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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.

This paper solves a new class of optimization problems under uncertainty which optimizes an objective function of decision variables and subjects to a set of probability contour constraints (PCC). The proposed PCC logically means that an optimal solution should satisfy a set of algebraic constraints for all possible high-probability realizations of the uncertain parameters. The PCC is an alternative to the conventional chance constraint while the latter cannot guarantee the solution's feasibility to high-probability realizations of uncertainty. Given that the existing solution methods of the conventional chance-constrained optimization are not suitable for solving the proposed probability contour constrained optimization (PCCO), we develop a novel data-based solution paradigm that uses historical measurements of the uncertain parameters as input samples. This solution paradigm is conceptually simple and allows us to develop effective data-reduction schemes which reduces computational burden while reserves high accuracy., Comment: 26 pages

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.

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., Comment: Subtitle: Applications in Smart Grid Optimization under Both Exogenous and Endogenous Uncertainty

This paper presents a novel optimization framework of formulating the three-phase optimal power flow that involves uncertainty. The proposed uncertainty-aware optimization (UaO) framework is: 1) a deterministic framework that is less complex than the existing optimization frameworks involving uncertainty, and 2) convex such that it admits polynomial-time algorithms and mature distributed optimization methods. To construct this UaO framework, a methodology of learning-aided uncertainty-aware modeling, with prediction errors of stochastic variables as the measurement of uncertainty, and a theory of data-driven convexification are proposed. Theoretically, the UaO framework is applicable for modeling general optimization problems under uncertainty., Comment: Accepted for pubication in the IEEE Transactions on Power Systems

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., Comment: Accepted for publication ahead of the PES General Meeting 2019

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

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 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

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., Comment: Submitted to IEEE Transactions on Automatic Control

A branch flow model (BFM) is used to formulate the AC power flow in general networks. For each branch/line, the BFM contains a nonconvex quadratic equality. A mathematical formulation of its convex hull is proposed, which is the tightest convex relaxation of this quadratic equation. The convex hull formulation consists of a second order cone inequality and a linear inequality within the physical bounds of power flows. The convex hull formulation is analytically proved and geometrically validated. An optimal scheduling problem of distributed energy storage (DES) in radial distribution systems with high penetration of photovoltaic resources is investigated in this paper. To capture the performance of both the battery and converter, a second-order DES model is proposed. Following the convex hull of the quadratic branch flow equation, the convex hull formulation of the nonconvex constraint in the DES model is also derived. The proposed convex hull models are used to generate a tight convex relaxation of the DES optimal scheduling (DESOS) problem. The proposed approach is tested on several radial systems. A discussion on the extension to meshed networks is provided., Comment: 9 pages, 7 figures, transaction

An optimization problem considering AC power flow constraints and integer decision variables can usually be posed as a mixed-integer quadratically constrained quadratic program (MIQCQP) problem. In this paper, first, a set of valid linear equalities are applied to strengthen the semidefinite program (SDP) relaxation of the MIQCQP problem without significantly increasing the problem dimension so that an enhanced mixed-integer SDP (MISDP) relaxation, which is a mixed-integer convex problem, is obtained. Then, the enhanced MISDP relaxation is reformulated as a disjunctive programming (DP) problem which is tighter than the former one, since the disjunctions are designed to capture the disjunctive nature of the terms in the rank-1 constraint about the integral variables. The DP relaxation is then equivalently convert-ed back into a MISDP problem the feasible set of whose continu-ous relaxation is the convex hull of feasible region of the DP prob-lem. Finally, globally optimal solution of the DP problem which is the tightest relaxation for the MIQCQP proposed in the paper is obtained by solving the resulting MISDP problem using a branch-and-bound (B&B) algorithm. Computational efficiency of the B&B algorithm is expected to be high since feasible set of the continuous relaxation of a MISDP sub-problem is the convex hull of that of the corresponding DP sub-problem. To further reduce the dimension of the resulting MISDP problem, a compact for-mulation of this problem is proposed considering the sparsity. An optimal placement problem of smart PV inverter in distribution systems integrated with high penetration of PV, which is an MIQCQP problem, is studied as an example. The proposed ap-proach is tested on an IEEE distribution system. The results show that it can effectively improve the tightness and feasibility of the SDP relaxation., Comment: 8 pages with 5 figures