In this paper, we present an algorithm to automatically detect meaningful modes in a histogram. The proposed method is based on the behavior of local minima in a scale-space representation. We show that the detection of such meaningful modes is equivalent in a two classes clustering problem on the length of minima scale-space curves. The algorithm is easy to implement, fast and does not require any parameter. We present several results on histogram and spectrum segmentation, grayscale image segmentation and color image reduction.

Abstract-Typical coordination schemes for future power grids require two-way communications. Since the number of end power-consuming devices is large, the bandwidth requirements for such two-way communication schemes may be prohibitive. Motivated by this observation, we study distributed coordination schemes that require only one-way limited communications. In particular, we investigate how dual descent distributed optimization algorithm can be employed in power networks using one-way communication. In this iterative algorithm, system coordinators broadcast coordinating (or pricing) signals to the users/devices who update power consumption based on the received signal. Then system coordinators update the coordinating signals based on the physical measurement of the aggregate power usage. We provide conditions to guarantee the feasibility of the aggregated power usage at each iteration so as to avoid blackout. Furthermore, we prove the convergence of algorithms under these conditions, and establish its rate of convergence. We illustrate the performance of our algorithms using numerical simulations. These results show that one-way limited communication may be viable for coordinating/operating the future smart grids.

Distributed optimization increasingly plays a central role in economical and sustainable operation of cyber-physical systems. Nevertheless, the complete potential of the technology has not yet been fully exploited in practice due to communication limitations posed by the real-world infrastructures. This work investigates fundamental properties of distributed optimization based on gradient methods, where gradient information is communicated using limited number of bits. In particular, a general class of quantized gradient methods are studied where the gradient direction is approximated by a finite quantization set. Sufficient and necessary conditions are provided on such a quantization set to guarantee that the methods minimize any convex objective function with Lipschitz continuous gradient and a nonempty and bounded set of optimizers. A lower bound on the cardinality of the quantization set is provided, along with specific examples of minimal quantizations. Convergence rate results are established that connect the fineness of the quantization and the number of iterations needed to reach a predefined solution accuracy. Generalizations of the results to a relevant class of constrained problems using projections are considered. Finally, the results are illustrated by simulations of practical systems.

Abstract-As electric power system operators shift from conventional energy to renewable energy sources, power distribution systems will experience increasing fluctuations in supply. These fluctuations present the need to not only design online decentralized power allocation algorithms, but also characterize how effective they are given fast-changing consumer demand and generation. In this paper, we present an Online Decentralized Dual Descent (OD3) power allocation algorithm and determine (in the worst case) how much of observed social welfare can be explained by fluctuations in generation capacity and consumer demand. Convergence properties and performance guarantees of the OD3 algorithm are analyzed by characterizing the difference between the online decision and the optimal decision. We demonstrate validity and accuracy of the theoretical results in the paper through numerical experiments using real power generation data.

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As independent service providers shift from conventional energy to renewable energy sources, the power distribution system will likely experience increasingly significant fluctuation in supply, given the uncertain and intermittent nature of renewable sources like wind and solar energy. These fluctuations in power generation, coupled with time-varying consumer demands of electricity and the massive scale of power distribution networks present the need to not only design real-time decentralized power allocation algorithms, but also characterize how effective they are given fast-changing consumer demands and power generation capacities. In this paper, we present an Online Decentralized Dual Descent (OD3) power allocation algorithm and determine (in the worst case) how much of observed social welfare and price volatility can be explained by fluctuations in generation capacity and consumer demand. Convergence properties and performance guarantees of the OD3 algorithm are analyzed by characterizing the difference between the online decision and the optimal decision. The theoretical results in the paper are validated and illustrated by numerical experiments using real data.