Optimizing heliostat arrays

Every optimized heliostat array embodies a phase transition between a packed inner zone (solid phase) and a dispersed outer zone (gas phase.)

When a heliostat array is optimized, heliostats arrange themselves into two phases. Close to the tower, heliostats are closely packed (the solid phase.) Far from the tower, heliostats are spaced apart (the gas phase.) The gas phase wastes land area; the solid phase wastes thermodynamic availability because these heliostats are too close to the target—from the receiver's point of view the sun's image fails to fill the mirror.  Only at the boundary between these two phases (the condensation line) do heliostats work with maximal return on investment, wasting neither land nor thermodynamic availability.

That the heliostats at the boundary are working optimally is evidenced by the fact that, given the opportunity to spread out a little to improve performance, they choose not to spread out at all!

Low-aiming, target points, and optimization

Cross-section of an array of telescopic heliostats showing the target point for all heliostats lying at the same azimuth to the field center. The field radius is 14 times the height of the beam-down optics—about twice the comparable ratio for conventional heliostats.

Aiming slightly low in the outer portion of the heliostat field, and slightly high in the inner portion of the field, offers the advantage of making the focal zone more compact. This variable aiming can be idealized as aiming all the heliostats that lie on the same azimuth from the origin toward a single target point behind the focal zone. When heliostats are targeted in this way, the locus of the geometric constraint—the ideal shape of the beam-down optics—is no longer a family of confocal parabolas, instead it is a family of ellipses having as their foci the target point and the oculus center.

Whatever happens near the inner radius of the field is of minor importance due to the relatively small number of heliostats involved. On the other hand, what happens with heliostats near the outer radius of the field is important because there are the most numerous rows of heliostats.

Low-aiming of heliostats at the periphery can be carried too far. With constant aiming, the lowest ray in the beam from a secondary rises at only beta/2 (1:36 with 6x telescopic heliostats.) Lowering aim by a mere beta/2 would send the lowest rays traveling horizontally—guaranteeing that they would hit the back of another secondary. In fact, any lowering of the aiming angle will entail either some blocking by other secondaries (should we keep the primaries close-packed) or a decrease in land utilization (if we space the primaries out.) An optimized system will always have a bit of every possible kind of loss: all the possible losses come out to play in the big trade-off game. For the purpose of a simple baseline design we can be arbitrary: we will insist on zero blocking loss, but we will permit the land utilization in the outermost row to fall to 2/3.

The circles are the primary mirrors viewed from above; the ellipses are the shadows cast on a horizontal plane by the secondaries when illuminated from the direction of the lowest ray in there beam. 

In a system with 6x telescopic heliostats, the beam spread is 3°, and the nominal beaming angle is 3.1°. That leaves the lowest rays in the beam directed 1.6° above horizontal, or a run-over-rise of 36:1. The ellipses in the figure above are 9:1, they would need to be elongated to 36:1, but still with the same surface area, to represent the design condition. Aiming lower while keeping blocking at zero means that the ellipses must be further elongated, but this time the dimension of the minor axis is kept constant and the surface area is allowed to increase. The surface area of the circles (which represent the primary mirrors) are not allowed to change when we aim lower because we are using the same design of heliostats over the whole field. Therefore, the area increase of the ellipses associated with low-aiming is balanced by extra empty space between primaries.

At the last row we allow 2/3 land utilization, meaning that low-aiming is permitted to increase the surface area of the ellipses by 50%: their aspect ratio can increase from 36:1 to 54:1. So, at the last row, the lowest ray can have a run-over-rise of 54:1, which is an elevation angle of 1.1°. Recalling that the beam spread is 3°, the middle ray will have an elevation angle of 2.6°, or a run-over-rise of 22:1; the top ray will have an elevation angle of 4.1°, or a run-over-rise of 14 (putting that in a short hand, the outer heliostat operates at: 14-22-54, in run-over-rise numbers, which are also the reciprocals of the elevation angles measured in radians.) By comparison, a heliostat at its design condition operates at 12.4-18.5-36.

The unique row of heliostats that operates with both design condition aiming and chock-a-block packing should be located somewhere near the median radius of the field (outer radius / sqrt 2.) The top image diagrams such a system. The target point turns out to be little bit higher than the beam-down optics, and about one field radius behind the center of the field.

Calculation of target position, where γ is the ratio of design beaming angle, β, to the beaming angle, β_e at the outer radius, r_o; and r is the target point's radial distance from the center of the field. Heliostats operating at the design condition are assumed to lie at the median radius (0.707 * r_o) from the center of the field. In this example γ = 22/18.5 = 1.19, so r/r_o = 0.84. The height of the target point is then r_o * (1 + 0.84) / 22 = .084 r_o .

The run-over-rise number for the top ray of the outermost heliostats—in this example, 14—is also the ratio of the field radius to the tower height (or height at the top of the beam-down optics.) Telescopic heliostats give twice the value for conventional heliostat fields, but with much better land utilization.

In what sense are telescopic heliostats telescopic?

A pair of binoculars focussed at infinity and aimed at the sun emit intense beams from the eyepieces.

Definition: A telescopic heliostat is a heliostat in which the emergent rays are increased in concentration and divergence.

A telescopic heliostat is not telescopic in the sense that it can throw a beam of sunlight farther. If that were the only consideration, a plane-mirror heliostat would be the best choice. A telescopic heliostat concentrates sunlight while keeping the emergent rays as parallel as possible—which is precisely the action of a telescope focussed at infinity. An inevitable consequence of the concentration, is that the emerging light rays are more divergent than a natural sunbeam, so beam spread from a telescopic heliostat is greater than from a conventional heliostat.

An optical cross-section of a telescopic heliostat shows it to be an afocal, off-axis, Cassegrain telescope. Both mirrors are parabolic.

The object of a heliostat is to hit a target of low angular elevation. (Note how much sunlight falls to the ground in the process.) Image quoted from NRG Energy.

If hitting a distant target were the object of a heliostat we would stick with plane mirrors; but the object of a heliostat is to hit a low target. There are practical limits to the height that a boiler or beam-down optics can be elevated above the ground, so the goal of a heliostat is to redirect sunlight to a target of low angular elevation.

When 100% of the sunlight incident on a horizontal surface at angle of incidence θ_i emerges redirected at an angle of reflection of θ_r, the sunlight has been concentrated by a factor of cosθ_i / cosθ_r . The Second Law of Thermodynamics forbids this concentration without a compensating increase in divergence. Since plane mirrors cannot accomplish the required increase in divergence, they cannot accomplish the desired efficient redirection of light.

Consider, for example, the Ivanpah Solar Project in California. According to the 2010 environmental impact statement for this project, each power tower is 140 m tall measured to the top of the boiler. Ivanpah units 1 and 2 each have 55,000 heliostats, with each heliostat carrying 14 m² of mirror. Let's say 1 m² of mirror typically shades 1.5 m² of ground. Each unit occupies about 915 acres (3.7 million m²) of land, but only 55,000 x 14 x 1.5 = 1.2 million m2 are typically shaded, a land utilization of only a third. To achieve full utilization we would need to squeeze three times as many heliostats into the view above—but they would only block each others' view!

Mutual blocking is worst at the farthest edge of a heliostat field, so land utilization there is really poor. Ultimately it makes more sense to build another tower than to extend a low-utilization field to greater radius. Approximating an Ivanpah unit as a circle, it has a radius of 1085 m. From the farthest edge of the field the top of the boiler has an angular elevation of arctan(140/1085) = 7.4°.