Asymptotic Optimal Online Energy Distribution in the Smart Grid ... utility, cost, pricing functions, a general model that can .... He is on the Edito...

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provider, should be jointly considered. In this paper, we take a holistic approach, to incorporate the key design factors including user’s utility and cost, grid load smoothing, dynamic pricing, and energy provisioning cost in a problem formulation. To solve the real-time energy distribution problem, we first present an offline algorithm that can produce optimal solutions but assuming that the future user and grid information are known in advance. Based on the offline algorithm, we then develop an online algorithm that does not require any future information. As the name suggests, an online algorithm operates in an online setting, where the complete input is not known a priori [9]. It is very useful for solving problems with uncertainties. We find the online algorithm particularly suitable in addressing the lack of accurate mathematical models and the lack of future information for electricity demand and supply in this problem. We also prove that the online algorithm converges to the optimal offline algorithm almost surely.

Fig. 1. Illustration of the key elements and interactions in the smart grid. The proposed framework is quite general. It does not require any specific models for the electricity demand and supply processes, and only have some mild assumptions on the utility, cost, and price functions (e.g., convex and differentiable). The proposed algorithm can thus be applied to many different scenarios. The online algorithm also does not require any future information, making it easy to be implemented in a real smart grid system. It is also asymptotically optimal, a highly desirable property.

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IEEE COMSOC MMTC E-Letter The proposed algorithm is evaluated with trace-driven simulation using energy consumption traces recorded in the field. It outperforms a benchmark scheme that assumes global information.

asymptotically convergent to the offline optimal solution, i.e., asymptotically optimal. The online energy distribution algorithm consists of the following three steps.

2. Problem Statement and Main Results We aim to minimize the load variance in the grid while maximizing user satisfaction. Large load variance is undesirable for grid operation. It brings about uncertainties that affect not only user satisfaction but also the stability of the power system. Furthermore, the energy provisioning cost should be bounded and users’ necessary power needs should be guaranteed. We first consider an offline scenario where the DCC distributes the power to users during time t = [1, 2, ···, T], and all the information on users’ flexibility ωi(t) and provider’s budget c(t) are assumed to be known in advance. Let Pi (t) denote the power usage for user i at time t. In this paper, we use upper case P in the offline problem, where all the necessary constraints are known a priori. In the corresponding online problem, we use lower case p for the corresponding variables. A vector with subscript i is used to denote a time sequence, e.g., RS for the power usage by user i for t = {1, 2, ···, T}. The offline problem Prob-OFF can be formulated as follows.

Similar to Prob-OFF, problem Prob-ON is also a convex optimization problem satisfying Slater’s condition. We have the following theorem. Please see [11] for a detailed proof. Theorem 1. The online optimal solution converges asymptotically and almost surely to the offline optimal solution. 3. Performance Evaluation

where Var(·) is the variance of the total power, U(·) is the user utility function, f(·) is the price function, α is a nonnegative parameter to trade-off between user satisfaction and grid load variance, ,O T U is user i’s minimum demand at time t, C(·) is the energy provisioning cost function. See [11] for details. We show that Prob-OFF is a convex problem, as given in the following Lemma. Lemma 1. Prob-OFF is a convex optimization problem and has a unique solution. We next develop an online algorithm for energy distribution, and prove that the online solution is

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We evaluate the proposed online algorithm with tracedriven simulations. The simulation data and parameters are acquired from the traces of power consumption in the Southern California Edison (SCE) area recorded in 2011 [10]. We compare the online algorithm with the Optimal Real-time Pricing Algorithm (ORPA) presented in [8] as a Benchmark. The total power consumption of the different algorithms are plotted in Fig. 2. From the aspect of smoothness, we could see clearly that the online optimal real-time energy distribution algorithm with α=1 (termed OORA(1)) achieves the best performance. The figure also shows that the online algorithm with α=0.01 (termed OORA(0.01)) also outperforms the benchmark ORPA. All the three algorithms achieve smoother total loads than the real consumption (RC). The peak reductions over RC are 35% for OORA(1), 28% for OORA(0.01), and 12.5% for ORPA. Therefore, OORA(1) achieves the largest peak reduction, while OORA(0.01) still outperforms ORPA with

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IEEE COMSOC MMTC E-Letter Power Systems, vol. 27, no. 4, pp. 1926–1940, Nov. 2012.

considerable gains. Please check out [11] for more simulation results and discussions.

[5] D. O’Neill, M. Levorato, A. Goldsmith, and U. Mitra, “Residential demand response using reinforcement learning,” in Proc. IEEE SmartGridComm’10, Gaithersburg, MD, Oct. 2010, pp. 409–414.

[6] P. Samadi, A. H. Mohsenian-Rad, R. Schober, and V. W. Wong, “Advanced demand side management for the future smart grid using mechanism design,” IEEE Trans. Smart Grid, vol. 3, no. 3, pp. 1170–1180, Sept. 2012.

[7] A. H. Mohsenian-Rad and A. Leon-Garcia, “Optimal residential load control with price prediction in real-time electricity pricing environments,” IEEE Trans. Smart Grid, vol. 1, no. 2, pp. 120–133, Sept. 2010.

[8] P. Samadi, A. H. Mohsenian-Rad, R. Schober, V. W. Wong, and J. Jatskevich, “Optimal real-time pricing algorithm based on utility maximization for smart grid,” in Proc. IEEE SmartGridComm’10, Gaithersburg, MD, Oct. 2010, pp. 415–420.

Fig. 2. Total power consumption for OORA(1), OORA(0.01), ORPA and RC. 4. Conclusion

[9] S. Albers, “Online algorithms: A survey,” Mathematical In this paper, we present a study of optimal real-time energy distribution in smart grid. With a formulation that captures the key design factors of the system, we first present an offline algorithm that can solve the problem with optimal solutions. We then develop an online algorithm that requires no future information about users and the grid. We also show that the online solution converges to the offline optimal solution asymptotically and almost surely. The proposed online algorithm is evaluated with trace-driven simulations. ACKNOWLEDGMENT This work is supported in part by the US National Science Foundation (NSF) under Grant CNS-0953513, and through the NSF Broadband Wireless Access and Applications Center (BWAC) Site at Auburn University. References [1] X. Fang, S. Misra, G. Xue, and D. Yang, “Smart grid the new and improved power grid: A survey,” IEEE Commun. Surveys & Tutorials, vol. PP, no. 99, pp. 1– 37, Dec. 2011.

Programming, vol. 97, pp. 3–26, 2003.

[10] SCE, “Regulatory information–SCE load profiles–2011 static load profiles,” 2011, [online] Available: http://www.sce.com/005regulinfo/eca/DOMSM11.DLP.

[11] Y. Wang, S. Mao, and R. M. Nelms, "An online algorithm for optimal real-time energy distribution in smart grid," IEEE Transactions on Emerging Topics in Computing, vol.1, no.1, pp.10–21, July 2013.

YU WANG (S’13) received the M.E. degree in instrument science and technology and the B.E. degree in measuring and control technology and instrumentation from Southeast University, Nanjing, China, in 2011 and 2008, respectively. Since 2011, he has been pursuing the Ph.D. degree with the Department of Electrical and Computer Engineering, Auburn University, Auburn, AL, USA. His current research interests include smart grid and optimization.

[2] Y. Huang, S. Mao, and R. M. Nelms, “Adaptive electricity scheduling in microgrids,” in Proc. IEEE INFOCOM’13, Turin, Italy, Apr. 2013, pp. 1–9.

[3] T. Logenthiran, D. Srinivasan, and T. Z. Shun, “Demand side management in smart grid using heuristic optimization,” IEEE Trans. Smart Grid, vol. 3, no. 3, pp. 1244–1252, Sept. 2012.

[4] M. Roozbehani, M. Dahleh, and S. Mitter, “Volatility of power grids under real-time pricing,” IEEE Trans.

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SHIWEN MAO (S'99-M'04SM'09) received the Ph.D. degree in electrical and computer engineering from Polytechnic University, Brooklyn, NY, USA. Currently, he is the McWane Associate Professor with the Department of Electrical and Computer Engineering, Auburn University, Auburn, AL, USA. His current

Vol.9, No.4, July 2014

IEEE COMSOC MMTC E-Letter research interests include cross-layer optimization of wireless networks and multimedia communications, with current focus on cognitive radios, femtocells, 60 GHz mmWave networks, free space optical networks, and smart grid. He is on the Editorial Board of the IEEE Transactions on Wireless Communications, IEEE Internet of Things Journal, IEEE Communications Surveys and Tutorials, and several other journals. He received the 2013 IEEE ComSoc MMTC Outstanding Leadership Award and the NSF CAREER Award in 2010. He is a co-recipient of the IEEE ICC 2013 Best Paper Award and the 2004 IEEE Communications Society Leonard G. Abraham Prize in the Field of Communications Systems.

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R. M. NELMS (F'04) received the B.E.E. and M.S. degrees in electrical engineering from Auburn University, AL, USA, in 1980 and 1982, respectively. He received the Ph.D. degree in electrical engineering from Virginia Polytechnic Institute and State University, Blacksburg, VA, USA, in 1987. He is currently a Professor and Chair with the Department of Electrical and Computer Engineering, Auburn University. His current research interests include power electronics, power systems, and electric machinery. In 2004, he was named an IEEE Fellow "for technical leadership and contributions to applied power electronics." He is a Registered Professional Engineer in Alabama.

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