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%.

Connectivity among populations and habitats is important for a wide range of ecological processes. Understanding, preserving, and restoring connectivity in complex landscapes requires connectivity models and metrics that are reliable, efficient, and process based. We introduce a new class of ecological connectivity models based in electrical circuit theory. Although they have been applied in other disciplines, circuit-theoretic connectivity models are new to ecology. They offer distinct advantages over common analytic connectivity models, including a theoretical basis in random walk theory and an ability to evaluate contributions of multiple dispersal pathways. Resistance, current, and voltage calculated across graphs or raster grids can be related to ecological processes (such as individual movement and gene flow) that occur across large population networks or landscapes. Efficient algorithms can quickly solve networks with millions of nodes, or landscapes with millions of raster cells. Here we review basic circuit theory, discuss relationships between circuit and random walk theories, and describe applications in ecology, evolution, and conservation. We provide examples of how circuit models can be used to predict movement patterns and fates of random walkers in complex landscapes and to identify important habitat patches and movement corridors for conservation planning.

Conservation practitioners have long recognized ecological connectivity as a global priority for preserving biodiversity and ecosystem function. In the early years of conservation science, ecologists extended principles of island biogeography to assess connectivity based on source patch proximity and other metrics derived from binary maps of habitat. From 2006 to 2008, the late Brad McRae introduced circuit theory as an alternative approach to model gene flow and the dispersal or movement routes of organisms. He posited concepts and metrics from electrical circuit theory as a robust way to quantify movement across multiple possible paths in a landscape, not just a single least-cost path or corridor. Circuit theory offers many theoretical, conceptual, and practical linkages to conservation science. We reviewed 459 recent studies citing circuit theory or the open-source software Circuitscape. We focused on applications of circuit theory to the science and practice of connectivity conservation, including topics in landscape and population genetics, movement and dispersal paths of organisms, anthropogenic barriers to connectivity, fire behavior, water flow, and ecosystem services. Circuit theory is likely to have an effect on conservation science and practitioners through improved insights into landscape dynamics, animal movement, and habitat-use studies and through the development of new software tools for data analysis and visualization. The influence of circuit theory on conservation comes from the theoretical basis and elegance of the approach and the powerful collaborations and active user community that have emerged. Circuit theory provides a springboard for ecological understanding and will remain an important conservation tool for researchers and practitioners around the globe.Aplicaciones de la Teoría de Circuitos a la Conservación y a la Ciencia de la Conectividad Resumen: Quienes practican la conservación han reconocido durante mucho tiempo que la conectividad ecológica es una prioridad mundial para la preservación de la biodiversidad y el funcionamiento del * email: brett@csp-inc.org Article impact statement: Uses of circuit theory to understand connectivity have had a durable and global impact on conservation science and practice.

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The conservation field is experiencing a rapid increase in the amount, variety, and quality of spatial data that can help us understand species movement and landscape connectivity patterns. As interest grows in more dynamic representations of movement potential, modelers are often limited by the capacity of their analytic tools to handle these datasets. Technology developments in software and high-performance computing are rapidly emerging in many fields, but uptake within conservation may lag, as our tools or our choice of computing language can constrain our ability to keep pace. We recently updated Circuitscape, a widely used connectivity analysis tool developed by Brad McRae and Viral Shah, by implementing it in Julia, a high-performance computing language. In this initial re-code (Circuitscape 5.0) and later updates, we improved computational efficiency and parallelism, achieving major speed improvements, and enabling assessments across larger extents or with higher resolution data. Here, we reflect on the benefits to conservation of strengthening collaborations with computer scientists, and extract examples from a collection of 572 Circuitscape applications to illustrate how through a decade of repeated investment in the software, applications have been many, varied, and increasingly dynamic. Beyond empowering continued innovations in dynamic connectivity, we expect that faster run times will play an important role in facilitating co-production of connectivity assessments with stakeholders, increasing the likelihood that connectivity science will be incorporated in land use decisions.