# Playing Pollution Games with Thermal Electricity Generators

- 72 Downloads

## Abstract

This paper discusses the economic and environmental implications of a stylised electricity market with transmission grid constraints and shared temporal pollution standards that restrict the joint strategy space of the agents. These are problematic to enforce if individual monitoring is impossible or very expensive. For such situations, we propose a time-dependent (or “open-loop dynamic”), game-theoretic model capable of analysing coupled constraints equilibria, also known as generalised Nash. We compute these equilibria for thermal generators subjected to annual pollution limits and instantaneous grid restrictions for a three-node dc model with a two-period load duration curve. The model illustrates the possibility that well-meaning environmental regulation might harm consumer surplus. It also highlights the cost to efficiency of regulatory attempts to ease the burden of compliance.

## Keywords

Coupled constraints games Generalised Nash equilibrium Deregulated electric industry Electricity transmission Pollution constraints## References

- 1.Expósito, A. G., Conejo, A. J., & Cañizares, C. (2008).
*Electric energy systems: Analysis and operation*. Boca Raton: CRC.Google Scholar - 2.Boucekkine, R., Krawczyk, J. B., & Vallée, T. (2010). Towards an understanding of tradeoffs between regional wealth, tightness of a common environmental con- straint and the sharing rules.
*Journal of Economic Dynamics and Control, 34*(9), 1813–1835.Google Scholar - 3.Cabe, R., & Herriges, J. (1992). The regulation of non-point-source pollution under imperfect and asymmetric information.
*Journal of Environmental Economics and Management, 22*(2), 134–146.Google Scholar - 4.Chen, Y., Sijm, J., Hobbs, B., & Lise, W. (2008). Implications of CO
_{2}emissions trading for short-run electricity market outcomes in Northwest Europe.*Journal of Regulatory Economics, 34*, 251–281.Google Scholar - 5.Chu, K. C., Jamshidi, M., & Levitan, R. (1977). An approach to on-line power dispatch with ambient air pollution constraints.
*IEEE Transactions on Automatic Control, 22*(3), 385–396.Google Scholar - 6.Contreras, J., Klusch, M., & Krawczyk, J. B. (2004). Numerical solutions to Nash Cournot equilibria in coupled constraint electricity markets.
*IEEE Transactions on Power Systems, 19*(1), 195–206.Google Scholar - 7.Contreras, J., Krawczyk, J. B., Zuccollo, J. (2008).
*The invisible polluter: can regulators save consumer surplus?*(pp. 1–32) In: 13th International Symposium of the International Society of Dynamic Games, Wroclaw, Poland, URL http://hdl.handle.net/10063/528. - 8.Contreras, J., Krawczyk, J. B., Zuccollo, J., & Garcıa, J. (2013). Competition of thermal electricity generators with coupled transmission and emission constraints.
*Journal of Energy Engineering, 139*(4), 239–252. https://doi.org/10.1061/(ASCE)EY.1943-7897.0000113.Google Scholar - 9.Contreras, J., Krawczyk, J. B., & Zuccollo, J. (2016). Economics of collective monitoring: a study of environmentally constrained electricity generators.
*J Comput Manag Sci, 13*(3), 349–369.Google Scholar - 10.Drouet, L., Haurie, A., Moresino, F., Vial, J. P., Vielle, M., & Viguier, L. (2008). An oracle based method to compute a coupled equilibrium in a model of international climate policy.
*Computational Management Science, 5*, 119–140.Google Scholar - 11.Du, S., Ma, F., Fu, Z., Zhu, L., & Zhang, J. (2015). Game-theoretic analysis for an emission-dependent supply chain in a ‘cap-and-trade’ system.
*Annals of Operations Research, 228*(1), 135–149.Google Scholar - 12.Du, S., Hu, L., & Song, M. (2016). Production optimization considering environmental performance and preference in the cap-and-trade system.
*Journal of Cleaner Production, 112*, 1600–1607.Google Scholar - 13.Facchinei, F., & Kanzow, C. (2007). Generalised Nash equilibrium problems.
*4OR: A Quartely Journal of Operations Research, 5*(3), 173–210.Google Scholar - 14.Facchinei, F., Fischer, A., & Piccialli, V. (2007). On generalised Nash games and variational inequalities.
*Operations Research Letters, 35*, 159–164.Google Scholar - 15.Facchinei, F., Fischer, A., & Piccialli, V. (2009). Generalised Nash equilibrium problems and newton methods.
*Mathematical Programming, Ser B, 117*, 163–194.Google Scholar - 16.Fukushima, M. (2011). Restricted generalised Nash equilibria and controlled penalty algorithm.
*Computational Management Science, 8*(3), 201–218.Google Scholar - 17.Harker, P. T. (1991). Generalised Nash games and quasivariational inequalities.
*European Journal of Operational Research, 4*, 81–94.Google Scholar - 18.Haurie, A. (1994). Environmental coordination in dynamic oligopolistic markets.
*Group Decision and Negotiation, 4*, 46–67.Google Scholar - 19.Haurie, A., & Krawczyk, J. B. (1997). Optimal charges on river effluent from lumped and distributed sources.
*Environmental Modelling and Assessment, 2*, 177–189.Google Scholar - 20.He, P., Zhang, W., Xu, X., & Bian, Y. (2015). Production lot-sizing and carbon emissions under cap-and-trade and carbon tax regulations.
*Journal of Cleaner Production, 103*, 241–248.Google Scholar - 21.von Heusingen, C., & Kanzow, A. (2009). Optimization reformulations of the generalised Nash equilibrium problem using Nikaido-Isoda-type functions.
*Computational Optimization and Applications, 43*(3), 353–377.Google Scholar - 22.Hobbs, B. E. (2001). Linear complementarity models of Nash-Cournot competition in bilateral and POOLCO power markets.
*IEEE Transactions on Power Systems, 16*(2), 194–202.Google Scholar - 23.Horan, R., Shortle, J., & Abler, D. (1998). Ambient taxes when polluters have multiple choices.
*Journal of Environmental Economics and Management, 36*(2), 186–199.Google Scholar - 24.Karp, L. (2005). Nonpoint source pollution taxes and excessive tax burden.
*Environmental and Resource Economics, 31*(2), 229–251.Google Scholar - 25.Krawczyk, J. B. (2007). Numerical solutions to coupled-constraint (or generalised) Nash equilibrium problems.
*Computational Management Science, 4*, 183–204. https://doi.org/10.1007/s10287-006-0033-9.Google Scholar - 26.Krawczyk, J. B., & Tidball, M. (2016). Economic problems with constraints: how efficiency relates to equilibrium.
*International Game Theory Review, 18*(04), 1650011. https://doi.org/10.1142/S0219198916500110.Google Scholar - 27.Krawczyk, J. B., Townsend, W. (2014). A MATLAB application which solves for couple-constraint Nash equilbria - NIRA-GUI Code. URL https://code.google.com/p/nira-gui/.
- 28.Krawczyk, J. B., Zuccollo, J. (2007). NIRA-3: an improved MATLAB package for finding Nash equilibria in infinite games. Munich Personal RePEc Archive, URL http://mpra.ub.uni-muenchen.de/1119/.
- 29.Krawczyk, J. B., Contreras, J., Zuccollo, J. (2008).
*Thermal electricity generators’ competition with coupled constraints*. In: Conference Abstracts of the International Conference on Modelling, Computation and Optimization, Indian Statistical Institute, Delhi Centre.Google Scholar - 30.Metzler, C., Hobbs, B., & Pang, J. S. (2003). Nash-Cournot equilibria in power markets on a linearized DC network with arbitrage: formulations and properties.
*Networks and Spatial Economics, 3*(2), 123–150.Google Scholar - 31.Muslu, M. (2004). Economic dispatch with environmental considerations: treadeoff curves and emission reduction rates.
*Electric Power Systems Research, 71*, 153–158.Google Scholar - 32.Pang, J. S., & Fukushima, M. (2005). Quasi-variational inequalities, generalised Nash equilibria and multi-leader-follower games.
*Computational Management Science, 1*, 21–56.Google Scholar - 33.Purchala, K., Meeus, L., van Dommelen, D., Belmans, R. (12–16 July, 2005).
*Usefulness of dc power flow for active power flow analysis*(pp. 454–459). In: Power & Energy Society General Meeting, San Francisco: IEEE.Google Scholar - 34.Ramanathan, R. (1994). Emission constrained economic dispatch.
*IEEE Transactions on Power Systems, 9*(4), 1994–2000.Google Scholar - 35.Rosen, J. B. (1965). Existence and uniqueness of equilibrium points for concave n-person games.
*Econometrica, 33*(3), 520–534.Google Scholar - 36.Segerson, K. (1988). Uncertainty and incentives for nonpoint pollution control.
*Journal of environmental economics and management, 15*(1), 87–98.Google Scholar - 37.Shortle, J., & Horan, R. (2001). The economics of nonpoint pollution control.
*Journal of Economic Surveys, 15*(3), 251–253.Google Scholar - 38.Stevenson, W. D. (1982).
*Elements of power system analysis*. New York: McGraw-Hill New York.Google Scholar - 39.Wei, J. Y., & Smeers, Y. (1999). Spatial oligopolistic electricity models with Cournot generators and regulated transmission prices.
*Operations Research, 47*(1), 102–112.Google Scholar