Ricardo Marquez

Chapter 8 – Overview of Solar-Forecasting Methods and a Metric for Accuracy Evaluation

This chapter provides an introduction to and overview of the subsequent chapters, which discuss specific solar-forecasting technologies and time horizons. Solar-forecasting methods are classified by technique, time horizon, and application. Advantages and disadvantages of deterministic and stochastic forecasting approaches are laid out and discussed in the context of solar forecasting based on numerical weather prediction, satellite data, and ground measurements. Metrics to evaluate solar-forecasting techniques are then presented and a time horizon–invariant metric is introduced that allows comparing forecast errors across time horizons, geographical regions, and time steps. Finally, the metric is demonstrated with hour-ahead forecasts based on stochastic-learning and satellite cloud-motion vector techniques.

Comparisons of Radiative Heat Transfer Calculations in a Jet Diffusion Flame Using Spherical Harmonics and k-Distributions

A new nongray radiation modeling library for combustion gases has been implemented in OpenFOAM. The spectral models for single species include gray, correlation tables and full spectrum k-distributions (FSK) assembled from a narrow-band database. Mixing models for k-distributions include the multiplication and uncorrelated mixture models. Radiative transfer equation solvers for the library include spherical harmonics such as P1, P3, SP3 and SP5 as well as the optically thin approximation. The performance of the different solution methods is compared for accuracy and speed as a tool for future model strategy selection.

Implementation of High-Order Spherical Harmonics Methods for Radiative Heat Transfer on openfoam

Wenjun Ge School of Engineering, University of California, Merced, CA 95343 e-mail: wge@ucmerced.edu Ricardo Marquez School of Engineering, University of California, Merced, CA 95343 e-mail: rmarquez3@ucmerced.edu Michael F. Modest 1 Professor Fellow ASME School of Engineering, University of California, Merced, CA 95343 e-mail: mmodest@ucmerced.edu Somesh P. Roy School of Engineering, University of California, Merced, CA 95343 e-mail: sroy3@ucmerced.edu Implementation of High-Order Spherical Harmonics Methods for Radiative Heat Transfer on OPENFOAM A general formulation of the spherical harmonics (P N ) methods was developed recently to expand the method to high orders of P N . The set of N(N þ 1)/2 three-dimensional second-order elliptic PDEs formulation and their Marshak boundary conditions for arbi- trary geometries are implemented in the OPENFOAM finite volume based CFD software. The results are verified for four cases, including a 1D slab, a 2D square enclosure, a 3D cylindrical enclosure, and an axisymmetric flame. All cases have strongly varying radia- tive properties, and the results are compared with exact solutions and solutions from the photon Monte Carlo method (PMC). [DOI: 10.1115/1.4029546] Keywords: radiative heat transfer, RTE solvers, spherical harmonics, computer implementation Introduction The study of radiative heat transfer in a multidimensional ge- ometry with a strongly varying participating medium has become increasingly important in various practical applications like com- bustion, manufacturing, and environmental systems. The radiative transfer equation (RTE) is an integro-differential equation in five independent variables (three in space and two in direction), which is exceedingly difficult to solve. Many approximate methods have been developed over time. The most widely used approximate methods today are the discrete ordinates method (DOM) [1,2] or its finite volume version (FVM) [3], the PMC [4], and the spheri- cal harmonics method (SHM) [5]. The DOM/FVM method discre- tizes the entire solid angle by finite ordinate directions and is relatively simple to implement within modern CFD software. But an iterative solution is required for scattering media or reflecting surfaces, and computational cost is high for optically thick media. The method may also suffer from ray effects and false scattering due to the angular discretization [6]. The PMC method statisti- cally provides the exact solution with sufficient photon bundles, but accurate solutions are computationally expensive. The spheri- cal harmonics P N approximation is potentially more accurate than DOM/FVM at comparable computational cost, but higher order P N are mathematically very complex and difficult to implement. The P N method decouples spatial and directional dependencies by expanding the radiative intensity into a series of spherical har- monics. The lowest order of the P N family, the P 1 approximation, has been extensively applied to radiative transfer problems. Corresponding author. Contributed by the Heat Transfer Division of ASME for publication in the J OURNAL OF H EAT T RANSFER . Manuscript received August 19, 2014; final manuscript received December 29, 2014; published online February 10, 2015. Assoc. Editor: Zhuomin Zhang. Journal of Heat Transfer However, it loses accuracy when intensity is directionally very ani- sotropic [7], as is often the case in optically thin media. Applications of higher order SHM methods were limited to one-dimensional cases for a long time, because of the cumbersome mathematics. Recently, Modest and Yang [8,9] and Modest [10] have derived a general three-dimensional P N formulation consisting of N(N þ 1)/ 2 second-order elliptic partial differential equations (PDEs) and their Marshak boundary conditions for arbitrary geometries. The main purpose of this paper is to present the procedure of implementing high-order P N formulations within the OPENFOAM [11] open source libraries, and a preliminary version was pre- sented in Ref. [12]. The numerical methods used are summarized along with four example cases. The results of high-order P N meth- ods are found to be very close to the exact solution of the RTE and accurately predict the incident radiation and radiative heat source. Also, the time cost of the P N methods is summarized for different examples and orders. Formulation 2.1 Governing Equations. In the spherical harmonics approximation, the radiative intensity is expanded into a sum of spherical harmonics Iðs; ^ s Þ ¼ N X n X I n m ðsÞY n m ð^ s Þ n¼0 m¼n where s ¼ b r dr is an optical coordinate, and b r is the extinction coefficient. Y n m ð^ s Þ are the spherical harmonics and the upper limit N is the order of the approximation. The set of N(N þ 1)/2 elliptic PDEs [10] of the P N method for isotropic scattering in 3D Carte- sian coordinates are as follows: C 2015 by ASME Copyright V MAY 2015, Vol. 137 / 052701-1 Downloaded From: http://heattransfer.asmedigitalcollection.asme.org/ on 02/25/2015 Terms of Use: http://asme.org/terms

Spectral Photon Monte Carlo With Energy Splitting Across Phases for Gas–Particle Mixtures

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A Novel Metric for Evaluation of Solar Forecasting Models

This work presents an alternative, time-window invariant metric for evaluating the quality of solar forecasting models. Conventional approaches use statistical quantities such as the root-mean-square-error and/or the correlation coefficients to evaluate model quality. The straightforward use of statistical quantities to assign forecasting quality can be misleading because these metrics do not convey a measure of the variability of the time-series included in the solar irradiance data. In contrast, the quality metric proposed here, which is defined as the ratio of solar uncertainty to solar variability, compares forecasting error with solar variability directly. By making the forecasting error to variability comparisons for different time windows, we show that this ratio is essentially a statistical invariant for each forecasting model employed, i. e., the ratio is preserved for widely different time horizons.Copyright

Forecasting of Global Horizontal Irradiance Using Sky Cover Indices

This work discusses the relevance of three Sky Cover (SC) indices for solar radiation modeling and forecasting. The three indices are global in the sense that they integrate relevant information from the whole sky and thus encode cloud cover information. However, the three indices also emphasize different specific meteorological processes and sky radiosity components. The three indices are derived from the observed cloud cover via Total Sky Imager (TSI), via measurements of the Infrared Radiation (IR), and via pyranometer measurements of Global Horizontal Irradiance (GHI). We enhance the correlations between these three indices by choosing optimal expressions that are benchmarked against the GHI SC index. The similarity of the three indices allows for a good qualitative approximation of the GHI irradiance when using any of the other two indices. An Artificial Neural Network (ANN) algorithm is employed to improve the quantitative modeling of the GHI sky cover index, thus improving significantly the forecasting details of GHI when all three indices are used.Copyright

Implementation of the PN-Approximation for Radiative Heat Transfer on OpenFOAM

This work presents an OpenFOAM implementation of the PN approximation for radiative heat transfer, including higher orders P3, P5, and P7. Also described is a procedure which enables the sequential numerical computations of the coupled partial differential equations (PDEs) by re-expressing the boundary conditions in matrix form so that individual boundary conditions can be associated with each PDE. The implementation of the software programs are verified with derived analytical solutions for 1-D slabs with constant and variable properties, and are also tested with various orientations in order to demonstrate the geometric invariance properties of the 3-dimensional PN formulation. A few examples taken from the literature are also considered in this work and could be taken as benchmark solutions for the PN approximations.Copyright

Characterization and Cost Analysis for the UC Merced Campus Load Including Effects of Solar Farm Variability

Variability and uncertainty in renewable energy generation has a direct impact on the cost of electricity prices because of the potential impacts on the operation of existing power generating units. The purpose of this paper is to present a case study detailing the variability added to the UC Merced campus load after including a 1-MW photovolataic (PV) plant to supply a significant amount (25–30% of maximum demand) of the energy load. Cost estimates of the required mechanisms to compensate for the variability are calculated based on utility prices and the PV power output fluctuations. The resulting cost estimates are then compared with solar power variability metrics in order to show the economic impacts of solar variability. Additionally, a simulation is performed for a integrated battery storage system to mitigate the PV fluctuations by solving a revenues optimization formulation. The simulated costs are also correlated with three variability metrics including the standard deviation of the fluctuations and the mean absolute values of the fluctuations. In order to extrapolate results from the case study to a more general scenario, we show that the obtained correlations of normalized variables can be useful for providing estimates on the financial impacts of variable generation resources based on more widely available solar irradiance data.Copyright