Storage-like devices in load leveling: Complementarity constraints and new and exact relaxation method

Significance Statement

  • A load-leveling problem with distributed storage-like devices (batteries, electric vehicles, ice storages and so on), which are being more and more prevalent in smart grids, is a focused issue.
  • It is extremely difficult to solve the problem, because the complementarity constraints brought out by the non-simultaneous-charging-and-discharging features of storages are introduced into the optimization model. Mathematically, the load-leveling problem with complementary constraints is the so-called “mathematical programs with equilibrium constraints” (MPEC) for which regular nonlinear programming (NLP) methods are invalid. What is worse, though several methods are developed for MPEC, such as mixed-integer programming (MIP) methods, smoothing methods, and regularization relaxation methods, they may all result in long solution time due to additional integer variables or excessive iterations.
  • To this end, this paper develops a new exact relaxation method which directly removes the complementarity constraints from the load-leveling model while still keeping the optimal solution of the relaxed model satisfying the non-simultaneous-charging-and-discharging constraints under two conditions:
    • The first condition requires that a storage-like device should enjoy a lower charging price it pays to grid than the discharging price the grid pays to it.
    • The second condition requires that the load-leveling objective should dominate the other optional objectives in the load-leveling problem.

Obviously, the above two conditions are always true for load-leveling problems. Hence, the complementary constraints resulting in the solution difficulty can be directly removed from the models in most cases, and regular NLP methods start to work on the relaxed load-leveling problems.

  • Compared with other methods to solve MPEC, exhaustive simulations have shown that the proposed exact relaxation method guarantees the optimality of the solution and enhances the computational efficiency by tens and hundreds of times. In the case of 100 storage-like devices, the MIP method takes 242.836 seconds to reach the optimum while the exact relaxation method only takes 1.114 seconds. The speedup ratio is 218.
  • Moreover, based on the relaxation method, decentralized optimization methods can be further used to solve the relaxed model in a decentralized manner with guaranteed optimality and convergence. The proposed method can also be extended to other applied-energy-field problems with distributed storage-like devices involved.

Storage-like devices in load leveling Complementarity constraints and a new and exact relaxation method .Renewable Energy Global Innovations

About the author

Zhengshuo Li (S’12) received his Bachelor degree from the Department of Electrical Engineering, Tsinghua University, Beijing, China, in 2011. He is pursuing the Ph.D. degree at Dept. of Electrical Engineering at Tsinghua University. From February 2014 to May 2014, he was a visiting student with the Decision and Information Sciences Division at Argonne National Laboratory, Argonne, IL, USA. His research interests include economic dispatch and security analysis of smart transmission and distribution grids, distributed storage utilization and demand response in smart grids.

About the author

Qinglai Guo (SM’2014) was born in Jilin City, Jilin Province in China on Mar. 6, 1979. He graduated from the Department of Electrical Engineering, Tsinghua University, Beijing, China, in 2000 with B.S degree. He received his PhD degree from Tsinghua University in 2005 where he is now an associate professor. He is a member of CIGRE C2.13 Task Force on Voltage/Var support in System Operations. His special fields of interest include smart grids, cyber-physical systems and electrical power control center applications. 

About the author

Hongbin Sun (SM’ 2012) received his double B.S. degrees from Tsinghua University in 1992, the Ph.D from Dept. of E.E., Tsinghua University in 1997. He is now Changjiang Scholar of Education Ministry of China, full professor of electrical engineering in Tsinghua Univ. and assistant director of State Key Laboratory of Power Systems in China. From 2007.9 to 2008.9, he was a visiting professor with School of EECS at the Washington State University in Pullman. He is a Fellow of IET. He is a member of IEEE PES CAMS Cascading Failure Task Force and CIGRE C2.13 Task Force on Voltage/Var support in System Operations. In recent 15 years, he led a research group in Tsinghua University to develop a commercial system-wide automatic voltage control systems, which has been applied to PJM interconnection, the largest regional power grid in USA, and to more than 60 large-scale power grids in China. He published more than 300 academic papers. He won the China National Technology Innovation Award in 2008, the National Distinguished Teacher Award in China in 2009, and the National Science Fund for Distinguished Young Scholars of China in 2010. His research interests include smart grids, renewable generation integration, and electrical power control center applications.

About the author

Jianhui Wang (M’07-SM’12) received the Ph.D. degree in electrical engineering from Illinois Institute of Technology, Chicago, IL, USA, in 2007. Presently, he is the Section Lead for Advanced Power Grid Modeling at the Energy Systems Division at Argonne National Laboratory, Argonne, IL, USA. Dr. Wang is the secretary of the IEEE Power & Energy Society (PES) Power System Operations Committee. He is an associate editor of Journal of Energy Engineering and an editorial board member of Applied Energy. He is also an affiliate professor at Auburn University and an adjunct professor at University of Notre Dame. He has held visiting positions in Europe, Australia and Hong Kong including a VELUX Visiting Professorship at the Technical University of Denmark (DTU). Dr. Wang is the Editor-in-Chief of the IEEE Transactions on Smart Grid and an IEEE PES Distinguished Lecturer. He is also the recipient of the IEEE PES Power System Operation Committee Prize Paper Award in 2015. 

Journal Reference

Applied Energy, Volume 151, 2015, Pages 13–22

Zhengshuo Li1, Qinglai Guo1, Hongbin Sun1, Jianhui Wang2

Show Affiliations
  1. Department of Electrical Engineering, Tsinghua University, 100084 Beijing, China
  2. Argonne National Laboratory, Argonne, IL 60439, USA


Storage-like devices (SLDs), which include energy storage systems as well as devices with similar properties such as electric vehicles, can be exploited for load leveling. However, to prevent simultaneous charging and discharging of an Storage-like devices, complementarity constraints should be included in the optimization model, which makes the problem strongly non-convex. Mixed-integer programming (MIP) methods are commonly used to solve such problems; however, this results in long solution time to achieve an approximate optimal solution. Therefore, a method to efficiently find optimal solutions of load-leveling problems with SLDs is desirable. Here, we report a load-leveling optimization model for a system with Storage-like devices and show that the complementarity constraints can be exactly relaxed under two sufficient conditions so that a convex relaxed model can be solved instead. Moreover, the exactness of the relaxation can be determined prior to solving the relaxed model, and the sufficient conditions are usually satisfied in practical situations. The numerical studies verify the theoretical analysis and show that an equally good optimal solution of the load-leveling problem with Storage-like devices can be obtained far more efficiently by using the proposed method than a commonly used MIP method.

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