S T < min costs THEN

  8. 8.

             best tree ← S T

  9. 9.

             min costs ← costs of ST

  10. 10.

         END IF

  11. 11.

    END FOR

  12. 12.

    RETURN best tree

A.4 [SUBSTITUTE] – Procedure to Replace Part of the Network by a Better Subtree

The procedure needs as input the overall network connecting all terminals and a subtree connecting a subset T of terminals. The output of the procedure is the new network in which the relevant part of the network is replaced.

  1. 1.

    Find the subtree in the old network that connects the terminals in T

  2. 2.

    Replace this subtree by the new one

  3. 3.

    Update the capacities of the new edges in the network

  4. 4.

    RETURN updated network

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Heijnen, P., Ligtvoet, A., Stikkelman, R. et al. Maximising the Worth of Nascent Networks. Netw Spat Econ 14, 27–46 (2014). https://doi.org/10.1007/s11067-013-9199-1

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Keywords

  • Graph theory
  • Steiner tree
  • Uncertainty
  • Infrastructural networks and constraints
  • Maximising worth