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Since the celebrated work by Cartan, distributions with \nobreak{small} growth vector $(2,3,5)$ have been studied extensively. In the holomorphic setting, there is a natural correspondence between holomorphic $(2,3,5)$-distributions and nondegenerate lines on holomorphic contact manifolds of dimension 5. We generalize this correspondence to higher dimensions by studying nondegenerate lines on holomorphic contact manifolds and the corresponding class of distributions of small growth vector $(2m, 3m, 3m+2)$ for any positive integer $m$., Comment: To appear in Nagoya Mathematical Journal

This paper considers endogenous uncertainty (EnU) in the stochastic economic dispatch (SED) problem, where the endogenous uncertainty means decision dependent uncertainty. In this problem, demand response (DR) commitment is the source of the EnU. Nevertheless, EnU is not well considered in existing literature. Our first contribution is to build up an optimization model of DR-involved SED under EnU (SED-DR-EnU). This is a computational challenging problem due to the EnU. Our second contribution is introducing a coupled learning enabled optimization algorithm which can effectively solve the proposed SED-DR-EnU problem. This strategy is tested on the IEEE 14 bus, and IEEE 39 bus systems, and the results showed the importance of considering EnU in the DR-involved SED problem.

To each complex composition algebra $\mathbb{A}$, there associates a projective symmetric manifold $X(\mathbb{A})$ of Picard number one, which is just a smooth hyperplane section of the following varieties ${\rm Lag}(3,6), {\rm Gr}(3,6), \mathbb{S}_6, E_7/P_7.$ In this paper, it is proven that these varieties are rigid, namely for any smooth family of projective manifolds over a connected base, if one fiber is isomorphic to $X(\mathbb{A})$, then every fiber is isomorphic to $X(\mathbb{A})$.

Pasquier and Perrin discovered that the ${\rm G}_2$-horospherical manifold ${\bf X}$ of Picard number 1 can be realized as a smooth specialization of the rational homogeneous space parameterizing the lines on the 5-dimensional hyperquadric, in other words, it can be deformed nontrivially to the rational homogeneous space. We show that ${\bf X}$ is the only smooth projective variety with this property. This is obtained as a consequence of our main result that ${\bf X}$ can be recognized by its VMRT, namely, a Fano manifold of Picard number 1 is biregular to ${\bf X}$ if and only if its VMRT at a general point is projectively isomorphic to that of ${\bf X}$. We employ the method the authors developed to solve the corresponding problem for symplectic Grassmannians, which constructs a flat Cartan connection in a neighborhood of a general minimal rational curve. In adapting this method to ${\bf X}$, we need an intricate study of the positivity/negativity of vector bundles with respect to a family of rational curves, which is subtler than the case of symplectic Grassmannians because of the nature of the differential geometric structure on ${\bf X}$ arising from VMRT., Comment: To appear in Transformation Groups

This paper defines a logic model of optimization under uncertainty which optimizes the expectation of a uncertainty-perturbed objective function and subjects to a new type of constraints--the probabilistically robust constraints (PRC). This novel logic model is an alternative to, but more general than, the existing ones like the stochastic, robust, and chance-constrained models. The novelty mainly resides in the newly defined PRC which logically requires that an optimal solution should be feasible to high-probability realizations of the uncertain variables. Given that the existing methods of obtaining deterministic approximations are either inapplicable or inefficient to this new logic model, we propose a novel solution paradigm. First, the logic model is approximated by a data-driven deterministic program following an alpha-process of the input data (scenario) set. A sufficient condition on the relation between the accuracy of deterministic approximation and the needed number of input scenarios is provided. Second, the concept of strategic scenario selection (S^3) is developed to figure out a limited number of active scenarios as the input to the data-driven deterministic approximation. Three S^3 algorithms are designed for the cases of discrete, continuous, and mixed-integer uncertain variables respectively, which are not sensitive to the continuity of the decision variables. Numerical experiments showed that the S^3-based data-driven program can accurately approximate the new logic model with low computational complexity., Comment: 36 pages

This paper presents a novel solution paradigm of general optimization under both exogenous and endogenous uncertainties. This solution paradigm consists of a probability distribution (PD)-free method of obtaining deterministic equivalents and an innovative approach of scenario reduction. First, dislike the existing methods that use scenarios sampled from pre-known PD functions, the PD-free method uses historical measurements of uncertain variables as input to convert the logical models into a type of deterministic equivalents called General Scenario Program (GSP). Our contributions to the PD-free deterministic equivalent construction reside in generalization (making it applicable to general optimization under uncertainty rather than just chance-constrained optimization) and extension (enabling it to the problems under endogenous uncertainty via developing an iterative and a non-iterative frameworks). Second, this paper reveals some unknown properties of the PD-free deterministic equivalent construction, such as the characteristics of active scenarios and repeated scenarios. Base on this discoveries, we propose a concept and methods of strategic scenario selection which can effectively reduce the required number of scenarios as demonstrated in both mathematical analysis and numerical experiments. Numerical experiments are conducted on two typical smart grid optimization problems under exogenous and endogenous uncertainties., Subtitle: Applications in Smart Grid Optimization under Both Exogenous and Endogenous Uncertainty

The uncertainty of multiple power loads and renewable energy generations (PLREG) in power systems increases the complexity of power flow analysis for decision-makers. The chance-constrained method can be applied to model the optimization problems of power flow under uncertainty. This paper develops a novel solution approach for chance-constrained AC optimal power flow (CCACOPF) problem based on the data-driven convexification of power flow and a fast algorithm for scenario technique (FAST). This method is computationally effective for mainly two reasons. First, the original nonconvex AC power flow (ACPF) constraints are approximated by a set of learning-based quadratic convex ones. Second, FAST is an advanced scenario-based solution method (SSM) that doesn't rely on the pre-assumed probability distribution, using far less scenarios than the conventional SSM. Eventually, the CCACOPF is converted into a computationally tractable convex optimization problem. The simulation results on IEEE test cases indicate that 1) the proposed solution method can outperform the conventional SSM in computational efficiency, 2) the data-driven convexification of power flow is effective in approximating original complex AC power flow.

For a complex connected semisimple linear algebraic group $G$ of adjoint type and of rank $n$, De Concini and Procesi constructed its wonderful compactification $\bar{G}$, which is a smooth Fano $G \times G$-variety of Picard number $n$ enjoying many interesting properties. In this paper, it is shown that the wonderful compactification $\bar{G}$ is rigid under Fano deformation. Namely, for any regular family of Fano manifolds over a connected base, if one fiber is isomorphic to $\bar{G}$, then so are all other fibers. This answers a question raised by Bien and Brion in their work on the local rigidity of wonderful varieties., 42 pages, to appear in Journal of Differential Geometry

Energy storage has been proven to yield positive effects on planning, operation and control of electric grids. It has become a crucial task to properly model the energy storage systems (ESS) under the framework of grid optimization on transmission and distribution networks including microgrids. This paper presents a review on mathematical models and test cases of ESSs used for grid optimization studies, where the network constraints of power systems are included. The existing ESS models are mainly classified into two categories; linear and nonlinear models. The two main categories are further divided into several subcategories respectively; such as mixed integer linear and convex nonlinear subcategories. Based on the review and discussions, this paper aims at providing suggestions for choosing proper ESS models for specific grid optimization studies considering the chosen power network model., Accepted for publication ahead of the PES General Meeting 2019

This paper investigates the water network's potential ability to provide demand response services to the power grid under the framework of a distribution-level water-energy nexus (micro-WEN). In particular, the hidden controllability of water loads, such as irrigation systems, was closely studied to improve the flexibility of electrical grids. A optimization model is developed for the demand-side management (DSM) of micro-WEN, and the simulation results assert that grid flexibility indeed benefits from controllable water loads. Although the proposed optimal DSM model is an intractable mixed-integer nonlinear programming (MINLP) problem, quasi-convex hull techniques were developed to relax the MINLP into a mixed-integer convex programming (MICP) problem. The numerical study shows that the quasi-convex hull relaxation is tight and that the resulting MICP problem is computationally efficient.

This paper presents a novel scalable framework to solve the optimization of a nonlinear system with differential algebraic equation (DAE) constraints that enforce the asymptotic stability of the underlying dynamic model with respect to certain disturbances. Existing solution approaches to analogous DAE-constrained problems are based on discretization of DAE system into a large set of nonlinear algebraic equations representing the time-marching schemes. These approaches are not scalable to large size models. The proposed framework, based on LaSalle's invariance principle, uses convex Lyapunov functions to develop a novel stability certificate which consists of a limited number of algebraic constraints. We develop specific algorithms for two major types of nonlinearities, namely Lur'e, and quasi-polynomial systems. Quadratic and convex-sum-of-square Lyapunov functions are constructed for the Lur'e-type and quasi-polynomial systems respectively. A numerical experiment is performed on a 3-generator power network to obtain a solution for transient-stability-constrained optimal power flow., Comment: 10 pages, 7 figures

Let X be a connected family of complex Fano manifolds. We show that if some fiber is the product of two manifolds of lower dimensions, then so is every fiber. Combining with previous work of Hwang and Mok, this implies immediately that if a fiber is (possibly reducible) Hermitian symmetric space of compact type, then all fibers are isomorphic to the same variety., Comment: 5 pages. Comments are welcome

For many nonlinear control systems, the chosen equilibrium determines both the steady-state efficiency and the dynamic performance. This paper addresses the issue of obtaining an optimal equilibrium in terms of some steady-state operation criteria for a Lur'e-type system and such an equilibrium can also guarantee a sufficiently large stability region in the dynamic domain such that the system can tolerate some given disturbance. For this purpose, a set of computationally tractable algebraic constraints, which can properly represent the stability certificate under the optimization framework, are proposed. The existing methods formulate the dynamic performance under the optimization framework by discretizing the differential-algebraic equations, which are computationally intractable for large-scale Lur'e systems like power grids. Dissimilarly, the introduced approach first constructs the stability region based on quadratic Lyapunov functions. Then, a novel method is proposed to project the stability region onto the feasible domain of the optimization problem such that the stability certificate can be incorporated into the optimization framework easily. In the transient stability-constrained optimal power flow (TSCOPF) problem of power systems, researchers look for a steady-state operating point with the minimum generation costs that can maintain system stability under some given transient disturbances. The proposed approach is applied to develop a scalable TSCOPF framework for power systems. The TSCOPF model is tested on the IEEE 118-Bus power system., Submitted to IEEE Transactions on Automatic Control

This paper introduces a micro-nexus of water and energy which can be considered as one of the physical infrastructures of the future building/city/village systems. For the electricity side, an alternating current (AC) power flow model integrated with battery energy storage and renewable generation is adopted. The nonlinear hydraulic characteristics in pipe networks is also considered in the proposed micro water-energy nexus (WEN) model. Integer variables are involved to represent the on/off state of pumps. Base on the proposed nexus model, a co-optimization framework of water and energy networks is developed. The overall co-optimization model is a mixed-integer nonlinear programming problem which is tested on a waterenergy nexus which consists of the IEEE 13-bus distribution system and a 8-node water distribution network. The simulation results demonstrate that the cost-efficiency of the co-optimization framework is higher than optimizing two systems independently., Comment: Submitted to IEEE Power & Energy Society 2018 General Meeting