Relative Rota-Baxter operators of nonzero weights on 33-Lie algebras and 3-post-Lie algebras

In this paper, first we introduce the notion of a relative Rota-Baxter operator of nonzero weights on a $3$-Lie algebra with respect to an action on another 3-Lie algebra, which can be characterized by graphs of the semidirect product 3-Lie algebra constructed from the action. %We show that a relative Rota-Baxter operator of weight $\lambda$ induces a new $3$-Lie algebra structure, which is called the descendent 3-Lie algebra. Then we introduce a new algebraic structure, which is called a $3$-post-Lie algebra. A 3-post-Lie algebra consists of a 3-Lie algebra structure and a ternary operation such that some compatibility conditions are satisfied. We show that a relative Rota-Baxter operator of nonzero weights induces a $3$-post-Lie algebra naturally. Thus $3$-post-Lie algebras can be viewed as the underlying algebraic structures of relative Rota-Baxter operators of nonzero weights on $3$-Lie algebras. Moreover, a $3$-post-Lie algebra also gives rise to a new 3-Lie algebra, which is called the subadjacent 3-Lie algebra, and an action on the original 3-Lie algebra. Next we construct an $L_\infty$-algebra from an action of 3-Lie algebras whose Maurer-Cartan elements are relative Rota-Baxter operators of nonzero weights. Consequently, we obtain the twisted $L_\infty$-algebra that controls deformations of a given relative Rota-Baxter operator of nonzero weights on 3-Lie algebras. Finally, we construct a cohomology theory for a relative Rota-Baxter operator of nonzero weights on $3$-Lie algebras and use the second cohomology group to classify infinitesimal deformations.Comment: 22 pages. arXiv admin note: text overlap with arXiv:2107.13950, arXiv:2108.0262

Two-Stage Submodular Optimization of Dynamic Thermal Rating for Risk Mitigation Considering Placement and Operation Schedule

Cascading failure causes a major risk to society currently. To effectively mitigate the risk, dynamic thermal rating (DTR) technique can be applied as a cost-effective strategy to exploit potential transmission capability. From the perspectives of service life and Braess paradox, it is important and challenging to jointly optimize the DTR placement and operation schedule for changing system state, which is a two-stage combinatorial problem with only discrete variables, suffering from no approximation guarantee and dimension curse only based on traditional models. Thus, the present work proposes a novel two-stage submodular optimization (TSSO) of DTR for risk mitigation considering placement and operation schedule. Specifically, it optimizes DTR placement with proper redundancy in first stage, and then determines the corresponding DTR operation for each system state in second stage. Under the condition of the Markov and submodular features in sub-function of risk mitigation, the submodularity of total objective function of TSSO can be proven for the first time. Based on this, a state-of-the-art efficient solving algorithm is developed that can provide a better approximation guarantee than previous studies by coordinating the separate curvature and error form. The performance of the proposed algorithms is verified by case results

Dynamic Game-based Maintenance Scheduling of Integrated Electric and Natural Gas Grids with a Bilevel Approach

This paper proposes a dynamic game-based maintenance scheduling mechanism for the asset owners of the natural gas grid and the power grid by using a bilevel approach. In the upper level, the asset owners of the natural gas grid and the power grid schedule maintenance to maximize their own revenues. This level is modeled as a dynamic game problem, which is solved by the backward induction algorithm. In the lower level, the independent system operator (ISO) dispatches the system to minimize the loss of power load and natural gas load in consideration of the system operating conditions under maintenance plans from the asset owners in the upper level. This is modeled as a mixed integer linear programming problem. For the model of the natural gas grid, a piecewise linear approximation associated with the big-M approach is used to transform the original nonlinear model into the mixed integer linear model. Numerical tests on a 6-bus system with a 4-node gas grid show the effectiveness of the proposed model.Comment: accepted by IEEE Transactions on Power System