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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References
Alt H, Welzl E (1988) Visibility graphs and obstacle-avoiding shortest paths. Math Meth Oper Res 32:145–164
Asadi A, Razzazi M (2007) Euclidean steiner minimal tree inside simple polygon avoiding obstacles. In: Proceedings of the international conference computational science and its applications (ICCSA ’07), pp 201–207
Bertsimas DJ (1990) The probabilistic minimum spanning tree problem. Networks 20(3):245–275
Conde E, Candia A (2007) Minimax regret spanning arborescences under uncertain costs. Eur J Oper Res 182:561–577
de Neufville R, de Weck O, Lin X, Scholtes S (2009) Identifying real options to improve the design of engineering systems, chap. In: Real options in engineering design, operations, and management. CRC Press
Eberly D (1998) Triangulation by ear clipping. http://www.geometrictools.com/
Gilbert E (1967) Minimum cost communication networks. Bell System Tech J 46:2209–2227
Gilbert E, Pollak H (1968) Steiner minimal trees. SIAM J Appl Math 16(1):1–29
Heijnen PW, Stikkelman RM, Ligtvoet A, Herder PM (2011) Using gilbert networks to reveal uncertainty in the planning of multi-user infrastructures. In: IEEE international conference on networking, sensing and control (ICNSC), pp 371–376 doi:10.1109/ICNSC.2011.5874945
Herder P, de Joode J, Ligtvoet A, Schenk S, Taneja P (2011) Buying real options – valuing uncertainty in infrastructure planning. Futures 43(9):961–969. doi:10.1016/j.futures.2011.06.005
Hwang F, Richards D, Winter P (1992) The steiner tree problem. Ann discret math 53. Elsevier Scientific Publishers, Amsterdam
Klinke A, Renn O (2002) A new approach to risk evaluation and management: risk-based, precaution-based, and discourse-based strategies. Risk Anal 22(6):1071–1094
Koreneef S (2012) Sustainable fuel from digested manure system for the salland region. Master’s thesis. Delft University of Technology
Kruskal J (1956) On the shortest spanning subtree of a graph and the traveling salesman problem. Proc Am Math Soc 7:48–50
Lee D, Preparata F (1984) Euclidean shortest paths in the presence of rectilinear barriers. Networks 14:393–410
Ligtvoet A (2013) Chapter 3: energy networks case studies. In: Images of cooperation – a methodological exploration in energy networks. Next Generation Infrastructures/Delft University of Technology
Melzak A (1961) On the problem of Steiner. Can Math Bull 4:143–148
Prim R (1957) Shortest connection networks and some generalizations. Bell Syst Tech J 36:1389–1401
Thomas D, Weng J (2006) Minimum cost flow-dependent communication networks. Networks 48(1):39–46
Trietsch D (1985) Minimal Euclidean networks with ow-dependent cost – the generalized Steiner case. Discussion paper 655, The Center for Mathematical Studies in Economics and Management Science, Northwestern University
Walker WE, Rahman SA, Cave J (2001) Adaptive policies, policy analysis, and policy-making. Eur J Oper Res 128:282–289
Weng JF, MacGregor Smith J (2001) Steiner minimal trees with one polygonal obstacle. Algorithmica 29(4):638–648
Winter P (1993) Euclidean steiner minimal trees with obstacles and steiner visibility graphs. Discret Appl Math 47(2):187–206
Winter P, Zachariasen M, Nielsen J (2002) Short trees in polygons. Discret Appl Math 118(1-2):55–72
Xue G, Lillys T, Dougherty D (1999) Computing the minimum cost pipe network interconnecting one sink and many sources. SIAM J Optim 10(2):22–42
Zachariasen M (1999) Local search for the steiner tree problem in the euclidean plane. Eur J Oper Res 119(2):282–300
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This research was partly supported by the Next Generation Infrastructures Foundation.
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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.
1.1 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.
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FOR all edges in network DO
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Replace edge by shortest path in polygon
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END FOR
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Remove cycles from the network
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Remove obsolete Steiner and corner points
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RETURN rerouted network
1.2 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.
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minimum angle ← ∞
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FOR all nodes V in the network DO
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FOR all pair of vertices (A, B) adjacent to node V DO
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Angle ← calculate angle between edge A, V and V, B using cosinus rule
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IF angle < minimum angle THEN
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angle nodes ← A, V, B
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minimum angle ← angle
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END IF
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END FOR
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END FOR
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RETURN angle nodes
1.3 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.
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Determine minimal cost spanning tree for set T of terminals
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min costs ← cost of minimal cost spanning tree
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best tree ← minimal cost spanning tree
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FOR all possible full Steiner topologies ST of terminals in T DO
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Find optimal location of the Steiner points in ST
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Add capacity to the edges in ST
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IF costs of S T < min costs THEN
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best tree ← S T
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min costs ← costs of ST
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END IF
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END FOR
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RETURN best tree
1.4 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.
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Find the subtree in the old network that connects the terminals in T
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Replace this subtree by the new one
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Update the capacities of the new edges in the network
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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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DOI: https://doi.org/10.1007/s11067-013-9199-1