Showing papers on "Linear programming published in 2018"

Fractional Programming for Communication Systems—Part I: Power Control and Beamforming

Abstract: Fractional programming (FP) refers to a family of optimization problems that involve ratio term(s). This two-part paper explores the use of FP in the design and optimization of communication systems. Part I of this paper focuses on FP theory and on solving continuous problems. The main theoretical contribution is a novel quadratic transform technique for tackling the multiple-ratio concave–convex FP problem—in contrast to conventional FP techniques that mostly can only deal with the single-ratio or the max-min-ratio case. Multiple-ratio FP problems are important for the optimization of communication networks, because system-level design often involves multiple signal-to-interference-plus-noise ratio terms. This paper considers the applications of FP to solving continuous problems in communication system design, particularly for power control, beamforming, and energy efficiency maximization. These application cases illustrate that the proposed quadratic transform can greatly facilitate the optimization involving ratios by recasting the original nonconvex problem as a sequence of convex problems. This FP-based problem reformulation gives rise to an efficient iterative optimization algorithm with provable convergence to a stationary point. The paper further demonstrates close connections between the proposed FP approach and other well-known algorithms in the literature, such as the fixed-point iteration and the weighted minimum mean-square-error beamforming. The optimization of discrete problems is discussed in Part II of this paper.

Trajectory Design for Completion Time Minimization in UAV-Enabled Multicasting

Abstract: This paper studies an unmanned aerial vehicle (UAV)-enabled multicasting system, where a UAV is dispatched to disseminate a common file to a set of ground terminals (GTs). We aim to design the UAV trajectory to minimize its mission completion time, while ensuring that each GT successfully recovers the file with a desired high probability. The formulated problem is nonconvex and difficult to be solved in its original form. Therefore, we first derive an effective lower bound for the success file recovery probability of each GT. The problem is then reformulated in a more tractable form, where the UAV trajectory only needs to be designed to ensure the minimum connection time constraint with each GT, during which their distance is below a certain threshold. We show that without loss of optimality, the UAV trajectory consists of connected line segments only, which can be obtained by determining the optimal set of waypoints as well as the UAV speed along the path connecting the waypoints. We propose efficient schemes for the waypoint design based on a novel concept of virtual base station placement and by applying convex optimization. Furthermore, for fixed waypoints, the optimal UAV speed is efficiently obtained by solving a linear programming problem. Numerical results show that the proposed UAV-enabled multicasting with optimized trajectory design achieves significant performance gains over other benchmark schemes.

Output Range Analysis for Deep Feedforward Neural Networks

Abstract: Given a neural network (NN) and a set of possible inputs to the network described by polyhedral constraints, we aim to compute a safe over-approximation of the set of possible output values. This operation is a fundamental primitive enabling the formal analysis of neural networks that are extensively used in a variety of machine learning tasks such as perception and control of autonomous systems. Increasingly, they are deployed in high-assurance applications, leading to a compelling use case for formal verification approaches. In this paper, we present an efficient range estimation algorithm that iterates between an expensive global combinatorial search using mixed-integer linear programming problems, and a relatively inexpensive local optimization that repeatedly seeks a local optimum of the function represented by the NN. We implement our approach and compare it with Reluplex, a recently proposed solver for deep neural networks. We demonstrate applications of our approach to computing flowpipes for neural network-based feedback controllers. We show that the use of local search in conjunction with mixed-integer linear programming solvers effectively reduces the combinatorial search over possible combinations of active neurons in the network by pruning away suboptimal nodes.

End-to-End Waveform Utterance Enhancement for Direct Evaluation Metrics Optimization by Fully Convolutional Neural Networks

Abstract: Speech enhancement model is used to map a noisy speech to a clean speech. In the training stage, an objective function is often adopted to optimize the model parameters. However, in the existing literature, there is an inconsistency between the model optimization criterion and the evaluation criterion for the enhanced speech. For example, in measuring speech intelligibility, most of the evaluation metric is based on a short-time objective intelligibility (STOI) measure, while the frame based mean square error (MSE) between estimated and clean speech is widely used in optimizing the model. Due to the inconsistency, there is no guarantee that the trained model can provide optimal performance in applications. In this study, we propose an end-to-end utterance-based speech enhancement framework using fully convolutional neural networks (FCN) to reduce the gap between the model optimization and the evaluation criterion. Because of the utterance-based optimization, temporal correlation information of long speech segments, or even at the entire utterance level, can be considered to directly optimize perception-based objective functions. As an example, we implemented the proposed FCN enhancement framework to optimize the STOI measure. Experimental results show that the STOI of a test speech processed by the proposed approach is better than conventional MSE-optimized speech due to the consistency between the training and the evaluation targets. Moreover, by integrating the STOI into model optimization, the intelligibility of human subjects and automatic speech recognition system on the enhanced speech is also substantially improved compared to those generated based on the minimum MSE criterion.

Towards Fast Computation of Certified Robustness for ReLU Networks

Abstract: Verifying the robustness property of a general Rectified Linear Unit (ReLU) network is an NP-complete problem [Katz, Barrett, Dill, Julian and Kochenderfer CAV17]. Although finding the exact minimum adversarial distortion is hard, giving a certified lower bound of the minimum distortion is possible. Current available methods of computing such a bound are either time-consuming or delivering low quality bounds that are too loose to be useful. In this paper, we exploit the special structure of ReLU networks and provide two computationally efficient algorithms Fast-Lin and Fast-Lip that are able to certify non-trivial lower bounds of minimum distortions, by bounding the ReLU units with appropriate linear functions Fast-Lin, or by bounding the local Lipschitz constant Fast-Lip. Experiments show that (1) our proposed methods deliver bounds close to (the gap is 2-3X) exact minimum distortion found by Reluplex in small MNIST networks while our algorithms are more than 10,000 times faster; (2) our methods deliver similar quality of bounds (the gap is within 35% and usually around 10%; sometimes our bounds are even better) for larger networks compared to the methods based on solving linear programming problems but our algorithms are 33-14,000 times faster; (3) our method is capable of solving large MNIST and CIFAR networks up to 7 layers with more than 10,000 neurons within tens of seconds on a single CPU core. In addition, we show that, in fact, there is no polynomial time algorithm that can approximately find the minimum $\ell_1$ adversarial distortion of a ReLU network with a $0.99\ln n$ approximation ratio unless $\mathsf{NP}$=$\mathsf{P}$, where $n$ is the number of neurons in the network.

Equity of Attention: Amortizing Individual Fairness in Rankings

Abstract: Rankings of people and items are at the heart of selection-making, match-making, and recommender systems, ranging from employment sites to sharing economy platforms. As ranking positions influence the amount of attention the ranked subjects receive, biases in rankings can lead to unfair distribution of opportunities and resources such as jobs or income. This paper proposes new measures and mechanisms to quantify and mitigate unfairness from a bias inherent to all rankings, namely, the position bias which leads to disproportionately less attention being paid to low-ranked subjects. Our approach differs from recent fair ranking approaches in two important ways. First, existing works measure unfairness at the level of subject groups while our measures capture unfairness at the level of individual subjects, and as such subsume group unfairness. Second, as no single ranking can achieve individual attention fairness, we propose a novel mechanism that achieves amortized fairness, where attention accumulated across a series of rankings is proportional to accumulated relevance. We formulate the challenge of achieving amortized individual fairness subject to constraints on ranking quality as an online optimization problem and show that it can be solved as an integer linear program. Our experimental evaluation reveals that unfair attention distribution in rankings can be substantial, and demonstrates that our method can improve individual fairness while retaining high ranking quality.

A Linear Algorithm for Optimization Over Directed Graphs With Geometric Convergence

Abstract: In this letter, we study distributed optimization, where a network of agents, abstracted as a directed graph, collaborates to minimize the average of locally known convex functions. Most of the existing approaches over directed graphs are based on push-sum (type) techniques, which use an independent algorithm to asymptotically learn either the left or right eigenvector of the underlying weight matrices. This strategy causes additional computation, communication, and nonlinearity in the algorithm. In contrast, we propose a linear algorithm based on an inexact gradient method and a gradient estimation technique. Under the assumptions that each local function is strongly convex with Lipschitz-continuous gradients, we show that the proposed algorithm geometrically converges to the global minimizer with a sufficiently small step-size. We present simulations to illustrate the theoretical findings.

Distributed Optimal Consensus Over Resource Allocation Network and Its Application to Dynamical Economic Dispatch

Abstract: The resource allocation problem is studied and reformulated by a distributed interior point method via a $\theta$ - logarithmic barrier. By the facilitation of the graph Laplacian, a fully distributed continuous-time multiagent system is developed for solving the problem. Specifically, to avoid high singularity of the $\theta$ - logarithmic barrier at boundary, an adaptive parameter switching strategy is introduced into this dynamical multiagent system. The convergence rate of the distributed algorithm is obtained. Moreover, a novel distributed primal–dual dynamical multiagent system is designed in a smart grid scenario to seek the saddle point of dynamical economic dispatch, which coincides with the optimal solution. The dual decomposition technique is applied to transform the optimization problem into easily solvable resource allocation subproblems with local inequality constraints. The good performance of the new dynamical systems is, respectively, verified by a numerical example and the IEEE six-bus test system-based simulations.

ADD-OPT: Accelerated Distributed Directed Optimization

Abstract: In this paper, we consider distributed optimization problems where the goal is to minimize a sum of objective functions over a multiagent network. We focus on the case when the interagent communication is described by a strongly connected, directed graph. The proposed algorithm, Accelerated Distributed Directed OPTimization (ADD-OPT), achieves the best known convergence rate for this class of problems, $O(\mu ^{k}),0 , given strongly convex, objective functions with globally Lipschitz-continuous gradients, where $k$ is the number of iterations. Moreover, ADD-OPT supports a wider and more realistic range of step sizes in contrast to existing work. In particular, we show that ADD-OPT converges for arbitrarily small (positive) step sizes. Simulations further illustrate our results.

Deep neural networks and mixed integer linear optimization

Abstract: Deep Neural Networks (DNNs) are very popular these days, and are the subject of a very intense investigation. A DNN is made up of layers of internal units (or neurons), each of which computes an affine combination of the output of the units in the previous layer, applies a nonlinear operator, and outputs the corresponding value (also known as activation). A commonly-used nonlinear operator is the so-called rectified linear unit (ReLU), whose output is just the maximum between its input value and zero. In this (and other similar cases like max pooling, where the max operation involves more than one input value), for fixed parameters one can model the DNN as a 0-1 Mixed Integer Linear Program (0-1 MILP) where the continuous variables correspond to the output values of each unit, and a binary variable is associated with each ReLU to model its yes/no nature. In this paper we discuss the peculiarity of this kind of 0-1 MILP models, and describe an effective bound-tightening technique intended to ease its solution. We also present possible applications of the 0-1 MILP model arising in feature visualization and in the construction of adversarial examples. Computational results are reported, aimed at investigating (on small DNNs) the computational performance of a state-of-the-art MILP solver when applied to a known test case, namely, hand-written digit recognition.

Linear Convergence in Optimization Over Directed Graphs With Row-Stochastic Matrices

Abstract: This paper considers a distributed optimization problem over a multiagent network, in which the objective function is a sum of individual cost functions at the agents. We focus on the case when communication between the agents is described by a directed graph. Existing distributed optimization algorithms for directed graphs require at least the knowledge of the neighbors’ out-degree at each agent (due to the requirement of column-stochastic matrices). In contrast, our algorithm requires no such knowledge. Moreover, the proposed algorithm achieves the best known rate of convergence for this class of problems, $O(\mu ^k)$ for $0 , where $k$ is the number of iterations, given that the objective functions are strongly convex and have Lipschitz-continuous gradients. Numerical experiments are also provided to illustrate the theoretical findings.

A decomposition-based iterative optimization algorithm for traveling salesman problem with drone

Abstract: This study investigates a new delivery problem that has emerged after the attempts of several e-commerce and logistics firms to deploy drones in their operations to increase efficiency and reduce delivery times. In this problem, a delivery truck that carries a drone on its roof serves customers in coordination with a drone. The drone is considered to complement the truck due to its cost-efficiency and ability to access difficult terrains and to travel without exposure to congestion. This study presents an iterative algorithm that is based on a decomposition approach to minimize delivery completion time. In the first stage of the proposed methodology, the truck route and the customers assigned to the drone are determined. In the second stage, a mixed-integer linear programming model is solved to optimize the drone route by fixing the routing and the assignment decisions that are made in the first stage. Beginning with the shortest truck route, the assignment and the routing decisions are iteratively improved. The solution times of our algorithm are compared with the solution times of the state-of-the-art formulations that are solved by CPLEX. The results demonstrate that our algorithm yields shorter solution times for the instances that we generated with the specified parameters. An optimization-based heuristic algorithm, which obtains solutions for medium-sized instances, is developed by reducing the feasible search area.

Mixed-Integer Linear Programming for Optimal Scheduling of Autonomous Vehicle Intersection Crossing

Abstract: We propose an urban traffic management scheme for an all connected vehicle environment. If all the vehicles are autonomous, for example, in smart city projects or future's dense city centers, then such an environment does not need a physical traffic signal. Instead, an intersection control server processes data streams from approaching vehicles, periodically solves an optimization problem, and assigns to each vehicle an optimal arrival time that ensures safety while significantly reducing number of stops and intersection delays. The scheduling problem is formulated as a mixed-integer linear program (MILP), and is solved by IBM CPLEX optimization package. The optimization outputs (scheduled access/arrival times) are sent to all approaching vehicles. The autonomous vehicles adjust their speed accordingly by a proposed trajectory planning algorithm with the aim of accessing the intersection at their scheduled times. A customized traffic microsimulation environment is developed to determine the potentials of the proposed solution in comparison to two baseline scenarios. In addition, the proposed MILP-based intersection control scheme is modified and simulated for a mixed traffic consisting of autonomous and human-controlled vehicles, all connected through a wireless communication to the intersection controller of a signalized intersection.

Optimal Scheduling for Electric Vehicle Charging With Discrete Charging Levels in Distribution Grid

Abstract: To accommodate the increasing electric vehicle (EV) penetration in distribution grid, coordinated EV charging has been extensively studied in the literature. However, most of the existing works optimistically consider the EV charging rate as a continuous variable and implicitly ignore the capacity limitation in distribution transformers, which both have great impact on the efficiency and stability of practical grid operation. Towards a more realistic setting, this paper formulates the EV coordinated discrete charging problem as two successive binary programs. The first one is designed to achieve a desired aggregate load profile (e.g., valley-filling profile) at the distribution grid level while taking into account the capacity constraints of distribution transformers. Leveraging the properties of separable convex function and total unimodularity, the problem is transformed into an equivalent linear program, which can be solved efficiently and optimally. The second problem aims to minimize the total number of on-off switchings of all the EVs’ charging profiles while preserving the optimality of the former problem. We prove the second problem is NP-hard and propose a heuristic algorithm to approximately achieve our target in an iterative manner. Case studies confirm the validity of our proposed scheduling methods and indicate our algorithm’s potential for real-time implementations.

Domain Generalization and Adaptation Using Low Rank Exemplar SVMs

Abstract: Domain adaptation between diverse source and target domains is challenging, especially in the real-world visual recognition tasks where the images and videos consist of significant variations in viewpoints, illuminations, qualities, etc. In this paper, we propose a new approach for domain generalization and domain adaptation based on exemplar SVMs. Specifically, we decompose the source domain into many subdomains, each of which contains only one positive training sample and all negative samples. Each subdomain is relatively less diverse, and is expected to have a simpler distribution. By training one exemplar SVM for each subdomain, we obtain a set of exemplar SVMs. To further exploit the inherent structure of source domain, we introduce a nuclear-norm based regularizer into the objective function in order to enforce the exemplar SVMs to produce a low-rank output on training samples. In the prediction process, the confident exemplar SVM classifiers are selected and reweigted according to the distribution mismatch between each subdomain and the test sample in the target domain. We formulate our approach based on the logistic regression and least square SVM algorithms, which are referred to as low rank exemplar SVMs (LRE-SVMs) and low rank exemplar least square SVMs (LRE-LSSVMs), respectively. A fast algorithm is also developed for accelerating the training of LRE-LSSVMs. We further extend Domain Adaptation Machine (DAM) to learn an optimal target classifier for domain adaptation, and show that our approach can also be applied to domain adaptation with evolving target domain, where the target data distribution is gradually changing. The comprehensive experiments for object recognition and action recognition demonstrate the effectiveness of our approach for domain generalization and domain adaptation with fixed and evolving target domains.

The Fastest Known Globally Convergent First-Order Method for Minimizing Strongly Convex Functions

Abstract: We design and analyze a novel gradient-based algorithm for unconstrained convex optimization. When the objective function is $m$ -strongly convex and its gradient is $L$ -Lipschitz continuous, the iterates and function values converge linearly to the optimum at rates $\rho $ and $\rho ^{2}$ , respectively, where $\rho = 1-\sqrt {m/L}$ . These are the fastest known guaranteed linear convergence rates for globally convergent first-order methods, and for high desired accuracies the corresponding iteration complexity is within a factor of two of the theoretical lower bound. We use a simple graphical design procedure based on integral quadratic constraints to derive closed-form expressions for the algorithm parameters. The new algorithm, which we call the triple momentum method, can be seen as an extension of methods such as gradient descent, Nesterov’s accelerated gradient descent, and the heavy-ball method.

Exploiting integrality in the global optimization of mixed-integer nonlinear programming problems with BARON

Abstract: In this paper, we present recent developments in the global optimization software BARON to address problems with integer variables. A primary development was the addition of mixed-integer linear programming relaxations to BARON's portfolio of linear and nonlinear programming relaxations, aiming to improve dual bounds and offer good starting points for primal heuristics. Since such relaxations necessitate the solution of NP-hard problems, their introduction to a branch-and-bound algorithm raises many practical issues regarding their effective implementation. In addition to describing BARON's dynamic strategy for deciding under what conditions to activate integer programming relaxations in the course of branch-and-bound, the paper also describes cutting plane and probing techniques that originate from the literature of integer linear programming and have been adapted in BARON to solve nonlinear problems. Finally, we describe BARON's primal heuristics for finding good solutions of mixed-integer nonlinear pro...

Optimal Coordination of Directional Overcurrent Relays in Microgrids by Using Cuckoo-Linear Optimization Algorithm and Fault Current Limiter

Abstract: Nowadays, distributed generations (DGs) and microgrids play a significant role in development of power systems, but DGs produce several protection problems especially in optimal coordination of directional overcurrent relays (DORs). Optimal coordination of DORs is a highly constrained nonlinear programming problem, which has attracted attention of many research studies in recent years. Cuckoo optimization algorithm (COA) is introduced as an optimization tool based on behavior patterns of Cuckoos. In this paper, COA and linear programming (LP) are combined as a new hybrid COA-LP optimization algorithm in order to optimize the coordination protection of DORs in microgrids and find the optimum value of fault current limiter at the point of common coupling. The optimum settings of DORs are obtained for both modes of operation: 1) grid-connected and 2) island mode. The proposed algorithm is applied to Canadian distribution benchmark and a modified IEEE 14 bus networks and compared with particle swarm optimization (PSO) and genetic algorithm (GA). The obtained results show that the computational efficiency of the proposed COA-LP is better than PSO and GA in terms of speed and total operating time of DORs.

A Distributionally Robust Optimization Model for Unit Commitment Based on Kullback–Leibler Divergence

Abstract: This paper proposes a new distance-based distributionally robust unit commitment (DB-DRUC) model via Kullback–Leibler (KL) divergence, considering volatile wind power generation. The objective function of the DB-DRUC model is to minimize the expected cost under the worst case wind distributions restricted in an ambiguity set. The ambiguity set is a family of distributions within a fixed distance from a nominal distribution. The distance between two distributions is measured by KL divergence. The DB-DRUC model is a “min-max-min” programming model; thus, it is intractable to solve. Applying reformulation methods and stochastic programming technologies, we reformulate this “min-max-min” DB-DRUC model into a one-level model, referred to as the reformulated DB-DRUC (RDB-DRUC) model. Using the generalized Benders decomposition, we then propose a two-level decomposition method and an iterative algorithm to address the RDB-DRUC model. The iterative algorithm for the RDB-DRUC model guarantees global convergence within finite iterations. Case studies are carried out to demonstrate the effectiveness, global optimality, and finite convergence of a proposed solution strategy.

Additive consistency analysis and improvement for hesitant fuzzy preference relations

Abstract: Hesitant fuzzy preference relation (HFPR) is an effective tool to elicit decision makers’ hesitant preference information over alternatives, and consistency analysis is of great importance for an HFPR since inconsistent judgments may result in unreasonable results. In this paper, the best additive consistency index, the worst additive consistency index and the average additive consistency index are defined to measure the consistency level of an HFPR. To improve the additive consistency of an HFPR, some mixed 0–1 linear programming models which aim to minimize the overall adjustment amount and the number of the elements that need to be adjusted are established. Moreover, the proposed models are extended to improve the additive consistency and impute the missing elements for incomplete HFPRs. Some numerical examples are presented to show the characteristics of the proposed models. The results demonstrate that the proposed models can improve the consistency of an HFPR effectively.

Bi-Level Two-Stage Robust Optimal Scheduling for AC/DC Hybrid Multi-Microgrids

Abstract: In view of the plurality of ac and dc microgrids connected to the power grid, this paper proposes a bi-level two-stage robust optimal scheduling model for ac/dc hybrid multi-microgrids (HMMs). In this model, the ac/dc HMM system is divided into two interest bodies, called the utility level and supply level, and two-stage robust (also called adaptive robust) optimal scheduling is carried out considering the uncertainties in each level. For the mutual interaction between the two bodies, power constraints and deviation penalties of the interaction lines are introduced in the model to realize bi-level coordinated scheduling and obtain a unified robust dispatch scheme of ac/dc HMMs. The column-and-constraint generation algorithm is used to convert the Min-Max-Min problem of each level into a two-stage mixed-integer linear programming problem, which can be solved quickly and effectively. Case studies verify the rationality and validity of the proposed model and the solving procedure.

High-Quality Nonparallel Voice Conversion Based on Cycle-Consistent Adversarial Network

Abstract: Although voice conversion (VC) algorithms have achieved remarkable success along with the development of machine learning, superior performance is still difficult to achieve when using nonparallel data. In this paper, we propose using a cycle-consistent adversarial network (CycleGAN) for nonparallel data-based VC training. A CycleGAN is a generative adversarial network (GAN) originally developed for unpaired image-to-image translation. A subjective evaluation of inter-gender conversion demonstrated that the proposed method significantly outperformed a method based on the Merlin open source neural network speech synthesis system (a parallel VC system adapted for our setup) and a GAN-based parallel VC system. This is the first research to show that the performance of a nonparallel VC method can exceed that of state-of-the-art parallel VC methods.

Optimal Covariance Control for Stochastic Systems Under Chance Constraints

Abstract: This letter addresses the optimal covariance control problem for stochastic discrete-time linear systems subject to chance constraints. To the best of our knowledge, covariance steering problems with probabilistic chance constraints have not been discussed previously in the literature, although their treatment seems to be a natural extension. In this letter, we first show that, unlike the case with no chance constraints, the covariance steering problem with chance constraints cannot be decoupled to mean and covariance steering sub-problems. We then propose an approach to solve the covariance steering problem with chance constraints by converting it to a convex programming problem. The proposed algorithm is verified using a numerical example.

Iterative Parameter Estimation for Signal Models Based on Measured Data

Abstract: This paper studies the modeling of multi-frequency signals based on measured data. With the use of the hierarchical identification principle and the iterative search, several iterative parameter estimation algorithms are derived for the signal models with the known frequencies and the unknown frequencies. For the multi-frequency signals, the hierarchical estimation algorithms are derived by means of parameter decomposition. Through the decomposition, the original optimization problem is transformed into the combination of the nonlinear optimization and the linear optimization problems. The simulation results show that the proposed hierarchical algorithms have better performance than the overall estimation algorithms without parameter decomposition.

A Linear Programming Method Based on an Improved Score Function for Interval-Valued Pythagorean Fuzzy Numbers and Its Application to Decision-Making

Abstract: The present paper proposes an improved score function for solving multi-criteria decision-making (MCDM) problem with partially known weight information, In it, the preferences related to criteria a

Prediction-Correction Interior-Point Method for Time-Varying Convex Optimization

Abstract: In this paper, we develop an interior-point method for solving a class of convex optimization problems with time-varying objective and constraint functions. Using log-barrier penalty functions, we propose a continuous-time dynamical system for tracking the (time-varying) optimal solution with an asymptotically vanishing error. This dynamical system is composed of two terms: a correction term consisting of a continuous-time version of Newton's method, and a prediction term able to track the drift of the optimal solution by taking into account the time-varying nature of the objective and constraint functions. Using appropriately chosen time-varying slack and barrier parameters, we ensure that the solution to this dynamical system globally asymptotically converges to the optimal solution at an exponential rate. We illustrate the applicability of the proposed method in two applications: a sparsity promoting least squares problem and a collision-free robot navigation problem.

A Distributionally Robust Chance-Constrained MILP Model for Multistage Distribution System Planning With Uncertain Renewables and Loads

Abstract: Successful transition to active distribution networks (ADNs) requires a planning methodology that includes an accurate network model and accounts for the major sources of uncertainty. Considering these two essential features, this paper proposes a novel model for the multistage distribution expansion planning (MDEP) problem, which is able to jointly expand both the network assets (feeders and substations) and renewable/conventional distributed generators. With respect to network characteristics, the proposed planning model employs a convex conic quadratic format of ac power flow equations that is linearized using a highly accurate polyhedral-based linearization method. Furthermore, a chance-constrained programming approach is utilized to deal with the uncertain renewables and loads. In this regard, as the probability distribution functions of uncertain parameters are not perfectly known, a distributionally robust (DR) reformulation is proposed for the chance constraints that guarantees the robustness of the expansion plans against all uncertainty distributions defined within a moment-based ambiguity set. Effective linearization techniques are also devised to eliminate the nonlinearities of the proposed DR reformulation, which yields a distributionally robust chance-constrained mixed-integer linear programming model for the MDEP problem of ADNs. Finally, the 24-node and 138-node test systems are used to demonstrate the effectiveness of the proposed planning methodology.