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Parametric determination of heliostat minimum cost per unit area
Available online at www.sciencedirect.com
ScienceDirect Solar Energy 97 (2013) 342–349 www.elsevier.com/locate/solener
Parametric determination of heliostat minimum cost per unit area James B. Blackmon ⇑ Department of Mechanical and Aerospace Engineering, Propulsion Research Center, S233 Technology Hall, University of Alabama in Huntsville, Huntsville, AL 35899, United States Received 12 March 2013; received in revised form 14 August 2013; accepted 21 August 2013 Available online 18 September 2013 Communicated by: Associate Editor Lorin Vant-Hull
Abstract A parametric analysis illustrates how heliostat costs per unit area can be minimized by distributing the costs into categories having different cost dependence on area. Heliostat costs are distributed among three basic categories:
Category 1, costs that on a per unit area basis are constant, such as mirrors. Category 2, costs that are dependent on the imposed loads, and thus area, such as the structure, pedestal, and drive units. Category 3, fixed costs per heliostat, irrespective of heliostat area, such as controllers, and position sensors. Using the 150 m2 United States Department of Energy (DOE) baseline heliostat design and its cost of approximately $200/m2, cost reductions of the order of 50% are determined for a range of much smaller areas, primarily as a function of decreased Category 3 costs. For achievable Category 3 costs, the minimum cost per unit area is projected to meet the DOE 2020 goal of $75/m2. Ó 2013 Elsevier Ltd. All rights reserved. Keywords: Central receiver systems; Heliostat cost minimization; Optimum heliostat area; Parametric cost analysis
1. Introduction Minimum heliostat costs per unit area can be determined for any particular design by analyzing parametric cost estimating relationships in terms of the heliostat area for costs distributed among three basic categories. The projected $200/m2 installed hardware cost for the 150 m2 DOE baseline elevation-azimuth heliostat is used as an example, with a range of reasonable fractions of these costs distributed into the three categories; this parametric method does not require a detailed treatment of the specific component costs for the DOE baseline or any other design. In the parametric analysis, whatever the design and configuration, it is assumed that materials, performance requirements, and configuration are the same for the range of ⇑ Tel.: +1 (256) 824 5106.
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areas analyzed. Category 1 costs for the heliostat hardware are primarily for mirrors, which are essentially constant on a per unit area basis irrespective of heliostat area for a given type, thickness, etc. and total field reflector area. Category 1 can include other costs that are essentially constant on a per unit area basis, such as site preparation. Category 2 costs are associated with the torques imposed on the load-bearing structural/mechanical assemblies, primarily from winds; these costs are also associated with constant performance requirements, such as strength margins of safety, stiffness, rotational rate, and tracking accuracy. Category 2 costs are dependent on imposed moments, in accordance with the so-called “area to the three-halves law” for uniform wind speed, or its variant for non-uniform wind speed; these costs increase on a per unit area basis as the size increases. Category 3 costs are associated primarily with controllers, position sensors, processors, etc., which are the same irrespective of heliostat size.
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Reducing Category 3 costs results in a direct savings and also enables a smaller heliostat with additional reductions in the cost per unit area due to the Category 2 cost dependence on imposed loads. The differing cost dependencies on heliostat area of Category 2 and 3 determine a minimum in total cost per unit area as a function of area for given design and performance requirements. For areas less than that optimum, the Category 3 costs rise inversely with area and quickly become inordinately costly as the number of heliostats increases for a particular total field area. Conversely, as the heliostat size increases above the optimum, the higher loads on a per unit area increase the weight and cost of these Category 2 load-dependent components. Results based on the DOE baseline cost of $200/m2 at 150 m2 show heliostats significantly smaller offer approximately a 50% or greater reduction in cost per unit area, for Category 3 costs that should be achievable. 2. Analysis The parametric approach was in part developed under a Phase 1 DOE-funded heliostat development program (Kusek, 2012) as well as prior work (Blackmon, 2012). Those analyses used specific component costs for the el/ az, glass–metal, pedestal mounted DOE baseline heliostat (Kolb et al., 2007). These are the only detailed, openly available cost data available. Other size-dependent aspects were considered, such as the effect of optical performance improvements of smaller heliostats on the total reflector field area required, learning curve benefits, and the use of net present value (NPV) of operations and maintenance operations. These analyses also considered a fourth category for the field wiring cost estimating relationships that did not fit well into the three basic categories. The following parametric method determines a range of optimum heliostat areas and associated minimum costs per unit area, but does not require a treatment of the individual component costs as conducted in the previous analyses. This approach is also not dependent on a particular design or configuration. The structural–mechanical design, materials, configuration, control system design, and performance requirements are assumed to remain unchanged over the range of sizes, whatever the design. The analysis shows the sensitivity and leveraging effect of Category 3 cost reductions on heliostat area and cost per unit area. One observation is that if the developer were to further reduce their Category 3 costs, then it would not be sufficient to simply apply that cost reduction directly. To fully achieve the minimum cost per unit area, it would also be important to re-determine the optimum area of their initial “point design”. Costs of manufactured goods comprise fixed costs and costs per unit of good produced. Fixed costs are typically associated with such factors as management, overhead, plant maintenance, debt service, retirement obligations, leases, and utilities, which normally are approximately
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the same irrespective of the annual number of units produced and total sales; these costs are typically embedded in the total price. Per-unit costs include labor, materials, and other costs directly associated with production and are determined by such design aspects as load, and torque. At high production rates these per-unit costs can often be approximated as a constant for a particular item, but with other factors, such as percentage of plant capacity, used to determine the breakeven point and sales price. There are also potential learning curve effects as more units are produced and the manufacturing process, supply chain costs, worker productivity, etc., improve. These aspects are treated in many different business administration texts. Here, a simplified analysis is used to illustrate the general parametric principles that result in an optimum area at the minimum cost per unit area. For this analysis, a representative heliostat field cost is based on a fixed cost and on a per heliostat cost, with the total heliostat hardware cost determined by individual heliostat area and number of heliostats. Learning curve and optical performance benefits are neglected. Also, the Net Present Value of Operations and Maintenance costs are not included in the cost distribution. Total cost of a particular heliostat field, CT(N), is a function of a fixed cost, F, the number of heliostats, N, and the costs per heliostat, B. The total heliostat field cost for a given field reflector area is approximated as the sum of the fixed cost and the per-heliostat hardware cost and number of heliostats: C T ðN Þ ¼ F þ BN :
ð1Þ
The value F represents the cumulative costs for the given total reflector area of the field, separate from the hardware costs per heliostat. The costs in F can include factors such as civil engineering site preparation and other costs that can be associated with the heliostat field, but separate from costs of the other central receiver subsystems; examples include heliostat field management cost, net present value (NPV) of certain Operations and Maintenance costs, permitting costs, some aspects of debt service, etc., which are assumed to increase roughly in proportion to the overall size of the field but are essentially independent of the heliostat size. The hardware cost per heliostat is the cumulative costs of its various Category 1, 2 and 3 components. A major example of Category 1 hardware costs for the heliostat is the mirror costs per unit area, which are essentially constant irrespective of heliostat size. Category 2 costs are for load-bearing components and Category 3 is primarily for electronics. Load-bearing components can be treated using either a uniform or non-uniform wind speed. First the uniform wind speed case is considered. 2.1. Uniform wind speed The uniform wind speed “three-halves law” is used for imposed torque, which relates the equivalent point-load force based on the distribution of wind pressure, P, times
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heliostat area, AH, and the moment arm associated with the force components that produce the moment. The force is proportional to AH. The moment arm is proportional to 1=2 AH , and thus the moment applied is proportional to 3=2 AH . It is further assumed that this torque relationship with area adequately describes torques used to determine the design requirements for the drive unit, structural support, pedestal, etc. and that, in particular, the costs are linearly proportional to torque, and thus these costs also vary as 3=2 AH . This assumption is shown to hold for structures (e.g., Kolb et al., 2007), but it is also supported by many different types of motors, gear drives, etc. (e.g., Blackmon, 2012; Kusek, 2012). To enable a consistent comparison for the basic DOE baseline design and configuration, the aerocoefficients and operational requirements (tracking accuracy, slew rates, etc.) are assumed to be constant over the range of areas for both the uniform and non-uniform wind conditions. With these assumptions, it is not necessary to develop the detailed loads and design conditions for each size. Instead, the loads are based on those for the ATS heliostat (Heller and Peters, 1989) used for the DOE baseline; the parametric dependence is sufficient to determine the cost/area as a function of area. Reflector support structure costs have been shown to 3=2 vary as heliostat area to the three-halves, AH (Kolb et al., 2007) for uniform wind speed. In addition, all of the load-bearing components, such as the drive unit, pedestal, and foundation costs exhibit costs that vary linearly with torque or imposed moment, and thus these costs are 3=2 proportional to AH for a uniform wind (Kusek, 2012; Blackmon, 2012). The linear relationship of price vs. torque is supported by hundreds of examples for many different types of motors and drive units (Kusek, 2012; Blackmon, 2012). Those curve fits showed that there is a constant cost as well as linear dependence on torque; for limited orders, such as catalogue prices, those constant costs are significant, due in part to costs for handling, inventory, overhead and administration, pricing policy, etc. But for large orders these costs would be distributed over a far larger number, which would greatly reduce the fixed cost as a percentage of the total hardware cost. The unit hardware cost B for a heliostat of area AH is the sum of the costs for each category. The Category 1 cost for the heliostat is C1, and this cost is based on a constant cost per unit area, C1A, times the heliostat area, AH. Category 2 costs, C2, for a heliostat are represented by f2 þ kAH3=2 , where f2 is the aggregated fixed cost associated with the load-bearing components and k is the appropriate aggregated coefficient. Category 3 costs per heliostats are f3, and are independent of size. The hardware cost, B, per heliostat is thus 3=2
B ¼ C 1A AH þ f2 þ kAH þ f3
ð2Þ
The number of heliostats, N, is approximately the total area of reflectors in the field divided by the heliostat area, AF/AH; optical efficiency improvements with small
heliostats are low and are not included in this analysis. Inserting these expressions into Eq. (1), the total heliostat field costs are: 3=2
C T ðN Þ ¼ F þ ðC 1A AH þ f2 þ kAH þ f3 ÞAF =AH :
ð3Þ
Simplifying, total heliostat field cost per unit reflector area becomes: 1=2
C T ðN Þ=AF ¼ F =AF þ C 1A þ f2 =AH þ kAH þ f3 =AH :
ð4Þ
Heliostat cost per unit area is typically considered as an installed hardware cost, although if field wiring costs are included, then there would be some intrinsic site preparation aspects, such as trenching. A comprehensive heliostat cost would include all costs for the field, as well as for the heliostat hardware and installation costs. For example, net present value of operations and maintenance have been included in the total hardware cost, with distributions of these costs into the three categories (Kusek, 2012; Blackmon, 2012); these costs have aspects that would increase or decrease as a function of heliostat size, as well as having some fraction as a constant cost per unit area. But since only installed hardware costs are readily available at this time and the other factors are typically reported separately, it is more practical to incorporate terms to have a representative cost made up of generalized Category 1, 2, and 3 costs, however they are defined in a detailed cost analysis. Therefore, the first two terms in (4) are combined into a composite term representing whatever identifiable costs are available from a particular cost analysis that are constant on a per unit area basis, represented as CCat1. With f = f2 + f3, a simplified Category 3 expression results, f/AH. The cost per unit area is then represented for uniform wind as: 1=2
C T ðN Þ=AF ¼ C Cat1 þ kAH þ f =AH :
ð5Þ
The optimum area, AH|opt, for the minimum cost per unit area is found by setting the derivative of Eq. (5) equal to zero and solving, giving: AH jopt ¼ ð2f =kÞ
2=3
ð6Þ
:
Eq. (6) shows that as f approaches zero, for very low Category 3 “constant” or “fixed” costs per heliostat, then the minimum cost heliostat size approaches zero. This illustrates the importance of these constant costs whatever the costs are for mirrors (and any other costs that are a constant per unit area) and the costs associated with the drive units, support structures, pedestals, etc. For f increasing, corresponding primarily to higher Category 3 costs, then the optimum area increases. Substituting the optimum area from (6) into the total cost per unit area from (5) gives the minimum cost per unit area, C T ðN Þ=AF jmin ¼ C Cat1 þ kð2f =kÞ
1=3
þ f 1=3 =ð2=kÞ
2=3
:
ð7Þ
The cost per unit area of the field can now be considered in terms of its dependence on the Category 2 and 3 relative costs and on heliostat area. If the costs for a particular
J.B. Blackmon / Solar Energy 97 (2013) 342–349
heliostat design and size are known and have been distributed into the three categories, then the cost per unit area for the field can be illustrated parametrically as a function of area. Thus, a heliostat developer could initially select a size for a detailed “point-design” that determined the total mass-produced costs for that heliostat on a per unit area basis, but then use this parametric analysis to estimate the optimum size for the minimum cost per unit area. Treating the total cost on a per unit area basis, and using a range of reasonable percentages of these costs that are in the three categories, also provides insight into the effectiveness of reducing costs in each of the three categories. As a specific example, installed hardware costs can be approximated in current dollars as CT(N)/AF = $200/m2 for the DOE baseline (Kolb et al., 2011), which is close to the $211/m2 noted by DOE (DOE FOA, 2009). The analysis does not depend on the design or configuration, and is thus not limited to the DOE baseline or its specific component costs. With 20% for Category 1 representing costs of mirrors and other properly apportioned constant costs on a per unit area basis for the 150 m2 heliostat, the effect of Category 3 costs on the optimum area and on the cost per unit area can be shown. (Note, Kolb et al., 2007 show mirror costs of the order of $23/m2, which is of the order of 18% of the total cost of their heliostat. The point here is to have a representative percentage for additional constant cost/area aspects for illustration purposes.) The heliostat area is 150 m2; the remaining 80% is distributed into Categories 2 and 3. Eq. (5) becomes: kð150Þ1=2 þ f =150 ¼ 160$=m2 :
ð8Þ
Four examples are shown for a range of Category 3 costs. Details are developed in Excel. First, if the Category 3 cost per unit area of the 150 m2 heliostat is 10% of the total cost per unit area, then f =150 ¼ 0:1 200 ¼ $20=m2 and f ¼ $3000: That is, for the 150 m2 heliostat the cost primarily of the electronics (processors, encoders, limit switches, etc.), as well as any other relatively low fixed costs per heliostat for other components (junction boxes, connectors, etc.), remain constant at $3000 as the area changes. That cost is demonstrably extremely high. For example, from Kolb et al. (2007) the drive motors and limit switches were $1.78/m2 and the controls costs were $1.90/m2 for a total of $3.68/m2. Their heliostat field wiring costs were $7.40/ m2. The first two give a cost for the 148 m2 ATS heliostat of $545. However, drive motor costs are determined by motor power and torque, and thus some fraction of these costs should be in Category 2. If field wiring is also included, then the cost rises to $11.08/m2 or $1640. Distributing all field wiring costs only into Category 3 would be an over-simplification, since field wiring also has costs that should be distributed among the three categories and depends on the heliostat size. For example, with smaller heliostats, there are more rows and total length for wiring
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and trenching, and more connections. Also, heliostat area, and thus imposed wind moments, determines motor torques and horsepower, and this affects the wiring gauge and costs. Alternatives to field wiring such as heliostats powered by photovoltaic arrays with wireless communication, offer substantial potential cost savings (Kusek, 2012). Therefore, the assumption of $3000 in Category 3 for the DOE baseline is unrealistically high, even if it were assumed to include a substantial fraction of field wiring and fixed costs from Category 2, but it at least establishes an upper boundary. The remaining cost for this example is for the structure, pedestal, foundation, and drive units, or 70% of $200/m2, giving $140/m2. Then, k(150)1/2 = 140, and k = 11.43. Heliostat field cost per unit area for this case is then: 1=2
C T ðN Þ=AF ¼ C Cat1 þ kAH þ f =AH 1=2
¼ 40 þ 11:44AH þ 3000=AH :
ð9Þ
For the second case, assume 5% of the total cost per unit area or $1500 is in Category 3. Then, Category 2 is 75% of the total cost, or $150/m2, f = 1500 and k = 150/ 1500.5 = 12.25. For the third case, assume 1% of the total cost or $2/m2 or $300 is for Category 3. Then, the load dependent part of Category 2 is 79%, or $158/m2 with f = 300 and k = 158/ 1500.5 = 12.90. Finally, assume that the Category 3 costs are reduced to 0.25% of the total cost per unit area for that heliostat, or $0.5/m2, or $75 per heliostat. Thus, Category 2 is now 79.75%, or $159.50/m2 with f = 75 and k = 159.5/ 1500.5 = 13.02. The family of curves in Fig. 1 shows that relatively small changes in the relatively small percentage of fixed costs per heliostat for the 150 m2 size have a significant effect on the optimum heliostat area and the minimum cost per unit area. Finding ways to reduce Category 3 electronic/control costs results in a direct savings and in further cost reductions from the lower size of the optimum area, and thus the effect of imposed loads.
Fig. 1. Cost comparisons vs. area for uniform wind.
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Comparing the two lowest cost cases provides insight into the leveraging effect. The 1% case corresponds to a cost of $300 per heliostat. The 0.25% case corresponds to $75 per heliostat, for a direct savings of $225 per heliostat. However, the lower cost produces a minimum cost per unit area that is $85/m2 at about 5 m2, compared to $110/m2 at 13.5 m2 for the 1% case. Thus, the heliostat cost per unit area difference is $25/m2. If the direct savings had been used, but the area had not been reduced from 13.5 m2 to 5 m2, then the total field cost for about 1,000,000 m2 would be 106(110–225/13.5) = $93.3M, vs. the cost without the direct savings of $110M, or a savings of $16.7M. If in addition the heliostat size were reduced, then the total field cost would be $85M, for a total cost reduction of $25M, with the additional $8.3M in savings from further reducing the heliostat area from 13.5 m2 to the new optimum at about 5 m2. 2.2. Non-uniform wind speed case For non-uniform wind the speed, V, at height h, relative to the meteorological wind speed, V0, at height h0, is V/ V0 = (h/h0)0.15 where the meteorological winds are typically measured at 10 m (Heller and Peters, 1989). Assume the distance from the pivot point at which the center of pressure is exerted is h, and that this is proportional to the length of a side of the heliostat, and thus proportional to A0.5. Using this wind speed variation with height the torque exerted by the wind pressure is approximately related to area to the 1.65 power, not the 1.5 power for uniform wind speed. The effect is that smaller heliostatsare exposed to lower wind speeds and thus wind torques, ignoring severe effects such as downbursts and dust devils. Using the 1.65 exponent for torque, or 0.65 for torque per unit area, and performing the same basic analysis as for the “threehalves law”, a similar family of curves results. The total heliostat cost per unit area expression from (5) is now modified to incorporate the higher non-uniform wind speed exponent: C T ðN Þ=AF ¼ C Cat1 þ kA0:65 H þ f =AH :
ð10Þ
Table 1 Category 3 cost summary values for non-uniform wind. Category 3 cost
f Value
k Value
Optimum area (m2)
Minimum cost/ area ($/m2)
3000 1500 300 75
5.39 5.78 6.08 6.14
59.9 37.7 13.8 5.9
167.16 140.96 95.22 72.18
(Percentage of DOE baseline) 0.1 0.05 0.01 0.0025
The values for f from the uniform wind speed examples remain the same, only the values for k change, due to the different exponent. The values for the four non-uniform wind speed cases are summarized in Table 1. The dependency of cost per unit area for non-uniform wind is shown in Fig. 2. From Eq. (11) the minimum area is 5.9 m2 and the minimum cost per unit area is 72.18/m2, for the $75 or 0.25% Category 3 cost assumption. This value is slightly below the 2020 DOE goal of $75/m2, and illustrates the importance of considering cost dependence on size and the leveraging effect of low fixed costs for all of the heliostat components, especially electronic costs in Category 3. Insight into the relatively low, but non-negligible fixed cost per unit for the load-bearing components, is also needed. With the total Category 3 cost of $300, then from Table 1 the total heliostat installed hardware cost is $95.22/ m2 13.8 m2 = $1314. Subtracting the mirror cost of $40/ m2 13.8m2 = $552 and the Category 3 cost of $300 gives $462 for Category 2, but without inclusion of the fixed cost for those load-bearing components, since these were incorporated into Category 3 for convenience. If the Category 2 fixed cost were to include, say, 5–10% of the load-bearing cost, or about $23 to $46, then the electronic costs in Category 3 would be the difference, or about $277 to $254 and the total load-bearing costs in Category 2 would be $485 to $508. Costs are now: Category 1, $552, Category 2, $485 to $508, and Category 3, $277 to $254. But to achieve this optimum area requires that all Category 3 costs be
Again taking the derivative with respect to AH, setting equal to zero, and solving for AH|opt: AH jopt ¼ ðf =0:65kÞ
1=1:65
ð11Þ
Substituting the optimum area into the total cost per unit area from (10) and simplifying, C T ðN Þ=AF jmin ¼ C Cat1 þ kðf =0:65kÞ þ f =ðf =0:65kÞ1=1:65 :
0:65=1:65
ð12Þ
Eq. (12) gives the minimum cost per unit area as a function of the parameters f and k for the non-uniform wind. Using the same value of 20% for Category 1, Eq. (12) becomes slightly modified from the uniform wind expression in Eq. (8), with the higher exponent: kð150Þ0:65 þ f =150 ¼ 160$=m2 :
ð13Þ
Fig. 2. Cost comparisons vs. area for non-uniform wind.
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decreased, and that includes f2, the fixed costs for the loadbearing components. If, for example, a custom drive design incurs high non-recurring development costs that are fully amortized over the first procurement, the total fixed cost in that case would increase and this would increase the theoretical optimum area, but that area would not correspond to the optimum area for later procurements. The strategy of how to select the optimum area and account for future sales is best determined by the heliostat developer. 3. Practical considerations Parametric analysis projections should be tempered by practical considerations. These considerations include broader aspects of the design, such as environmental aspects, trade-offs of life cycle cost vs. initial cost, availability of mass produced, low cost hardware at specific torque values and design safety factors, and use of mass-produced standard mirror sizes. Environmental aspects include ice loads, blowing sand and dust, hail impact, temperature extremes, and combined effects. But they can also include requirements imposed by regulatory agencies, such as minimal environmental impact, including limited disturbance of the desert soil, stresses on plant and animal life, etc. These aspects could impose limits on site preparation, and this could result in consideration of ground clearance. For example, if the desert ecological system were to be preserved, then it would be necessary to avoid the heliostats encroaching on and damaging desert plants as well as being damaged by plants in moving back and forth from stow to track or by having shading and blocking from the plants. Site disturbance may pose other problems. For example, blowing sand and dust at very low levels, less than about one meter, can cause severe surface damage to mirrors. This was noted by the author in 1973 when building windows were inspected at China Lake Naval Weapons Center in support of the NSF study (Blackmon, 1975; Vant-Hull and Easton, 1975) conducted by McDonnell Douglas as a subcontractor to the University of Houston. Windows close to the ground were severely abraded, but above about 50 or so, there was negligible degradation. Those buildings at that time were several decades old and were in areas that had experienced various degrees of soil disturbance, and thus may have had a higher degree of blowing sand and dust than undisturbed desert areas. Several specimens of glass panes were removed and specular transmissivity measurements were made; these showed very significant loss for the abraded samples near the ground, which in some cases were so severe that little specular transmission occurred. These observations helped establish a ground clearance of 0.5 m or so for the heliostats in the vertical position. That height also decreased interference with typical desert shrubs such as sagebrush. Later, as part of a McDonnell Douglas Independent Research and Development (IRAD) program, sand and dust collection cones spaced from ground level to 200 were installed at various potential sites (China Lake,
347
Sandia National Laboratories, Albuquerque, and a site in the Central Mojave Desert), and these showed several surprisingly large rocks/pebbles (of the order of 0.5–1.5 cm) collected in the lowest cones over a year or so. These cones also had substantial amounts of sand exceeding about 2000 cm3 up to about 0.3 m. Higher up, above approximately 1–1.25 m, no rocks/pebbles were found and the amount of sand and dust was substantially less. Access issues include washing and repair and/or replacement of heliostats. For very small pedestal-mounted heliostats optimally placed in the field, access could be limited, especially for wash trucks and installation and maintenance equipment, but for small heliostats, there is less need for large special handling equipment. The minimum separation distance between pedestal-mounted heliostats is estimated in the RCELL field layout optimization code (Lipps and Vant-Hull, 1978; Lipps, 1981; Pitman and Vant-Hull, 1988) as being twice the distance from the rotation center to the farthest corner (to allow for off-center mirrors), plus some practical distance for ensuring that the corners do not contact each other. Clearance below the heliostat in a horizontal stow position is also a consideration, but for small heliostats, this is too low for standard equipment access. Whatever the access distances are estimated to be, they should account for the possibility of installation errors, rough ground, movement of heliostats due to winds during installation, unforeseen interference, as with ice on the heliostat, personnel working space and access, or heliostat movement during a maintenance operation, as well as to ensure adequate access for maintenance equipment, such as cranes, attachment of spreader arms to lift the reflectors, etc. Another consideration is maneuverability of the installation and maintenance equipment and ability to access these in the event of mechanical problems, etc. If separation distance at the corners for pedestalmounted heliostats is 0.3 m, then the minimum pedestal separation distance between two centered square heliostats 1=2 is 0:3 þ 21=2 AH . To first order, with a vertical stow ground clearance of 0.5 m, the height of the reflector is 1=2 0:5 þ 0:5AH . Assuming heliostats can be oriented vertically, then access between the two rows is approximated by the minimum separation distance, less some additional distance to account for interference from drive unit components, such as protruding linear actuators. If that separation distance were to be increased for very small heliostats to accommodate equipment, then the ground coverage percentage is less and the access paths remove a fraction of the heliostats; this effect on the field layout produces some degree of overall system performance degradation. Determination of that effect would likely require additional field layout optimization to include access pathways/roads. A reasonable estimate of the required separation distance can be made by assuming that for conventional trucks, etc., a minimum “one lane” width is needed of approximately 3–4.5 m. Fig. 3 shows that for the diagonal separation clearance of 0.3 m, the separation distance between pedestals for
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Fig. 3. Minimum pedestal separation distance for access with vertically stowed heliostats and horizontal stow clearance.
vertically stowed heliostats is about 3.5 m for heliostat areas of about 5 m2 and about 4.8 m for 10 m2. Thus inner-field pedestal-mounted heliostats above about 5 m2 should have sufficient access width. Much less separation than this could cause access issues for standard equipment in the near-field for both width and height and could impose non-optimum spacing for wider access ways/roads. Farther away from the tower, the heliostat separation distance of course increases due to blocking and shadowing effects, but in the inner near-field, the heliostats are approximately at this minimum separation distance. Therefore, while “ultra small” heliostats may be predicted by this type of cost analysis, if very low Category 3 costs can be achieved, practical aspects could result in heliostats below some minimum practical area not producing the overall system cost optimum. Fig. 3 also shows the approximate height of the reflector; assuming the ground clearance is 0.5 m, total clearance is about 1.5 m for the 5 m2 heliostat and about 2 m for the 10 m2 heliostat. Normally, access for trucks and equipment can assume heliostats are near-vertical, such as for washing, and height restrictions would not be an issue. For other functions, such as inspection or corrective maintenance, the heliostats could be stowed horizontally or vertically, but the smaller heliostats could conveniently allow these operations without special lifts or scaffolding. Of course, multiple heliostats on a structure, common for very small heliostats, must be considered separately. For example, Rogers et al. (2013) discuss the eSolar dense layout of heliostats for modular fields, which are packed so close that “semi-automated cleaning systems can be designed to wash many heliostat reflectors in one pass without manual intervention.” They further state that “. . .high density heliostat fields provide less clearance. . .which needs to be taken into careful account. . .” Hardware availability and type can also play a role in determining the minimum cost. For example, there are limits on ready availability of standard mass produced motor and drive unit torque capabilities. Theoretical cost savings of a smaller, non-custom glass area would be at least to some extent offset by the additional glass cutting and breakage costs. Smaller heliostats, with shorter support spans, could allow thinner mirrors to be used and meet requirements such as hail impact; these could enable a
Fig. 4. Low sensitivity of cost per unit area vs. area near the optimum.
reduction in cost that is not considered in this analysis. Availability of certain sizes of support structure (channel width and thickness, etc.) can also limit the actual size, compared to the theoretically ideal size, since custom structures for non-standard specifications could be more costly compared to items already in mass production and would likely involve amortization costs for special tooling, production, etc. Fig. 4 shows cost per unit area in the vicinity of the optimum for the $75 Category 3 cost is relatively insensitive to the area; there is less than about 7% variation in cost for areas from about 3–12 m2 and the upper range of 12 m2 is reasonably close to the optimum area for the $300 Category 3 cost. This allows practical requirements to be considered in determining a final size that can be somewhat different from the analytical optimum, with only a minor impact on cost. For example, standard mirror sizes may be selected, rather than having a custom-cut size. 4. Conclusion These examples illustrate the parametric analysis technique for cost minimization but are not meant to define a particular heliostat size or cost per unit area. The examples shown are based on the $200/m2 reported hardware cost of the 150 m2 DOE baseline heliostat, but the approach is not limited to that configuration. The analysis does not address a wide range of cost details that would be required to fully assess heliostat total cost nor does the analysis address the specific component costs. These details can cause substantial differences in total heliostat costs by including or excluding costs that may be separate from typical installed hardware costs. For example, some aspects of site preparation are included in field wiring costs, which include trenching, but civil engineering site preparation for the heliostat field is typically not included in the hardware cost. Other costs that could be associated with the heliostat, such as management, debt service, net present value for certain O&M costs, etc., are also not included, but these too would have costs that could be distributed into the basic catego-
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ries. Other aspects, such as learning curve and optical performance vs. size are neglected here as well, but are covered in prior work noted earlier. With these uncertainties, the resulting costs per unit area vs. area relationships presented here provide only a partial assessment of the actual cost dependency on size. However, the cost relationships illustrate the important leveraging effect of reducing those costs that are independent of size. They also show that the optimum area has a more uniformly distributed cost for the three categories. The low total cost per unit area associated with the relatively small optimized area that results from low Category 3 costs also shows that mirror costs now become a more substantial fraction; efforts to reduce these costs therefore can provide additional, relatively important, cost savings. Finally, the analysis supports the strategy of developing smaller heliostats. A heliostat developer can apply this technique to the costs associated with a particular “point design” and area and from this quickly approximate the optimum size for that design. This technique also shows that if the heliostat developer were able to achieve an additional fixed cost reduction by further reducing, say, the electronics costs, then what appeared to be the optimum heliostat size for their initial design would decrease. It would be important to consider the leveraging effect of the cost reduction by allowing the initial design area to be further reduced. Achieving a direct fixed cost reduction without also considering the effect on area would not take full advantage of that cost reduction. Optimizing heliostat area is important, but it is also necessary to consider size effects on total system costs and field layout, including limitations due to practical requirements, in order to determine minimum system cost per unit area and minimum levelized cost of energy (LCOE). Acknowledgments This work was supported by the United States Department of Energy (DOE) SunShot Initiative, under award
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DE-FOA-0003593 to HiTek Services, Inc. and a subsequent subcontract to the University of Alabama in Huntsville, purchase order number 0212295. References Blackmon, James B., 1975. Design, fabrication, and test of a heliostat for a central receiver solar thermal power plant, NSF Grant GI-39456 Final Report, September 1975, conducted under University of Houston P.O. Number 12397 (1973–1975). Blackmon, J.B., 2012. Heliostat Size Analysis, in: Lovegrove, Keith and Stein, Wes, Concentrating solar power technology – Principles, developments and applications. Woodhead Publishing Series in Energy No. 21. DOE FOA, 2009. DOE Financial Assistance Funding Opportunity Announcement, Baseload Concentrating Solar Power Generation, DE-FOA-0000104, CFDA Number 81.087, issued 07/15/2009. Heller, Werner H., Joseph S., Peters, 1989. Development of a low-cost drive mechanism for solar heliostat. Peerless Winsmith Report prepared for Sandia National Laboratories, Albuquerque, New Mexico, Document Number 90-5753, February 1989. Kolb, G.J., et al., 2007. Heliostat cost reduction study, Sandia Report SAND2007-3293, June 2007. Kolb, G.J., et al., 2011. Power tower technology roadmap and cost reduction plan, Sandia Report, SAND2011-2419, Printed April 2011. Kusek, S., 2012. Phase 1 Final Report, Low Cost Heliostat Development. Award Number: EE0003593, October 1, 2010–March, 30 2012. Lipps, F.W., Vant-Hull, L.L., 1978. A cell-wise method for the optimization of large central receiver systems. Solar Energy 20 (6), 505–516. Lipps, F.W., 1981. Theory of cell-wise optimizations for solar central receiver systems. Technical Report SAN/1637-1, University of Houston, Houston, TX. Pitman, C.L., Vant-Hull L.L., 1988. The university of houston solar central receiver code system: concepts, updates, and start-up kits. Technical Report, SAND 88-7029 (SNLA). Dale, Rogers, Michael, Slack, Bill, Cassity, 2013. Addressing the Challenges Associated with eSolar’s Unique Approach to Central Receiver Power Plants, SolarPaces 2013. Vant-Hull, L., Easton C.R., 1975. Solar thermal power systems based on optical transmission, October 21, 1975, Final Report (June 15, 1973– September 30, 1975), NSF Grant G(-39456, ERDA Grant AER 7307950.
ScienceDirect Solar Energy 97 (2013) 342–349 www.elsevier.com/locate/solener
Parametric determination of heliostat minimum cost per unit area James B. Blackmon ⇑ Department of Mechanical and Aerospace Engineering, Propulsion Research Center, S233 Technology Hall, University of Alabama in Huntsville, Huntsville, AL 35899, United States Received 12 March 2013; received in revised form 14 August 2013; accepted 21 August 2013 Available online 18 September 2013 Communicated by: Associate Editor Lorin Vant-Hull
Abstract A parametric analysis illustrates how heliostat costs per unit area can be minimized by distributing the costs into categories having different cost dependence on area. Heliostat costs are distributed among three basic categories:
Category 1, costs that on a per unit area basis are constant, such as mirrors. Category 2, costs that are dependent on the imposed loads, and thus area, such as the structure, pedestal, and drive units. Category 3, fixed costs per heliostat, irrespective of heliostat area, such as controllers, and position sensors. Using the 150 m2 United States Department of Energy (DOE) baseline heliostat design and its cost of approximately $200/m2, cost reductions of the order of 50% are determined for a range of much smaller areas, primarily as a function of decreased Category 3 costs. For achievable Category 3 costs, the minimum cost per unit area is projected to meet the DOE 2020 goal of $75/m2. Ó 2013 Elsevier Ltd. All rights reserved. Keywords: Central receiver systems; Heliostat cost minimization; Optimum heliostat area; Parametric cost analysis
1. Introduction Minimum heliostat costs per unit area can be determined for any particular design by analyzing parametric cost estimating relationships in terms of the heliostat area for costs distributed among three basic categories. The projected $200/m2 installed hardware cost for the 150 m2 DOE baseline elevation-azimuth heliostat is used as an example, with a range of reasonable fractions of these costs distributed into the three categories; this parametric method does not require a detailed treatment of the specific component costs for the DOE baseline or any other design. In the parametric analysis, whatever the design and configuration, it is assumed that materials, performance requirements, and configuration are the same for the range of ⇑ Tel.: +1 (256) 824 5106.
E-mail address: [email protected] 0038-092X/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.solener.2013.08.032
areas analyzed. Category 1 costs for the heliostat hardware are primarily for mirrors, which are essentially constant on a per unit area basis irrespective of heliostat area for a given type, thickness, etc. and total field reflector area. Category 1 can include other costs that are essentially constant on a per unit area basis, such as site preparation. Category 2 costs are associated with the torques imposed on the load-bearing structural/mechanical assemblies, primarily from winds; these costs are also associated with constant performance requirements, such as strength margins of safety, stiffness, rotational rate, and tracking accuracy. Category 2 costs are dependent on imposed moments, in accordance with the so-called “area to the three-halves law” for uniform wind speed, or its variant for non-uniform wind speed; these costs increase on a per unit area basis as the size increases. Category 3 costs are associated primarily with controllers, position sensors, processors, etc., which are the same irrespective of heliostat size.
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Reducing Category 3 costs results in a direct savings and also enables a smaller heliostat with additional reductions in the cost per unit area due to the Category 2 cost dependence on imposed loads. The differing cost dependencies on heliostat area of Category 2 and 3 determine a minimum in total cost per unit area as a function of area for given design and performance requirements. For areas less than that optimum, the Category 3 costs rise inversely with area and quickly become inordinately costly as the number of heliostats increases for a particular total field area. Conversely, as the heliostat size increases above the optimum, the higher loads on a per unit area increase the weight and cost of these Category 2 load-dependent components. Results based on the DOE baseline cost of $200/m2 at 150 m2 show heliostats significantly smaller offer approximately a 50% or greater reduction in cost per unit area, for Category 3 costs that should be achievable. 2. Analysis The parametric approach was in part developed under a Phase 1 DOE-funded heliostat development program (Kusek, 2012) as well as prior work (Blackmon, 2012). Those analyses used specific component costs for the el/ az, glass–metal, pedestal mounted DOE baseline heliostat (Kolb et al., 2007). These are the only detailed, openly available cost data available. Other size-dependent aspects were considered, such as the effect of optical performance improvements of smaller heliostats on the total reflector field area required, learning curve benefits, and the use of net present value (NPV) of operations and maintenance operations. These analyses also considered a fourth category for the field wiring cost estimating relationships that did not fit well into the three basic categories. The following parametric method determines a range of optimum heliostat areas and associated minimum costs per unit area, but does not require a treatment of the individual component costs as conducted in the previous analyses. This approach is also not dependent on a particular design or configuration. The structural–mechanical design, materials, configuration, control system design, and performance requirements are assumed to remain unchanged over the range of sizes, whatever the design. The analysis shows the sensitivity and leveraging effect of Category 3 cost reductions on heliostat area and cost per unit area. One observation is that if the developer were to further reduce their Category 3 costs, then it would not be sufficient to simply apply that cost reduction directly. To fully achieve the minimum cost per unit area, it would also be important to re-determine the optimum area of their initial “point design”. Costs of manufactured goods comprise fixed costs and costs per unit of good produced. Fixed costs are typically associated with such factors as management, overhead, plant maintenance, debt service, retirement obligations, leases, and utilities, which normally are approximately
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the same irrespective of the annual number of units produced and total sales; these costs are typically embedded in the total price. Per-unit costs include labor, materials, and other costs directly associated with production and are determined by such design aspects as load, and torque. At high production rates these per-unit costs can often be approximated as a constant for a particular item, but with other factors, such as percentage of plant capacity, used to determine the breakeven point and sales price. There are also potential learning curve effects as more units are produced and the manufacturing process, supply chain costs, worker productivity, etc., improve. These aspects are treated in many different business administration texts. Here, a simplified analysis is used to illustrate the general parametric principles that result in an optimum area at the minimum cost per unit area. For this analysis, a representative heliostat field cost is based on a fixed cost and on a per heliostat cost, with the total heliostat hardware cost determined by individual heliostat area and number of heliostats. Learning curve and optical performance benefits are neglected. Also, the Net Present Value of Operations and Maintenance costs are not included in the cost distribution. Total cost of a particular heliostat field, CT(N), is a function of a fixed cost, F, the number of heliostats, N, and the costs per heliostat, B. The total heliostat field cost for a given field reflector area is approximated as the sum of the fixed cost and the per-heliostat hardware cost and number of heliostats: C T ðN Þ ¼ F þ BN :
ð1Þ
The value F represents the cumulative costs for the given total reflector area of the field, separate from the hardware costs per heliostat. The costs in F can include factors such as civil engineering site preparation and other costs that can be associated with the heliostat field, but separate from costs of the other central receiver subsystems; examples include heliostat field management cost, net present value (NPV) of certain Operations and Maintenance costs, permitting costs, some aspects of debt service, etc., which are assumed to increase roughly in proportion to the overall size of the field but are essentially independent of the heliostat size. The hardware cost per heliostat is the cumulative costs of its various Category 1, 2 and 3 components. A major example of Category 1 hardware costs for the heliostat is the mirror costs per unit area, which are essentially constant irrespective of heliostat size. Category 2 costs are for load-bearing components and Category 3 is primarily for electronics. Load-bearing components can be treated using either a uniform or non-uniform wind speed. First the uniform wind speed case is considered. 2.1. Uniform wind speed The uniform wind speed “three-halves law” is used for imposed torque, which relates the equivalent point-load force based on the distribution of wind pressure, P, times
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heliostat area, AH, and the moment arm associated with the force components that produce the moment. The force is proportional to AH. The moment arm is proportional to 1=2 AH , and thus the moment applied is proportional to 3=2 AH . It is further assumed that this torque relationship with area adequately describes torques used to determine the design requirements for the drive unit, structural support, pedestal, etc. and that, in particular, the costs are linearly proportional to torque, and thus these costs also vary as 3=2 AH . This assumption is shown to hold for structures (e.g., Kolb et al., 2007), but it is also supported by many different types of motors, gear drives, etc. (e.g., Blackmon, 2012; Kusek, 2012). To enable a consistent comparison for the basic DOE baseline design and configuration, the aerocoefficients and operational requirements (tracking accuracy, slew rates, etc.) are assumed to be constant over the range of areas for both the uniform and non-uniform wind conditions. With these assumptions, it is not necessary to develop the detailed loads and design conditions for each size. Instead, the loads are based on those for the ATS heliostat (Heller and Peters, 1989) used for the DOE baseline; the parametric dependence is sufficient to determine the cost/area as a function of area. Reflector support structure costs have been shown to 3=2 vary as heliostat area to the three-halves, AH (Kolb et al., 2007) for uniform wind speed. In addition, all of the load-bearing components, such as the drive unit, pedestal, and foundation costs exhibit costs that vary linearly with torque or imposed moment, and thus these costs are 3=2 proportional to AH for a uniform wind (Kusek, 2012; Blackmon, 2012). The linear relationship of price vs. torque is supported by hundreds of examples for many different types of motors and drive units (Kusek, 2012; Blackmon, 2012). Those curve fits showed that there is a constant cost as well as linear dependence on torque; for limited orders, such as catalogue prices, those constant costs are significant, due in part to costs for handling, inventory, overhead and administration, pricing policy, etc. But for large orders these costs would be distributed over a far larger number, which would greatly reduce the fixed cost as a percentage of the total hardware cost. The unit hardware cost B for a heliostat of area AH is the sum of the costs for each category. The Category 1 cost for the heliostat is C1, and this cost is based on a constant cost per unit area, C1A, times the heliostat area, AH. Category 2 costs, C2, for a heliostat are represented by f2 þ kAH3=2 , where f2 is the aggregated fixed cost associated with the load-bearing components and k is the appropriate aggregated coefficient. Category 3 costs per heliostats are f3, and are independent of size. The hardware cost, B, per heliostat is thus 3=2
B ¼ C 1A AH þ f2 þ kAH þ f3
ð2Þ
The number of heliostats, N, is approximately the total area of reflectors in the field divided by the heliostat area, AF/AH; optical efficiency improvements with small
heliostats are low and are not included in this analysis. Inserting these expressions into Eq. (1), the total heliostat field costs are: 3=2
C T ðN Þ ¼ F þ ðC 1A AH þ f2 þ kAH þ f3 ÞAF =AH :
ð3Þ
Simplifying, total heliostat field cost per unit reflector area becomes: 1=2
C T ðN Þ=AF ¼ F =AF þ C 1A þ f2 =AH þ kAH þ f3 =AH :
ð4Þ
Heliostat cost per unit area is typically considered as an installed hardware cost, although if field wiring costs are included, then there would be some intrinsic site preparation aspects, such as trenching. A comprehensive heliostat cost would include all costs for the field, as well as for the heliostat hardware and installation costs. For example, net present value of operations and maintenance have been included in the total hardware cost, with distributions of these costs into the three categories (Kusek, 2012; Blackmon, 2012); these costs have aspects that would increase or decrease as a function of heliostat size, as well as having some fraction as a constant cost per unit area. But since only installed hardware costs are readily available at this time and the other factors are typically reported separately, it is more practical to incorporate terms to have a representative cost made up of generalized Category 1, 2, and 3 costs, however they are defined in a detailed cost analysis. Therefore, the first two terms in (4) are combined into a composite term representing whatever identifiable costs are available from a particular cost analysis that are constant on a per unit area basis, represented as CCat1. With f = f2 + f3, a simplified Category 3 expression results, f/AH. The cost per unit area is then represented for uniform wind as: 1=2
C T ðN Þ=AF ¼ C Cat1 þ kAH þ f =AH :
ð5Þ
The optimum area, AH|opt, for the minimum cost per unit area is found by setting the derivative of Eq. (5) equal to zero and solving, giving: AH jopt ¼ ð2f =kÞ
2=3
ð6Þ
:
Eq. (6) shows that as f approaches zero, for very low Category 3 “constant” or “fixed” costs per heliostat, then the minimum cost heliostat size approaches zero. This illustrates the importance of these constant costs whatever the costs are for mirrors (and any other costs that are a constant per unit area) and the costs associated with the drive units, support structures, pedestals, etc. For f increasing, corresponding primarily to higher Category 3 costs, then the optimum area increases. Substituting the optimum area from (6) into the total cost per unit area from (5) gives the minimum cost per unit area, C T ðN Þ=AF jmin ¼ C Cat1 þ kð2f =kÞ
1=3
þ f 1=3 =ð2=kÞ
2=3
:
ð7Þ
The cost per unit area of the field can now be considered in terms of its dependence on the Category 2 and 3 relative costs and on heliostat area. If the costs for a particular
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heliostat design and size are known and have been distributed into the three categories, then the cost per unit area for the field can be illustrated parametrically as a function of area. Thus, a heliostat developer could initially select a size for a detailed “point-design” that determined the total mass-produced costs for that heliostat on a per unit area basis, but then use this parametric analysis to estimate the optimum size for the minimum cost per unit area. Treating the total cost on a per unit area basis, and using a range of reasonable percentages of these costs that are in the three categories, also provides insight into the effectiveness of reducing costs in each of the three categories. As a specific example, installed hardware costs can be approximated in current dollars as CT(N)/AF = $200/m2 for the DOE baseline (Kolb et al., 2011), which is close to the $211/m2 noted by DOE (DOE FOA, 2009). The analysis does not depend on the design or configuration, and is thus not limited to the DOE baseline or its specific component costs. With 20% for Category 1 representing costs of mirrors and other properly apportioned constant costs on a per unit area basis for the 150 m2 heliostat, the effect of Category 3 costs on the optimum area and on the cost per unit area can be shown. (Note, Kolb et al., 2007 show mirror costs of the order of $23/m2, which is of the order of 18% of the total cost of their heliostat. The point here is to have a representative percentage for additional constant cost/area aspects for illustration purposes.) The heliostat area is 150 m2; the remaining 80% is distributed into Categories 2 and 3. Eq. (5) becomes: kð150Þ1=2 þ f =150 ¼ 160$=m2 :
ð8Þ
Four examples are shown for a range of Category 3 costs. Details are developed in Excel. First, if the Category 3 cost per unit area of the 150 m2 heliostat is 10% of the total cost per unit area, then f =150 ¼ 0:1 200 ¼ $20=m2 and f ¼ $3000: That is, for the 150 m2 heliostat the cost primarily of the electronics (processors, encoders, limit switches, etc.), as well as any other relatively low fixed costs per heliostat for other components (junction boxes, connectors, etc.), remain constant at $3000 as the area changes. That cost is demonstrably extremely high. For example, from Kolb et al. (2007) the drive motors and limit switches were $1.78/m2 and the controls costs were $1.90/m2 for a total of $3.68/m2. Their heliostat field wiring costs were $7.40/ m2. The first two give a cost for the 148 m2 ATS heliostat of $545. However, drive motor costs are determined by motor power and torque, and thus some fraction of these costs should be in Category 2. If field wiring is also included, then the cost rises to $11.08/m2 or $1640. Distributing all field wiring costs only into Category 3 would be an over-simplification, since field wiring also has costs that should be distributed among the three categories and depends on the heliostat size. For example, with smaller heliostats, there are more rows and total length for wiring
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and trenching, and more connections. Also, heliostat area, and thus imposed wind moments, determines motor torques and horsepower, and this affects the wiring gauge and costs. Alternatives to field wiring such as heliostats powered by photovoltaic arrays with wireless communication, offer substantial potential cost savings (Kusek, 2012). Therefore, the assumption of $3000 in Category 3 for the DOE baseline is unrealistically high, even if it were assumed to include a substantial fraction of field wiring and fixed costs from Category 2, but it at least establishes an upper boundary. The remaining cost for this example is for the structure, pedestal, foundation, and drive units, or 70% of $200/m2, giving $140/m2. Then, k(150)1/2 = 140, and k = 11.43. Heliostat field cost per unit area for this case is then: 1=2
C T ðN Þ=AF ¼ C Cat1 þ kAH þ f =AH 1=2
¼ 40 þ 11:44AH þ 3000=AH :
ð9Þ
For the second case, assume 5% of the total cost per unit area or $1500 is in Category 3. Then, Category 2 is 75% of the total cost, or $150/m2, f = 1500 and k = 150/ 1500.5 = 12.25. For the third case, assume 1% of the total cost or $2/m2 or $300 is for Category 3. Then, the load dependent part of Category 2 is 79%, or $158/m2 with f = 300 and k = 158/ 1500.5 = 12.90. Finally, assume that the Category 3 costs are reduced to 0.25% of the total cost per unit area for that heliostat, or $0.5/m2, or $75 per heliostat. Thus, Category 2 is now 79.75%, or $159.50/m2 with f = 75 and k = 159.5/ 1500.5 = 13.02. The family of curves in Fig. 1 shows that relatively small changes in the relatively small percentage of fixed costs per heliostat for the 150 m2 size have a significant effect on the optimum heliostat area and the minimum cost per unit area. Finding ways to reduce Category 3 electronic/control costs results in a direct savings and in further cost reductions from the lower size of the optimum area, and thus the effect of imposed loads.
Fig. 1. Cost comparisons vs. area for uniform wind.
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Comparing the two lowest cost cases provides insight into the leveraging effect. The 1% case corresponds to a cost of $300 per heliostat. The 0.25% case corresponds to $75 per heliostat, for a direct savings of $225 per heliostat. However, the lower cost produces a minimum cost per unit area that is $85/m2 at about 5 m2, compared to $110/m2 at 13.5 m2 for the 1% case. Thus, the heliostat cost per unit area difference is $25/m2. If the direct savings had been used, but the area had not been reduced from 13.5 m2 to 5 m2, then the total field cost for about 1,000,000 m2 would be 106(110–225/13.5) = $93.3M, vs. the cost without the direct savings of $110M, or a savings of $16.7M. If in addition the heliostat size were reduced, then the total field cost would be $85M, for a total cost reduction of $25M, with the additional $8.3M in savings from further reducing the heliostat area from 13.5 m2 to the new optimum at about 5 m2. 2.2. Non-uniform wind speed case For non-uniform wind the speed, V, at height h, relative to the meteorological wind speed, V0, at height h0, is V/ V0 = (h/h0)0.15 where the meteorological winds are typically measured at 10 m (Heller and Peters, 1989). Assume the distance from the pivot point at which the center of pressure is exerted is h, and that this is proportional to the length of a side of the heliostat, and thus proportional to A0.5. Using this wind speed variation with height the torque exerted by the wind pressure is approximately related to area to the 1.65 power, not the 1.5 power for uniform wind speed. The effect is that smaller heliostatsare exposed to lower wind speeds and thus wind torques, ignoring severe effects such as downbursts and dust devils. Using the 1.65 exponent for torque, or 0.65 for torque per unit area, and performing the same basic analysis as for the “threehalves law”, a similar family of curves results. The total heliostat cost per unit area expression from (5) is now modified to incorporate the higher non-uniform wind speed exponent: C T ðN Þ=AF ¼ C Cat1 þ kA0:65 H þ f =AH :
ð10Þ
Table 1 Category 3 cost summary values for non-uniform wind. Category 3 cost
f Value
k Value
Optimum area (m2)
Minimum cost/ area ($/m2)
3000 1500 300 75
5.39 5.78 6.08 6.14
59.9 37.7 13.8 5.9
167.16 140.96 95.22 72.18
(Percentage of DOE baseline) 0.1 0.05 0.01 0.0025
The values for f from the uniform wind speed examples remain the same, only the values for k change, due to the different exponent. The values for the four non-uniform wind speed cases are summarized in Table 1. The dependency of cost per unit area for non-uniform wind is shown in Fig. 2. From Eq. (11) the minimum area is 5.9 m2 and the minimum cost per unit area is 72.18/m2, for the $75 or 0.25% Category 3 cost assumption. This value is slightly below the 2020 DOE goal of $75/m2, and illustrates the importance of considering cost dependence on size and the leveraging effect of low fixed costs for all of the heliostat components, especially electronic costs in Category 3. Insight into the relatively low, but non-negligible fixed cost per unit for the load-bearing components, is also needed. With the total Category 3 cost of $300, then from Table 1 the total heliostat installed hardware cost is $95.22/ m2 13.8 m2 = $1314. Subtracting the mirror cost of $40/ m2 13.8m2 = $552 and the Category 3 cost of $300 gives $462 for Category 2, but without inclusion of the fixed cost for those load-bearing components, since these were incorporated into Category 3 for convenience. If the Category 2 fixed cost were to include, say, 5–10% of the load-bearing cost, or about $23 to $46, then the electronic costs in Category 3 would be the difference, or about $277 to $254 and the total load-bearing costs in Category 2 would be $485 to $508. Costs are now: Category 1, $552, Category 2, $485 to $508, and Category 3, $277 to $254. But to achieve this optimum area requires that all Category 3 costs be
Again taking the derivative with respect to AH, setting equal to zero, and solving for AH|opt: AH jopt ¼ ðf =0:65kÞ
1=1:65
ð11Þ
Substituting the optimum area into the total cost per unit area from (10) and simplifying, C T ðN Þ=AF jmin ¼ C Cat1 þ kðf =0:65kÞ þ f =ðf =0:65kÞ1=1:65 :
0:65=1:65
ð12Þ
Eq. (12) gives the minimum cost per unit area as a function of the parameters f and k for the non-uniform wind. Using the same value of 20% for Category 1, Eq. (12) becomes slightly modified from the uniform wind expression in Eq. (8), with the higher exponent: kð150Þ0:65 þ f =150 ¼ 160$=m2 :
ð13Þ
Fig. 2. Cost comparisons vs. area for non-uniform wind.
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decreased, and that includes f2, the fixed costs for the loadbearing components. If, for example, a custom drive design incurs high non-recurring development costs that are fully amortized over the first procurement, the total fixed cost in that case would increase and this would increase the theoretical optimum area, but that area would not correspond to the optimum area for later procurements. The strategy of how to select the optimum area and account for future sales is best determined by the heliostat developer. 3. Practical considerations Parametric analysis projections should be tempered by practical considerations. These considerations include broader aspects of the design, such as environmental aspects, trade-offs of life cycle cost vs. initial cost, availability of mass produced, low cost hardware at specific torque values and design safety factors, and use of mass-produced standard mirror sizes. Environmental aspects include ice loads, blowing sand and dust, hail impact, temperature extremes, and combined effects. But they can also include requirements imposed by regulatory agencies, such as minimal environmental impact, including limited disturbance of the desert soil, stresses on plant and animal life, etc. These aspects could impose limits on site preparation, and this could result in consideration of ground clearance. For example, if the desert ecological system were to be preserved, then it would be necessary to avoid the heliostats encroaching on and damaging desert plants as well as being damaged by plants in moving back and forth from stow to track or by having shading and blocking from the plants. Site disturbance may pose other problems. For example, blowing sand and dust at very low levels, less than about one meter, can cause severe surface damage to mirrors. This was noted by the author in 1973 when building windows were inspected at China Lake Naval Weapons Center in support of the NSF study (Blackmon, 1975; Vant-Hull and Easton, 1975) conducted by McDonnell Douglas as a subcontractor to the University of Houston. Windows close to the ground were severely abraded, but above about 50 or so, there was negligible degradation. Those buildings at that time were several decades old and were in areas that had experienced various degrees of soil disturbance, and thus may have had a higher degree of blowing sand and dust than undisturbed desert areas. Several specimens of glass panes were removed and specular transmissivity measurements were made; these showed very significant loss for the abraded samples near the ground, which in some cases were so severe that little specular transmission occurred. These observations helped establish a ground clearance of 0.5 m or so for the heliostats in the vertical position. That height also decreased interference with typical desert shrubs such as sagebrush. Later, as part of a McDonnell Douglas Independent Research and Development (IRAD) program, sand and dust collection cones spaced from ground level to 200 were installed at various potential sites (China Lake,
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Sandia National Laboratories, Albuquerque, and a site in the Central Mojave Desert), and these showed several surprisingly large rocks/pebbles (of the order of 0.5–1.5 cm) collected in the lowest cones over a year or so. These cones also had substantial amounts of sand exceeding about 2000 cm3 up to about 0.3 m. Higher up, above approximately 1–1.25 m, no rocks/pebbles were found and the amount of sand and dust was substantially less. Access issues include washing and repair and/or replacement of heliostats. For very small pedestal-mounted heliostats optimally placed in the field, access could be limited, especially for wash trucks and installation and maintenance equipment, but for small heliostats, there is less need for large special handling equipment. The minimum separation distance between pedestal-mounted heliostats is estimated in the RCELL field layout optimization code (Lipps and Vant-Hull, 1978; Lipps, 1981; Pitman and Vant-Hull, 1988) as being twice the distance from the rotation center to the farthest corner (to allow for off-center mirrors), plus some practical distance for ensuring that the corners do not contact each other. Clearance below the heliostat in a horizontal stow position is also a consideration, but for small heliostats, this is too low for standard equipment access. Whatever the access distances are estimated to be, they should account for the possibility of installation errors, rough ground, movement of heliostats due to winds during installation, unforeseen interference, as with ice on the heliostat, personnel working space and access, or heliostat movement during a maintenance operation, as well as to ensure adequate access for maintenance equipment, such as cranes, attachment of spreader arms to lift the reflectors, etc. Another consideration is maneuverability of the installation and maintenance equipment and ability to access these in the event of mechanical problems, etc. If separation distance at the corners for pedestalmounted heliostats is 0.3 m, then the minimum pedestal separation distance between two centered square heliostats 1=2 is 0:3 þ 21=2 AH . To first order, with a vertical stow ground clearance of 0.5 m, the height of the reflector is 1=2 0:5 þ 0:5AH . Assuming heliostats can be oriented vertically, then access between the two rows is approximated by the minimum separation distance, less some additional distance to account for interference from drive unit components, such as protruding linear actuators. If that separation distance were to be increased for very small heliostats to accommodate equipment, then the ground coverage percentage is less and the access paths remove a fraction of the heliostats; this effect on the field layout produces some degree of overall system performance degradation. Determination of that effect would likely require additional field layout optimization to include access pathways/roads. A reasonable estimate of the required separation distance can be made by assuming that for conventional trucks, etc., a minimum “one lane” width is needed of approximately 3–4.5 m. Fig. 3 shows that for the diagonal separation clearance of 0.3 m, the separation distance between pedestals for
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Fig. 3. Minimum pedestal separation distance for access with vertically stowed heliostats and horizontal stow clearance.
vertically stowed heliostats is about 3.5 m for heliostat areas of about 5 m2 and about 4.8 m for 10 m2. Thus inner-field pedestal-mounted heliostats above about 5 m2 should have sufficient access width. Much less separation than this could cause access issues for standard equipment in the near-field for both width and height and could impose non-optimum spacing for wider access ways/roads. Farther away from the tower, the heliostat separation distance of course increases due to blocking and shadowing effects, but in the inner near-field, the heliostats are approximately at this minimum separation distance. Therefore, while “ultra small” heliostats may be predicted by this type of cost analysis, if very low Category 3 costs can be achieved, practical aspects could result in heliostats below some minimum practical area not producing the overall system cost optimum. Fig. 3 also shows the approximate height of the reflector; assuming the ground clearance is 0.5 m, total clearance is about 1.5 m for the 5 m2 heliostat and about 2 m for the 10 m2 heliostat. Normally, access for trucks and equipment can assume heliostats are near-vertical, such as for washing, and height restrictions would not be an issue. For other functions, such as inspection or corrective maintenance, the heliostats could be stowed horizontally or vertically, but the smaller heliostats could conveniently allow these operations without special lifts or scaffolding. Of course, multiple heliostats on a structure, common for very small heliostats, must be considered separately. For example, Rogers et al. (2013) discuss the eSolar dense layout of heliostats for modular fields, which are packed so close that “semi-automated cleaning systems can be designed to wash many heliostat reflectors in one pass without manual intervention.” They further state that “. . .high density heliostat fields provide less clearance. . .which needs to be taken into careful account. . .” Hardware availability and type can also play a role in determining the minimum cost. For example, there are limits on ready availability of standard mass produced motor and drive unit torque capabilities. Theoretical cost savings of a smaller, non-custom glass area would be at least to some extent offset by the additional glass cutting and breakage costs. Smaller heliostats, with shorter support spans, could allow thinner mirrors to be used and meet requirements such as hail impact; these could enable a
Fig. 4. Low sensitivity of cost per unit area vs. area near the optimum.
reduction in cost that is not considered in this analysis. Availability of certain sizes of support structure (channel width and thickness, etc.) can also limit the actual size, compared to the theoretically ideal size, since custom structures for non-standard specifications could be more costly compared to items already in mass production and would likely involve amortization costs for special tooling, production, etc. Fig. 4 shows cost per unit area in the vicinity of the optimum for the $75 Category 3 cost is relatively insensitive to the area; there is less than about 7% variation in cost for areas from about 3–12 m2 and the upper range of 12 m2 is reasonably close to the optimum area for the $300 Category 3 cost. This allows practical requirements to be considered in determining a final size that can be somewhat different from the analytical optimum, with only a minor impact on cost. For example, standard mirror sizes may be selected, rather than having a custom-cut size. 4. Conclusion These examples illustrate the parametric analysis technique for cost minimization but are not meant to define a particular heliostat size or cost per unit area. The examples shown are based on the $200/m2 reported hardware cost of the 150 m2 DOE baseline heliostat, but the approach is not limited to that configuration. The analysis does not address a wide range of cost details that would be required to fully assess heliostat total cost nor does the analysis address the specific component costs. These details can cause substantial differences in total heliostat costs by including or excluding costs that may be separate from typical installed hardware costs. For example, some aspects of site preparation are included in field wiring costs, which include trenching, but civil engineering site preparation for the heliostat field is typically not included in the hardware cost. Other costs that could be associated with the heliostat, such as management, debt service, net present value for certain O&M costs, etc., are also not included, but these too would have costs that could be distributed into the basic catego-
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ries. Other aspects, such as learning curve and optical performance vs. size are neglected here as well, but are covered in prior work noted earlier. With these uncertainties, the resulting costs per unit area vs. area relationships presented here provide only a partial assessment of the actual cost dependency on size. However, the cost relationships illustrate the important leveraging effect of reducing those costs that are independent of size. They also show that the optimum area has a more uniformly distributed cost for the three categories. The low total cost per unit area associated with the relatively small optimized area that results from low Category 3 costs also shows that mirror costs now become a more substantial fraction; efforts to reduce these costs therefore can provide additional, relatively important, cost savings. Finally, the analysis supports the strategy of developing smaller heliostats. A heliostat developer can apply this technique to the costs associated with a particular “point design” and area and from this quickly approximate the optimum size for that design. This technique also shows that if the heliostat developer were able to achieve an additional fixed cost reduction by further reducing, say, the electronics costs, then what appeared to be the optimum heliostat size for their initial design would decrease. It would be important to consider the leveraging effect of the cost reduction by allowing the initial design area to be further reduced. Achieving a direct fixed cost reduction without also considering the effect on area would not take full advantage of that cost reduction. Optimizing heliostat area is important, but it is also necessary to consider size effects on total system costs and field layout, including limitations due to practical requirements, in order to determine minimum system cost per unit area and minimum levelized cost of energy (LCOE). Acknowledgments This work was supported by the United States Department of Energy (DOE) SunShot Initiative, under award
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DE-FOA-0003593 to HiTek Services, Inc. and a subsequent subcontract to the University of Alabama in Huntsville, purchase order number 0212295. References Blackmon, James B., 1975. Design, fabrication, and test of a heliostat for a central receiver solar thermal power plant, NSF Grant GI-39456 Final Report, September 1975, conducted under University of Houston P.O. Number 12397 (1973–1975). Blackmon, J.B., 2012. Heliostat Size Analysis, in: Lovegrove, Keith and Stein, Wes, Concentrating solar power technology – Principles, developments and applications. Woodhead Publishing Series in Energy No. 21. DOE FOA, 2009. DOE Financial Assistance Funding Opportunity Announcement, Baseload Concentrating Solar Power Generation, DE-FOA-0000104, CFDA Number 81.087, issued 07/15/2009. Heller, Werner H., Joseph S., Peters, 1989. Development of a low-cost drive mechanism for solar heliostat. Peerless Winsmith Report prepared for Sandia National Laboratories, Albuquerque, New Mexico, Document Number 90-5753, February 1989. Kolb, G.J., et al., 2007. Heliostat cost reduction study, Sandia Report SAND2007-3293, June 2007. Kolb, G.J., et al., 2011. Power tower technology roadmap and cost reduction plan, Sandia Report, SAND2011-2419, Printed April 2011. Kusek, S., 2012. Phase 1 Final Report, Low Cost Heliostat Development. Award Number: EE0003593, October 1, 2010–March, 30 2012. Lipps, F.W., Vant-Hull, L.L., 1978. A cell-wise method for the optimization of large central receiver systems. Solar Energy 20 (6), 505–516. Lipps, F.W., 1981. Theory of cell-wise optimizations for solar central receiver systems. Technical Report SAN/1637-1, University of Houston, Houston, TX. Pitman, C.L., Vant-Hull L.L., 1988. The university of houston solar central receiver code system: concepts, updates, and start-up kits. Technical Report, SAND 88-7029 (SNLA). Dale, Rogers, Michael, Slack, Bill, Cassity, 2013. Addressing the Challenges Associated with eSolar’s Unique Approach to Central Receiver Power Plants, SolarPaces 2013. Vant-Hull, L., Easton C.R., 1975. Solar thermal power systems based on optical transmission, October 21, 1975, Final Report (June 15, 1973– September 30, 1975), NSF Grant G(-39456, ERDA Grant AER 7307950.