Internal Time Step Research Articles

Uneven internal time-step adjustment for fast power system dynamic simulations based on Trapezoidal integration of elementary transfer function blocks

With the development of renewable energy sources, the scale and complexity of power systems are continuously increasing, resulting in a high requirement for the efficiency of dynamic simulation. Increasing the simulation time-step is a possible measure for the fixed time-step (FTS) method to accelerate the simulation, yet also a potential cause for deteriorating the simulation accuracy. To improve the overall power system dynamic simulation speed without jeopardizing the expected accuracy, this paper proposes a novel Uneven Internal Time-Step Adjustment method based on Trapezoidal integration of elementary transfer function blocks (ETFBs). For ETFBs with different time constants, three different integration modes are defined, including single-step integration, multiple-step integration, and static approximation. Each ETFB can select an appropriate integration mode and adjust its internal time-step, automatically expanding the size of the simulation time-step. Implementation of static approximation for ETFBs with small time constants is given. Discussion on the determination of threshold values of internal time-step is provided, which mainly affect the accuracy of the system. The proposed method is applied to the open-source Simulation Toolkit for Electrical Power Systems (STEPS). Different cases are tested to obtain the maximum simulation time-step that can maintain numerical stability and validate the efficiency and accuracy of the proposed method.

On discretizing sea-ice dynamics on triangular meshes using vertex, cell or edge velocities

Discretization of the equations of Viscous Plastic and Elastic Viscous Plastic (EVP) sea ice dynamics on triangular meshes can be done by placing discrete velocities at vertices, cells or edges. Since there are more cells and edges than vertices, the cell- and edge-based discretizations simulate more linear kinematic features at the same mesh than the vertex discretization. However, the discretizations based on cell and edge velocities may suffer from non-trivial kernels in the strain rate or stress divergence operators and need either special strain rate computations for cell velocities, or stabilization for edge velocities. An elementary Fourier analysis clarifies how kernels are removed, and also shows that cell and edge velocity placement leads to spurious branches of stress divergence operator with large negative eigenvalues. Although spurious branches correspond to fast decay and are not expected to distort sea ice dynamics, they demand either smaller internal time steps or higher stability parameters in explicit EVP-like methods.

On the Damping Time Scale of EVP Sea Ice Dynamics

AbstractWe propose to make the damping time scale, which governs the decay of pseudo‐elastic waves in the Elastic Viscous Plastic (EVP) sea‐ice solvers, independent of the external time step and large enough to warrant numerical stability for a moderate number of internal time steps. A necessary condition is that the forcing on sea ice varies slowly on the damping time scale, in which case an EVP solution may still approach a Viscous Plastic one, but on a time scale longer than a single external time step. In this case, the EVP method becomes very close to the recently proposed modified EVP (mEVP) method in terms of stability and simulated behavior. In a simple test case dealing with sea ice breaking under the forcing of a moving cyclone, the EVP method with an enlarged damping time scale can simulate linear kinematic features which are very similar to those from the traditional EVP implementation, although a much smaller number of internal time steps is used. There is more difference in sea‐ice thickness and linear kinematic features simulated in a realistic Arctic configuration between using the traditional and our suggested choices of EVP damping time scales, but it is minor considering model uncertainties associated with choices of many other parameters in sea‐ice models.

Modeling Climate Change Impacts on Rangeland Productivity and Livestock Population Dynamics in Nkayi District, Zimbabwe

Smallholder farmers in semi-arid areas depend on both cropping and livestock as the main sources of livelihoods. Rangeland productivity varies on both spatial and temporal scales and provides the major source of feed for livestock. Rangeland productivity is expected to decline with climate change thereby reducing livestock feed availability and consequently livelihoods that depend on livestock. This study was carried out to assess the impacts of climate change on rangeland productivity and consequently livestock population dynamics using a 30-year simulation modeling approach. The climate scenarios used in the simulations are built from the localized predictions by General Circulation Models (GCMs). The primary climate variables under consideration are rainfall (+/−7% change), carbon dioxide (CO2 up to 650 ppm) and temperature (+4 °C change). This was done by applying the SAVANNA ecosystem model which simulates rangeland processes and demographic responses of herbivores on a temporal and spatial scale using a weekly internal time step and monthly spatial and temporal outputs. The results show that rainfall levels of less than 600 mm/year have the largest negative effect on herbaceous biomass production. The amount of biomass from the woody layer does not change much during the year. The carbon dioxide (CO2) effects are more influential on the tree and shrub layers (C3 plants) than the herbaceous layer (C4 grasses). The CO2 effect was more dominant than the effects of rainfall and temperature. In the baseline simulations, the shrub plant layer increased significantly over 30 years while there is a three-fold increase in the woody plant layer (trees and shrubs) where biomass increased from a 1980 production to that of 2010. The biomass of the herbaceous layer was stable over the historical period (1980 to 2010) with values fluctuating between 200 and 400 g/m2. Grass green biomass has a variable distribution where most production occurred in the fields and cleared areas while lower levels of production were found in the forested areas. The spatial distribution of shrub green biomass was less directly linked to yearly rainfall. Shrub biomass was mostly found in forested areas, and it showed a steady increase in production. Cattle, donkey, and goat populations rose slowly from 1980 but the rise was disrupted by a dry period during the late 1980s to the early 1990s causing a decline in all populations primarily due to grass unavailability. The populations of cattle goats and donkeys started to rise again from 1995 onwards due to improvements in rainfall. Cattle and donkey populations were rising faster than that of goats while sheep population was not changing much for most of the simulation period, otherwise they declined significantly during the drought of 2002. Similar changes in simulated grass biomass (g/m2) were observed in almost all climate scenarios, except for the peak and low years. The livestock population simulation showed few variations in livestock population under all scenarios. The main conclusion from the study is that CO2 effects on rangeland productivity are much more dominant than the localized effects of rainfall and temperature. This has implications of favoring the growth of the tree and shrub layers over herbaceous layer, which meant that in the long run, the species that are able to use tree and shrub layers may be kept as a livelihood source as they will have a feed source.

Non-intrusive hierarchical coupling strategies for multi-scale simulations in gravitational dynamics

Hierarchical code coupling strategies make it possible to combine the results of individual numerical solvers into a self-consistent symplectic solution. We explore the possibility of allowing such a coupling strategy to be non-intrusive. In that case, the underlying numerical implementation is not affected by the coupling itself, but its functionality is carried over in the interface. This method is efficient for solving the equations of motion for a self-gravitating system over a wide range of scales. We adopt a dedicated integrator for solving each particular part of the problem and combine the results to a self-consistent solution. In particular, we explore the possibilities of combining the evolution of one or more microscopic systems that are embedded in a macroscopic system. The here presented generalizations of Bridge include higher-order coupling strategies (from the classic 2nd order up to 10th-order), but we also demonstrate how multiple bridges can be nested and how additional processes can be introduced at the bridge time-step to enrich the physics, for example by incorporating dissipative processesor. Such augmentation allows for including additional processes in a classic Newtonian N-body integrator without alterations to the underlying code. These additional processes include for example the Yarkovsky effect, dynamical friction or relativistic dynamics. Some of these processes operate on all particles whereas others apply only to a subset.The presented method is non-intrusive in the sense that the underlying methods remain operational without changes to the code (apart from adding the get- and set-functions to enable the bridge operator). As a result, the fundamental integrators continue to operate with their internal time step and preserve their local optimizations and parallelism. Multiple bridges can be nested and coupled hierarchically, allowing for the construction of a complex environment of multiple nested augmented bridges. While the coupling topology may become rather complicated, we introduce the hierarchical coupling language (HCL), a meta language in which complex bridge topologies can be described. The meta language is meant for stimulating the discussion on even more complex hierarchies in which the bridge operators are introduced as patternsWe present example applications for several of these cases and discuss the conditions under which these integrators can be applied. Typical applications range over 10 orders of magnitude in temporal and spatial scales when we apply the method to simulating planetary systems (au spatial and year-temporal scale) in a star cluster that orbits in the Galaxy (100 kpc-spatial and 10 Gyr-temporal scale).

Deep multiscale model learning

The objective of this paper is to design novel multi-layer neural networks for multiscale simulations of flows taking into account the observed fine data and physical modeling concepts. Our approaches use deep learning techniques combined with local multiscale model reduction methodologies to predict flow dynamics. Using reduced-order model concepts is important for constructing robust deep learning architectures since the reduced-order models provide fewer degrees of freedom. We consider flow dynamics in porous media as multi-layer networks in this work. More precisely, the solution (e.g., pressures and saturation) at the time instant n+1 depends on the solution at the time instant n and input parameters, such as permeability fields, forcing terms, and initial conditions. One can regard the solution as a multi-layer network, where each layer, in general, is a nonlinear forward map and the number of layers relates to the internal time steps. We will rely on rigorous model reduction concepts to define unknowns and connections between layers. It is critical to use reduced-order models for this purpose, which will identify the regions of influence and the appropriate number of variables. Furthermore, due to the lack of available observed fine data, the reduced-order model can provide us sufficient inexpensive data as needed. The designed deep neural network will be trained using both coarse simulation data which is obtained from the reduced-order model and observed fine data. We will present the main ingredients of our approach and numerical examples. Numerical results show that using deep learning with data generated from multiscale models as well as available observed fine data, we can obtain an improved forward map which can better approximate the fine scale model.

Global soil consumption of atmospheric carbon monoxide: an analysis using a process-based biogeochemistry model

Abstract. Carbon monoxide (CO) plays an important role in controlling the oxidizing capacity of the atmosphere by reacting with OH radicals that affect atmospheric methane (CH4) dynamics. We develop a process-based biogeochemistry model to quantify the CO exchange between soils and the atmosphere with a 5 min internal time step at the global scale. The model is parameterized using the CO flux data from the field and laboratory experiments for 11 representative ecosystem types. The model is then extrapolated to global terrestrial ecosystems using monthly climate forcing data. Global soil gross consumption, gross production, and net flux of the atmospheric CO are estimated to be from −197 to −180, 34 to 36, and −163 to −145 Tg CO yr−1 (1 Tg = 1012 g), respectively, when the model is driven with satellite-based atmospheric CO concentration data during 2000–2013. Tropical evergreen forest, savanna and deciduous forest areas are the largest sinks at 123 Tg CO yr−1. The soil CO gross consumption is sensitive to air temperature and atmospheric CO concentration, while the gross production is sensitive to soil organic carbon (SOC) stock and air temperature. By assuming that the spatially distributed atmospheric CO concentrations (∼ 128 ppbv) are not changing over time, the global mean CO net deposition velocity is estimated to be 0.16–0.19 mm s−1 during the 20th century. Under the future climate scenarios, the CO deposition velocity will increase at a rate of 0.0002–0.0013 mm s−1 yr−1 during 2014–2100, reaching 0.20–0.30 mm s−1 by the end of the 21st century, primarily due to the increasing temperature. Areas near the Equator, the eastern US, Europe and eastern Asia will be the largest sinks due to optimum soil moisture and high temperature. The annual global soil net flux of atmospheric CO is primarily controlled by air temperature, soil temperature, SOC and atmospheric CO concentrations, while its monthly variation is mainly determined by air temperature, precipitation, soil temperature and soil moisture.

An error evaluation on the vertical velocity algorithm in POM

A time splitting technique is common to many free surface ocean models. The different truncation errors in the equations of the internal and external modes require a numerical adjustment to make sure that algorithms correctly satisfy continuity equations and conserve tracers quantities. The princeton ocean model (POM) has applied a simple method of adjusting the vertical mean of internal velocities to external velocities at each internal time step. However, due to the Asselin time filter method adopted to prevent the numerical instability, the method of velocity adjustment used in POM can no longer guarantee the satisfaction of the continuity equation in the internal mode, though a special treatment is used to relate the surface elevation of the internal mode with that of the external mode. The error is proved to be a second-order term of the coefficient in the Asselin filter. One influence of this error in the numerical model is the failure of the kinetic boundary condition at the sea floor. By a regional experiment and a quasi-global experiment, the magnitudes of this error are evaluated, and several sensitivity tests of this error are performed. The characteristic of this error is analyzed and two alternative algorithms are suggested to reduce the error.

Estimation of the impact of short-term fluctuations in inputs on temporally aggregated outputs of process-oriented models

Process-oriented models driven by highly resolved meteorological inputs and comprising a short internal time step are sometimes used to predict substance fluxes in air, soil and water over fairly long periods of time. To ascertain whether regression-based input–output analyses in such cases can provide adequate parametric models of the impact of daily and monthly fluctuations in inputs on annual outputs, we studied the SOIL/SOILN model of vertical transport of heat, water and nitrogen through arable soils. Annual leaching of nitrate from the root zone was regarded as the response variable, and regressors were selected from among the set of all linear combinations of daily or monthly values of five different meteorological inputs. We found that, although several of the underlying processes described by the SOIL/SOILN model are non-linear, both ordinary and partial least squares regression (OLS and PLS) identified the subsets of input variables with the strongest influence on the model output, and the dominating time lags between model inputs and outputs. Furthermore, highly resolved explanatory variables were a prerequisite for good performance of linear predictors of temporally aggregated outputs and, to discern the full dynamic behaviour of the model, it was necessary to analyse the response to artificially generated daily meteorological data representing a very large number of different weather conditions. PLS had one advantage over OLS: a smooth pattern in the regression coefficients facilitated physical interpretation of the derived impulse–response weights.

Incorporation of a permafrost model into a large‐scale ecosystem model: Evaluation of temporal and spatial scaling issues in simulating soil thermal dynamics

This study evaluated whether a model of permafrost dynamics with a 0.5‐day resolution internal time step that is driven by monthly climate inputs is adequate for representing the soil thermal dynamics in a large‐scale ecosystem model. An extant version of the Goodrich model was modified to develop a soil thermal model (STM) with the capability to operate with either 0.5‐hour or 0.5‐day internal time steps and to be driven with either daily or monthly input data. The choice of internal time step had little effect on the simulation of soil thermal dynamics of a black spruce site in Alaska. The use of monthly climate inputs to drive the model resulted in an error of less than 1°C in the upper organic soil layer and in an accurate simulation of seasonal active layer dynamics. Uncertainty analyses of the STM driven with monthly climate inputs identified that soil temperature estimates of the upper organic layer were most sensitive to variability in parameters that described snow thermal conductivity, moss thickness, and moss thermal conductivity. The STM was coupled to the Terrestrial Ecosystem Model (TEM), and the performance of the coupled model was verified for the simulation of soil temperatures in applications to a black spruce site in Canada and to white spruce, aspen, and tundra sites in Alaska. A 1°C error in the temperature of the upper organic soil layer had little influence on the carbon dynamics simulated for the black spruce site in Canada. Application of the model across the range of black spruce ecosystems in North America demonstrated that the STM‐TEM has the capability to operate over temporal and spatial domains that consider substantial variation in surface climate given that spatial variability in key structural characteristics and physical properties of the soil thermal regime are described.

A Correction to the Baroclinic Pressure Gradient Term in the Princeton Ocean Model

Abstract An error in the calculation of the baroclinic pressure gradient term in the Princeton Ocean Model (POM) was identified while modeling the M2 tidal current near its critical latitude in the southern Weddell Sea. The error arises from the present calculation of density, which involves the subtraction of a background density profile from the density field calculated at each internal time step. The small displacement of sigma surface depths relative to the surface, as surface elevation changes, causes a slight error in the calculation of the vertical and horizontal gradients of potential density. The error is largest at the seabed over rapidly changing bathymetry such as the continental slope. The baroclinic pressure gradient error is typically much smaller than the Coriolis term in the momentum equations and, therefore, usually unimportant. Close to the critical latitude, however, near-resonance between the error and Coriolis terms can cause an energetic and spatially complex spurious inertial mode ...

A real-time, unsupervised neural network for the low-level control of a mobile robot in a nonstationary environment

This article introduces a real-time, unsupervised neural network that learns to control a two-degree-of-freedom mobile robot in a nonstationary environment. The neural controller, which is termed neural NETwork MObile Robot Controller (NETMORC), combines associative learning and Vector Associative Map (VAM) learning to generate transformations between spatial and velocity coordinates. As a result, the controller learns the wheel velocities required to reach a target at an arbitrary distance and angle. The transformations are learned in an unsupervised training phase, during which the robot moves as a result of randomly selected wheel velocities. The robot learns the relationship between these velocities and the resulting incremental movements. Aside from being able to reach stationary or moving targets, the NETMORC structure also enables the robot to perform successfully in spite of disturbances in the environment, such as wheel slippage, or changes in the robot's plant, including changes in wheel radius, changes in interwheel distance, or changes in the internal time step of the system. Finally, the controller is extended to include a module that learns an internal odometric transformation, allowing the robot to reach targets when visual input is sporadic or unreliable.