Patent Publication Number: US-2011067689-A1

Title: Primary concentrator with adjusted etendue combined with secondaries associated to multiple receivers and with convection reduction

Description:
FIELD OF THE INVENTION 
     The present invention relates to primary and secondary solar radiation concentrators. The invention relates to new primary etendue adjusted concentrators, combined with secondary ones, for simple or multiple receivers, able to reach the highest concentration value possible. The present invention also relates to devices to reduce thermal losses due to convection at the receivers. 
     BACKGROUND OF THE INVENTION 
     Large-scale solar power plants can produce large quantities of electric power. This means that large quantities of sunlight must be collected. 
     Some of these plants may simply collect sunlight without concentrating it, as in the case of using flat photovoltaic panels exposed to the sunlight. However, when using high-efficiency solar cells or thermodynamic cycles, some degree of concentration is needed to increase efficiency and, therefore, typically also some tracking of the sun. 
     Sunlight is concentrated using optics. There are cases in which a large number of small optics are placed side by side, each one of them with its own receiver. That is the of, say, a set of parabolic primaries with solar cells at the foci, or adding kaleidoscopes to improve irradiance uniformity on the cell [Daniel Feuermann and Jeffrey M. Gordon,  Solar Fiber - optic mini - dishes: a new approach to the efficient collection of sunlight , Solar Energy Vol. 65, No. 3, pp. 159-170, 1999]. In other cases, a smaller number of larger receivers is used, but then large optics are needed. The problem with large optics is that they are hard to assemble and move to track the sun. One possible way around this problem is to replace the large optic by a large number of small optics that mimic its behaviour. One example of this process is found in the tower power plants which have a large number of small mirrors called heliostats that reflect the light to a large receiver. In this case, instead of a large parabolic primary, these plants have a “Fresnel” primary composed of many small mirrors. [J. I. Ortega, J. I. Burgaleta, F. M. Tellez,  Central Receiver System  ( CRS )  Solar Power Plant using Molten Salts as Heat Transfer Fluid , Proceedings 13th International Symposium on Solar Power and Chemical Energy Technologies ISBN 84-7834-519-1, Edit. M. Romero, D. Martinez, V. Ruiz, M. Silva, M. Brown, M. Snachez, M. Romero,  Methodology for generation of heliostat field layout in central receiver system based on yearly normalized energy surfaces , Solar Energy 80, pp 861-874, 2006]. 
     A further possibility is to have trough receivers onto which light is concentrated using trough optics. Also in this case parabolic primaries may be used [E. Rojas, A. Fernandez, E. Zarza,  Theoretical evaluation of parabolic trough designs for industrial applications , Proceedings 13th International Symposium on Solar Power and Chemical Energy Technologies ISBN 84-7834-519-1, Edit. M. Romero, D. Martinez, V. Ruiz, M. Silva, M. Brown], or alternatively “Fresnel” primaries composed of long linear heliostats running in the direction of the receiver [Patente U.S. Pat. No. 4,131,336: Miller et al.,  Primary reflector for solar energy collection system , 1978,  Solar thermal power plants , Renewable Energy World 06/2003 pp. 109-113]. The heliostats may track the sun keeping the receiver illuminated by concentrated sunlight. However, the heliostats shade each other, especially those further away from the receiver, and the light that is shaded is lost. The concentration of these primaries may be increased by secondary optics, such as the TERC secondaries [J. M. Gordon and Harald Ries,  Tailored edge - ray concentrators as ideal second stages for Fresnel reflectors , Applied Optics, Vol. 32, No. 13, pp. 2243-2251, 1993]. These concentrators will, however, only approach the theoretical maximum concentration in the limit case of a primary composed of infinitesimal heliostats, a severe practical limitation. 
     In the prior art, an improvement over a simple “Fresnel” primary is to intersect two heliostat fields in an arrangement know as CLFR (Compact Linear Fresnel Reflector) [David R. Mills and Graham L. Morrison,  Compact linear Fresnel reflector solar thermal power plants , Solar Energy Vol. 68, No. 3, pp. 263-283, 2000; U.S. Pat. No. 5,899,199: David Mills,  Solar Energy Collector system , 1999, U.S. Pat. No. 6,131,565: David Mills,  Solar Energy Collector system , 2000]. In this arrangement, instead of a single receiver there are several receivers. The heliostats are all the same size and those closer to a first receiver redirect the light to it. Those more spaced apart, alternatively redirect the light to the first receiver and to a second receiver. This creates a W shaped heliostat field in the areas more spaced apart from the receivers where the odd heliostats reflect light to one receiver, while the even heliostats reflect light to the other receiver. This approach, however, still does not adjust the etendue of the incoming radiation with that reflected to the receivers and, therefore, there will always be either some shading of light or areas of the heliostat field not fully illuminated when seen from the receivers. 
     This is a fundamental limitation of these optics and is independent of the size or shape of the heliostats. The concentrations attained by these optics are much lower than the theoretical limit. 
     To solve the etendue mismatch problem between the etendue of the light received by the primary and the etendue the primary should ideally redirect towards the receivers, new primaries are needed. 
     The present invention discloses two different ways of improving the primary: changing its overall shape and changing the size and shape of its heliostats. To increase concentration and approach the theoretical limit, the new primaries must be combined with new secondary optics. 
     When changing the overall shape of the primary, the heliostats are placed on a wave shaped trough surface and the size and shape of the heliostats is a function of the position in the heliostat field. The heliostats may also be flat, in which case the smaller the heliostats, the higher the concentration the primary can provide. 
     The size and shape of the heliostats may be adjusted in order to increase concentration, which can further be augmented with the use of a secondary. The heliostats constitute a discontinuous primary and, in order to design a continuous secondary, a continuous primary is developed. The heliostats are inter connected by flow lines resulting in a continuous primary (broken line) for which a continuous secondary can be designed. The portions of primary along flow lines can then be removed, leaving the initial heliostats present at the start of the procedure. In this concept the primary is conceived (for the purpose of secondary design) has a continuous but broken mirror, i.e. in steps, portions of which follow flow lines and other portions are transverse to those flow lines [Pablo Benitez, Juan Carlos Minano, Maikel Hernandez,  On the analysis of microstructured surfaces , SPIE Proceedings, Vol. 5529, Nonimaging Optics and Efficient Illumination Systems, pp. 186-197, 2004]. This type of design is common in Fresnel Lens design, which can also be combined with secondaries to increase their concentration [M. Collares Pereira, A. Rabl and R. Winston, Lens-mirror combination with maximal concentration, Applied Optics, Vol. 16, No. 10, pp. 2677-2683, 1977, M. Collares Pereira,  High temperature solar collector with optimal concentration: non - focusing Fresnel lens with a secondary concentrator , Solar Energy, Vol. 23, pp. 40-9420, 1979, Ralf Leutz, Akio Suzuki, Atsushi Akisawa and Takao Kashiwagi,  Design of a nonimaging lens for solar concentrators , Solar Energy, Vol. 65, No. 6, pp. 379-387, 1999]. Other more elaborated types of stepped optics are also possible [Julio Chaves, Manuel Collares-Pereira,  Ultra flat ideal concentrators of high concentration , Solar Energy Vol. 69, No. 4, pp. 269-281, 2000, Julio Chaves and Manuel Collares-Pereira,  Ideal concentrators with gaps , Applied Optics, Vol. 41, No. 7, pp, 1267-1276, 2002 Julio Chaves,  Introduction to Nonimaging Optics , CRC Press, Taylor and Francis Group, 2008]. Continuous secondaries may also be directly designed from discontinuous primaries (a set of heliostats) in which case joining the primary heliostats by flow lines is not required. In this version the heliostats may be on the wave shaped surface or on a flat one. The secondary concentrators conceived for these new primaries are continuous, from portion to portion, in accordance with the piece wise nature of the primary. 
     In the case of a single receiver, the concentrators thus obtained compare favourably to traditional combinations of Fresnel reflectors and secondaries, increasing the concentration on the receiver close to the theoretical maximum, even for primaries composed of large size heliostats. In the case of multiple receivers, these new concentrators also compare favourably to CLFRs, once again achieving concentrations on the receivers close to the theoretical maximum, while having lower losses. 
     The secondary mirrors typically touch the receiver and a method for preventing/solve such problem is also disclosed. 
     The invention further includes devices for reducing convection losses in the receiver. These include specially shaped mirrors and transparent covers. 
     SUMMARY OF THE INVENTION 
     The present invention relates to an optical system with primary concentrator of the Fresnel type and secondary concentrator adjacent to the receiver, the secondary-receiver set being above the primary and characterized by the primary concentrator containing a stepped flow-line optics which shape verifies the fact that it reflects a set of edge rays tangent to the receiver and the other set of edge rays into the direction of the secondary concentrator which has a shape that (in turn) reflects those edge rays tangent to the receiver producing theoretically there maximum concentration and in which the secondary concentrator is truncated. 
     In one embodiment, the optical system is characterized by the receiver having a convex face and also a planar one in which the secondary receiver touches only in one point and in which the planar face may be substituted with a concave one resulting in a secondary that does not touch the receiver. 
     In another embodiment, the referred optical system is characterized by the absence of some or all of the stepped optics that go with the flow lines, leaving only those portions that cross the flow lines, this is, the heliostats and in which the eventual continuous (not step like) portions of the primary concentrator may also be divided into heliostats. 
     In another embodiment, the referred optical system is characterized by the heliostats being able to rotate about themselves to track the apparent diurnal motion of the sun. 
     In another embodiment, the referred optical system is characterized by combining at least two optical systems as those referred above thus forming a concentrator with multiple receivers. 
     In an embodiment the referred optical system is characterized by the fact that the heliostats being placed on a wave like surface, with cylindrical geometry, substantially corresponding to a curve that optically conserves etendue. In another preferred embodiment, the referred optical system is characterized by the mirrors of the primary concentrator being curved or flat. Still in another preferred embodiment the referred optical system is characterized y the global form of the primary being flat. 
     Preferably, the referred optical system is characterized by all receivers being substantially at the same height and the angles measured from the vertical that passes through one of the receivers substantially being about 33±10° with the line that, leaving the receiver, passes through the point on the primary that marks the transition between one and two receivers, and about 61±5° with the line that goes through the intermediate point of the primary optics and about 71±5° with the line that passes in another transition point from one to two receivers. In a most preferred way the previous optical system is characterized by the angles as measured from the vertical that goes through one of the receivers being: 32.6° with the line that leaving the receiver, passes through the point on the primary that marks the transition between one and two receivers, of 61.2° with the line that goes through the intermediate point of the primary optics and 71.2° with the line that passes in another transition point from one to two receivers. In a more preferred embodiment the previous optical system is characterized by not comprising secondary concentrators by the receivers. 
     In another embodiment said optical system is characterized by the receivers having a larger size than they would if sized to correspond to the ideal or maximum concentration. 
     The present invention also relates to an optical system characterized by comprising transparent covers, substantially perpendicular two flow lines and mirrors (mirrored on both sides) that substantially follow the flow lines, to reduce the convection by the receiver. 
     In a preferred embodiment of the present invention the referred optical system is characterized by comprising mirrors and/or transparent covers with the shape of broken lines that substantially follow flow lines or are transverse to them and in which the covers may be simple or double. 
     The present invention refers to the use of any optical system as defined above, characterized by the optical system being intended to concentrating the solar radiation. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The above and other aspects, features and advantages of the present invention will be apparent from the following more particular description thereof, presented in conjunction with the following drawings wherein: 
         FIG. 1  shows a parabolic reflector. 
         FIG. 2  shows a flat mirror reflecting radiation total angular aperture 2θ. 
         FIG. 3  shows a parabolic reflector cut into several portions and those portions moved down to form a Fresnel reflector. The orientation of these small mirrors then must be adjusted. 
         FIG. 4  shows a Fresnel reflector. 
         FIG. 5  shows a schematic view of a Fresnel reflector with an infinite number of infinitely small mirrors. 
         FIG. 6  shows shading occurring in a Fresnel reflector due to etendue mismatch between the incoming and outgoing radiation. 
         FIG. 7  shows an incoming beam of light being divided and reflected onto two separate receivers. 
         FIG. 8  shows the geometry of dividing a beam of incoming radiation into two of outgoing radiation by a tilted surface. 
         FIG. 9  shows several etendue conserving curves when the incoming light is redirected to two different receivers. 
         FIG. 10  shows the most compact curve from those shown in  FIG. 9 . 
         FIG. 11  shows the possible geometry for a radiation splitter optic, dividing a beam of incoming light into two beams of outgoing light. In this example, the radiation splitter optic is composed of two heliostats. 
         FIG. 12  shows the shading that may occur within two contiguous radiation splitters, such as that shown in  FIG. 11 . 
         FIG. 13  shows the curve in  FIG. 10  with radiation splitters in it. 
         FIG. 14  shows the tangency point on the etendue conserving curve beyond which light redirected towards the left receiver is shaded. 
         FIG. 15  shows a combination of a parabola and an etendue conserving curve. 
         FIG. 16  shows the curve in  FIG. 15  with simple heliostats and radiation splitters on it. 
         FIG. 17  shows a comparison between the geometry of the optic in  FIG. 16  and a parabola for the same purpose. 
         FIG. 18A  and  FIG. 18B  show a Fresnel primary with a different ray assignment. 
         FIG. 19  shows a secondary mirror for a Fresnel primary as shown in  FIG. 18 . 
         FIG. 20  shows a central V receiver with a truncated secondary mirror and a Fresnel primary. 
         FIG. 21  shows the overall geometry of the heliostat field, determined by etendue considerations. 
         FIG. 22  shows a primary-secondary system for a small acceptance angle. 
         FIG. 23  shows a detail of  FIG. 22  around the receiver area. 
         FIG. 24  shows a Fresnel lens with different prisms on the right and left halves. 
         FIG. 25  shows an ideal stepped flow-line concentrator with a linear receiver. 
         FIG. 26  shows a combination of a Fresnel primary and a truncated flow-line secondary. 
         FIG. 27A  shows an ideal stepped flow-line concentrator with a V receiver. 
         FIG. 27B  shows the construction method of the concentrator in  FIG. 27A . 
         FIG. 28  shows a concentrator for several receivers with Fresnel primary and truncated secondary mirrors. 
         FIG. 29  shows a concentrator similar to the one in  FIG. 28 , but designed for a smaller acceptance angle. 
         FIG. 30  shows a concentrator producing maximum concentration on the receiver, with a discontinuous primary mirror. 
         FIG. 31  shows the geometry of an ideal (infinite number of infinitesimal heliostats) flat Fresnel primary for two receivers. 
         FIG. 32  shows the ideal relative dimensions of an ideal flat Fresnel primary for two receivers. 
         FIG. 33  shows the etendue mismatch across the primary for an ideal flat Fresnel primary for two receivers and with ideal relative dimensions. 
         FIG. 34  shows the geometry of a concentrator for an acceptance angle 2θ and flat receiver, showing that the receiver is not large enough to accommodate all the etendue emitted by the primary. 
         FIG. 35  shows an ideal primary-secondary concentrator for a V receiver. 
         FIG. 36  shows an ideal primary-secondary concentrator for a U receiver. 
         FIG. 37  shows an ideal primary-secondary concentrator for a receiver made of circular tubes. 
         FIG. 38  shows a concentrator for a V receiver whose primary has finite size mirrors. 
         FIG. 39  shows a concentrator for two V receivers whose primary has finite size mirrors and a truncated secondary mirror. 
         FIG. 40  shows a concentrator for small acceptance angle and two V receivers, whose primary has finite size mirrors and a truncated secondary mirror. 
         FIG. 41  shows a modification of a V receiver and a secondary mirror that results in a design in which the mirror does not touch the receiver. 
         FIG. 42  shows a concentrator for an inverted Δ receiver with a smooth secondary minor (with a continuous derivative). 
         FIG. 43  shows a concentrator for a larger than ideal V receiver. The secondary mirror no longer extends all the way to the end of the primary. 
         FIG. 44  shows an ideal concentrator for a V receiver and the flow lines (or G-lines) and their perpendicular, or F-lines, inside it. 
         FIG. 45  shows a primary-secondary in which the receiver is protected by an F-line shaped transparent cover (perpendicular to the flow lines) and with internal mirrors along the flow lines. Both these components reduce internal convection thermal losses. 
         FIG. 46  shows a V receiver with thermal insulation on the back. The secondary mirror may have a heat dissipater to prevent overheating. 
     
    
    
     A better understanding of the characteristics and advantages of the present invention will be obtained with reference to the detailed description of the invention and corresponding figures, which are illustrative of the way in which the principles behind the invention are used. 
     DETAILED DESCRIPTION OF THE INVENTION 
     The present invention relates to new etendue-adjusted primary concentrators combined with secondary concentrators, for simple or multiple receptors, able to reach the maximal possible concentration. The present invention also relates to devices able to reduce convective thermal losses from the receivers. 
     In prior art these concentrators may have only one receiver (as in the case Linear Reflectors of the Fresnel type-LFR) or they may have multiple reflectors (as in the case of Compact Linear Fresnel Reflectors-CLFR). The present invention shows improvements in the general primary shape in the case of multiple receivers. It further shows new concentrators of the primary-secondary type, coming very close to the theoretical concentration limit, even when the primary is formed by large heliostats (something impossible in prior art). These new concentrators are also applicable in the case of multiple receivers. The present invention further shows devices to reduce thermal convective losses from the receiver(s). 
     The present invention describes a new type of Fresnel primaries so-called “etendue adjusted”. These primary shapes have a wave type configuration, conserving “etendue” and are characterized by the fact that the etendue of the incident radiation is perfectly adjusted to the etendue of the reflected radiation towards the receivers (in particular in the case that only one receiver exists the shape is no longer a wave but a parabola (prior art)). 
     The corresponding reflector mirror corresponds to a large number of small structures that follow the etendue conserving curve and reflects the light/radiation to the receivers. A limiting case of these new primaries occurs (as in prior art) when they are formed by an infinite number of heliostats. New secondaries can then be designed for these primaries coming close to the theoretical limit for concentration. Although these new sets represent a step ahead when compared with prior art, because of their reduced etendue mismatch, they have a reduced practical interest. As discussed below the present invention also shows geometries for primary structures with large (finite) and practical sizes. 
     The present invention also describes new primary-secondary concentrators for single receivers. The primary is composed by a set of heliostats (movable mirrors able to track the sun). To increase primary concentration it may be combined with a secondary one. To facilitate the design of a continuous secondary, the heliostats are interconnected by flow lines, resulting in a continuous primary with a stepped shape, a continuous and broken line. A continuous secondary may then be designed for this now continuous primary. A continuous primary is not a necessary condition and a continuous secondary may also be designed even for a discontinuous primary (this type of optics is also described in the patent. However a continuous primary makes the design more intuitive. Once the secondary designed, the primary portions along flow lines can be removed, leaving the primary just with the initial heliostats. 
     The primary reflector is thus conceived to be a flow line stepped reflector, consisting of continuous of portions, as a broken line, which parts either follow or cross flow lines. These individual portions are curved in general, but some may be flat. The underlying shape of the primary, as a whole, may be curved or flat. 
     In particular configurations of the present invention, the resulting optics, primary-secondary, comes close to the theoretical concentration limit on the receiver. In contrast, prior art could only (theoretically) reach maximum concentration on the receiver only with a primary of infinitesimal structures. In the primary-secondary set, the primary is conceived to reflect one set of edge rays into directions tangent to the receiver. The other set of edge rays is reflected by the primary into the direction of the secondary and this one, in turn, redirects them in a way that they also end up tangent to the receiver. 
     In case, for instance, of large solar systems, the sections of stepped reflectors along flow lines, may be eliminated, thus resulting a discontinuous reflector, containing only the portions that cross the flow lines. These portions may now be considered as heliostats that “track” the apparent motion of the Sun. This set by itself is not ideal and the heliostats may be extended a little to recuperate some of the lost radiation. The secondary initially designed for the continuous reflector can now be used with the resulting curved heliostats, constituting high efficiency optics, coming close to the theoretical concentration limit on the receiver. 
     Some of these concentrators for a single receiver may now be combined, originating concentrators for multiple receivers. The process is similar to that of combining LFR to get CLFR, in which several LFR juxtapose on each other, intersecting and forming a CLFR. In a simpler solution consists on having all heliostats on a straight line. A more elaborated solution the underlying primary shape may be different, just as a curve that conserves etendue. In these optics the heliostats can be placed on a wave shaped line (surface), reflecting solar radiation towards different receivers. The size and shape of the heliostats are a function of their position on the wave line (surface). This method permits matching the incident radiation etendue with the one reflected towards the different receivers. The result is a Fresnel primary with very little blocking, and thus with very small optical losses. In contrast, prior art does not contemplate the matching of the incident etendues to the reflected ones at and by the primary and so the resulting optics are less efficient due to radiation blocking between heliostats. 
     The present invention also explores the relative dimensions of a reflector optimized for two receivers, in which all components (portions) are placed on a straight line (flat surface). For a final optimal (in the theoretical limit) concentration of the present invention, obtained with the help of the secondary concentrators described in the present invention, the receiver can no longer be flat and have only one absorbing side, as it can no longer accommodate all the reflected radiation etendue from the primary towards it. 
     The present invention further describes a methodology to create a separation (gap) between the mirror and the receiver, to prevent a thermal bridge and the consequent thermal losses. 
     The present invention further describes ways to reduce the convective losses around the receiver, through the use of covers/transparent surfaces, substantially perpendicular to flow lines and mirrors that substantially follow those flow lines. 
     The present invention relates to optical design with two-dimensional geometry, that may be implemented in a practical way applying to these designs a translation symmetry (or a rotational one when only a design with a flat receiver exits). 
     The term “étendue” of the radiation that crosses a curve (two-dimensional geometry) relates to the integral of the projected length with the angular aperture of the radiation: U=∫∫dxcosθdθ in which U is the étendue, dx is an infinitesimal length along the curve, θ is the angle the propagation direction makes with the normal to dx and dθ the angular aperture occupied by the radiation that crosses dx. In a more general way the curve may be immersed in a medium with refractive index n, and in that case etendue is defined by U=∫∫ndxcosθdθ. 
     In the present invention the term “etendue matching curve” relates to a curve to which small structures may be added and which redirect the light towards one or more receivers and for which the incident light etendue coincides with the “etendue” of the redirected light towards the receivers. 
     Additionally, the term “stepped optics” or “optics in steps”, relates to optics that consists of a mirror that follows a flow line, followed by one that crosses the flow lines which captures and re-directs a portion of light and then another mirror along a flow line, followed by another mirror crossing the flow lines capturing and redirecting the light and so on. A particular case occurs when the optics crossing flow lines is a simple mirror (curve or flat). In that case the stepped optics is a continuous mirror, continuous but broken, either following or crossing the flow lines. 
     The term “simple mirror” relates to any surface with only one reflecting face, and “double mirror” relates to any surface mirrored on both sides thereof. 
     Further, the term “double cover/transparent surface” relates to one that is formed by two single transparent surfaces. 
     The present invention will be further clarified with reference to the accompanying figures. 
       FIG. 1  shows a parabolic mirror  101 . Light ray  102  parallel to the axis of the parabola is reflected at point  103  with normal n in direction  104  towards focus  105 . The normal to the parabola at point  103  is in direction n. All rays parallel to  102  are also reflected by the parabola towards its focus  105 . 
       FIG. 2  shows a flat mirror  201  and the etendue balance of the light it receives and it reflects. Light comes in vertically with angular aperture 2θ, making an angle α 1  to the normal n to the mirror, and leaves it still with the same angular aperture 2θ, but now making an angle α 2  to its the normal. The middle ray  202  comes in vertically and leaves the mirror as ray  203  making an angle φ to the vertical. The etendue of the incoming radiation is given by dU 1 =2dlsinθcosα 1  where dl is the length of mirror  201 . Also, the etendue of the light leaving the mirror is given by dU 2 =2dlsinθcosα 2 . If the angular aperture 2θ is conserved at the mirror, then the conservation of etendue dU 1 =dU 2  results in α 1 =α 2 . Now suppose that this mirror is a very small portion of a curve, such as the one shown in  FIG. 1 . This condition (also the law of reflection) defines the direction of the normal n to the curve for each value of φ since the middle ray of the incoming radiation is vertical before reflection. If an initial point is given for the curve and the curve concentrates all the middle rays to a focus, then this condition α 1 =α 2  results in a parabolic shape. The parabola can then be seen as a shape that conserves the etendue of the radiation when the latter is concentrated to a focus. 
       FIG. 3  shows a parabola divided into sections. Each one of those sections can be brought down to the plane and adjusted to form a Fresnel reflector. 
       FIG. 4  shows the Fresnel reflector resulting from the construction in  FIG. 3 . Now, instead of a parabola, there are a set of Fresnel mirrors  401  on the plane that concentrate the light to the focus  402 . The Fresnel mirrors in this figure are large compared to the overall size of the optic and, if a sharp focus is needed, then the mirrors must, obviously, be curved. 
       FIG. 5  shows a schematic view of the case in which the Fresnel mirrors in  FIG. 4  get smaller and smaller. In the limit case of an infinite number of infinitely small mirrors, the Fresnel reflector would be a microstructure  502  that reflects vertical rays to the focus  503 . For each point  501  on the microstructure, the reflected ray makes an angle φ to the vertical. For a sharp focus of a set of vertical parallel rays onto focus  503 , as the Fresnel mirrors get smaller and smaller, their curvature may decrease. In the limit case shown in the figure, these mirrors would be flat. 
       FIG. 6  shows the etendue mismatch for the incoming and outgoing radiation in a Fresnel reflector and the consequent and inevitable shading of light. This is the situation at point  501  of the Fresnel reflector in  FIG. 5 . The etendue of the incoming radiation is dU 1 =2dlsinθ while that of the outgoing radiation is dU 2 =2dlsinθcosφ where dl is the length of aperture  602  (the projection of  602  in direction φ is da=dlcosφ). We can then see that etendue is no longer conserved. The lost etendue corresponds to shading by adjacent Fresnel mirrors. Only the vertical light rays between  606  and  607  are reflected towards the receiver, and those between  605  and  606  are shaded by the mirror to the left. From these considerations, it can be concluded that not all the light received at point  501  in  FIG. 5  can be redirected towards receiver  503 . This means that either part of the light is lost (to shading/blocking) or it must be redirected somewhere else. In order to redirect the excess light somewhere else we need another receiver and, therefore, we are lead to consider a multi-receiver Fresnel reflector. 
       FIG. 7  shows the geometry of a point  707  on a Fresnel reflector with two receivers  703  and  704 . An incoming vertical beam of light hits point  707  on the Fresnel reflector and is split in two separate beams: a beam  705  redirected to the left towards receiver  703  and another beam  706  redirected to the right towards another receiver  704 . These beams make, respectively, angles φ 1  and φ 2  to the vertical. 
       FIG. 8  shows a detail of what occurs at point  707  in  FIG. 7 . In a neighbourhood of that point, the Fresnel reflector can be seen shaped as an infinitesimal flat line  801  of length dl tilted by an angle α to the horizontal. Just like we did above for the case of the parabola, also here we consider that the incoming radiation has angular aperture 2θ and that also the redirected beams have the same angular aperture. The etendue of the radiation  804  received by  801  (incoming radiation) is given by dU 0 =2dlsinθcosα. The etendue of the radiation  805  redirected to the left is dU 1 =2dlsinθcos(φ 1 −α) and that of the radiation  806  redirected to the right is dU 2 =2dlsinθcos(φ 2 +α). Conservation of etendue can be written in this case as dU 0 =dU 1 +dU 2  or cos(φ 1 −α)+cos(φ 2 +α)=cosα. Now, from the geometry in  FIG. 7  and the position of point  707 , angles φ 1  and φ 2  can be determined. From the previous equation it is then possible to determine angle α that the Fresnel reflector makes to the horizontal at that point. Given an initial point for the Fresnel reflector, a curve can be obtained that conserves the etendue of the light redirected towards receivers  703  and  704 . This is a similar process to that used to define the parabola above (in the case of a single receiver). 
     In the case in which the incoming radiation  804  has angular aperture 2θ 0 , the radiation  805  redirected to the left has angular aperture 2θ 1  and the radiation  806  redirected to the right has angular aperture 2θ 2  the shape of the etendue-conserving curve is governed by the equation for the conservation of etendue that can now be written as: sinθ 1 cos(φ 1 −α)+sinθ 2 cos(φ 2 +α)=sinθ 0 cosα. 
       FIG. 9  shows three etendue conserving curves  902 ,  903  and  904  calculated according to the method described in  FIG. 7  and  FIG. 8  for three different initial points  905 ,  906  and  907 , all on the vertical of the receiver  909 . For midpoint  901  the condition φ 1 =φ 2  is verified. The equation for etendue conservation cos(φ 1 −α)+cos(φ 2 +α)=cosα results in α=0 and also in φ 1 =φ 2 =α°. Point  901  may thus be obtained if the distance between receivers  908  and  909  is given (in this example they are assumed to be at the same height). The most compact etendue-conserving curve is  902  starting at point  905  at the same height as  901 . 
       FIG. 10  shows curve  1001  that is the most compact etendue-conserving curve of  FIG. 9 . It also shows the geometry of a vertical incident beam which splits at point  1006  into the beam  1004  redirected to the left towards receiver  1002  and into the beam  1005  redirected to the right towards receiver  1003 . The angles these redirected beams make with the vertical are φ 1  and φ 2  respectively. 
       FIG. 11  shows a radiation splitter optic, splitting the incident beam in two beams. This optic may now be applied in point  707  of  FIG. 7 , a detail of which is shown in  FIG. 8 , allowing for the incident beam to be split in two reflected beams. The incident light is split in two parts:  1104  and  1105 . The light in  1104  is reflected by mirror  1102  to the left in direction  1106 . Besides, the light in  1105  is reflected by the mirror  1103  to the right in direction  1107 . The inclined line  1101  projects up as a combination of lengths  1104  and  1105 . The area  1104  of light after reflection is the same area  1104  before reflection. Likewise, area  1105  of the light after reflection is the same as the area  1105  before reflection. 
       FIG. 12  shows two radiation splitters on a straight line. Each radiation splitter is composed of a mirror  1201  and a mirror  1202 . This geometry produces a small blocking/shading  1203  for the light reflected by mirrors  1201  to the left. There is also some blocking/shading for the light reflected by mirrors  1202  to the right. This combination of splitting optics may also be applied over an etendue conserving curve. 
       FIG. 13  shows the same etendue-conserving curve of  FIG. 10  but now with radiation splitters  1303  on it. These optics redirect light towards receivers  1301  and  1302 . 
       FIG. 14  shows etendue-conserving curve  1401 , the same as shown in  FIG. 10 . There is point  1404  on this curve which tangent intersects receiver  1402 . To the right of this point, light reflected towards  1402  is shaded by the curve to the left. This defines a maximum rim angle  1405  at receiver  1402 . The situation is symmetrical for receiver  1403 . The portion of the curve between points  1404  and  1407  (like the rest of the curve) was conceived so that the radiation impinging on it was redirected to receivers  1402  and  1403 . However, because of the referred blocking/shading effect, the radiation that would be reflected towards receiver  1402  would not reach it. It would then make more sense to have the portion of the curve between  1404  and  1407  reflect radiation towards receiver  1403 . This change originates a readjustment of the design of the whole reflecting primary profile. 
       FIG. 15  shows a combination of a parabola and an etendue conserving curve for two receivers. This combination is a consequence of the arguments presented in  FIG. 14 . Since the parabola is an etendue conserving curve for one receiver, the hole curve conserves etendue. The parabolic portion of the curve reflects light onto a single receiver while the remaining etendue conserving curve reflects light to two receivers. Portion  1501  of the curve between points  1507  and  1508  is etendue conserving for two receivers  1502  and  1503 . Portion of the curve to the right of  1505  is a parabola with focus  1503  and portion of the curve to the left of point  1507  is a parabola with focus  1502 . Accordingly a vertical light beam hitting point  1505  on the parabolic portion of the curve is reflected towards receiver  1503  while a light beam hitting point  1504  on the etendue conserving curve for two receivers is split into two beams redirected towards receivers  1502  and  1503 . The rim angle of the primary as seen from the receivers is angle  1509 . There is a small angle  1506  of the Fresnel reflector spanning an angle of 1.5 deg that is not visible from the receiver. In a particular case, the parabola to the right of point  1508  is such that its tangent at that point intersects the receptor  1502 . The point  1508  serves as an initial point for the construction of the remaining etendue conserving curve for two receivers, which extends from here to point  1507 . 
       FIG. 16  shows the curve of  FIG. 15  with heliostats and radiation splitters  1601  on it. The parabolic portion of the curve has simple heliostats while the etendue conserving for two receivers has radiation splitters. Receivers are at positions  1602  and  1603 . 
       FIG. 17  shows a comparison between the curve in  FIG. 15  and the corresponding parabola. The etendue conserving curve  1701  has a height  1705  for receivers  1702  and  1703  at a height  1704 . As a comparison, parabola  1706 , for the same rim angle as the etendue conserving curve, has a height  1707 . Height  1705  is 15% of height  1704  of the receivers and is 14% of the height  1707  of the corresponding parabola. 
       FIG. 18A  and  FIG. 18B  show a different design for a Fresnel primary with a different ray assignment. The right edge rays parallel to  1810  are reflected in directions tangent to the receiver. In this example, the receiver is V-shaped with end points  1801  and  1803  and vertex  1802 . The description below, however, is also valid in the case in which the receiver has some other convex shape, provided that the shape of the curves in the Fresnel reflector is defined accordingly. 
     This is a common procedure when designing nonimaging optics. The right receiver  1831  has the same shape as the left one (bound by points  1801 ,  1802  and  1803 ). 
     For this particular shape of receiver, the Fresnel reflector curve starts with a parabola  1804  with axis parallel to  1810  and focus  1801 . The vertical though point  1805  is the same as through point  1802 . The Fresnel reflector then continues between points  1806  and  1808  as another parabola  1807 , also with axis parallel to  1810 , but now with focus  102 . The central portion of the Fresnel receiver is an etendue conserving curve  1821  for two receivers, extending from point  1808  to its symmetrical  1822 . With reference to a generic point  1823  on this curve, the light reflected to the left receiver  1830  is bound by ray  1826  tangent to the receiver at point  1802  and by the other edge ray  1827 . The light reflected to the right receiver  1831  is bound by ray  1824  tangent to the receiver at point  1820  and by the other edge ray  1825 . This portion of the curve is calculated using a similar geometry to that shown in  FIG. 8 , but where now rays  807  and  808  correspond to rays  1824  and  1826 . 
     As point  1823  moves towards point  1808  where curve  1821  ends, ray  1824  tends to ray  1828 . This ray  1828  is tangent to curve  1821  at point  1808 . Choosing a point  1808  further up on curve  1807  to start curve  1821  would result in less compact Fresnel primary. On the other hand, choosing a point  1808  further down on curve  1807  to start curve  1821  would result in light losses due to shading of some light by curve  1821  (in this case, ray  1828  would intersect curve  1821 ). 
     It is possible to determine how much etendue is reflected by the Fresnel mirror in the direction of the left receiver  1830  and how much is reflected in the direction of the right receiver  1831 . All the light falling on parabolas  1804  and  1807  is redirected in the direction of  1830  (although in the present configuration not all this light will hit the receiver). On the other hand, for the light hitting curve  1821  between points  1808  and  1822 , half of it is reflected towards  1830  and the other half towards  1831 . This means that the light redirected in the direction of  1830  by the Fresnel reflector on  1821  corresponds to the etendue of the light falling on half the curve  1821 . The total etendue of the light redirected towards  1830  is then given by U=2Rsinθ where R is the horizontal distance between point  1805  and midpoint  1829  of curve  1821 . 
       FIG. 19  shows a secondary mirror  1901  for a Fresnel primary, such as the one in  FIG. 18 , further increasing its concentration. The secondary mirror  1901  is designed in such a way that left edge rays  1809 , after being reflected at the primary, are redirected by the secondary in a direction tangent to the receiver  1830 . In this example of a V receiver, this means that these edge rays are redirected towards the edge  1802  of receiver  1830 . This condition defines each point  1902  on the secondary mirror. The secondary mirror starts at edge  1803  of receiver  1830  and extends all the way to the endpoint  1822  of curve  1821 . The concentration this optic produces on the receiver is the maximum allowed by conservation of etendue and, therefore, the concentrator is ideal. These secondaries are known as TERC (Tailored Edge Ray Concentrators). Unfortunately, in this geometry, the secondary completely shades the primary and it must, therefore, be truncated to be usable, otherwise no light would reach the receiver. 
       FIG. 20  shows the complete shape of a primary Fresnel reflector for a V receiver  1830 , combined with a truncated secondary mirror  2001 . The Fresnel primary is shaped as curves  1804 ,  1807  and  1821  to the right and by their symmetrical  2002 ,  2003  and  2004  to the left of receiver. The receiver receives light from the points on the primary between point  1822  on the right and its symmetrical  2005  on the left. 
     Due to secondary truncation, now some light is lost because part of the secondary mirror is no longer there to redirect it towards the receiver. The final concentration of the optic also decreases accordingly. The receiver and the secondary mirror also shade the primary, further decreasing the final concentration. For small acceptance angles such as those needed for the collection of sunlight, these losses are rather small. 
     If the system had no losses, the etendue of the light reaching receiver  1830  would equal that of the light falling on the primary between point  1829  and its symmetrical  2006 . 
     This optical system is extended left and right by reflection symmetry with  2007  and  2008  as the axes of symmetry. 
       FIG. 21  shows the geometry used to derive the overall dimensions of the primary-secondary optic. The positions of points  1801  and  1803  are first defined (in this example they are at the same height). Given the total acceptance angle 2θ, point  1805  can be determined. The geometry now has two unknowns: the length L of each side of the V receiver and the distance R from point  1805  to the midpoint  1829  of the heliostat field. The condition is imposed that point  1829  is at the same height as  1805  to ensure primary compactness. These two unknowns L and R can be determined by imposing the equations of etendue balance at point  1829  and between the heliostat field and the receiver. Just like in the case of the primary in  FIG. 9 , also here the condition of conservation of etendue at point  1829  results in φ=60° and, therefore, β=60°−θ. The conservation of etendue between the heliostat field and the receiver is 2L=2Rsinθ. These two conditions determine L and R. 
       FIG. 22  shows an optic similar to that in  FIG. 20 , but now designed for an acceptance angle of ±0.01 rad (total acceptance angle of 1.15°. The primary is now curve  2201  and the secondary is curve  2202 . 
       FIG. 23  shows a detail of  FIG. 22 , showing the V receiver  2301  and truncated secondary mirror  2202 , also shown in  FIG. 22 . The concentration of the optic in  FIG. 22  is 87% of the theoretical maximum with an efficiency of also 87%. These results already account for the shading that the receiver  2301  and secondary mirror  2202  produce on the primary. The optics shown from  FIG. 18  to  FIG. 23  assume the primary is formed by an infinite number of infinitesimal structures. This is a severe practical limitation. However, it is possible to design primary-secondary combinations in which the structures on the primary have a finite dimension. 
       FIG. 24  shows a Fresnel lens which left and right halves are different. Similar principles to those used in the design of this lens will be used in the following figures for the design of the finite size structures to place over the form of the primary. To the right of the vertical line  2401  the bottom surface of the lens has the form of a continuous but broken line, which portions  2402  follow flow lines  2403  of the incident radiation, while portions  2404  cross those flow lines. 
     To the left of the vertical line  2401 , the bottom surface is also a broken line but now designed in a different way. The lines  2412  are parallel to the edge rays  2413  inside the lens (and, thus, have no optical function) while lines  2411  cross the flow lines of the incident radiation. 
       FIG. 25  shows a primary-secondary concentrator with an acceptance angle 2θ and a single receiver  2501  with edges  2502  and  2503 . 
     In this example, the primary is formed of a parabolic mirror  2504 , a flat flow-line mirror  2505 , and another mirror composed of two sections: a flat section  2506  and a parabolic section  2507 . These two sections ( 2506  and  2507 ) share a common derivative at point  2513 . Both parabolas ( 2504  and  2507 ) in the primary have axes parallel to edge rays  2517  and focus  2502 . 
     The secondary is formed of three sections  2508 ,  2509  and  2510 . An edge ray coming from the left is reflected at a point  2511  on parabolic section  2504  in the primary towards a point  2512  on the secondary. This point is calculated in such a way that it reflects that ray towards the edge  2502  of the receiver. For the reflected edge rays at point  2513  of the primary, one goes into direction  2514 , while the other is reflected again by the flow-line  2505  into the direction  2515 , parallel to  2514 . The parabolic arc  2509  with axis parallel to  2514  and  2515  concentrates these edge rays towards edge point  2502  of the receiver. Points  2516  in section  2510  of the secondary are calculated in such a way that they reflect towards the edge  2502  of the receiver the edge ray they get from the primary parabolic arc  2507 . Primary and secondary meet at point  2518 . 
     Flow-line  2505  and section  2506  may be given different shapes according to the anidolic optics principles (nonimaging optics). In this case the secondary section corresponding to the primary will no longer be parabolic, but will be calculated with the edge ray principle of anidolic optics. 
     This optic produces the maximal concentration on the receiver. Radiation incident directly on the mirror that follows the flow line  2505  is reflected into directions other that those of the receiver, but all other light is ideally concentrated on the receiver. 
     As in the above-mentioned cases, the secondary must be truncated so that radiation may reach the primary. 
     This combination of primary and secondary is a stepped flow-line optic with walls following flow lines, as the primary is obtained with mirrors generated and placed along flow lines, in a successive way and alternating with other that cross the flow lines. In the limiting case these structures, portions, of primary become infinitesimal and the secondary becomes what is known as a TERC for that primary. 
     This method for producing the primary shares principles similar to those used in designing the right half of the Fresnel lens, shown in  FIG. 24 . 
       FIG. 26  shows a primary-secondary concentrator with one receiver only  2601 . The primary is similar to the one shown in  FIG. 25 , only with more steps. Mirrors  2505  along the flow lines have been removed, leaving only mirrors  2603  in the primary to reflect light/radiation. Each one of these mirrors  2603  is composed of two sections as in  FIG. 25 . The secondary  2602  was also truncated so that light/radiation may reach the primary. 
     Since mirrors  2505  along the flow-lines have been removed, section  2506  of the mirror of the primary (as shown in  FIG. 25 ) can now be slightly extended to the left to recuperate some light/radiation. Once the mirrors along the flow-lines are eliminated, the primary is no longer ideal. 
       FIG. 27A  shows the same construction as  FIG. 25  in which receiver  2720  has, in this example, V-shape in which the primary portions  2701  follow the curve  2721 . In this example the curve has the same shape as that of  FIG. 19 , and conserves the etendue of light/radiation reflected towards two receivers. The primary is formed by mirrors  2702  along the flow lines and mirrors  2703  across the flow lines. These mirrors  2703  are also formed of two sections, just as in  FIG. 25 . The secondary mirror  2704  starts at the tip  2705  of the receiver  2720  and ends at point  2706  where it meets the primary. 
       FIG. 27B  shows the construction of the primary-secondary mirrors, the same as in  FIG. 25 . An edge ray coming from the right and incident on one of the top points of a mirror  2703  is reflected as ray  2712 , tangent to the receiver at point  2708 . The edge ray coming from the left and reflected at the same point of the primary, generates ray  2713 , reflected in turn at the secondary mirror at point  2714  and in a direction tangent to the receiver at point  2708 . 
     An edge ray coming from the left and hitting a low point on another primary mirror  2703  is reflected in direction  2709 . On the other hand, the edge ray coming from the right and reflected at the same point on the primary is once again reflected by mirror  2702  along the flow line in a direction  2710 , parallel to  2709 . Both these rays  2709  and  2710  are redirected towards directions tangent to the receiver at point  2708  by parabolic arc  2711  on the secondary mirror. 
       FIG. 28  shows a primary-secondary optic with two receivers. It is obtained from the construction in  FIG. 27A . Flow line mirrors  2702  are eliminated, leaving only the primary mirrors  2703 . Mirrors  2703  are then duplicated by reflection symmetry with  2707  as symmetry axis, thus originating the  2801  primary section. Section  2803  may also be divided into smaller mirrors, simply by cutting it with vertical lines. The concentration produced by the primary mirror is augmented by secondary  2802 . 
     This optical system is also extended left and right by reflection symmetry with  2804  and  2805  as symmetry axes. 
       FIG. 29  shows a best embodiment of the present invention. It shows a concentrator similar to that of  FIG. 28 , but designed for a smaller acceptance angle of ±0.01 rad. It has an efficiency of about 85% and attains 85% of the maximum possible concentration. This result already accounts for the shading the secondary and receiver produces on the primary. 
     Vertical line  2902  corresponds to vertical line  2707  in  FIG. 28 . Line  2901  corresponds to the symmetrical of line  2707  relative to symmetry axis  2804 . This optical system is extended left and right by reflection symmetry with  2901  and  2902  as the axes of symmetry. It is to be noted that if  2901  and  2902  are vertical flat mirrors, this optic behaves like a concentrator for a V receiver. 
       FIG. 30  shows another example of the present invention with a primary alternative to that of  FIG. 25 . The design in both figures is quite similar, except for the small flat mirror  2506  that is now replaced with a larger flat mirror  3001  and for the flow line  2505  that is not shown in this  FIG. 30 . 
     This optic still produces maximum concentration on the receiver  3002 . All light falling on the space between end points  3003  and  3004  of the primary mirrors is lost. Also, all the light hitting a point  3005  on mirror  3001  and reflected in directions contained between  3006  and  3007  is shaded by mirror  3008  and lost. The size of the receiver is such that the etendue it can receive matches that reflected by the whole primary (left and right halves) towards it. When that happens, the secondary touches the primary at its end point  3009 . 
     The geometry in this figure may also be considered an alternative to that in  FIG. 25  for the design of the secondary. It may be noted that the secondary obtained for this new discontinuous primary is the same as the one obtained for the continuous primary of  FIG. 25 . It can therefore be seen that a continuous primary is not a necessary condition for the design of a continuous secondary. 
     Like in the case of the secondary in  FIG. 25 , portion  3010  of the secondary mirror is a parabola with axis parallel to rays  3012  and focus at the opposing edge  3011  of the receiver. Portions  3013  and  3014  of the secondary are calculated the same way as in  FIG. 25 . 
     This construction of the primary shares similar principles to those used in the design the left half of the Fresnel lens shown in  FIG. 24 , in which now the surface without optical function  2412  correspond to mirrors without optical function and have been removed. 
       FIG. 31  shows a concentrator for two receivers, but in which the primary instead of being wave-shaped as in the two previous cases is now straight or flat. This Figure shows an ideal flat Fresnel primary extending from  3103  to  3109  for two point receivers  3101  and  3102 . By ideal it is meant a primary with an infinite number of infinitesimal structures. From point  3103  to point  3105  light is reflected towards receiver  3101  only. Between points  3105  and  3108 , light is reflected towards both receivers  3101  and  3102 . In this example, it is considered that the optical system is symmetric relative to the vertical through midpoint  3107 . 
     Reflection of light by the Fresnel mirror is such that the bisector to the edge rays points towards points towards  3101  (or  3102 ). 
     For a point  3104  between points  3103  and  3105  on the reflector, the etendue mismatch between that of the incoming radiation and that of the reflected light towards receiver  3101  is given by ΔU 1 =2sinθ(1−cosφ 1 )dx. For another point  3106  between points  3105  and  3108 , the etendue mismatch between that of the incoming light and that ideally reflected towards receivers  3101  and  3102  is |ΔU 2 |=2sinθ|1−cosφ 1 +cosφ 2 |dx where |α| is the absolute value of α. The absolute value in the expression for ΔU 2  ensures that the etendue mismatch is always accounted for as a positive quantity when integrating it across the Fresnel reflector. From these expressions it can be seen that the etendue of the incoming radiation does not match what ideally the Fresnel reflector should emit towards the receivers. The parameters of the optical system must then be adjusted in such a way as to minimize this etendue mismatch. The height of the receivers (distance from point  3103  to  3101 ) may be considered as a scale factor of the whole system and, therefore, it can be made equal to one. Being the receivers at a different height, the whole system would be scaled accordingly. The parameters that must now be adjusted are the distance between receivers (distance from  3101  to  3102 ) and the horizontal coordinate x T  of the transition point from one to two receivers. 
     The total etendue mismatch is proportional to the integral of ΔU 1 +|ΔU 2 | from x 0  (point  3103 ) to x M  (point  3107 ). This integral must be minimized relative to the parameters of the optic: horizontal coordinate x T and distance between receivers. 
       FIG. 32  shows the overall relative dimensions of an optimized Fresnel primary when all the reflectors are on the same plane. Angle α T  from the vertical through the receiver  3101  to the transition point  3105  from one to two receivers is α T =32.647°. Angle α M  from the vertical through the receiver  3101  to the middle point  3107  of the primary is α M =61.2353°. This makes D=3.643 H where H is the height of receivers  3101  and  3102  and D the distance between them. Receiver  3101  receives light from the right side from point  3103  to point  3108 . The rim angle α R  measured from the vertical through the receiver  3101  to point  3108  is given by α R =71.58°. 
       FIG. 33  shows the etendue mismatch across the primary for the optimized geometry in  FIG. 32 , Horizontal axis  3301  has coordinate x representing the distance across the primary and vertical axis  3302  has coordinate U representing the etendue mismatch for each point on the primary. This figure represents etendue mismatch per unit incoming etendue. The available etendue per available unit etendue is clearly unity across the whole Fresnel primary and is represented as horizontal line  3303  with U=1. 
     For points  3104  on the primary between point  3103  and  3105 , the etendue that the Fresnel reflector should emit towards receiver  3101  is given by 2sinθ cosφ 1 dx and the etendue of the incoming radiation is 2sinθdx. Therefore, per unit incoming etendue, the primary should emit an etendue of cosφ 1 . This is represented by curve  3304 . 
     For points  3106  on the primary between point  3105  and  3108 , the etendue that the Fresnel reflector should emit towards receivers  3101  and  3102  is given by 2sinθ(cosφ 1 +cosφ 2 )dx and the etendue of the incoming radiation is 2sinθdx. Therefore, per unit incoming etendue, the primary should emit a etendue of cosφ 1 +cosφ 2 . This is represented by curve  3306 . 
     Curve  3305  is symmetrical to  3304  relative to the midpoint of the Fresnel reflector. 
     Comparing the curves  3304  and  3306  representing the etendue of the light the Fresnel reflector should emit towards the respective receivers with straight line  3303  representing the available etendue at each point, it can be seen that there are points with excessive etendue available and other points with etendue deficit. For the points to the left of  3105 , the etendue that the reflector should emit towards receiver  3101  is less than the etendue available (line U=1). This means that some etendue must be lost at the reflector and there will be shading between the microstructures of the primary (infinitesimal heliostats). On the other hand, for points on the Fresnel reflector between  3105  and  3108 , curve  3306  sometimes is above line  3303 . This means that the etendue the Fresnel reflector should be emitting towards receivers  3101  and  3102  is more than the etendue available. This means that, as seen from the receivers, the Fresnel reflector will have “dark holes” that do not emit light. For the region in which curve  3306  is below  3302 , the needed etendue is less than what is available and there is shading of light at the Fresnel reflector. 
     Point  3105  for which there is a transition from one to two receivers, the vertical distance between curve  3306  and  3303  and between curve  3304  and  3303  are equal to each other and given by  3307 , with a value of 0.158 (or 15.8%). 
       FIG. 34  shows the geometry of a concentrator for an acceptance angle 2θ. Once the overall geometry of the optical system has been determined (as shown in  FIG. 32 ), the acceptance angle 2θ of the optic determines the width of the receiver as the distance between points  3401  and  3402  (in general, the edge rays are tangent to the receiver). If the receiver was flat between those two points, it could not accommodate all the etendue emitted by a large Fresnel reflector spanning from  3103  to  3108  and its symmetrical. This means that the receiver cannot be flat, but must have some kind of convex shape so that its length is able to accommodate all the etendue emitted towards it by the Fresnel reflector. A possibility (although by no means the only one) is to have a V shaped receiver. 
       FIG. 35  shows a concentrator for a V receiver  3501  formed of an ideal primary and a secondary. The primary is made of an infinite number of infinitesimal structures and spans from point  3103  to  3108 . The secondary  3502  starts at the edge of the receiver and extends to the end  3108  of the primary. The paths of some edge rays  3503  are also shown. 
       FIG. 36  shows a concentrator similar to the one shown in  FIG. 35 , but now for a U receiver. The receiver has flat walls  3601  and curved bottom  3602 . Edge rays  3603  are reflected in directions tangent to the receiver. 
       FIG. 37  shows another concentrator similar to the one in  FIG. 36  but in which the receiver is now made of a set of circular tubes. The tubes are tangent to the U shaped receiver  3701 . Other shapes of receivers are possible using the same design methods. 
       FIG. 38  shows a concentrator for a V receiver  3801  composed of a primary  3802  and a secondary  3803 . The overall geometry of the optic is based on that in  FIG. 35 , but now the primary has finite size structures, designed according to the method shown in  FIG. 25 . Those structures can also be designed according to the method shown in  FIG. 30 . The secondary  3803  starts at the edge of the receiver and extends to the end of the primary. 
     Flow lines  3805  can now be removed and mirrors  3806  given mirror symmetry about vertical line  3804  thought midpoint  3107 . 
     The primary reflects one set of edge rays tangent to the receiver. In the case of a V receiver, this means that to the left of point  3807  one set of edge rays is concentrated to point  3808 , while to the right of  3807 , this same set of edge rays is reflected towards point  3809 . The other set of edge rays is concentrated to point  3809  after reflection on the secondary  3803 . Point  3807  is in this example on the straight line through points  3808  and  3809 . 
       FIG. 39  shows the concentrator obtained by removing the flow lines and mirroring the remaining mirrors about the vertical through point  3107 . In this process there are mirrors that intersect or shade other mirrors. The portions of the mirrors above the intersections or portion of mirrors that shade other mirrors are trimmed. This results in a primary which heliostats do not intersect or shade each other. 
     The primary  3901  now reflects light to both V receivers and the secondary  3902  was truncated to allow light to reach the primary. 
       FIG. 40  shows a concentrator similar to that in  FIG. 39 , but designed for a smaller acceptance angle of ±0.01 rad. 
       FIG. 41  shows a secondary mirror and a receiver.  FIG. 41A  shows one such configuration with a secondary mirror  4101  and a V receiver  4102 . The mirror touches the receiver at its end points  4103 . This may be a problem since the receiver may get very hot during operation of the solar concentrator and this contact may heat excessively the mirror damaging it and also creates a heat sink through which heat can escape, reducing the efficiency of the system. 
       FIG. 41B  shows a modification of the geometry in  FIG. 41A  in which the V receiver is now replaced with an equivalent inverted Δ receiver  4106 . The secondary mirrors are now extended by two circular arcs  4105  with centres  4104  and its symmetrical. Now there is only one point of contact between the secondary mirror and the receiver at the midpoint of the top surface  4107 . The secondary for triangular receiver has a discontinuous derivative (kink) at point  4103 . 
       FIG. 41C  shows a modification of the geometry in  FIG. 41B  in which the top flat surface  4107  of the receiver was removed and replaced with a concave surface  4109 . This new surface may be circular with centre  4110 . Now there is no contact between the secondary mirror and the receiver. Concentration, however, is smaller because the surface area of the receiver is larger. If, however, the receiver was a perfect back body, the emission of concave surface  4109  would be equivalent to that of flat surface  4107  and therefore the concentrator would behave as if it had ideal concentration (considering only radiative losses). 
       FIG. 42  shows another embodiment of the present invention, a concentrator for an inverted Δ receiver but with a smooth secondary mirror (continuous derivative). The portion of the primary between points  4201  and  4202  concentrates one set of edge rays to point  4204  above the secondary mirror. These edge rays are then concentrated to lower tip  4205  of the inverted Δ receiver by hyperbolic arc  4206  with foci  4204  and  4205 . Hyperbolic arc  4209  is symmetrical to  4206  relative to the vertical line through the lower tip  4205  of the receiver. The secondary mirror also has a circular arc  4211  with centre in  4207 . 
     The other set of edge rays reflected by the primary between points  4201  and  4202  is concentrated to the lower tip  4205  of the receiver by a portion of the secondary to the right of hyperbolic arc  4209 . This portion of the secondary mirror is designed in the same way as the portion of the secondary mirror touching receiver  3501  in  FIG. 35 . 
     The portion of the primary between points  4202  and  4203  concentrates a set of edge rays to lower tip  4205  of the receiver, while the other set of edge rays is reflected by the secondary mirror towards the lower tip  4205  of the receiver. Again this is similar to the design of the right portion of the secondary mirror in  FIG. 35 . 
       FIG. 43  shows a concentrator similar to that in  FIG. 39 . Now the lower tip  4301  of the V receiver was moved down from its ideal position, increasing the size of the receiver. The resulting secondary  4302  no longer extends all the way to the end of the primary. Depending on how much the receiver size is increased, it may or may not be necessary to truncate the secondary mirror for optimum operation. The resulting concentrator has a higher efficiency, since more light is now collected by the secondary, but at the cost of a lower concentration, since the receiver has now increased in size. 
       FIG. 44  shows a concentrator similar to that in  FIG. 35  for a V receiver  3501 . This figure shows flow lines (or G-lines)  4401  and their perpendicular, or F-lines,  4402 . G-lines bisect the edge rays at each point and point in the direction of propagation of light. F-lines also bisect the edge rays at each point, but point in the direction perpendicular to the propagation of light and are, therefore, also perpendicular to the G-lines at each point. 
       FIG. 45  shows a preferred embodiment of the present invention and shows the concentrator of  FIG. 44  with a truncated secondary  4502 . Receiver  3501  is now protected by a transparent cover  4503  shaped as an F-line. This shape minimizes the incidence angles of light on the transparent cover and, therefore, minimizes Fresnel reflection losses. F-lines may sometimes be well approximated by simple shapes, such as circumferences. 
     Between the transparent cover and the receiver, there are mirrors  4504 , mirrored on both sides, shaped as G-lines (or flow lines). These mirrors do not affect the flow of light and, therefore, do not affect the optical behaviour of the optic. G-lines may sometimes be well approximated by simple shapes, such as straight lines. 
     Both the transparent cover  4503  and the internal mirrors  4504  help reduce the convection around the receiver, reducing thermal losses. 
     The primary extends from point  3108  on the right to point  4501  on the left. These edge points of the primary are symmetrical relative to midpoint  3103 . 
       FIG. 46  shows another embodiment of the present invention. It shows the V receiver of  FIG. 45 , but with the curved transparent cover  4503  now replaced by a facetted transparent cover. In this example, this faceting has only two facets, resulting in a V transparent cover  4601 . The tilt of these flat transparent covers is calculated in such a way as to minimize the average angle  4602  between G-lines (flow line)  4404  and the normal  4603  to transparent cover  4601 . 
     Also internal mirrors (mirrored on both sides)  4504  placed along the G-lines help reduce internal convection. 
     Thermal losses are further reduced by using thermal insulation  4604  on the back of the V receiver. Overheating of the secondary mirrors may be prevented by using heat dissipaters  4605 . At point  4606  there should also be some thermal insulation between the receiver and the secondary mirror to prevent a thermal bridge between the two, resulting in a thermal loss. 
     The preceding description of the presently contemplated best mode of practicing the invention is not to be taken in a limiting sense, but is made merely for the purpose of describing the general principles of the invention. The full scope of the invention should be determined with reference to the claims.