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Optimal storage and transmission investments in a bilevel electricity market model

  • S.I. : Game theory and optimization
  • Published:
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Abstract

This paper analyzes the interplay of transmission and storage investments in a multistage game that we translate into a bilevel market model. In particular, on the first level we assume that a transmission system operator chooses optimal line investments and a corresponding optimal network fee. On the second level we model competitive firms that trade energy on a zonal market with limited transmission capacities and decide on their optimal storage facility investments. To the best of our knowledge, we are the first to analyze interdependent transmission and storage facility investments in a zonal market environment that accounts for the described hierarchical decision structure. As a first best benchmark, we also present an integrated, single-level problem that may be interpreted as a long-run nodal pricing model. Our numerical results show that adequate storage facility investments of firms may in general have the potential to reduce the amount of line investments of the transmission system operator. However, our bilevel zonal pricing model may yield inefficient investments in storages, which may be accompanied by suboptimal network facility extensions as compared to the nodal pricing benchmark. In this context, the chosen zonal configuration of the network will highly influence the equilibrium investment outcomes including the size and location of the newly invested facilities. As zonal pricing is used for instance in Australia or Europe, our models may be seen as valuable tools for evaluating different regulatory policy options in the context of long-run investments in storage and network facilities.

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Notes

  1. In contrast to simple two-period models [see, e.g., Sioshansi (2010) or Sioshansi (2014)] our model is able to capture more than two time periods. Ultimately, this may help to increase the accuracy of the proposed storage model.

  2. Observe that for an infinitely small slope, such an affine investment cost function will converge to a constant investment cost function.

  3. For applications of this zonal pricing formulation, see for instance Bjørndal et al. (2003), Ehrenmann and Smeers (2005), Bjørndal and Jørnsten (2007), Weibelzahl (2017), or Weibelzahl and Märtz (2017).

  4. We note that the time to build new transmission lines will in general be much larger than the time to build (battery) storage facilities. In turn, investments in storage facilities will have a larger time horizon than the market clearing interval. From this point of view, our bilevel model may be (re)interpreted as some kind of trilevel problem. On the first level, again the TSO chooses an optimal line extension plan with a corresponding network fee. The TSO anticipates optimal storage investments of perfectly competitive firms on the second level and competitive market outcomes of a zonal market on the third level. Assuming perfectly competitive firms, the objective functions of the second level and of the third level will be affine-equivalent as described in Grimm et al. (2016a). In particular, the objective of the second level will correspond to the aggregated difference between consumer surplus, variable generation cost, network charges, and storage investment cost. On the opposite, on the third level the spot market welfare objective can be expressed as the difference between consumer surplus, network fees, and variable production cost. From a mathematical point of view, we can equivalently subtract storage investment costs from the objective function of the third level without changing the optimal solution. This implies that the discussed hierarchical trilevel model may be reformulated and solved as the bilevel maximization problem introduced in this section. On the other hand, our proposed bilevel model may be reinterpreted using three levels that correspond to long-term line investment, medium-term storage investment, and short-term market clearing.

  5. An alternative approach would be to use an explicit formulation of the strong duality equality; see, e.g., Garcés et al. (2009b).

  6. Note that for the case of presentation, we neglect supply functions in (20).

References

  • Alguacil, N., Motto, A. L., & Conejo, A. J. (2003). Transmission expansion planning: A mixed-integer LP approach. IEEE Transactions on Power Systems, 18(3), 1070–1077. https://doi.org/10.1109/TPWRS.2003.814891.

  • Baringo, L., & Conejo, A. J. (2012). Transmission and wind power investment. IEEE Transactions on Power Systems, 27(2), 885–893. https://doi.org/10.1109/TPWRS.2011.2170441.

  • Bjørndal, E., Bjørndal, M., & Cai, H. (2014). Nodal pricing in a coupled electricity market. In 2014 11th international conference on the European energy market (EEM) (pp. 1–6). IEEE.

  • Bjørndal, M., & Jørnsten, K. (2001). Zonal pricing in a deregulated electricity market. The Energy Journal, 22(1), 51–73.

    Article  Google Scholar 

  • Bjørndal, M., & Jørnsten, K. (2007). Benefits from coordinating congestion management: The Nordic power market. Energy Policy, 35(3), 1978–1991.

    Article  Google Scholar 

  • Bjørndal, M., Jørnsten, K., & Pignon, V. (2003). Congestion management in the nordic power market: Counter purchasers and zonal pricing. Journal of Network Industries, 4(3), 271–292.

    Google Scholar 

  • Bohn, R. E., Caramanis, M. C., & Schweppe, F. C. (1984). Optimal pricing in electrical networks over space and time. The RAND Journal of Economics, 15, 360–376.

    Article  Google Scholar 

  • Boucher, J., & Smeers, Y. (2001). Alternative models of restructured electricity systems, part 1: No market power. Operations Research, 49(6), 821–838. https://doi.org/10.1287/opre.49.6.821.10017.

  • Boyd, S., & Vandenberghe, L. (2004). Convex optimization. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Bucksteeg, M., Trepper, K., & Weber, C. (2015). Impacts of RES-generation and demand pattern on net transfer capacity: Implications for effectiveness of market splitting in Germany. Generation Transmission and Distribution (Vol. 9, pp. 1510–1518)

  • Bunn, D. W., & Oliveira, F. S. (2003). Evaluating individual market power in electricity markets via agent-based simulation. Annals of Operations Research, 121(1), 57–77.

    Article  Google Scholar 

  • Campêlo, M., & Scheimberg, S. (2005). A simplex approach for finding local solutions of a linear bilevel program by equilibrium points. Annals of Operations Research, 138(1), 143–157.

    Article  Google Scholar 

  • Chao, H.-P., & Peck, S. (1996). A market mechanism for electric power transmission. Journal of Regulatory Economics, 10(1), 25–59.

    Article  Google Scholar 

  • Chao, H.-P., & Peck, S. (1998). Reliability management in competitive electricity markets. Journal of Regulatory Economics, 14(2), 189–200. https://doi.org/10.1023/A:1008061319181.

  • Colson, B., Marcotte, P., & Savard, G. (2007). An overview of bilevel optimization. Annals of Operations Research, 153(1), 235–256.

    Article  Google Scholar 

  • Conejo, A. J., Cheng, Y., Zhang, N., & Kang, C. (2017). Long-term coordination of transmission and storage to integrate wind power. CSEE Journal of Power and Energy Systems, 3(1), 36–43.

    Article  Google Scholar 

  • CPLEX. (2013). User’s manual for CPLEX, 12.6 edition. Armonk: IBM Corporation.

  • David, A., & Wen, F. (2001). Transmission planning and investment under competitive electricity market environment. In Power engineering society summer meeting, 2001 (Vol. 3, pp. 1725–1730). IEEE.

  • Daxhelet, O., & Smeers, Y. (2007). The EU regulation on cross-border trade of electricity: A two-stage equilibrium model. European Journal of Operational Research, 181(3), 1396–1412. https://doi.org/10.1016/j.ejor.2005.12.040.

  • Dempe, S. (2002). Foundations of bilevel programming. Springer.

  • Dempe, S. (2003). Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization, 52(3), 333–359.

    Article  Google Scholar 

  • Dempe, S., & Zemkoho, A. B. (2012). Bilevel road pricing: Theoretical analysis and optimality conditions. Annals of Operations Research, 196(1), 223–240.

    Article  Google Scholar 

  • Dijk, J., & Willems, B. (2011). The effect of counter-trading on competition in electricity markets. Energy Policy, 39(3), 1764–1773.

    Article  Google Scholar 

  • Ehrenmann, A., & Smeers, Y. (2005). Inefficiencies in European congestion management proposals. Utilities policy, 13(2), 135–152. https://doi.org/10.1016/j.jup.2004.12.007.

  • Fan, H., Cheng, H., & Yao, L. (2009). A bi-level programming model for multistage transmission network expansion planning in competitive electricity market. In Power and energy engineering conference, 2009. APPEEC 2009. Asia-Pacific (pp. 1–6). IEEE.

  • Fortuny-Amat, J., & McCarl, B. (1981). A representation and economic interpretation of a two-level programming problem. Journal of the operational Research Society, 32(9), 783–792.

    Article  Google Scholar 

  • Galiana, F. D., Conejo, A. J., & Gil, H. A. (2003). Transmission network cost allocation based on equivalent bilateral exchanges. IEEE Transactions on Power Systems, 18(4), 1425–1431.

    Article  Google Scholar 

  • Gallego, R. A., Monticelli, A., & Romero, R. (1998). Transmission system expansion planning by an extended genetic algorithm. IEEE Proceedings-Generation, Transmission and Distribution, 145(3), 329–335. https://doi.org/10.1049/ip-gtd:19981895.

  • Garcés, L. P., Conejo, A. J., García-Bertrand, R., & Romero, R. (2009a). A bilevel approach to transmission expansion planning within a market environment. IEEE Transactions on Power Systems, 24(3), 1513–1522.

    Article  Google Scholar 

  • Garcés, L. P., Conejo, A. J., García-Bertrand, R., & Romero, R. (2009b). A bilevel approach to transmission expansion planning within a market environment. IEEE Transactions on Power Systems, 24(3), 1513–1522.

    Article  Google Scholar 

  • Gast, N., Le Boudec, J.-Y., Proutière, A., & Tomozei, D.-C. (2013). Impact of storage on the efficiency and prices in real-time electricity markets. In Proceedings of the fourth international conference on future energy systems (pp. 15–26). ACM.

  • German-Transmission-System-Operators. (2017). Grid development plan electricity 2030. https://www.netzentwicklungsplan.de/sites/default/files/paragraphs-files/NEP_2030_1_Entwurf_Teil1.pdf. Accessed March 2017

  • Gil, H. A., Da Silva, E. L., & Galiana, F. D. (2002). Modeling competition in transmission expansion. IEEE Transactions on Power Systems, 17(4), 1043–1049.

    Article  Google Scholar 

  • Glachant, J.-M., & Pignon, V. (2005). Nordic congestion’s arrangement as a model for europe? Physical constraints vs. economic incentives. Utilities Policy, 13(2), 153–162.

    Article  Google Scholar 

  • Grimm, V., Martin, A., Schmidt, M., Weibelzahl, M., & Zöttl, G. (2016a). Transmission and generation investment in electricity markets: The effects of market splitting and network fee regimes. European Journal of Operational Research, 254(2), 493–509.

    Article  Google Scholar 

  • Grimm, V., Martin, A., Weibelzahl, M., & Zöttl, G. (2016b). On the long-run effects of market splitting: Why more price zones might decrease welfare. Energy Policy, 94, 453–467.

    Article  Google Scholar 

  • Hirst, E., & Kirby, B. (2001). Key transmission planning issues. The Electricity Journal, 14(8), 59–70.

    Article  Google Scholar 

  • Hogan, W. (1992). Contract networks for electric power transmission. Journal of Regulatory Economics, 4(3), 211–242.

    Article  Google Scholar 

  • Hornnes, K. S., Grande, O. S., & Bakken, B. H. (2000). Main grid development planning in a deregulated market regime. In Power engineering society winter meeting, 2000 (Vol. 2, pp. 845–849). IEEE.

  • Hu, X., & Ralph, D. (2007). Using EPECs to model bilevel games in restructured electricity markets with locational prices. Operations Research, 55(5), 809–827. https://doi.org/10.1287/opre.1070.0431.

  • Huppmann, D., & Egerer, J. (2015). National-strategic investment in european power transmission capacity. European Journal of Operational Research, 247(1), 191–203.

    Article  Google Scholar 

  • Ishizuka, Y., & Aiyoshi, E. (1992). Double penalty method for bilevel optimization problems. Annals of Operations Research, 34(1), 73–88.

    Article  Google Scholar 

  • Jenabi, M., Ghomi, S. M. T. F., & Smeers, Y. (2013). Bi-level game approaches for coordination of generation and transmission expansion planning within a market environment. IEEE Transactions on Power Systems, 28(3), 2639–2650. https://doi.org/10.1109/TPWRS.2012.2236110.

  • Jeroslow, R. G. (1985). The polynomial hierarchy and a simple model for competitive analysis. Mathematical Programming, 32(2), 146–164. https://doi.org/10.1007/BF01586088.

  • Kirschen, D., & Strbac, G. (2005). Investing in transmission. Fundamentals of Power System Economics (pp. 227–264).

  • Koch, T. (2005). Rapid mathematical programming. Dissertation for the Doctoral Degree. Berlin: Technische Universitay.

    Google Scholar 

  • Kuznia, L., Zeng, B., Centeno, G., & Miao, Z. (2013). Stochastic optimization for power system configuration with renewable energy in remote areas. Annals of Operations Research, 210(1), 411–432.

    Article  Google Scholar 

  • Mangasarian, O. (1988). A simple characterization of solution sets of convex programs. Operations Research Letters, 7(1), 21–26.

    Article  Google Scholar 

  • Neuhoff, K., Barquin, J., Bialek, J. W., Boyd, R., Dent, C. J., Echavarren, F., et al. (2013). Renewable electric energy integration: Quantifying the value of design of markets for international transmission capacity. Energy Economics, 40, 760–772.

    Article  Google Scholar 

  • Oggioni, G., & Smeers, Y. (2013). Market failures of market coupling and counter-trading in Europe: An illustrative model based discussion. Energy Economics, 35, 74–87. https://doi.org/10.1016/j.eneco.2011.11.018.

  • Pozo, D., Sauma, E., & Contreras, J. (2017). Basic theoretical foundations and insights on bilevel models and their applications to power systems. Annals of Operations Research, 254(1–2), 303–334.

    Article  Google Scholar 

  • Rious, V., Glachant, J.-M., Perez, Y., & Dessante, P. (2008). The diversity of design of tsos. Energy Policy, 36(9), 3323–3332.

    Article  Google Scholar 

  • Sariddichainunta, P., & Inuiguchi, M. (2017). Global optimality test for maximin solution of bilevel linear programming with ambiguous lower-level objective function. Annals of Operations Research, 256(2), 285–304.

    Article  Google Scholar 

  • Sauma, E. E., & Oren, S. S. (2006). Proactive planning and valuation of transmission investments in restructured electricity markets. Journal of Regulatory Economics, 30(3), 261–290.

    Article  Google Scholar 

  • Sioshansi, R. (2010). Welfare impacts of electricity storage and the implications of ownership structure. The Energy Journal, 31(2), 173–198.

    Article  Google Scholar 

  • Sioshansi, R. (2014). When energy storage reduces social welfare. Energy Economics, 41, 106–116.

    Article  Google Scholar 

  • Sioshansi, R., Denholm, P., Jenkin, T., & Weiss, J. (2009). Estimating the value of electricity storage in PJM: Arbitrage and some welfare effects. Energy Economics, 31(2), 269–277.

    Article  Google Scholar 

  • Sioshansi, R., Madaeni, S. H., & Denholm, P. (2014). A dynamic programming approach to estimate the capacity value of energy storage. IEEE Transactions on Power Systems, 29(1), 395–403.

    Article  Google Scholar 

  • Steinke, F., Wolfrum, P., & Hoffmann, C. (2013). Grid vs. storage in a 100% renewable europe. Renewable Energy, 50, 826–832.

    Article  Google Scholar 

  • Von Stackelberg, H. (2010). Market structure and equilibrium. Springer.

  • Weibelzahl, M. (2017). Nodal, zonal, or uniform electricity pricing: How to deal with network congestion? Frontiers in Energy, 11(2), 210–232.

    Article  Google Scholar 

  • Weibelzahl, M., & Märtz, A. (2017). On the effects of storage facilities on optimal zonal pricing in electricity markets. Energy Policy, 113(2), 778–794.

    Google Scholar 

  • Zambrano, C., & Olaya, Y. (2017). An agent-based simulation approach to congestion management for the colombian electricity market. Annals of Operations Research, 258(2), 217–236.

    Article  Google Scholar 

  • Zare, M. H., Borrero, J. S., Zeng, B., & Prokopyev, O. A. (2017). A note on linearized reformulations for a class of bilevel linear integer problems. Annals of Operations Research, 1–19.

  • Zugno, M., Morales, J. M., Pinson, P., & Madsen, H. (2013). A bilevel model for electricity retailers’ participation in a demand response market environment. Energy Economics, 36, 182–197.

    Article  Google Scholar 

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Acknowledgements

We thank Claudia Ehrig, Arie M.C.A. Koster, Katja Kutzer, Paul Schott and Nils Spiekermann for their valuable comments and discussions. In addition, we highly acknowledge the good cooperation with Veronika Grimm, Alexander Martin, Martin Schmidt, Christian Sölch, and Gregor Zöttl at the Friedrich-Alexander-University Erlangen-Nuremberg in the past years.

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Authors and Affiliations

  1. FIM Research Center, Wittelsbacherring 10, 95444, Bayreuth, Germany

    Martin Weibelzahl

  2. University of Bayreuth, Wittelsbacherring 10, 95444, Bayreuth, Germany

    Martin Weibelzahl

  3. Chair of Energy Economics, Institute for Industrial Production (IIP), Karlsruhe Institute of Technology, Hertzstrasse 16, 76187, Karlsruhe, Germany

    Alexandra Märtz

  4. Discrete Optimization, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 11, 91058, Erlangen, Germany

    Alexandra Märtz

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  1. Martin Weibelzahl
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Correspondence to Martin Weibelzahl.

Appendix: Sets, parameters, and variables

Appendix: Sets, parameters, and variables

Tables 4, 5, and 6 summarize the main sets, parameters, and variables used in this paper.

Table 4 Sets
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Table 5 Parameters
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Table 6 Variables and Derived Quantities
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Weibelzahl, M., Märtz, A. Optimal storage and transmission investments in a bilevel electricity market model. Ann Oper Res 287, 911–940 (2020). https://doi.org/10.1007/s10479-018-2815-1

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