# American Institute of Mathematical Sciences

September  2019, 9(3): 327-348. doi: 10.3934/naco.2019022

## Distributed optimization algorithms for game of power generation in smart grid

 1 School of Mechanical Engineering, Gwangju Institute of Science and Technology, Gwangju 500-712, Republic of Korea 2 Center for Advanced Power Systems, Florida State University, 2000 Levy Ave, Tallahassee, FL 32310, USA

*Corresponding author: Hyo-Sung Ahn

Received  April 2018 Revised  April 2019 Published  May 2019

Fund Project: This work was supported by the National Research Foundation (NRF) of Korea under the grant NRF-2017R1A2B3007034

In this paper, we consider a problem of finding optimal power generation levels for electricity users in Smart Grid (SG) with the purpose of maximizing each user's benefit selfishly. As the starting point, we first develop a generalized model based on the framework of IEEE 118 bus system, then we formulate the problem as an aggregative game, where its Nash Equilibrium (NE) is considered as the collection of optimal levels of generated powers. This paper proposes three distributed optimization strategies in forms of singularly perturbed systems to tackle the problem under limited control authority concern, with rigorous analyses provided by game theory, graph theory, control theory, and convex optimization. Our analysis shows that without constraints in power generation, the first strategy provably exponentially converges to the NE from any initializations. Moreover, under the constraint consideration, we achieve locally exponential convergence result via the other proposed algorithms, one of them is more generalized. Numerical simulations in the IEEE 118 bus system are carried out to verify the correctness of the proposed algorithms.

Citation: Chuong Van Nguyen, Phuong Huu Hoang, Hyo-Sung Ahn. Distributed optimization algorithms for game of power generation in smart grid. Numerical Algebra, Control & Optimization, 2019, 9 (3) : 327-348. doi: 10.3934/naco.2019022
##### References:
 [1] H. S. Ahn, B. Y. Kim, Y. H. Lim, B. H. Lee and K. K. Oh, Distributed coordination for optimal energy generation and distribution in cyber-physical energy networks, IEEE Trans. on Cybernetics, 48 (2018), 941-954. [2] I. Barkana, Defending on the beauty of the invariance principle, Int. Journal of Control, 87 (2014), 186-206. doi: 10.1080/00207179.2013.826385. [3] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge Univ. Press, Cambridge, U.K., 2004. doi: 10.1017/CBO9780511804441. [4] S. Boyd, A. Ghosh, B. Prabhakar and D. Shah, Randomized gossip algorithms, IEEE Trans. Inf. Theory, 52 (2006), 2508-2530. doi: 10.1109/TIT.2006.874516. [5] H. Chen, Y. Li, R. Louie and B. Vucetic, Autonomous demand side management based on energy consumption scheduling and instantaneous load billing: an aggregative game approach, IEEE Trans. Smart Grid, 5 (2014), 1744-1754. [6] A. Cherukuri and J. Cortés, Distributed generator coordination for initialization and anytime optimization in economic dispatch, IEEE Trans. on Control of Network Systems, 2 (2015), 226-237. doi: 10.1109/TCNS.2015.2399191. [7] H. B. Dürr, E. Saka and C. Ebenbauer, A smooth vector field for quadratic programming, in 51st IEEE Conf. on Decision and Control, (2012), 2515–2520. doi: 10.1109/CDC.2012.6426496. [8] A. Freeman, P. Yang and K. M. Lynch, Stability and convergence properties of dynamic average consensus estimators, in IEEE Int. Conf. on Decision and Control, (2006), 398–403. doi: 10.1109/CDC.2006.377078. [9] IEEE 118 bus, Available from: http://motor.ece.iit.edu/data/. [10] M. K. Jensen, Aggregative games and best-reply potentials, Econom. Theory, 43 (2010), 45-66. doi: 10.1007/s00199-008-0419-8. [11] H. K. Khalil, Nonlinear Systems, 3$^{nd}$ edition, Prentice Hall, New Jersey, 2002. [12] S. S. Kia, J. Cortés and S. Martínez, Dynamic average consensus under limited control authority and privacy requirement, International Journal of Robust and Nonlinear Control, 25 (2015), 1941-1966. doi: 10.1002/rnc.3178. [13] S. S. Kia, J. Cortés and S. Martínez, Singularly perturbed algorithms for dynamic average consensus, in European Control Conference, (2013), 1758–1763. [14] J. Koshal, A. Nedić and Uday V. Shanbhag, A gossip algorithm for aggregative games on graphs, in 51st IEEE Conf. on Decision and Control, (2012), 4840–4845. doi: 10.1109/CDC.2012.6426136. [15] N. Li and J. Marden, Designing games for distributed optimization, IEEE Journal of Selected Topics in Signal Processing, 7 (2013), 230-242. doi: 10.1109/CDC.2011.6161053. [16] K. Ma, G. Hu and C. J. Spanos, Distributed energy consumption control via real-time pricing feedback in smart grid, IEEE Trans. on Control System, 22 (2014), 1907-1914. doi: 10.1109/tcst.2014.2299959. [17] A. H. Mohsenian-Rad, V. W. S. Wong, J. Jatskevich, R. Schober and A. Leon-Garcia, Autonomous demand-side management based on game-theoretic energy consumption scheduling for the future smart grid, IEEE Trans. Smart Grid, 1 (2016), 320-331. doi: 10.1109/TSG.2010.2089069. [18] C. V. Nguyen, P. H. Hoang, M. H. Trinh, B-H Lee and H-S. Ahn, Distributed Nash equilibrium seeking of an aggregative game by a singular perturbed algorithm, in Australian and New Zealand Control Conference, (2017), 145–150. [19] C. V. Nguyen, P. H. Hoang, H. K. Kim and H-S. Ahn, Distributed learning in a multi-agent potential game, in 2017 Int. Conf. on Control, Automation and Systems, (2017), 266–271. [20] R. Olfati-Saber, J. A. Fax and R. M. Murray, Consensus and cooperation in networked multi-agent systems, Proc. of the IEEE, 95 (2007), 215-233. doi: 10.1109/jproc.2006.887293. [21] M. A. A. Pedrasa, T. D. Spooner and I. F. MacGill, Scheduling of demand-side resources using binary particle swarm optimization, IEEE Trans. Power Syst., 24 (2009), 1173-1181. doi: 10.1109/TPWRS.2009.2021219. [22] B. Ramanathan and V. Vittal, A framework for evaluation of advanced direct load control with minimum disruption, IEEE Trans. Power Syst., 23 (2008), 1681-1688. doi: 10.1109/TPWRS.2008.2004732. [23] W. Ren and R. W. Beard, Distributed Consensus in Multi-Vehicle Cooperative Control: Theory and Applications, Springer-Verlag, London, 2008. doi: 10.1007/978-1-84800-015-5. [24] J. B. Rosen, Existence and uniqueness of equilibrium points for concave $n$-person games, Econometrica, 33 (1965), 520-534. doi: 10.2307/1911749. [25] F. Salehisadaghiani and L. Pavel, Distributed Nash equilibrium seeking: a gossip-based algorithm, Automatica J. IFAC, 72 (2016), 209-216. doi: 10.1016/j.automatica.2016.06.004. [26] F. Salehisadaghiani and L. Pavel, Distributed Nash equilibrium seeking via the alternating direction method of multipliers, IFAC-Papers Online, 50 (2017), 6166-6171. [27] E. Shchepakina, V. Sobolev and M. P. Mortell, Singular Perturbations: Introduction to System Order Reduction Methods with Applications, Springer, Switzerland, 2014. doi: 10.1007/978-3-319-09570-7. [28] W. Shi and L. Pavel, LANA: an ADMM-like Nash equilibrium seeking algorithm in decentralized environment, in American Control Conference, (2017), 285–290. [29] M. Stanković, K. Johansson and D. Stipanović, Distributed seeking of Nash equilibria with applications to mobile sensor networks, IEEE Trans. on Automat. Control, 57 (2012), 904-919. doi: 10.1109/TAC.2011.2174678. [30] X. Wang, N. Xiao, T. Wongpiromsarn, L. Xie, E. Frazzoli and D. Rus, Distributed consensus in noncooperative congestion games: An application to road pricing, in 10th IEEE Conf. on Control and Automation, (2013), 1668–1673. doi: 10.1109/ICCA.2013.6565153. [31] M. Ye and G. Hu, Distributed Nash equilibrium seeking by a consensus based approach, IEEE Trans. on Automat. Control, 62 (2017), 4811-4818. doi: 10.1109/TAC.2017.2688452.

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##### References:
 [1] H. S. Ahn, B. Y. Kim, Y. H. Lim, B. H. Lee and K. K. Oh, Distributed coordination for optimal energy generation and distribution in cyber-physical energy networks, IEEE Trans. on Cybernetics, 48 (2018), 941-954. [2] I. Barkana, Defending on the beauty of the invariance principle, Int. Journal of Control, 87 (2014), 186-206. doi: 10.1080/00207179.2013.826385. [3] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge Univ. Press, Cambridge, U.K., 2004. doi: 10.1017/CBO9780511804441. [4] S. Boyd, A. Ghosh, B. Prabhakar and D. Shah, Randomized gossip algorithms, IEEE Trans. Inf. Theory, 52 (2006), 2508-2530. doi: 10.1109/TIT.2006.874516. [5] H. Chen, Y. Li, R. Louie and B. Vucetic, Autonomous demand side management based on energy consumption scheduling and instantaneous load billing: an aggregative game approach, IEEE Trans. Smart Grid, 5 (2014), 1744-1754. [6] A. Cherukuri and J. Cortés, Distributed generator coordination for initialization and anytime optimization in economic dispatch, IEEE Trans. on Control of Network Systems, 2 (2015), 226-237. doi: 10.1109/TCNS.2015.2399191. [7] H. B. Dürr, E. Saka and C. Ebenbauer, A smooth vector field for quadratic programming, in 51st IEEE Conf. on Decision and Control, (2012), 2515–2520. doi: 10.1109/CDC.2012.6426496. [8] A. Freeman, P. Yang and K. M. Lynch, Stability and convergence properties of dynamic average consensus estimators, in IEEE Int. Conf. on Decision and Control, (2006), 398–403. doi: 10.1109/CDC.2006.377078. [9] IEEE 118 bus, Available from: http://motor.ece.iit.edu/data/. [10] M. K. Jensen, Aggregative games and best-reply potentials, Econom. Theory, 43 (2010), 45-66. doi: 10.1007/s00199-008-0419-8. [11] H. K. Khalil, Nonlinear Systems, 3$^{nd}$ edition, Prentice Hall, New Jersey, 2002. [12] S. S. Kia, J. Cortés and S. Martínez, Dynamic average consensus under limited control authority and privacy requirement, International Journal of Robust and Nonlinear Control, 25 (2015), 1941-1966. doi: 10.1002/rnc.3178. [13] S. S. Kia, J. Cortés and S. Martínez, Singularly perturbed algorithms for dynamic average consensus, in European Control Conference, (2013), 1758–1763. [14] J. Koshal, A. Nedić and Uday V. Shanbhag, A gossip algorithm for aggregative games on graphs, in 51st IEEE Conf. on Decision and Control, (2012), 4840–4845. doi: 10.1109/CDC.2012.6426136. [15] N. Li and J. Marden, Designing games for distributed optimization, IEEE Journal of Selected Topics in Signal Processing, 7 (2013), 230-242. doi: 10.1109/CDC.2011.6161053. [16] K. Ma, G. Hu and C. J. Spanos, Distributed energy consumption control via real-time pricing feedback in smart grid, IEEE Trans. on Control System, 22 (2014), 1907-1914. doi: 10.1109/tcst.2014.2299959. [17] A. H. Mohsenian-Rad, V. W. S. Wong, J. Jatskevich, R. Schober and A. Leon-Garcia, Autonomous demand-side management based on game-theoretic energy consumption scheduling for the future smart grid, IEEE Trans. Smart Grid, 1 (2016), 320-331. doi: 10.1109/TSG.2010.2089069. [18] C. V. Nguyen, P. H. Hoang, M. H. Trinh, B-H Lee and H-S. Ahn, Distributed Nash equilibrium seeking of an aggregative game by a singular perturbed algorithm, in Australian and New Zealand Control Conference, (2017), 145–150. [19] C. V. Nguyen, P. H. Hoang, H. K. Kim and H-S. Ahn, Distributed learning in a multi-agent potential game, in 2017 Int. Conf. on Control, Automation and Systems, (2017), 266–271. [20] R. Olfati-Saber, J. A. Fax and R. M. Murray, Consensus and cooperation in networked multi-agent systems, Proc. of the IEEE, 95 (2007), 215-233. doi: 10.1109/jproc.2006.887293. [21] M. A. A. Pedrasa, T. D. Spooner and I. F. MacGill, Scheduling of demand-side resources using binary particle swarm optimization, IEEE Trans. Power Syst., 24 (2009), 1173-1181. doi: 10.1109/TPWRS.2009.2021219. [22] B. Ramanathan and V. Vittal, A framework for evaluation of advanced direct load control with minimum disruption, IEEE Trans. Power Syst., 23 (2008), 1681-1688. doi: 10.1109/TPWRS.2008.2004732. [23] W. Ren and R. W. Beard, Distributed Consensus in Multi-Vehicle Cooperative Control: Theory and Applications, Springer-Verlag, London, 2008. doi: 10.1007/978-1-84800-015-5. [24] J. B. Rosen, Existence and uniqueness of equilibrium points for concave $n$-person games, Econometrica, 33 (1965), 520-534. doi: 10.2307/1911749. [25] F. Salehisadaghiani and L. Pavel, Distributed Nash equilibrium seeking: a gossip-based algorithm, Automatica J. IFAC, 72 (2016), 209-216. doi: 10.1016/j.automatica.2016.06.004. [26] F. Salehisadaghiani and L. Pavel, Distributed Nash equilibrium seeking via the alternating direction method of multipliers, IFAC-Papers Online, 50 (2017), 6166-6171. [27] E. Shchepakina, V. Sobolev and M. P. Mortell, Singular Perturbations: Introduction to System Order Reduction Methods with Applications, Springer, Switzerland, 2014. doi: 10.1007/978-3-319-09570-7. [28] W. Shi and L. Pavel, LANA: an ADMM-like Nash equilibrium seeking algorithm in decentralized environment, in American Control Conference, (2017), 285–290. [29] M. Stanković, K. Johansson and D. Stipanović, Distributed seeking of Nash equilibria with applications to mobile sensor networks, IEEE Trans. on Automat. Control, 57 (2012), 904-919. doi: 10.1109/TAC.2011.2174678. [30] X. Wang, N. Xiao, T. Wongpiromsarn, L. Xie, E. Frazzoli and D. Rus, Distributed consensus in noncooperative congestion games: An application to road pricing, in 10th IEEE Conf. on Control and Automation, (2013), 1668–1673. doi: 10.1109/ICCA.2013.6565153. [31] M. Ye and G. Hu, Distributed Nash equilibrium seeking by a consensus based approach, IEEE Trans. on Automat. Control, 62 (2017), 4811-4818. doi: 10.1109/TAC.2017.2688452.
Simulation results for Algorithm 1
Simulation results for Algorithm 2
Simulation results for Algorithm 3

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