- 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.
Zhengshuo Li1, Qinglai Guo1, Hongbin Sun1, Jianhui Wang2Show Affiliations
- Department of Electrical Engineering, Tsinghua University, 100084 Beijing, China
- 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.Go To Applied Energy