scholarly journals Optimization of wind plant layouts using an adjoint approach

2017 ◽  
Vol 2 (1) ◽  
pp. 115-131 ◽  
Author(s):  
Ryan N. King ◽  
Katherine Dykes ◽  
Peter Graf ◽  
Peter E. Hamlington
Keyword(s):  
Steady State ◽  
Flow Model ◽  
Rotational Symmetry ◽  
Layout Optimization ◽  
Adjoint Equations ◽  
Wind Rose ◽  
Flow Models ◽  
Curvature Effects ◽  
Gradient Based ◽  
Adjoint Approach

Abstract. Using adjoint optimization and three-dimensional steady-state Reynolds-averaged Navier–Stokes (RANS) simulations, we present a new gradient-based approach for optimally siting wind turbines within utility-scale wind plants. By solving the adjoint equations of the flow model, the gradients needed for optimization are found at a cost that is independent of the number of control variables, thereby permitting optimization of large wind plants with many turbine locations. Moreover, compared to the common approach of superimposing prescribed wake deficits onto linearized flow models, the computational efficiency of the adjoint approach allows the use of higher-fidelity RANS flow models which can capture nonlinear turbulent flow physics within a wind plant. The steady-state RANS flow model is implemented in the Python finite-element package FEniCS and the derivation and solution of the discrete adjoint equations are automated within the dolfin-adjoint framework. Gradient-based optimization of wind turbine locations is demonstrated for idealized test cases that reveal new optimization heuristics such as rotational symmetry, local speedups, and nonlinear wake curvature effects. Layout optimization is also demonstrated on more complex wind rose shapes, including a full annual energy production (AEP) layout optimization over 36 inflow directions and 5 wind speed bins.

scholarly journals Adjoint Optimization of Wind Plant Layouts

2016 ◽  
Author(s):  
Ryan N. King ◽  
Katherine Dykes ◽  
Peter Graf ◽  
Peter E. Hamlington
Keyword(s):  
Flow Model ◽  
Rotational Symmetry ◽  
Three Dimensional ◽  
Layout Optimization ◽  
Adjoint Equations ◽  
Wind Rose ◽  
Flow Models ◽  
Curvature Effects ◽  
Optimization Heuristics ◽  
Gradient Based

Abstract. Using adjoint optimization and three-dimensional Reynolds-averaged Navier Stokes (RANS) simulations, we present a new gradient-based approach for optimally siting wind turbines within utility-scale wind plants. By solving the adjoint equations of the flow model, the gradients needed for optimization are found at a cost that is independent of the number of control variables, thereby permitting optimization of large wind plants with many turbine locations. Moreover, compared to the common approach of superimposing prescribed wake deficits onto linearized flow models, the computational efficiency of the adjoint approach allows the use of higher-fidelity RANS flow models which can capture nonlinear turbulent flow physics within a wind plant. The RANS flow model is implemented in the Python finite element package FEniCS and the derivation of the adjoint equations is automated within the dolfin-adjoint framework. Gradient-based optimization of wind turbine locations is demonstrated on idealized test cases that reveal new optimization heuristics such as rotational symmetry, local speedups, and nonlinear wake curvature effects. Layout optimization is also demonstrated on more complex wind rose shapes, including a full annual energy production (AEP) layout optimization over 36 inflow directions and 5 windspeed bins.

Steady-State Tilting-Pad Bearing Performance Under Reduced Oil Supply Flow Rates

2018 ◽  
Vol 140 (5) ◽  
Author(s):  
Bradley R. Nichols ◽  
Roger L. Fittro ◽  
Christopher P. Goyne
Keyword(s):  
Experimental Data ◽  
Steady State ◽  
Flow Rate ◽  
Flow Model ◽  
Leading Edge ◽  
Input Power ◽  
Flow Rates ◽  
Flow Models ◽  
Oil Supply ◽  
Tilting Pad

Reduced oil supply flow rates in fluid film bearings can cause cavitation, or lack of a fully developed hydrodynamic film layer, at the leading edge of the bearing pads. Reduced oil flow has the well-documented effects of higher bearing operating temperatures and decreased power losses and is commonly referred to as starvation. This study looks at the effects of oil supply flow rate on steady-state bearing performance and provides increased experimental data for comparison to computational predictions. Tests are conducted on a five-pad tilting-pad bearing positioned in a vintage, flooded housing with oil supply nozzles. Pad temperatures, sump temperature, journal operating position, and motor input power are measured at various operating speeds ranging from 2000 to 12,000 rpm and various oil supply flow rates. Predicted results are obtained from bearing modeling software based on thermoelastohydrodynamic (TEHD) lubrication theory. A starved flow model was previously developed as an improvement over the original flooded flow model to more accurately capture bearing behavior under reduced flow conditions. Experimental results are compared to both flow models. The starved bearing model predicts significantly higher journal operating positions than the flooded model and shows good correlation with the experimental data. Predicted pressure profiles from the starved bearing model show cavitation of the upper unloaded pads that increase in severity with increasing speed and decreasing oil supply flow rate. The progressive unloading of these top pads explains the rise in shaft centerline position and helps further validate the starvation model.

scholarly journals FLOWERS: An integral approach to engineering wake models

2021 ◽  
Author(s):  
Michael LoCascio ◽  
Christopher Bay ◽  
Majid Bastankhah ◽  
Garrett Barter ◽  
Paul Fleming ◽  
...  
Keyword(s):  
Wind Direction ◽  
Wind Farm ◽  
Solution Space ◽  
Layout Optimization ◽  
Weighted Averaging ◽  
Integral Approach ◽  
Power Calculation ◽  
Wind Rose ◽  
Gradient Based ◽  
Dynamics Simulations

Abstract. Annual energy production (AEP) is often the objective function in wind plant layout optimization studies. The conventional method to compute AEP for a wind farm is to first evaluate power production for each wind direction and speed using either computational fluid dynamics simulations or engineering wake models. The AEP is then calculated by weighted-averaging (based on the wind rose at the wind farm site) the power produced across all wind directions. We propose a novel formulation for time-averaged wake velocity that incorporates an analytical integral of a wake deficit model across every wind direction. This approach computes the average flow field more efficiently, and layout optimization is an obvious application to exploit this benefit. The clear advantage of this new approach is that the layout optimization produces solutions with comparable AEP performance yet is completed about 700 times faster. The analytical integral and the use of a Fourier expansion to express the wind speed and wind direction frequency create a more smooth solution space for the gradient-based optimizer to excel compared with the discrete nature of the existing weighted-averaging power calculation.

Mathematical modelling of salt purification by recrystallization. A cross-current flow model and its comparison with the countercurrent arrangement

1986 ◽  
Vol 51 (11) ◽  
pp. 2489-2501
Author(s):  
Benitto Mayrhofer ◽  
Jana Mayrhoferová ◽  
Lubomír Neužil ◽  
Jaroslav Nývlt
Keyword(s):  
Flow Model ◽  
Current Flow ◽  
Countercurrent Flow ◽  
Flow Models ◽  
Multi Stage ◽  
Stage Crystallization ◽  
The Cross ◽  
Current Flows ◽  
Set Up ◽  
Cross Current

A model is derived for a multi-stage crystallization with cross-current flows of the solution and the crystals being purified. The purity of the product is compared with that achieved in the countercurrent arrangement. A suitable function has been set up which allows the cross-current and countercurrent flow models to be compared and reduces substantially the labour of computation for the countercurrent arrangement. Using the recrystallization of KAl(SO4)2.12 H2O as an example, it is shown that, when the cross-current and countercurrent processes are operated at the same output, the countercurrent arrangement is more advantageous because its solvent consumption is lower.

scholarly journals On Using a Gradient-based Method for Heliostat Field Layout Optimization

2014 ◽  
Vol 49 ◽  
pp. 1429-1438 ◽  
Author(s):  
S.L. Lutchman ◽  
A.A. Groenwold ◽  
P. Gauché ◽  
S. Bode
Keyword(s):  
Layout Optimization ◽  
Gradient Based

Study of the Response of PW Traffic Flow Model to a Bottleneck on a Circular Road and its Suitability for Heterogeneous Traffic Conditions

2021 ◽  
Vol 9 (4) ◽  
pp. 460-463
Keyword(s):  
Fluid Flow ◽  
Traffic Flow ◽  
Flow Model ◽  
Equation Model ◽  
Heterogeneous Traffic ◽  
Flow Conditions ◽  
Flow Models ◽  
Traffic Conditions ◽  
Model Response ◽  
Traffic Flow Models

The traffic flow conditions in developing countries are predominantly heterogeneous. The early developed traffic flow models have been derived from fluid flow to capture the behavior of the traffic. The very first two-equation model derived from fluid flow is known as the Payne-Whitham or PW Model. Along with the traffic flow, this model also captures the traffic acceleration. However, the PW model adopts a constant driver behavior which cannot be ignored, especially in the situation of heterogeneous traffic.This research focuses on testing the PW model and its suitability for heterogeneous traffic conditions by observing the model response to a bottleneck on a circular road. The PW model is mathematically approximated using the Roe Decomposition and then the performance of the model is observed using simulations.

scholarly journals Beyond the bucket – Developing a global gradient-based groundwater model (G<sup>3</sup>M v1.0) for a global hydrological model from scratch

2018 ◽  
Author(s):  
Robert Reinecke ◽  
Laura Foglia ◽  
Steffen Mehl ◽  
Tim Trautmann ◽  
Denise Cáceres ◽  
...  
Keyword(s):  
Steady State ◽  
Capillary Rise ◽  
Hydrological Models ◽  
Grid Cells ◽  
Reservoir Model ◽  
Model Framework ◽  
Water Flows ◽  
Macro Scale ◽  
Gradient Based ◽  
The Impact

Abstract. To quantify water flows between groundwater (GW) and surface water (SW) as well as the impact of capillary rise on evapotranspiration by global hydrological models (GHMs), it is necessary to replace the bucket-like linear GW reservoir model typical for hydrological models with a fully integrated gradient-based GW flow model. Linear reservoir models can only simulate GW discharge to SW bodies, provide no information on the location of the GW table and assume that there is no GW flow among grid cells. A gradient-based GW model simulates not only GW storage but also hydraulic head, which together with information on SW table elevation enables the quantification of water flows from GW to SW and vice versa. In addition, hydraulic heads are the basis for calculating lateral GW flow among grid cells and capillary rise. G3M is a new global gradient-based GW model with a spatial resolution of 5' that will replace the current linear GW reservoir in the 0.5° WaterGAP Global Hydrology Model (WGHM). The newly developed model framework enables in-memory coupling to WGHM while keeping overall runtime relatively low, allowing sensitivity analyses and data assimilation. This paper presents the G3M concept and specific model design decisions together with results under steady-state naturalized conditions, i.e. neglecting GW abstractions. Cell-specific conductances of river beds, which govern GW-SW interaction, were determined based on the 30'' steady-state water table computed by Fan et al. (2013). Together with an appropriate choice for the effective elevation of the SW table within each grid cell, this enables a reasonable simulation of drainage from GW to SW such that, in contrast to the GW model of de Graaf et al. (2015, 2017), no additional drainage based on externally provided values for GW storage above the floodplain is required in G3M. Comparison of simulated hydraulic heads to observations around the world shows better agreement than de Graaf et al. (2015). In addition, G3M output is compared to the output of two established macro-scale models for the Central Valley, California, and the continental United States, respectively. As expected, depth to GW table is highest in mountainous and lowest in flat regions. A first analysis of losing and gaining rivers and lakes/wetlands indicates that GW discharge to rivers is by far the dominant flow, draining diffuse GW recharge, such that lateral flows only become a large fraction of total diffuse and focused recharge in case of losing rivers and some areas with very low GW recharge. G3M does not represent losing rivers in some dry regions. This study presents the first steps towards replacing the linear GW reservoir model in a GHM while improving on recent efforts, demonstrating the feasibility of the approach and the robustness of the newly developed framework.

scholarly journals Improving the FLORIS wind plant model for compatibility with gradient-based optimization

2017 ◽  
Vol 41 (5) ◽  
pp. 313-329 ◽  
Author(s):  
Jared J Thomas ◽  
Pieter MO Gebraad ◽  
Andrew Ning
Keyword(s):  
Steady State ◽  
Wind Turbine ◽  
Case Studies ◽  
Wind Farm ◽  
Optimization Problems ◽  
Algorithmic Differentiation ◽  
Function Calls ◽  
Gradient Based ◽  
Wake Model ◽  
Yaw Angle

The FLORIS (FLOw Redirection and Induction in Steady-state) model, a parametric wind turbine wake model that predicts steady-state wake characteristics based on wind turbine position and yaw angle, was developed for optimization of control settings and turbine locations. This article provides details on changes made to the FLORIS model to make the model more suitable for gradient-based optimization. Changes to the FLORIS model were made to remove discontinuities and add curvature to regions of non-physical zero gradient. Exact gradients for the FLORIS model were obtained using algorithmic differentiation. A set of three case studies demonstrate that using exact gradients with gradient-based optimization reduces the number of function calls by several orders of magnitude. The case studies also show that adding curvature improves convergence behavior, allowing gradient-based optimization algorithms used with the FLORIS model to more reliably find better solutions to wind farm optimization problems.

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