Equilibrium Seeking and Optimal Selection Algorithms in PeertoPeer Energy Markets
Abstract
:1. Introduction
1.1. Notation
1.2. Operator Theory
2. PeertoPeer Energy Markets as a Generalized Nash Equilibrium Problem
3. Market Clearing Mechanism with Improved Convergence Speed
3.1. Market Clearing Algorithms Based on the Preconditioned Proximal Point Method
Algorithm 1 PPPbased Market Clearing Mechanism 
Algorithm 2 Central update of DNO 
Step sizes: set ${\alpha}_{N+1}<1/(3+N{max}_{h\in \mathcal{H}}{d}_{h}^{\mathrm{mg}})$, ${\gamma}^{\mathrm{mg}}<1/N$, ${\beta}^{\mathrm{tg}}<\left(\right\mathcal{N}\phantom{\rule{0.166667em}{0ex}}+{\phantom{\rule{0.166667em}{0ex}}\left\mathcal{B}\right)}^{1}$, and ${\beta}_{y}^{\mathrm{pb}}<(1\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}2{\mathcal{N}}_{y}^{\mathrm{b}}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{\mathcal{B}}_{y}{\left\right)}^{1}$, for all busses $y\phantom{\rule{0.166667em}{0ex}}\in \phantom{\rule{0.166667em}{0ex}}\mathcal{B}$.

Algorithm 3 Local update of prosumer $i\in \mathcal{I}$ 
Step sizes: For each $i\in \mathcal{I}$, set ${\alpha}_{i}<1/(3+N{max}_{h\in \mathcal{H}}{d}_{h}^{\mathrm{mg}})$, ${\beta}_{(i,j)}^{\mathrm{tr}}={\beta}_{(j,i)}^{\mathrm{tr}}<1/2$, for all $j\in {\mathcal{N}}_{i}$.

 a.
 (Inertial PPP variant)${\tau}_{1}=1+\theta $,${\tau}_{2}=\theta $,$\theta \in (0,{\textstyle \frac{1}{3}})$, ${\nu}_{i}={u}_{i}$, for all$i\in {\mathcal{I}}^{+}$, and$({\varphi}^{\mathrm{tg}},{({\varphi}_{y}^{\mathrm{pb}})}_{y\in \mathcal{B}},{({({\varphi}_{(i,j)}^{\mathrm{tr}})}_{j\in {\mathcal{N}}_{i}})}_{i\in \mathcal{I}},{\varphi}^{\mathrm{mg}})=({\mu}^{\mathrm{tg}},{({\mu}_{y}^{\mathrm{pb}})}_{y\in \mathcal{B}},{({({\mu}_{(i,j)}^{\mathrm{tr}})}_{j\in {\mathcal{N}}_{i}})}_{i\in \mathcal{I}},{\lambda}^{\mathrm{mg}})=\mathbf{\rho}$;
 b.
 (Overrelaxed PPP variant)${\tau}_{1}=\theta $,${\tau}_{2}=1\theta $,$\theta \in (1,2)$,${\nu}_{i}={\tilde{u}}_{i}$, for all$i\in {\mathcal{I}}^{+}$, and$({\varphi}^{\mathrm{tg}},{({\varphi}_{y}^{\mathrm{pb}})}_{y\in \mathcal{B}},{({({\varphi}_{(i,j)}^{\mathrm{tr}})}_{j\in {\mathcal{N}}_{i}})}_{i\in \mathcal{I}},{\varphi}^{\mathrm{mg}})=({\tilde{\mu}}^{\mathrm{tg}},{({\tilde{\mu}}_{y}^{\mathrm{pb}})}_{y\in \mathcal{B}},{({({\tilde{\mu}}_{(i,j)}^{\mathrm{tr}})}_{j\in {\mathcal{N}}_{i}})}_{i\in \mathcal{I}},{\tilde{\lambda}}^{\mathrm{mg}})=\tilde{\mathbf{\rho}}$.
3.2. Rate Improvement Evaluation
4. Equilibrium Selection as Preferred by the Network Operator
4.1. Formulation of Optimal Equilibrium Selection Problem
4.2. Optimal Equilibrium Selection Algorithm
4.3. Equilibria That Minimize Line Congestion
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
DNO  Distribution network operator 
HSDM  Hybrid steepest descent 
GNE  Generalized Nash equilibrium 
GNEP  Generalized Nash equilibrium problem 
IEEE  Institute of Electrical and Electronics Engineers 
KKT  Karush–Kuhn–Tucker 
PPP  Preconditioned proximal point 
VI  Variational inequality 
Appendix A. Proof of Proposition 2
References
 Sousa, T.; Soares, T.; Pinson, P.; Moret, F.; Baroche, T.; Sorin, E. Peertopeer and communitybased markets: A comprehensive review. Renew. Sustain. Energy Rev. 2019, 104, 367–378. [Google Scholar] [CrossRef] [Green Version]
 Soto, E.A.; Bosman, L.B.; Wollega, E.; LeonSalas, W.D. Peertopeer energy trading: A review of the literature. Appl. Energy 2021, 283, 116268. [Google Scholar] [CrossRef]
 Tushar, W.; Saha, T.K.; Yuen, C.; Smith, D.; Poor, H.V. Peertopeer trading in electricity networks: An overview. IEEE Trans. Smart Grid 2020, 11, 3185–3200. [Google Scholar] [CrossRef] [Green Version]
 Tushar, W.; Yuen, C.; Saha, T.K.; Morstyn, T.; Chapman, A.C.; Alam, M.J.E.; Hanif, S.; Poor, H.V. Peertopeer energy systems for connected communities: A review of recent advances and emerging challenges. Appl. Energy 2021, 282, 116131. [Google Scholar] [CrossRef]
 Tushar, W.; Yuen, C.; MohsenianRad, H.; Saha, T.; Poor, H.V.; Wood, K.L. Transforming energy networks via peertopeer energy trading: The potential of gametheoretic approaches. IEEE Signal Process. Mag. 2018, 35, 90–111. [Google Scholar] [CrossRef] [Green Version]
 Noor, S.; Yang, W.; Guo, M.; van Dam, K.H.; Wang, X. Energy demand side management within microgrid networks enhanced by blockchain. Appl. Energy 2018, 228, 1385–1398. [Google Scholar] [CrossRef]
 Yang, X.; Wang, G.; He, H.; Lu, J.; Zhang, Y. Automated demand response framework in ELNs: Decentralized scheduling and smart contract. IEEE Trans. Syst. Man Cybern. Syst. 2020, 50, 58–72. [Google Scholar] [CrossRef]
 Bhatti, B.A.; Broadwater, R. Energy trading in the distribution system using a nonmodel based game theoretic approach. Appl. Energy 2019, 253, 113532. [Google Scholar] [CrossRef]
 Wang, Z.; Liu, F.; Ma, Z.; Chen, Y.; Jia, M.; Wei, W.; Wu, Q. Distributed generalized Nash equilibrium seeking for energy sharing games in prosumers. IEEE Trans. Power Syst. 2021, 36, 3973–3986. [Google Scholar] [CrossRef]
 Belgioioso, G.; Ananduta, W.; Grammatico, S.; OcampoMartinez, C. Energy management and peertopeer trading in future smart grids: A distributed gametheoretic approach. In Proceedings of the 2020 European Control Conference (ECC), Saint Petersburg, Russia, 12–15 May 2020; pp. 1324–1329. [Google Scholar]
 Belgioioso, G.; Ananduta, W.; Grammatico, S.; OcampoMartinez, C. Operationallysafe peertopeer energy trading in distribution grids: A gametheoretic marketclearing mechanism. IEEE Trans. Smart Grid 2022. [Google Scholar] [CrossRef]
 Belgioioso, G.; Yi, P.; Grammatico, S.; Pavel, L. Distributed generalized Nash equilibrium seeking: An operatortheoretic perspective. IEEE Control Syst. Mag. 2022, 42, 87–102. [Google Scholar] [CrossRef]
 Yi, P.; Pavel, L. An operator splitting approach for distributed generalized Nash equilibria computation. Automatica 2019, 102, 111–121. [Google Scholar] [CrossRef] [Green Version]
 Bianchi, M.; Belgioioso, G.; Grammatico, S. Fast generalized Nash equilibrium seeking under partialdecision information. Automatica 2022, 136, 110080. [Google Scholar] [CrossRef]
 Belgioioso, G.; Grammatico, S. Semidecentralized generalized Nash equilibrium seeking in monotone aggregative games. IEEE Trans. Autom. Control. 2021. [Google Scholar] [CrossRef]
 Gadjov, D.; Pavel, L. Singletimescale distributed GNE seeking for aggregative games over networks via forward–backward operator splitting. IEEE Trans. Autom. Control 2021, 66, 3259–3266. [Google Scholar] [CrossRef]
 Benenati, E.; Ananduta, W.; Grammatico, S. Optimal selection and tracking of generalized Nash equilibria in monotone games. arXiv 2022, arXiv:2203.07765. [Google Scholar]
 Benenati, E.; Ananduta, W.; Grammatico, S. On the optimal selection of generalized Nash equilibria in linearlycoupled aggregative games. In Proceedings of the 61st Conference on Decision and Control, Cancún, Mexico, 6–9 December 2022. to appear. [Google Scholar]
 Sorin, E.; Bobo, L.; Pinson, P. Consensusbased approach to peertopeer electricity markets With product differentiation. IEEE Trans. Power Syst. 2019, 34, 994–1004. [Google Scholar] [CrossRef] [Green Version]
 Atzeni, I.; Ordóñez, L.G.; Scutari, G.; Palomar, D.P.; Fonollosa, J.R. Demandside management via distributed energy generation and storage optimization. IEEE Trans. Smart Grid 2013, 4, 866–876. [Google Scholar] [CrossRef]
 Le Cadre, H.; Jacquot, P.; Wan, C.; Alasseur, C. Peertopeer electricity market analysis: From variational to generalized Nash equilibrium. Eur. J. Oper. Res. 2020, 282, 753–771. [Google Scholar] [CrossRef] [Green Version]
 Baroche, T.; Moret, F.; Pinson, P. Prosumer markets: A unified formulation. In Proceedings of the 2019 IEEE Milan PowerTech, Milan, Italy, 23–27 June 2019; pp. 1–6. [Google Scholar]
 Facchinei, F.; Pang, J.S. 12 Nash equilibria: The variational approach. In Convex Optimization in Signal Processing and Communications; Palomar, D.P., Eldar, Y.C., Eds.; Cambridge University Press: Cambridge, UK, 2010; pp. 443–491. [Google Scholar]
 Bauschke, H.H.; Combettes, P.L. Convex Analysis and Monotone Operator Theory in Hilbert Spaces; Springer: Berlin/Heidelberg, Germany, 2011. [Google Scholar]
 Polyak, B.T. Some methods of speeding up the convergence of iteration methods. USSR Comput. Math. Math. Phys. 1964, 4, 1–17. [Google Scholar] [CrossRef]
 Nesterov, Y. Introductory Lectures on Convex Optimization: A Basic Course; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2004; Volume 87. [Google Scholar]
 Ghadimi, E.; Feyzmahdavian, H.R.; Johansson, M. Global convergence of the heavyball method for convex optimization. In Proceedings of the 2015 European control conference (ECC), Linz, Austria, 15–17 July 2015; pp. 310–315. [Google Scholar]
 Yamada, I.; Ogura, N. Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasinonexpansive mappings. Numer. Funct. Anal. Optim. 2005, 25, 619–655. [Google Scholar] [CrossRef]
 Ogura, N.; Yamada, I. Nonstrictly convex minimization over the bounded fixed point set of a nonexpansive mapping. Numer. Funct. Anal. Optim. 2003, 24, 129–135. [Google Scholar] [CrossRef]
 Auslender, A.; Teboulle, M. Lagrangian duality and related multiplier methods for variational inequality problems. SIAM J. Optim. 2000, 10, 1097–1115. [Google Scholar] [CrossRef]
Test Case  $\mathit{\phi}$ (Normalized) of Algorithm 1  

Baseline  For Equilibrium Selection  On Modified Game  
37bus  $100\%$  $74.6\%$  $71.9\%$ 
123bus  $100\%$  $75.6\%$  $73.4\%$ 
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. 
© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Ananduta, W.; Grammatico, S. Equilibrium Seeking and Optimal Selection Algorithms in PeertoPeer Energy Markets. Games 2022, 13, 66. https://doi.org/10.3390/g13050066
Ananduta W, Grammatico S. Equilibrium Seeking and Optimal Selection Algorithms in PeertoPeer Energy Markets. Games. 2022; 13(5):66. https://doi.org/10.3390/g13050066
Chicago/Turabian StyleAnanduta, Wicak, and Sergio Grammatico. 2022. "Equilibrium Seeking and Optimal Selection Algorithms in PeertoPeer Energy Markets" Games 13, no. 5: 66. https://doi.org/10.3390/g13050066