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To accommodate the advent of microgrids (MG) managing distributed energy resources (DER) in distribution systems, an interactive two-stage joint retail electricity market mechanism is proposed to provide an effective platform for these prosumers to proactively join in retail transactions. Day-ahead stochastic energy trading between the distribution system operator (DSO) and MGs is conducted in the first stage of a centralized retail market, where a chance-constrained uncertainty distribution locational marginal price (CC-UDLMP) containing the cost of uncertainty precautions is used to settle transactions. In the second stage, a novel intra-day peer-to-peer-based (P2P) flexibility transaction pattern is implemented between MGs in local flexibility markets under the regulation of DSO to eliminate power imbalances caused by rolling-based estimates whilst considering systematic operations. A fully distributed iterative algorithm is presented to find the equilibrium solution of this two-stage sequential game framework. Moreover, in order to enhance the versatility of this algorithm, an improved Lp-box alternating direction methods of multipliers (ADMM) algorithm is used to efficiently resolve the first-stage stochastic economic dispatch problem with a mixed-integer second-order cone structure. It is verified that the proposed market mechanism can effectively improve the overall market efficiency under uncertainties.

The burgeoning integration of distributed energy resources (DER) poses new challenges for the economic and safe operation of the electricity system. The current distribution-side policy is largely based on mandatory regulations and incentives, rather than the design of a competitive market mechanism to arouse DERs to freely compete in the retail market. To address this issue, we proposed a transactive retail market mechanism to attract the active participation of profit-driven DER retailers in a deregulated way. Considering the limited competitive property of the retail market, a bi-level DSO-dominated framework is constructed to simulate the virtual game between distribution system operator (DSO) and DER retailers. Specifically, the flexible interval pricing of retailers is modeled as a series of binary revenue constraints to influence the DSO's centralized economic dispatch decisions, where the time-varying distribution locational marginal price is adopted to settle energy activities at different locations over the time horizon. Due to the uncertain power flow path during the coordinated decision-making process, we proposed a general undirected second-order cone-based AC radial power flow model and provided sufficient conditions to ensure its exactness. Also, we applied a series of approximation and relaxation techniques to transform the bi-level mixed-integer quadratic framework with highly discrete induced domains into a solvable mixed-integer semidefinite programming problem. It is demonstrated in the case study that the proposed mechanism can not only improve market efficiency but also eliminate market power and the resulting market failures.

It is well known that, whenever $k$ divides $n$, the complete $k$-uniform hypergraph on $n$ vertices can be partitioned into disjoint perfect matchings. Equivalently, the set of $k$-subsets of an $n$-set can be partitioned into parallel classes so that each parallel class is a partition of the $n$-set. This result is known as Baranyai's theorem, which guarantees the existence of \emph{Baranyai partitions}. Unfortunately, the proof of Baranyai's theorem uses network flow arguments, making this result non-explicit. In particular, there is no known method to produce Baranyai partitions in time and space that scale linearly with the number of hyperedges in the hypergraph. It is desirable for certain applications to have an explicit construction that generates Baranyai partitions in linear time. Such an efficient construction is known for $k=2$ and $k=3$. In this paper, we present an explicit recursive quadrupling construction for $k=4$ and $n=4t$, where $t \equiv 0,3,4,6,8,9 ~(\text{mod}~12)$. In a follow-up paper (Part II), the other values of~$t$, namely $t \equiv 1,2,5,7,10,11 ~(\text{mod}~12)$, will be considered.

Generalized balance tournament packings (GBTPs) extend the concept of generalized balanced tournament designs introduced by Lamken and Vanstone (1989). In this paper, we establish the connection between GBTPs and a class of codes called equitable symbol weight codes. The latter were recently demonstrated to optimize the performance against narrowband noise in a general coded modulation scheme for power line communications. By constructing classes of GBTPs, we establish infinite families of optimal equitable symbol weight codes with code lengths greater than alphabet size and whose narrowband noise error-correcting capability to code length ratios do not diminish to zero as the length grows.

The use of multiple frequency shift keying modulation with permutation codes addresses the problem of permanent narrowband noise disturbance in a power line communications system. In this paper, we extend this coded modulation scheme based on permutation codes to general codes and introduce an additional new parameter that more precisely captures a code's performance against permanent narrowband noise. As a result, we define a new class of codes, namely, equitable symbol weight codes, which are optimal with respect to this measure.

We study (symbol-pair) codes for symbol-pair read channels introduced recently by Cassuto and Blaum (2010). A Singleton-type bound on symbol-pair codes is established and infinite families of optimal symbol-pair codes are constructed. These codes are maximum distance separable (MDS) in the sense that they meet the Singleton-type bound. In contrast to classical codes, where all known q-ary MDS codes have length O(q), we show that q-ary MDS symbol-pair codes can have length \Omega(q^2). In addition, we completely determine the existence of MDS symbol-pair codes for certain parameters.

A cross-bifix-free code is a set of words in which no prefix of any length of any word is the suffix of any word in the set. Cross-bifix-free codes arise in the study of distributed sequences for frame synchronization. We provide a new construction of cross-bifix-free codes which generalizes the construction in Bajic (2007) to longer code lengths and to any alphabet size. The codes are shown to be nearly optimal in size. We also establish new results on Fibonacci sequences, that are used in estimating the size of the cross-bifix-free codes.