This letter investigates parallelism approaches for equation and Jacobian evaluation in power flow calculations. Two levels of parallelism are proposed and analyzed: inter-model parallelism, which evaluates models in parallel, and intra-model parallelism, which evaluates calculations within each model in parallel. Parallelism techniques such as multi-threading and single instruction multiple data (SIMD) vectorization are discussed, implemented, and benchmarked as six calculation workflows. Case studies on the 70,000-bus synthetic grid show that equation evaluations can be accelerated by ten times, and the overall Newton power flow outperforms MATPOWER by 20%.

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Julia is a programming language for the scientific community that combines features of productivity languages, such as Python or MATLAB, with characteristics of performance-oriented languages, such as C++ or Fortran. Julia's productivity features include: dynamic typing, automatic memory management, rich type annotations, and multiple dispatch. At the same time, Julia allows programmers to control memory layout and leverages a specializing just-in-time compiler to eliminate much of the overhead of those features. This paper details the design choices made by the creators of Julia and reflects on the implications of those choices for performance and usability. CCS Concepts: • Software and its engineering → Language features; General programming languages; Just-in-time compilers; Multiparadigm languages;

Programming languages that support multiple dispatch rely on an expressive notion of subtyping to specify method applicability. In these languages, type annotations on method declarations are used to select, out of a potentially large set of methods, the one that is most appropriate for a particular tuple of arguments. Julia is a language for scientific computing built around multiple dispatch and an expressive subtyping relation. This paper provides the first formal definition of Julia's subtype relation and motivates its design. We validate our specification empirically with an implementation of our definition that we compare against the existing Julia implementation on a collection of real-world programs. Our subtype implementation differs on 122 subtype tests out of 6,014,476. The first 120 differences are due to a bug in Julia that was fixed once reported; the remaining 2 are under discussion. CCS Concepts: • Software and its engineering → Data types and structures; Semantics;

Arrays are such a rich and fundamental data type that they tend to be built into a language, either in the compiler or in a large lowlevel library. Defining this functionality at the user level instead provides greater flexibility for application domains not envisioned by the language designer. Only a few languages, such as C++ and Haskell, provide the necessary power to define n-dimensional arrays, but these systems rely on compile-time abstraction, sacrificing some flexibility. In contrast, dynamic languages make it straightforward for the user to define any behavior they might want, but at the possible expense of performance.As part of the Julia language project, we have developed an approach that yields a novel trade-off between flexibility and compiletime analysis. The core abstraction we use is multiple dispatch. We have come to believe that while multiple dispatch has not been especially popular in most kinds of programming, technical computing is its killer application. By expressing key functions such as array indexing using multi-method signatures, a surprising range of behaviors can be obtained, in a way that is both relatively easy to write and amenable to compiler analysis. The compact factoring of concerns provided by these methods makes it easier for userdefined types to behave consistently with types in the standard library.