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In this paper, we consider the problem of scheduling shiftable loads, over multiple users, in smart electrical grids. We approach the problem, which is becoming increasingly pertinent in our present energy-thirsty society, using a novel distributed game-theoretic framework. In our specific instantiation, we consider the scenario when the power system has a local-area Smart Grid subnet comprising of a single power source and multiple customers. The objective of the exercise is to tacitly control the total power consumption of the customers' shiftable loads, so to approach the rigid power budget determined by the power source, but to simultaneously not exceed this threshold. As opposed to the “traditional”paradigm that utilizes a central controller to achieve the load scheduling, we seek to achieve this by pursuing a distributed approach that allows the users1 to make individual decisions by invoking negotiations with other customers. The decisions are essentially of the sort, where the individual users can choose whether they want to be supplied or not. From a modeling perspective, the distributed scheduling problem is formulated as a game, and in particular, a so-called “Potential”game. This game has at least one pure strategy Nash equilibrium (NE), and we demonstrate that the NE point is a global optimal point. The solution that we propose, which utilizes the theory of learning automata (LA), permits the total supplied loads to approach the power budget of the subnet once the algorithm has converged to the NE point. The scheduling is achieved by attaching a LA to each customer. The paper discusses the applicability of three different LA schemes, and in particular, the recently-introduced Bayesian learning automata. Numerical results, obtained from testing the schemes on numerous simulated data sets, demonstrate the speed and the accuracy of proposed algorithms in terms of their convergence to the game's NE point.

From the earliest studies in graph theory, the phenomenon of transitivity has been used to design and analyze problems that can be mapped onto graphs. Some of the practical examples of this phenomenon are the “Transitive Closure”algorithm, the multiplication of Boolean matrices, the determination of communicating states in Markov chains, and so on. The use of transitivity, however, to catalyze the partitioning problems is, to our knowledge, unreported, and it is by no means trivial considering the pairwise occurrences of the queries in the query stream. This paper pioneers such a mechanism. In particular, we consider the object migrating automaton (OMA) that has been used for decades to solve the Equi-Partitioning Problem where W objects are placed in R partitions of equal sizes so that objects accessed together fall in to the same partition. The OMA, which encountered certain deadlock configurations, was enhanced by Gale et al. to yield the enhanced OMA (EOMA). Both the OMA and the EOMA were significantly improved by incorporating into them, the recently-introduced “Pursuit”phenomenon from the field of learning automata (LA). In this paper1, we shall show that the Pursuit matrix that consists of the estimates of the probabilities of the pairs presented to the LA, possesses the property of transitivity akin to the property found in graph-related problems. By making use of this observation, transitive-closure-like arguments can be made to invoke reward and penalty operations on the Pursuit OMA and the PEOMA. This implies that objects can be moved toward their correct partitions even when the system is dormant, i.e., when the environment does not present any queries or partitioning information to the learning algorithm. The results that we present demonstrate that the newly designed transitive-based algorithms are about 20% faster than their non-transitive versions. As far as we know, these are the fastest partitioning algorithms to-date.

Traditional pattern classification works with the moments of the distributions of the features and involves the estimation of the means and variances. As opposed to this, more recently, research has indicated the power of using the quantiles of the distributions because they are more robust and applicable for nonparametric methods. The estimation of the quantiles is even more pertinent when one is mining data streams. However, the complexity of quantile estimation is much higher than the corresponding estimation of the mean and variance, and this increased complexity is more relevant as the size of the data increases. Clearly, in the context of infinite data streams, a computational and space complexity that is linear in the size of the data is definitely not affordable. In order to alleviate the problem complexity, recently, a very limited number of studies have devised incremental quantile estimators [1], [2]. Estimators within this class resort to updating the quantile estimates based on the most recent observation(s), and this yields updating schemes with a very small computational footprint-a constant-time (i.e., O(1)) complexity. In this paper, we pursue this research direction and present an estimator that we refer to as a higher-fidelity frugal [1] quantile estimator. First, it guarantees a substantial advancement of the family of Frugal estimators introduced in [1]. The highlight of the present scheme is that it works in the discretized space, and it is thus a pioneering algorithm within the theory of discretized algorithms.1 The convergence results that we have proven are based on the theory of stochastic point location [3], which we advocate as a new tool for solving a large class of online estimation problems. Extensive simulation results show that our estimator outperforms the original Frugal algorithm in terms of both speed and accuracy.