October  2021, 8(4): 331-358. doi: 10.3934/jdg.2021012

The entry and exit game in the electricity markets: A mean-field game approach

1. 

Université Paris Dauphine-PSL, France

2. 

King's College London, England

3. 

ENSAE, Institut Polytechnique de Paris, France

* Corresponding author

Received  April 2020 Published  October 2021 Early access  March 2021

Fund Project: Peter Tankov and René Aïd gratefully acknowledge financial support from the ANR (project EcoREES ANR-19-CE05-0042) and from the FIME Research Initiative

We develop a model for the industry dynamics in the electricity market, based on mean-field games of optimal stopping. In our model, there are two types of agents: the renewable producers and the conventional producers. The renewable producers choose the optimal moment to build new renewable plants, and the conventional producers choose the optimal moment to exit the market. The agents interact through the market price, determined by matching the aggregate supply of the two types of producers with an exogenous demand function. Using a relaxed formulation of optimal stopping mean-field games, we prove the existence of a Nash equilibrium and the uniqueness of the equilibrium price process. An empirical example, inspired by the UK electricity market is presented. The example shows that while renewable subsidies clearly lead to higher renewable penetration, this may entail a cost to the consumer in terms of higher peakload prices. In order to avoid rising prices, the renewable subsidies must be combined with mechanisms ensuring that sufficient conventional capacity remains in place to meet the energy demand during peak periods.

Citation: René Aïd, Roxana Dumitrescu, Peter Tankov. The entry and exit game in the electricity markets: A mean-field game approach. Journal of Dynamics and Games, 2021, 8 (4) : 331-358. doi: 10.3934/jdg.2021012
References:
[1]

R. AïdL. Li and M. Ludkovski, Capacity expansion games with application to competition in power generation investments, Journal of Economic Dynamics and Control, 84 (2017), 1-31.  doi: 10.1016/j.jedc.2017.08.002.

[2]

C. Alasseur, L. Campi, R. Dumitrescu and J. Zeng, MFG model with a long-lived penalty at random jump times: Application to demand side management for electricity contracts, arXiv: 2101.06031.

[3]

C. AlasseurI. B. Taher and A. Matoussi, An extended mean field game for storage in smart grids, Journal of Optimization Theory and Applications, 184 (2020), 644-670.  doi: 10.1007/s10957-019-01619-3.

[4]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, vol. 254, Clarendon Press Oxford, 2000.

[5]

F. Bagagiolo and D. Bauso, Mean-field games and dynamic demand management in power grids, Dynamic Games and Applications, 4 (2014), 155-176.  doi: 10.1007/s13235-013-0097-4.

[6]

M. Ben Alaya and A. Kebaier, Parameter estimation for the square root diffusions: Ergodic and nonergodic cases, Stochastic Models, 28 (2012), 609-634.  doi: 10.1080/15326349.2012.726042.

[7]

D. Benatia, Functional Econometrics of Multi-Unit Auctions: An Application to the New York Electricity Market, Working paper.

[8]

C. Bertucci, Optimal stopping in mean field games, an obstacle problem approach, Journal de Mathématiques Pures et Appliquées, 120 (2018), 165-194. doi: 10.1016/j.matpur.2017.09.016.

[9]

P. C. BhagwatA. MarcheselliJ. C. RichsteinE. J. Chappin and L. J. De Vries, An analysis of a forward capacity market with long-term contracts, Energy policy, 111 (2017), 255-267.  doi: 10.1016/j.enpol.2017.09.037.

[10]

G. BouveretR. Dumitrescu and P. Tankov, Mean-field games of optimal stopping: A relaxed solution approach, SIAM Journal on Control and Optimization, 58 (2020), 1795-1821.  doi: 10.1137/18M1233480.

[11]

C. ByersT. Levin and A. Botterud, Capacity market design and renewable energy: Performance incentives, qualifying capacity, and demand curves, The Electricity Journal, 31 (2018), 65-74.  doi: 10.1016/j.tej.2018.01.006.

[12]

P. Casgrain and S. Jaimungal, Mean-field games with differing beliefs for algorithmic trading, Mathematical Finance, 30 (2020), 995-1034.  doi: 10.1111/mafi.12237.

[13]

S. ClòA. Cataldi and P. Zoppoli, The merit-order effect in the Italian power market: The impact of solar and wind generation on national wholesale electricity prices, Energy Policy, 77 (2015), 79-88. 

[14]

R. Couillet, S. M. Perlaza, H. Tembine and M. Debbah, A mean field game analysis of electric vehicles in the smart grid, in 2012 Proceedings IEEE INFOCOM Workshops, IEEE, (2012), 79-84. doi: 10.1109/INFCOMW.2012.6193523.

[15]

Department of Business Energy and Industrial Strategy, Capacity Market Five-Year Review 2014-2019, Technical report, UK Government, 2019.

[16]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, vol. 282, John Wiley & Sons, 2009. doi: 10.1002/9780470316658.

[17]

N. Fabra, A primer on capacity mechanisms, Energy Economics, 75 (2018), 323-335.  doi: 10.1016/j.eneco.2018.08.003.

[18]

O. Féron, P. Tankov and L. Tinsi, Price formation and optimal trading in intraday electricity markets, arXiv: 2009.04786.

[19]

G. Fu, P. Graewe, U. Horst and A. Popier, A mean field game of optimal portfolio liquidation, Mathematics of Operations Research, Published Online.

[20]

M. Fujii and A. Takahashi, A mean field game approach to equilibrium pricing with market clearing condition, CARF Working Paper CARF-F-473, (2020), 26 pp. doi: 10.2139/ssrn.3549733.

[21]

D. Gomes and S. Patrizi, Obstacle mean-field game problem, Interfaces and Free Boundaries, 17 (2015), 55-68.  doi: 10.4171/IFB/333.

[22]

D. Gomes and J. Saúde, Mean field games models - a brief survey, Dynamic Games and Applications, 4 (2014), 110-154.  doi: 10.1007/s13235-013-0099-2.

[23]

D. Gomes and J. Saúde, A mean-field game approach to price formation, Dynamic Games and Applications, 11 (2021), 29-53.  doi: 10.1007/s13235-020-00348-x.

[24]

C. Gouriéroux and P. Valéry, Estimation of a Jacobi process, Preprint.

[25]

A. Henriot and J.-M. Glachant, Melting-pots and salad bowls: The current debate on electricity market design for integration of intermittent RES, Utilities Policy, 27 (2013), 57-64.  doi: 10.1016/j.jup.2013.09.001.

[26]

International Energy Agency, Energy technology prospectives report, 2017.

[27]

R. Kiesel and F. Paraschiv, Econometric analysis of 15-minute intraday electricity prices, Energy Economics, 64 (2017), 77-90. 

[28]

J.-M. Lasry and P.-L. Lions, Mean field games, Japanese Journal of Mathematics, 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8.

[29]

L. F. Jacobs, Electricity Generation Costs and Hurdle Rates, Technical report, Department of Energy and Climate Change, 2016.

[30]

T. Levin and A. Botterud, Electricity market design for generator revenue sufficiency with increased variable generation, Energy Policy, 87 (2015), 392-406.  doi: 10.1016/j.enpol.2015.09.012.

[31]

B. Murray, The paradox of declining renewable costs and rising electricity prices, Forbes, (2019).

[32]

V. RiousY. Perez and F. Roques, Which electricity market design to encourage the development of demand response?, Economic Analysis and Policy, 48 (2015), 128-138.  doi: 10.1016/j.eap.2015.11.006.

[33]

S. Schwenen, Strategic bidding in multi-unit auctions with capacity constrained bidders: The New York capacity market, The RAND Journal of Economics, 46 (2015), 730-750.  doi: 10.1111/1756-2171.12104.

[34]

A. Shrivats, D. Firoozi and S. Jaimungal, A mean-field game approach to equilibrium pricing, optimal generation, and trading in solar renewable energy certificate (srec) markets, preprint, arXiv: 2003.04938.

[35]

R. TakashimaM. GotoH. Kimura and H. Madarame, Entry into the electricity market: Uncertainty, competition, and mothballing options, Energy Economics, 30 (2008), 1809-1830.  doi: 10.1016/j.eneco.2007.05.002.

[36]

R. C. Thomson and G. P. Harrison, Life Cycle Costs and Carbon Emissions of Onshore Wind Power, Technical report, University of Edinburgh, 2015.

[37]

A. Weidlich and D. Veit, A critical survey of agent-based wholesale electricity market models, Energy Economics, 30 (2008), 1728-1759.  doi: 10.1016/j.eneco.2008.01.003.

show all references

References:
[1]

R. AïdL. Li and M. Ludkovski, Capacity expansion games with application to competition in power generation investments, Journal of Economic Dynamics and Control, 84 (2017), 1-31.  doi: 10.1016/j.jedc.2017.08.002.

[2]

C. Alasseur, L. Campi, R. Dumitrescu and J. Zeng, MFG model with a long-lived penalty at random jump times: Application to demand side management for electricity contracts, arXiv: 2101.06031.

[3]

C. AlasseurI. B. Taher and A. Matoussi, An extended mean field game for storage in smart grids, Journal of Optimization Theory and Applications, 184 (2020), 644-670.  doi: 10.1007/s10957-019-01619-3.

[4]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, vol. 254, Clarendon Press Oxford, 2000.

[5]

F. Bagagiolo and D. Bauso, Mean-field games and dynamic demand management in power grids, Dynamic Games and Applications, 4 (2014), 155-176.  doi: 10.1007/s13235-013-0097-4.

[6]

M. Ben Alaya and A. Kebaier, Parameter estimation for the square root diffusions: Ergodic and nonergodic cases, Stochastic Models, 28 (2012), 609-634.  doi: 10.1080/15326349.2012.726042.

[7]

D. Benatia, Functional Econometrics of Multi-Unit Auctions: An Application to the New York Electricity Market, Working paper.

[8]

C. Bertucci, Optimal stopping in mean field games, an obstacle problem approach, Journal de Mathématiques Pures et Appliquées, 120 (2018), 165-194. doi: 10.1016/j.matpur.2017.09.016.

[9]

P. C. BhagwatA. MarcheselliJ. C. RichsteinE. J. Chappin and L. J. De Vries, An analysis of a forward capacity market with long-term contracts, Energy policy, 111 (2017), 255-267.  doi: 10.1016/j.enpol.2017.09.037.

[10]

G. BouveretR. Dumitrescu and P. Tankov, Mean-field games of optimal stopping: A relaxed solution approach, SIAM Journal on Control and Optimization, 58 (2020), 1795-1821.  doi: 10.1137/18M1233480.

[11]

C. ByersT. Levin and A. Botterud, Capacity market design and renewable energy: Performance incentives, qualifying capacity, and demand curves, The Electricity Journal, 31 (2018), 65-74.  doi: 10.1016/j.tej.2018.01.006.

[12]

P. Casgrain and S. Jaimungal, Mean-field games with differing beliefs for algorithmic trading, Mathematical Finance, 30 (2020), 995-1034.  doi: 10.1111/mafi.12237.

[13]

S. ClòA. Cataldi and P. Zoppoli, The merit-order effect in the Italian power market: The impact of solar and wind generation on national wholesale electricity prices, Energy Policy, 77 (2015), 79-88. 

[14]

R. Couillet, S. M. Perlaza, H. Tembine and M. Debbah, A mean field game analysis of electric vehicles in the smart grid, in 2012 Proceedings IEEE INFOCOM Workshops, IEEE, (2012), 79-84. doi: 10.1109/INFCOMW.2012.6193523.

[15]

Department of Business Energy and Industrial Strategy, Capacity Market Five-Year Review 2014-2019, Technical report, UK Government, 2019.

[16]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, vol. 282, John Wiley & Sons, 2009. doi: 10.1002/9780470316658.

[17]

N. Fabra, A primer on capacity mechanisms, Energy Economics, 75 (2018), 323-335.  doi: 10.1016/j.eneco.2018.08.003.

[18]

O. Féron, P. Tankov and L. Tinsi, Price formation and optimal trading in intraday electricity markets, arXiv: 2009.04786.

[19]

G. Fu, P. Graewe, U. Horst and A. Popier, A mean field game of optimal portfolio liquidation, Mathematics of Operations Research, Published Online.

[20]

M. Fujii and A. Takahashi, A mean field game approach to equilibrium pricing with market clearing condition, CARF Working Paper CARF-F-473, (2020), 26 pp. doi: 10.2139/ssrn.3549733.

[21]

D. Gomes and S. Patrizi, Obstacle mean-field game problem, Interfaces and Free Boundaries, 17 (2015), 55-68.  doi: 10.4171/IFB/333.

[22]

D. Gomes and J. Saúde, Mean field games models - a brief survey, Dynamic Games and Applications, 4 (2014), 110-154.  doi: 10.1007/s13235-013-0099-2.

[23]

D. Gomes and J. Saúde, A mean-field game approach to price formation, Dynamic Games and Applications, 11 (2021), 29-53.  doi: 10.1007/s13235-020-00348-x.

[24]

C. Gouriéroux and P. Valéry, Estimation of a Jacobi process, Preprint.

[25]

A. Henriot and J.-M. Glachant, Melting-pots and salad bowls: The current debate on electricity market design for integration of intermittent RES, Utilities Policy, 27 (2013), 57-64.  doi: 10.1016/j.jup.2013.09.001.

[26]

International Energy Agency, Energy technology prospectives report, 2017.

[27]

R. Kiesel and F. Paraschiv, Econometric analysis of 15-minute intraday electricity prices, Energy Economics, 64 (2017), 77-90. 

[28]

J.-M. Lasry and P.-L. Lions, Mean field games, Japanese Journal of Mathematics, 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8.

[29]

L. F. Jacobs, Electricity Generation Costs and Hurdle Rates, Technical report, Department of Energy and Climate Change, 2016.

[30]

T. Levin and A. Botterud, Electricity market design for generator revenue sufficiency with increased variable generation, Energy Policy, 87 (2015), 392-406.  doi: 10.1016/j.enpol.2015.09.012.

[31]

B. Murray, The paradox of declining renewable costs and rising electricity prices, Forbes, (2019).

[32]

V. RiousY. Perez and F. Roques, Which electricity market design to encourage the development of demand response?, Economic Analysis and Policy, 48 (2015), 128-138.  doi: 10.1016/j.eap.2015.11.006.

[33]

S. Schwenen, Strategic bidding in multi-unit auctions with capacity constrained bidders: The New York capacity market, The RAND Journal of Economics, 46 (2015), 730-750.  doi: 10.1111/1756-2171.12104.

[34]

A. Shrivats, D. Firoozi and S. Jaimungal, A mean-field game approach to equilibrium pricing, optimal generation, and trading in solar renewable energy certificate (srec) markets, preprint, arXiv: 2003.04938.

[35]

R. TakashimaM. GotoH. Kimura and H. Madarame, Entry into the electricity market: Uncertainty, competition, and mothballing options, Energy Economics, 30 (2008), 1809-1830.  doi: 10.1016/j.eneco.2007.05.002.

[36]

R. C. Thomson and G. P. Harrison, Life Cycle Costs and Carbon Emissions of Onshore Wind Power, Technical report, University of Edinburgh, 2015.

[37]

A. Weidlich and D. Veit, A critical survey of agent-based wholesale electricity market models, Energy Economics, 30 (2008), 1728-1759.  doi: 10.1016/j.eneco.2008.01.003.

Figure 1.  Left: typical offer curve in the French electricity market. Right: evolution of the daily mean and standard deviation of the offers by gas-fired power plants in the French electricity market
Figure 2.  Electricity demand projections used in the simulation
Figure 3.  Convergence of the gain increase upon switching to the best response. The gain is given in million GBP for the entire sector
Figure 4.  Evolution of conventional and renewable installed capacity in the three simulations. Scenario 1: renewable subsidy. Scenario 2: renewable subsidy and capacity payments for conventional plants
Figure 5.  Evolution of the electricity price in the three simulations. Top left: peak price. Top right: peak price (zoom on the last 4 years). Bottom: base price. Scenario 1: renewable subsidy. Scenario 2: renewable subsidy and capacity payments for conventional plants
Figure 6.  Conventional and renewable supply for peak (left graphs) and off-peak (right graphs) periods. Top: baseline scenario; middle: scenario 1; bottom: scenario 2
Table 1.  UK Electricity installed generation capacity in 2017, GW. Conventional steam includes coal and gas. CCGT stands for combined cycle gas turbine. Wind and solar is approximately 20% solar and 80% wind, out of which there is about 60% onshore and 40% offshore. Source: UK Energy in Brief 2018
Conventional steam CCGT Nuclear Pumped storage Wind & Solar
18.0 32.9 9.4 2.7 40.6
Conventional steam CCGT Nuclear Pumped storage Wind & Solar
18.0 32.9 9.4 2.7 40.6
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