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DOE OSTI2023

A comparison of eight optimization methods applied to a wind farm layout optimization problem

Abstract. Selecting a wind farm layout optimization method is difficult. Comparisons between optimization methods in different papers can be uncertain due to the difficulty of exactly reproducing the objective function. Comparisons by just a few authors in one paper can be uncertain if the authors do not have experience using each algorithm. In this work we provide an algorithm comparison for a wind farm layout optimization case study between eight optimization methods applied, or directed, by researchers who developed those algorithms or who had other experience using them. We provided the objective function to each researcher to avoid ambiguity about relative performance due to a difference in objective function. While these comparisons are not perfect, we try to treat each algorithm more fairly by having researchers with experience using each algorithm apply each algorithm and by having a common objective function provided for analysis. The case study is from the International Energy Association (IEA) Wind Task 37, based on the Borssele III and IV wind farms with 81 turbines. Of particular interest in this case study is the presence of disconnected boundary regions and concave boundary features. The optimization methods studied represent a wide range of approaches, including gradient-free, gradient-based, and hybrid methods; discrete and continuous problem formulations; single-run and multi-start approaches; and mathematical and heuristic algorithms. We provide descriptions and references (where applicable) for each optimization method, as well as lists of pros and cons, to help readers determine an appropriate method for their use case. All the optimization methods perform similarly, with optimized wake loss values between 15.48 % and 15.70 % as compared to 17.28 % for the unoptimized provided layout. Each of the layouts found were different, but all layouts exhibited similar characteristics. Strong similarities across all the layouts include tightly packing wind turbines along the outer borders, loosely spacing turbines in the internal regions, and allocating similar numbers of turbines to each discrete boundary region. The best layout by annual energy production (AEP) was found using a new sequential allocation method, discrete exploration-based optimization (DEBO). Based on the results in this study, it appears that using an optimization algorithm can significantly improve wind farm performance, but there are many optimization methods that can perform well on the wind farm layout optimization problem, given that they are applied correctly.

17 WIND ENERGY↗

NASA NTRS1977

Stress-free end problem in layered materials

In this paper the plane elastostatic problem for a medium which consists of periodically arranged two sets of bonded dissimilar layers or strips is considered. First it is assumed that one set of strips contains a crack which crosses the bimaterial interfaces. Then, by letting the collinear cracks join, the stress-free end problem is formulated. The singular behavior of the solutions at the point on intersection of the stress-free boundary and the interfaces is examined and appropriate stress intensity factors are defined. The results of some numerical examples are then presented which include the cases of both plane stress and plane strain.

Erdogan, F.↗

NASA NTRS1981

Eigenvalues of the Rayleigh-Benard and Marangoni problems

The eigenvalues of the linear Benard-Marangoni stability problem are discussed. Pearson and Nield boundary conditions, which correspond to a rigid, isothermal lower boundary and a stress-free conducting upper boundary are considered. It is shown that although a critical value of the Marangoni number can be determined, the number is not, strictly speaking, an eigenvalue and cannot be used as an eigenvalue parameter for the determination of an eigenvector set.

Rosenblat, S.↗

NASA NTRS1986

The construction and use of divergence free vector expansions for incompressible fluid flow calculations

For incompressible fluids the law of mass conservation reduces to a constraint on the velocity vector, namely that it be divergence free. This constraint has long been a source of great difficulty to the numericist seeking to discretize the Navier-Stokes and Euler equations. A spectral method is discussed which overcomes this difficulty. Its efficacy is demonstrated on some simple problems. The velocity is approximated by a finite sum of divergence free vectors, each of which satisfies the same boundary conditions as the velocity. Projecting the governing equation onto the space of inviscid vector fields eliminates the pressure term and produces a set of ordinary differential equations that must be solved for the coefficents in the velocity. The pressure can then be recovered if it is needed.

Mhuiris, N. M. G.↗

DOE OSTI2021

Ocean Energy Systems Wave Energy Modeling Task 10.4: Numerical Modeling of a Fixed Oscillating Water Column

This paper reports on an ongoing international effort to establish guidelines for numerical modeling of wave energy converters, initiated by the International Energy Agency Technology Collaboration Program for Ocean Energy Systems. Initial results for point absorbers were presented in previous work, and here we present results for a breakwater-mounted Oscillating Water Column (OWC) device. The experimental model is at scale 1:4 relative to a full-scale installation in a water depth of 12.8 m. The power-extracting air turbine is modeled by an orifice plate of 1–2% of the internal chamber surface area. Measurements of chamber surface elevation, air flow through the orifice, and pressure difference across the orifice are compared with numerical calculations using both weakly-nonlinear potential flow theory and computational fluid dynamics. Both compressible- and incompressible-flow models are considered, and the effects of air compressibility are found to have a significant influence on the motion of the internal chamber surface. Recommendations are made for reducing uncertainties in future experimental campaigns, which are critical to enable firm conclusions to be drawn about the relative accuracy of the numerical models. It is well-known that boundary element method solutions of the linear potential flow problem (e.g., WAMIT) are singular at infinite frequency when panels are placed directly on the free surface. This is problematic for time-domain solutions where the value of the added mass matrix at infinite frequency is critical, especially for OWC chambers, which are modeled by zero-mass elements on the free surface. A straightforward rational procedure is described to replace ad-hoc solutions to this problem that have been proposed in the literature.

16 TIDAL AND WAVE POWER↗

NASA NTRS1984

Probabilistic radiative transfer

A computationally efficient method has been developed for highly nonlinear problems in which radiative transfer is an important aspect of the heating and cooling of the medium. An approximate probabilistic radiative transfer equation is derived for one-dimensional plane-parallel atmospheres of finite or semi-infinite extent, for both spectral lines and bound-free continua. Boundary conditions, accuracy, escape probabilities, and practical aspects of complete linearization, are discussed. The method is accurate to a few tens of percent of a wide variety of realistic problems in which frequency redistribution of scattered photons dominates the transfer and escape of radiation.

Canfield, R. C.↗

NASA NTRS1984

Surface cracks in a plate of finite width under extension or bending

In this paper the problem of a finite plate containing collinear surface cracks is considered. The problem is solved by using the line spring model with plane elasticity and Reissner's plate theory. The main purpose of the study is to investigate the effect of interaction between two cracks or between cracks and stress-free plate boundaries on the stress intensity factors and to provide extensive numerical results which may be useful in applications. First, some sample results are obtained and are compared with the existing finite element results. Then the problem is solved for a single (internal) crack, two collinear cracks and two corner cracks for wide range of relative dimensions. Particularly in corner cracks the agreement with the finite element solution is surprisingly very good. The results are obtained for semielliptic and rectangular crack profiles which may, in practice, correspond to two limiting cases of the actual profile of a subcritically growing surface crack.

Erdogan, F.↗

NASA NTRS1956

An Engineering Approach to the Variable Fluid Property Problem in Free Convection

An analysis is made for the variable fluid property problem for laminar free convection on an isothermal vertical flat plate. For a number of specific cases, solutions of the boundary layer equations appropriate to the variable property situation were carried out for gases and liquid mercury. Utilizing these findings, a simple and accurate shorthand procedure is presented for calculating free convection heat transfer under variable property conditions. This calculation method is well established in the heat transfer field. It involves the use of results which have been derived for constant property fluids, and of a set of rules (called reference temperatures) for extending these constant property results to variable property situations. For gases, the constant property heat transfer results are generalized to the variable property situation by replacing beta (expansion coefficient) by one over T sub infinity and evaluating the other properties at T sub r equals T sub w minus zero point thirty-eight (T sub w minus T sub infinity). For liquid mercury, the generalization may be accomplished by evaluating all the properties (including beta) at this same T sub r. It is worthwhile noting that for these fluids, the film temperature (with beta equals one over T sub infinity for gases) appears to serve as an adequate reference temperature for most applications. Results are also presented for boundary layer thickness and velocity parameters.

Gregg, J. L.↗

NASA NTRS1990

On the instability of hypersonic flow past a flat plate

The instability of hypersonic boundary-layer flows over flat plates is considered. The viscosity of the fluid is taken to be governed by Sutherland's law, which gives a much more accurate representation of the temperature dependence of fluid viscosity at hypersonic speeds than Chapman's approximate linear law; although at lower speeds the temperature variation of the mean state is less pronounced so that the Chapman law can be used with some confidence. Attention is focussed on the so-called (vorticity) mode of instability of the viscous hypersonic boundary layer. This is thought to be the fastest growing inviscid disturbance at hypersonic speeds; it is also believed to have an asymptotically larger growth rate than any viscous or centrifugal instability. As a starting point the instability of the hypersonic boundary layer which exists far downstream from the leading edge of the plate is investigated. In this regime the shock that is attached to the leading edge of the plate plays no role, so that the basic boundary layer is non-interactive. It is shown that the vorticity mode of instability of this flow operates on a significantly different lengthscale than that obtained if a Chapman viscosity law is assumed. In particular, it is found that the growth rate predicted by a linear viscosity law overestimates the size of the growth rate by O(M(exp 2). Next, the development of the vorticity mode as the wavenumber decreases is described, and it is shown that acoustic modes emerge when the wavenumber has decreased from it's O(1) initial value to O(M (exp -3/2). Finally, the inviscid instability of the boundary layer near the leading edge in the interaction zone is discussed and particular attention is focussed on the strong interaction region which occurs sufficiently close to the leading edge. It is found that the vorticity mode in this regime is again unstable, and that it is concentrated in the transition layer at the edge of the boundary layer where the temperature adjusts from its large, O(M(exp 2), value in the viscous boundary layer, to its O(1) free stream value. The existence of the shock indirectly, but significantly, influences the instability problem by modifying the basic flow structure in this layer.

Blackaby, Nicholas↗

NASA NTRS1991

The unsteady laminar boundary layer on an axisymmetric body subject to small-amplitude fluctuations in the free-stream velocity

The effect of small-amplitude, time-periodic, free-stream disturbances on an otherwise steady axisymmetric boundary layer on a circular cylinder is considered. Numerical solutions to the problem are presented, and an asymptotic solution to the flow, valid far downstream along the axis of the cylinder is detailed. Particular emphasis is placed on the unsteady eigensolutions that occur far downstream, which turn out to be very different from the analogous planar eigensolutions. These axisymmetric eigensolutions are computed numerically and also are described by asymptotic analyses valid for low and high frequencies of oscillation.

Duck, Peter W.↗

NASA NTRS1994

Role of pressure diffusion in non-homogeneous shear flows

A non-local model is presented for approximating the pressure diffusion in calculations of turbulent free shear and boundary layer flows. It is based on the solution of an elliptic relaxation equation which enables local diffusion sources to be distributed over lengths of the order of the integral scale. The pressure diffusion model was implemented in a boundary layer code within the framework of turbulence models based on both the kappa-epsilon-(bar)upsilon(exp 2) system of equations and the full Reynolds stress equations. Model computations were performed for mixing layers and boundary layer flows. In each case, the pressure diffusion model enabled the well-known free-stream edge singularity problem to be eliminated. There was little effect on near-wall properties. Computed results agreed very well with experimental and DNS data for the mean flow velocity, the turbulent kinetic energy, and the skin-friction coefficient.

Demuren, A. O.↗

NASA NTRS1963

INVESTIGATION OF PROBLEMS IN THERMAL CONVECTION

Numerical solution of thermal convection equations for velocity and temperature fluctuations as well as the stability analysis and heat transport for free boundary conditions

THERMAL CONVECTION↗

NASA NTRS1985

Vortex studies relating to boundary layer turbulence and noise

The present study considers the two-dimensional case of an array of N rectilinear, like-sign vortices above an infinite flat boundary. The method of images can be employed with this configuration to reduce the problem to that of 2N vortices in free space, constrained by 2N symmetry relations. This system is Hamiltonian and therefore certain invariants of the motion are known. Futher, from the Hamiltonian constant, the equations of motion are readily derived and may be integrated numerically to determine the vortex trajectories. This knowledge of the time-dependent vortex motion then allows the resulting noise radiation to be computed by standard aeroacoustic techniques.

Adelman, J. L.↗

NASA NTRS1985

Vortex Studies Relating to Boundary Layer Turbulence and Noise

The present study considers the two-dimensional case of an array of N rectilinear, like-sign vortices above an infinite flat boundary. The method of images can be employed with this configuration to reduce the problem to that of 2N vortices in free space, constrained by 2N symmetry relations. This system is Hamiltonian and therefore certain invariants of the motion are known. Further, from the Hamiltonian constant, the equations of motion are readily derived and may be integrated numerically to determine the vortex trajectories. This knowledge of the time-dependent vortex motion then allows the resulting noise radiation to be computed by standard aeroacoustic techniques.

Adelman, J. L.↗

NASA NTRS1988

Analysis of symmetric cross-ply laminated elastic plates using a higher-order theory. II - Buckling and free vibration

A previously developed higher-order plate theory and a technique based on the state space concept are used to investigate free vibration and buckling problems of rectangular cross-ply laminated plates. Unified results are presented for the case of arbitrary boundary conditions on two opposite edges. Good agreement is obtained with previous data for simply supported edge conditions. It is pointed out that classical laminated plate theory tends to overpredict both eigenfrequencies and buckling loads, leading to an increase of the degree of orthotropicity of individual layers and of the thickness ratio of the plate.

Khdeir, A. A.↗

NASA NTRS1988

Confidence set inference with a prior quadratic bound

In the uniqueness part of a geophysical inverse problem, the observer wants to predict all likely values of P unknown numerical properties z = (z sub 1,...,z sub p) of the earth from measurement of D other numerical properties y(0)=(y sub 1(0),...,y sub D(0)) knowledge of the statistical distribution of the random errors in y(0). The data space Y containing y(0) is D-dimensional, so when the model space X is infinite-dimensional the linear uniqueness problem usually is insoluble without prior information about the correct earth model x. If that information is a quadratic bound on x (e.g., energy or dissipation rate), Bayesian inference (BI) and stochastic inversion (SI) inject spurious structure into x, implied by neither the data nor the quadratic bound. Confidence set inference (CSI) provides an alternative inversion technique free of this objection. CSI is illustrated in the problem of estimating the geomagnetic field B at the core-mantle boundary (CMB) from components of B measured on or above the earth's surface. Neither the heat flow nor the energy bound is strong enough to permit estimation of B(r) at single points on the CMB, but the heat flow bound permits estimation of uniform averages of B(r) over discs on the CMB, and both bounds permit weighted disc-averages with continous weighting kernels. Both bounds also permit estimation of low-degree Gauss coefficients at the CMB. The heat flow bound resolves them up to degree 8 if the crustal field at satellite altitudes must be treated as a systematic error, but can resolve to degree 11 under the most favorable statistical treatment of the crust. These two limits produce circles of confusion on the CMB with diameters of 25 deg and 19 deg respectively.

Backus, George E.↗

NASA NTRS1990

FIRE aircraft observations of horizontal and vertical transport in marine stratocumulus

A major goal of research on marine stratocumulus is to try to understand the processes that generate and dissipate them. One approach to studying this problem is to investigate the boundary layer structure in the vicinity of a transition from a cloudy to a cloud-free region to document the differences in structure on each side of the transition. Since stratiform clouds have a major impact on the radiation divergence in the boundary layer, the transition from a cloudy to a clear boundary layer is a region of large horizontal inhomogeneity in air temperature and turbulence intensity. This leads to a considerable difference in horizontal and vertical transports between the cloudy and cloud-free regions. Measurements are used from the NCAR Electra aircraft during flights 5 (7 July 1987) and 10 (18 July 1987) of FIRE for this purpose. Flight 5 coincided with a LANDSAT overflight, and was designed to investigate the transition across a well-defined N-S cloud boundary, since the LANDSAT image can document the cloud cover in considerable detail. Turbulence legs were flown about 60 km on both sides of the cloud boundary. Flight 10 was flown at night in an area of scattered small cumuli and broken cloud patches.

Paluch, Ilga R.↗

NASA NTRS1992

The effect of gravity modulation on thermosolutal convection

In a gravitational field, the opposing effects of components of different diffusivities, for example, temperature and solute, in the density profile in a fluid may produce convective instabilities that exhibit a broad range of dynamical behavior. The effect of time periodic vertical gravity modulation on the onset of these instabilities in an infinite horizontal layer with stress free boundaries is examined. This work is viewed as a first step in expanding previous results in solidification to the full problem of characterizing the effects of gravity modulation in thermosolutal convection during the directional solidification of binary alloys. Calculations carried out both with and without steady background acceleration are presented, the latter results being relevant to microgravity conditions.

Saunders, Bonita V.↗