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Power grids are undergoing major changes due to rapid growth in renewable energy resources and improvements in battery technology. While these changes enhance sustainability and efficiency, they also create significant management challenges as the complexity of power systems increases. To tackle these challenges, decentralized Internet-of-Things (IoT) solutions are emerging, which arrange local communities into transactive microgrids. Within a transactive microgrid, "prosumers" (i.e., consumers with energy generation and storage capabilities) can trade energy with each other, thereby smoothing the load on the main grid using local supply. It is hard, however, to provide security, safety, and privacy in a decentralized and transactive energy system. On the one hand, prosumers' personal information must be protected from their trade partners and the system operator. On the other hand, the system must be protected from careless or malicious trading, which could destabilize the entire grid. This paper describes Privacypreserving Energy Transactions (PETra), which is a secure and safe solution for transactive microgrids that enables consumers to trade energy without sacrificing their privacy. PETra builds on distributed ledgers, such as blockchains, and provides anonymity for communication, bidding, and trading.

A backtracking algomthm for testing a pair of digraphs for isomorphism is presented The mformatlon contained m the distance matrix representation of a graph is used to estabhsh an lmtlal partltmn of the graph's vertices This distance matrix reformation is then apphed m a backtracking procedure to reduce the search tree ofposmble mappings While the algorithm is not guaranteed to run m polynomml time, it performs efficmntly for a large class of graphs KEY WORDS AND PHRASES backtrack programming, digraph, distance matrix, momorphlsm, partltmnmg CR CATEGORIES 5 32A new backtracking algomthm for testing a pair of directed graphs for isomorphism is presented in this paper. The algorithm is efficient for a large class of graphs and terminates either providing the isomorphism, or Indicating that there is no isomorphism for the given pair of graphs. Given two graphs, G 1 and G 2, the problem of determining an isomorphism, ff it exmts, is important. It has applicatmn m a varmty of fields including chemistry, network theory, and reformation retrieval. The problem can be solved by examining all N! permutations of N vertices. However, this approach is practical only for graphs with very small N.Although there have been attempts to find a mathematical function which will identify isomorphic graphs [1,2], no such function has been found which can be computed in polynomial time. The problem is known to be in the set ofNP (nondeterministm polynomml) problems but it is not known if it is NP-complete (see Karp [3]) Since heuristics have been employed successfully on a variety of problems which have exhaustive search solutmns of factorial or exponential order, they were applied to the graph isomorphism problem by Unger [4], who reported processing times for some 24 node graphs at 2 min on an IBM 7090. For some 12 node graphs, however, the time was up to 7½ min. Other heuristm algorithms have been proposed [5][6][7].

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