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Register a new userExtended Convex Hull-Based Distributed Operation of Integrated Electric-Gas Systems
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Distributed operation of integrated electricity and gas systems (IEGS) receives much attention since it respects data security and privacy between different agencies. This paper proposes an extended convex hull (ECH) based method to address the distributed optimal energy flow (OEF) problem in the IEGS. First, a multi-block IEGS model is obtained by dividing it into N blocks according to physical and regional differences. This multi-block model is then convexified by replacing the nonconvex gas transmission equation with its ECH-based constraints. The Jacobi-Proximal alternating direction method of multipliers (J-ADMM) algorithm is adopted to solve the convexified model and minimize its operation cost. Finally, the feasibility of the optimal solution for the convexified problem is checked, and a sufficient condition is developed. The optimal solution for the original nonconvex problem is recovered from that for the convexified problem if the sufficient condition is satisfied. Test results reveal that this method is tractable and effective in obtaining the feasible optimal solution for radial gas networks.
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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.
In this work, we propose a distributed algorithm for stochastic non-convex optimization. We consider a worker-server architecture where a set of $K$ worker nodes (WNs) in collaboration with a server node (SN) jointly aim to minimize a global, potentially non-convex objective function. The objective function is assumed to be the sum of local objective functions available at each WN, with each node having access to only the stochastic samples of its local objective function. In contrast to the existing approaches, we employ a momentum based single loop distributed algorithm which eliminates the need of computing large batch size gradients to achieve variance reduction. We propose two algorithms one with adaptive and the other with non-adaptive learning rates. We show that the proposed algorithms achieve the optimal computational complexity while attaining linear speedup with the number of WNs. Specifically, the algorithms reach an $epsilon$-stationary point $x_a$ with $mathbb{E}| abla f(x_a) | leq tilde{O}(K^{-1/3}T^{-1/2} + K^{-1/3}T^{-1/3})$ in $T$ iterations, thereby requiring $tilde{O}(K^{-1} epsilon^{-3})$ gradient computations at each WN. Moreover, our approach does not assume identical data distributions across WNs making the approach general enough for federated learning applications.
Given a subset $mathbf{S}={A_1, dots, A_m}$ of $mathbb{S}^n$, the set of $n times n$ real symmetric matrices, we define its {it spectrahull} as the set $SH(mathbf{S}) = {p(X) equiv (Tr(A_1 X), dots, Tr(A_m X))^T : X in mathbf{Delta}_n}$, where ${bf Delta}_n$ is the {it spectraplex}, ${ X in mathbb{S}^n : Tr(X)=1, X succeq 0 }$. We let {it spectrahull membership} (SHM) to be the problem of testing if a given $b in mathbb{R}^m$ lies in $SH(mathbf{S})$. On the one hand when $A_i$s are diagonal matrices, SHM reduces to the {it convex hull membership} (CHM), a fundamental problem in LP. On the other hand, a bounded SDP feasibility is reducible to SHM. By building on the {it Triangle Algorithm} (TA) cite{kalchar,kalsep}, developed for CHM and its generalization, we design a TA for SHM, where given $varepsilon$, in $O(1/varepsilon^2)$ iterations it either computes a hyperplane separating $b$ from $SH(mathbf{S})$, or $X_varepsilon in mathbf{Delta}_n$ such that $Vert p(X_varepsilon) - b Vert leq varepsilon R$, $R$ maximum error over $mathbf{Delta}_n$. Under certain conditions iteration complexity improves to $O(1/varepsilon)$ or even $O(ln 1/varepsilon)$. The worst-case complexity of each iteration is $O(mn^2)$, plus testing the existence of a pivot, shown to be equivalent to estimating the least eigenvalue of a symmetric matrix. This together with a semidefinite version of Caratheodory theorem allow implementing TA as if solving a CHM, resorting to the {it power method} only as needed, thereby improving the complexity of iterations. The proposed Triangle Algorithm for SHM is simple, practical and applicable to general SDP feasibility and optimization. Also, it extends to a spectral analogue of SVM for separation of two spectrahulls.
Large scale, non-convex optimization problems arising in many complex networks such as the power system call for efficient and scalable distributed optimization algorithms. Existing distributed methods are usually iterative and require synchronization of all workers at each iteration, which is hard to scale and could result in the under-utilization of computation resources due to the heterogeneity of the subproblems. To address those limitations of synchronous schemes, this paper proposes an asynchronous distributed optimization method based on the Alternating Direction Method of Multipliers (ADMM) for non-convex optimization. The proposed method only requires local communications and allows each worker to perform local updates with information from a subset of but not all neighbors. We provide sufficient conditions on the problem formulation, the choice of algorithm parameter and network delay, and show that under those mild conditions, the proposed asynchronous ADMM method asymptotically converges to the KKT point of the non-convex problem. We validate the effectiveness of asynchronous ADMM by applying it to the Optimal Power Flow problem in multiple power systems and show that the convergence of the proposed asynchronous scheme could be faster than its synchronous counterpart in large-scale applications.
Seeking the convex hull of an object is a very fundamental problem arising from various tasks. In this work, we propose two variational convex hull models using level set representation for 2-dimensional data. The first one is an exact model, which can get the convex hull of one or multiple objects. In this model, the convex hull is characterized by the zero sublevel-set of a convex level set function, which is non-positive at every given point. By minimizing the area of the zero sublevel-set, we can find the desired convex hull. The second one is intended to get convex hull of objects with outliers. Instead of requiring all the given points are included, this model penalizes the distance from each given point to the zero sublevel-set. Literature methods are not able to handle outliers. For the solution of these models, we develop efficient numerical schemes using alternating direction method of multipliers. Numerical examples are given to demonstrate the advantages of the proposed methods.
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