# Optimal electricity tariff design with demand-side investments

- 6 Downloads

## Abstract

This paper proposes a method for evaluating tariffs based on mathematical programming. In contrast to previous approaches, the technique allows comparisons between portfolios of rates while capturing complexities emerging in modern electricity sectors. Welfare analyses conducted with the method can account for interactions between intermittent renewable generation, distributed energy resources and tariff structures. We explore the theoretical and practical implications of the model that underlies the technique. Our analysis shows that a regulator may induce the welfare maximizing configuration of the demand by properly updating portfolios of tariffs. We exploit the structure of the model to construct a simple algorithm to find globally optimal solutions of the associated nonlinear optimization problem; a computational experiment suggests that the specialized procedure can outperform standard nonlinear programming techniques. To illustrate the practical relevance of the rate analysis method, we compare portfolios of tariffs with data from two electricity systems. Although portfolios with sophisticated rates create value in both, these improvements differ enough to advise different portfolios. This conclusion is beyond the reach of previous techniques to analyze rates, illustrating the importance of using model-based data-driven approaches in the design of rates in modern electricity sectors.

## Keywords

Rate design Energy systems modeling Mathematical programming## Notes

### Acknowledgements

We are grateful for the insightful comments and observations of Shmuel Oren of the Industrial Engineering and Operations Research Department, and Pravin Varaiya of the Electrical Engineering and Computer Science Department, both at UC Berkeley. We would also like to thank the anonymous reviewers. Their comments helped to significantly improve this work.

## References

- 1.Acton, J.P., Bridger, M.M.: Welfare analysis of electricity rate changes. Note report N-2010-HF/FF/NSF. RAND, Santa Monica, CA (1983). https://www.rand.org/pubs/notes/n2010.html
- 2.Allcott, H.: Rethinking real-time electricity pricing. Resource Energy Econ.
**33**(4), 820–842 (2011)Google Scholar - 3.Alsac, O., Bright, J., Prais, M., Stott, B.: Further developments in LP-based optimal power flow. IEEE Trans. Power Syst.
**5**(3), 697–711 (1990)Google Scholar - 4.Boiteux, M.: Peak-load pricing. J. Bus.
**33**(2), 157–179 (1960)Google Scholar - 5.Borenstein, S.: The long-run efficiency of real-time electricity pricing. Energy J.
**26**(3), 93–116 (2005)Google Scholar - 6.Borenstein, S., Holland, S.: On the efficiency of competitive electricity markets with time-invariant retail prices. RAND J. Econ.
**36**(3), 469–493 (2005)Google Scholar - 7.Boyd, S., Vandenberghe, L.: Interior-Point Methods. Convex Optimization. Cambridge University Press, Cambridge (2009). (illustrated, reprinted)Google Scholar
- 8.Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2009). (illustrated, reprint edn)zbMATHGoogle Scholar
- 9.Carlton, D.W.: Peak load pricing with stochastic demand. Am. Econ. Rev.
**67**(5), 1006–1010 (1977)Google Scholar - 10.Caves, D.W., Christensen, L.R., Schoech, P.E., Hendricks, W.: A comparison of different methodologies in a case study of residential time-of-use electricity pricing: costbenefit analysis. J. Econ.
**26**(12), 17–34 (1984)Google Scholar - 11.Chao, H.-P.: Peak load pricing and capacity planning with demand and supply uncertainty. Bell J. Econ.
**14**(1), 179–190 (1983)Google Scholar - 12.Chao, H.P.: Efficient pricing and investment in electricity markets with intermittent resources. Energy Policy
**39**(7), 3945–3953 (2011). (special section: renewable energy policy and development)Google Scholar - 13.Crew, M.A., Fernando, C.S., Kleindorfer, P.R.: The theory of peak-load pricing: a survey. J. Regul. Econ.
**8**(3), 215–248 (1995)Google Scholar - 14.Crew, M.A., Kleindorfer, P.R.: Peak load pricing with a diverse technology. Bell J. Econ.
**7**(1), 207–231 (1976)Google Scholar - 15.De Jonghe, C., Hobbs, B., Belmans, R.: Optimal generation mix with short-term demand response and wind penetration. IEEE Trans. Power Syst.
**27**(2), 830–839 (2012)Google Scholar - 16.DOE. Demand reductions from the application of advance metering infrastructure, pricing programs and customer-based systems. Tech. Rep., U.S. Department of Energy, Dec (2012)Google Scholar
- 17.DOE. Operations and maintenance savings from advance metering infrastructure. Tech. rep., U.S. Department of Energy, Dec (2012)Google Scholar
- 18.Drèze, J.H.: Some postwar contributions of French economists to theory and public policy: With special emphasis on problems of resource allocation. Am. Econ. Rev.
**54**(4), 2–64 (1964)Google Scholar - 19.EIA. Updated capital cost estimates for electricity generation plants. Tech. Rep., US Energy Information Administration, Apr (2013)Google Scholar
- 20.Faruqui, A., Sergici, S.: Arcturus: International evidence on dynamic pricing. Electricity J.
**26**(7), 55–65 (2013)Google Scholar - 21.FERC. Assesment of demand response & advanced metering. Staff report Docket AD06-2-000, Federal Energy Regulatory Commission, Aug (2006)Google Scholar
- 22.Gallant, A.R., Koenker, R.W.: Costs and benefits of peak-load pricing of electricity. J. Econom.
**26**(1), 83–113 (1984)zbMATHGoogle Scholar - 23.Hiriart-Urruty, J.B., Lemarchal, C.: Convex Analysis and Minimization Algorithms I: Fundamentals, vol. 305. Springer, Berlin (2013)Google Scholar
- 24.Holland, S.P., Mansur, E.T.: Is real-time pricing green? The environmental impacts of electricity demand variance. Rev. Econ. Stat.
**90**(3), 550–561 (2008)Google Scholar - 25.Howrey, E.P., Varian, H.R.: Estimating the distributional impact of time-of-day pricing of electricity. J. Econom.
**26**(1), 65–82 (1984)Google Scholar - 26.IEA. Secure and efficient electricity supply during the transition to low carbon power system. Tech. rep., OECD, Paris (2013)Google Scholar
- 27.Joskow, P., Tirole, J.: Retail electricity competition. RAND J. Econ.
**37**(4), 799–815 (2006)Google Scholar - 28.Joskow, P.L.: Contributions to the theory of marginal cost pricing. Bell J. Econ.
**7**(1), 197–206 (1976)Google Scholar - 29.Joskow, P.L.: Regulation of natural monopoly. Handb. lLaw Econ.
**2**, 1227–1348 (2007)Google Scholar - 30.Joskow, P.L., Wolfram, C.D.: Dynamic pricing of electricity. Am. Econ. Rev.
**102**(3), 381–85 (2012)Google Scholar - 31.Kök, A.G., Shang, K., Yücel, Ş.: Impact of Electricity Pricing Policies on Renewable Energy Investments and Carbon Emissions. Management Science (Dec. 2016)Google Scholar
- 32.Konnov, I.V.: Selective bi-coordinate variations for resource allocation type problems. Comput. Optim. Appl.
**64**(3), 821–842 (2016)MathSciNetzbMATHGoogle Scholar - 33.Lazar, J., Gonzalez, W.: Smart rate design for a smart future. Regulatory assistance project, Montpelier, VT (2015). http://www.raponline.org/document/download/id/7680
- 34.Lillard, L.A., Aigner, D.J.: Time-of-day electricity consumption response to temperature and the ownership of air conditioning appliances. J. Bus. Econ. Stat.
**2**(1), 40–53 (1984)Google Scholar - 35.Low, S.H.: Convex relaxation of optimal power flowpart I: formulations and equivalence. IEEE Trans. Control Netw. Syst.
**1**(1), 15–27 (2014)MathSciNetzbMATHGoogle Scholar - 36.Low, S.H.: Convex relaxation of optimal power flowpart II: exactness. IEEE Trans. Control Netw. Syst.
**1**(2), 177–189 (2014)MathSciNetzbMATHGoogle Scholar - 37.Luderer, B., Minchenko, L., Satsura, T.: Multivalued Analysis and Nonlinear Programming Problems with Perturbations. Springer, Berlin (2013)zbMATHGoogle Scholar
- 38.Luenberger, D.G., Ye, Y.: Quasi-Newton methods. Linear and Nonlinear Programming. Springer, Berlin (2008)Google Scholar
- 39.Mas-Colell, A., Whinston, M.D., Green, J.R.: Microeconomic Theory. Oxford University Press, New York (1995)zbMATHGoogle Scholar
- 40.NARUC. Manual on distributed energy resources rate design and compensation. The national association of regulatory utility commissioners, Washington, D.C. (2016). http://pubs.naruc.org/pub/19FDF48B-AA57-5160-DBA1-BE2E9C2F7EA0
- 41.O’Neill, R.P., Sotkiewicz, P.M., Hobbs, B.F., Rothkopf, M.H., Stewart Jr., W.R.: Efficient market-clearing prices in markets with nonconvexities. Eur. J. Oper. Res.
**164**(1), 269–285 (2005)zbMATHGoogle Scholar - 42.Overbye, T.J., Cheng, X., Sun, Y.: A comparison of the AC and DC power flow models for LMP calculations. In: 37th Annual Hawaii International Conference on System Sciences, 2004. Proceedings of the (Jan. 2004), pp. 9Google Scholar
- 43.Panzar, J.C.: A neoclassical approach to peak load pricing. Bell J. Econ.
**7**(2), 521–530 (1976)MathSciNetGoogle Scholar - 44.Parks, R.W., Weitzel, D.: Measuring the consumer welfare effects of time-differentiated electricity prices. J. Econ.
**26**(12), 35–64 (1984)Google Scholar - 45.RAP. : Electricity regulation in the US: A guide. Regulatory assistance project, Montpelier, VT (2011). http://www.raponline.org/wp-content/uploads/2016/05/rap-lazarelectricityregulationintheus-guide-2011-03.pdf
- 46.Sauma, E.E., Oren, S.S.: Proactive planning and valuation of transmission investments in restructured electricity markets. J. Regul. Econ.
**30**(3), 261–290 (2006)Google Scholar - 47.Sioshansi, R.: OR forum-modeling the impacts of electricity tariffs on plug-in hybrid electric vehicle charging, costs, and emissions. Oper. Res.
**60**(3), 506–516 (2012)MathSciNetzbMATHGoogle Scholar - 48.Stanton, T.: Distributed Energy Resources: Status Report on Evaluating Proposals and Practices for Electric Utility Rate Design. Status Report 15-08, National Regulatory Research Institute, 8611 Second Avenue, Suite 2C, Silver Spring, MD 20910, Oct. (2015)Google Scholar
- 49.Steiner, P.O.: Peak loads and efficient pricing. Q. J. Econ.
**71**(4), 585–610 (1957)Google Scholar - 50.Stoft, S.: Power System Economics: Designing Markets for Electricity. IEEE Press, Piscataway (2002)Google Scholar
- 51.Stott, B., Jardim, J., Alsac, O.: DC power flow revisited. IEEE Trans. Power Syst.
**24**(3), 1290–1300 (2009)Google Scholar - 52.Taylor, T.N., Schwarz, P.M., Cochell, J.E.: 24/7 hourly response to electricity real-time pricing with up to eight summers of experience. J. Regul. Econ.
**27**(3), 235–262 (2005)Google Scholar - 53.Vitina, A.: Wind energy development in Denmark. In: IEA Wind Task 26 - Wind Technology, Cost, and Performance Trends in Denmark, Germany, Ireland, Norway, the European Union, and the United States: 2007-2012, hand, m. m., ed. ed., vol. Chapter 1. National Renewable Energy Laboratory, Golden, CO, USA, June 2015, pp. 16–47Google Scholar
- 54.Willig, R.D.: Consumer’s surplus without apology. Am. Econ. Rev.
**66**(4), 589–597 (1976)Google Scholar - 55.Zöttl, G.: A framework of peak load pricing with strategic firms. Oper. Res.
**58**(6), 1637–1649 (2010)MathSciNetzbMATHGoogle Scholar