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Complexity of transmission network expansion planning

NP-hardness of connected networks and MINLP evaluation

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Transmission network expansion planning in its original formulation is NP-hard due to the subproblem Steiner trees, the minimum cost connection of an initially unconnected network with mandatory and optional nodes. By using electrical network theory we show why NP-hardness still holds when this subproblem of network design from scratch is omitted by considering already (highly) connected networks only. This refers to the case of extending a long working transmission grid for increased future demand. It will be achieved by showing that this case is computationally equivalent to \(3\)-SAT. Additionally, the original mathematical formulation is evaluated by using an appropriate state-of-the-art mixed integer non-linear programming solver in order to see how much effort in computation and implementation is really necessary to solve this problem in practice.

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  2. Notice that by this definition \(n_{ij}^{(0)} = n_{ji}^{(0)}, \bar{n}_{ij} = \bar{n}_{ji}\) etc.

  3. That is: the current from \(v_{C_j}\) to \(t_{C_j}\).

  4. \(t_0+t_1^+ + t_1^- +t_2 = 3\).

  5. At least if P \(\ne \) NP.



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The authors would like to thank Prof. D. Wagner and Dr. Ignaz Rutters from KIT for initiation and mentoring of this work, Jay Apt from CMU for valuable references, Arne Lüllmann from Fraunhofer ISI for the topic suggestion, Michael Poss from VUB/ULB for making the test data available, Dr. Matthias Oertel for initial support in electrical engineering, the NEOS project and interACT with the Baden-Württemberg Stipendium for funding this work through a scholarship program with CMU.

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Correspondence to David Oertel.

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Oertel, D., Ravi, R. Complexity of transmission network expansion planning. Energy Syst 5, 179–207 (2014).

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