Single freeform surface design for prescribed input wavefront and target irradiance

In beam shaping applications, the minimization of the number of necessary optical elements for the beam shaping process can benefit the compactness of the optical system and reduce its cost. The single freeform surface design for input wavefronts, which are neither planar nor spherical, is therefore of interest. In this work, the design of single freeform surfaces for a given zero-\'etendue source and complex target irradiances is investigated. Hence, not only collimated input beams or point sources are assumed. Instead, a predefined input ray direction vector field and irradiance distribution on a source plane, which has to be redistributed by a single freeform surface to give the predefined target irradiance, is considered. To solve this design problem, a partial differential equation (PDE) or PDE system, respectively, for the unknown surface and its corresponding ray mapping is derived from energy conservation and the ray-tracing equations. In contrast to former PDE formulations of the single freeform design problem, the derived PDE of Monge-Amp\`ere type is formulated for general zero-\'etendue sources in cartesian coordinates. The PDE system is discretized with finite differences and the resulting nonlinear equation system solved by a root-finding algorithm. The basis of the efficient solution of the PDE system builds the introduction of an initial iterate constuction approach for a given input direction vector field, which uses optimal mass transport with a quadratic cost function. After a detailed description of the numerical algorithm, the efficiency of the design method is demonstrated by applying it to several design examples. This includes the redistribution of a collimated input beam beyond the paraxial approximation, the shaping of point source radiation and the shaping of an astigmatic input wavefront into a complex target irradiance distribution.Comment: 11 pages, 10 figures version 2: Equation (7) was corrected; additional minor changes/improvement

Double freeform illumination design for prescribed wavefronts and irradiances

A mathematical model in terms of partial differential equations (PDE) for the calculation of double freeform surfaces for irradiance and phase control with predefined input and output wavefronts is presented. It extends the results of B\"osel and Gross [J. Opt. Soc. Am. A 34, 1490 (2017)] for the illumination design of single freeform surfaces for zero-\'etendue light sources to double freeform lenses and mirrors. The PDE model thereby overcomes the restriction to paraxiality or the requirement of at least one planar wavefront of the current design models in the literature. In contrast with the single freeform illumination design, the PDE system does not reduce to a Monge-Amp\`ere type equation for the unknown freeform surfaces, if nonplanar input and output wavefronts are assumed. Additionally, a numerical solving strategy for the PDE model is presented. To show its efficiency, the algorithm is applied to the design of a double freeform mirror system and double freeform lens system.Comment: Copyright 2018 Optical Society of America. One print or electronic copy may be made for personal use only. Systematic reproduction and distribution, duplication of any material in this paper for a fee or for commercial purposes, or modifications of the content of this paper are prohibite

Case study 2. Model validation using existing data from PV generation on selected New Zealand schools

Solar energy is abundant, free and non-polluting. Solar energy can offset the consumption of fossil fuels, greenhouse gas emission reduction targets and contribute to meeting the fast-growing energy demands. The use of solar energy for electricity generation from photovoltaic (PV) panels has increased but is still not a widely utilised technology in New Zealand. This research approximated the potential solar energy that could be harvested from the rooftops of existing residential buildings in a case study city. This research is divided into two work strands, each involving a case study. The first strand investigated if a model could be developed, using existing data sources to determine the solar harvesting potential from the rooftops of existing residential buildings. The second strand involved the validation of the solar PV prediction model proposed in the first strand of the research, to test the reliability of the modelling outcomes. Invercargill City was selected as the study city for case study 1. Invercargill is the southernmost city in New Zealand so represents a worst case scenario. The method involved merging computer-simulation of solar energy produced from PV modelling and mapping incoming solar radiation data from north facing residential rooftop area. The work utilised New Zealand statistical census map of population and dwelling data, as well as digital aerial map to quantify the efficient roof surface area available for PV installations. The solar PV potential was calculated using existing formulas to investigate the contribution of roof area to the solar PV potential in buildings using roof area and population relationship. The estimated solar PV potential was 82,947,315 kWh per year generated from the total solar efficient roof surface area of 740,504 m². This equates to approximately 60.8% of the residential electricity used in Invercargill’s urban area, based on the 7,700 kWh typical annual electricity consumptions per household. The result represents an immense opportunity to harvest sustainable energy from Invercargill’s residential rooftops. To verify the accuracy of the developed method for predicting the PV outputs, the model was applied to actual generation data from grid-connected solar photovoltaic (PV) systems that are installed in New Zealand schools under the Schoolgen programme (Case Study 2). A total of 66 Schoolgen PV rooftop models were incorporated in the analysis. At this stage, the actual system parameters including size, panel type and efficiency were included in the analysis. The performance prediction and analysis outcome showed the parameters and operating conditions that affect the amount of energy generated by the PV systems. This part of the research showed the area where the PV model can be improved. The predicted generation from the model was found to be lower than the actual generation data. Schoolgen systems operating at over 0.75 performance ratio were found to be underestimated. This indicated that most Schoolgen PV systems were operating at higher capacities than predicted by the default value of system losses. The analysis demonstrated the effects of PV technology type, site orientation, direction and tilted angle of the panels on the ability to generate expected amount of potential capacity based on solar resource availability in different site scenarios. This in turn has provided more in depth analysis of the research and served to expand the area for improvements in the design of the model

Identification of Photovoltaic Arrays' Maximum Power Extraction Point via Dynamic Regressor Extension and Mixing

This paper deals with the problem of identification of photovoltaic arrays' maximum power extraction point---information that is encrypted in the current-voltage characteristic equation. We propose a new parameterisation of the classical five parameter model of this function that, combined with the recently introduced identification technique of dynamic regressor extension and mixing, ensures a fast and accurate estimation of all unknown parameters. A concavity property of the current-voltage characteristic equation is then exploited to directly identify the desired voltage operating point. Realistic numerical examples via computer simulations are presented to assess the performance of the proposed approach.Comment: This paper is submitted to th International Journal of Adaptive Control and Signal Processin