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This paper studies the problem of constructing codes correcting deletions in arrays. Under this model, it is assumed that an $n \times n$ array can experience deletions of rows and columns. These deletion errors are referred to as $({t_{\mathrm {r}}}, {t_{\mathrm {c}}})$ -criss-cross deletions if ${t_{\mathrm {r}}}$ rows and ${t_{\mathrm {c}}}$ columns are deleted, while a code correcting these deletion patterns is called a $({t_{\mathrm {r}}}, {t_{\mathrm {c}}})$ -criss-cross deletion correction code. The definitions for criss-cross insertions are similar. It is first shown that when $t_{r}=t_{c}$ the problems of correcting criss-cross deletions and criss-cross insertions are equivalent. The focus of this paper lies on the case of (1, 1)-criss-cross deletions. A non-asymptotic upper bound on the cardinality of (1, 1)-criss-cross deletion correction codes is shown which assures that the redundancy is at least $2n-3+2\log n$ bits. A code construction with an existential encoding and an explicit decoding algorithm is presented. The redundancy of the construction is at most $2n+4 \log n + 7 +2 \log e$. A construction with explicit encoder and decoder is presented. The explicit encoder adds an extra $5\log n + 5$ bits of redundancy to the construction. [ABSTRACT FROM AUTHOR]

The first of this two-paper series addresses a practical day-ahead auction model, where generation revenue constraints are explicitly incorporated in the problem formulation, as routinely done in several national electricity markets across Europe. The revenue-constrained market-clearing procedure includes the effect of the transmission network, inter-temporal constraints associated with generation scheduling, demand-side bidding, and marginal pricing. This auction design is an instance of price-based market clearing which features two major complicating factors. First, locational marginal prices become decision variables of the optimization process. In addition, producer revenues are formulated as bilinear and highly nonconvex products of power outputs and market-clearing prices. The resulting problem is formulated as a mixed-integer nonlinear bilevel program with bilinear terms for which available solution techniques rely on heuristics, approximations, or modeling simplifications. This paper presents a novel and exact methodology whereby the original problem is recast as an equivalent single-level mixed-integer linear program. As a consequence, finite convergence to optimality is guaranteed and the use of standard commercial software is allowed. The proposed transformation is based on duality theory of linear programming, Karush-Kuhn-Tucker optimality conditions, and integer algebra results. In the second part of this two-paper series, numerical results from several case studies illustrate the effective performance of the proposed solution approach. [ABSTRACT FROM AUTHOR]

Interleaved Reed–Solomon codes admit efficient decoding algorithms which correct burst errors far beyond half the minimum distance in the random errors regime, e.g., by computing a common solution to the Key Equation for each Reed–Solomon code, as described by Schmidt et al. If this decoder does not succeed, it may either fail to return a codeword or miscorrect to an incorrect codeword, and good upper bounds on the fraction of error matrices for which these events occur are known. The decoding algorithm immediately applies to interleaved alternant codes as well, i.e., the subfield subcodes of interleaved Reed–Solomon codes, but the fraction of decodable error matrices differs, since the error is now restricted to a subfield. In this paper, we present new general lower and upper bounds on the fraction of error matrices decodable by Schmidt et al.’s decoding algorithm, thereby making it the only decoding algorithm for interleaved alternant codes for which such bounds are known. [ABSTRACT FROM AUTHOR]

Transitioning from fossil fuel dominated power systems to high penetrations of intermittent renewable generation is affecting classical electricity market designs. Here, a method is proposed to model and assess the absence of co-optimization of energy and reserve that prevails in the European system. Co-optimization is formulated through an optimization problem (COM), and its absence as an equilibrium problem (EQM) built on Karush-Kuhn-Tucker conditions of agents’ optimization and market clearing equations. EQM cannot be reformulated as a single optimization problem. Market distortions are identified by comparing the complementarity conditions of both models. These are then discussed on system with Feed in Premium to wind generation. Parameters in the models allow to represent different market configurations regarding: available wind generation, Feed in Premium to wind, generators’ risk aversion, and required reserve from wind generation. [ABSTRACT FROM AUTHOR]