Design of modular concrete heliostats using symmetry reduction methods

Funding information German Research Foundation, Grant/Award Number: 389020360 Abstract In Concentrated Solar Power (CSP) plants the incident solar radiation is focused onto a receiver by means of collectors. A fluid is heated up and in a downstream power block electricity is generated. In point-focusing solar towers, the solar concentration is achieved by so-called heliostats that are arranged to a solar field. In this contribution, the development of concrete heliostats with circular shapes and an aperture area of 30 m is presented. A high-performance concrete with high tensile and compressive strength values is used. The circular structure is dissolved into identical but symmetrically reduced modules derived from system reduction methods. For designing, the tensile strength of the concrete is restrictive to ensure linear-elastic material behavior and to avoid softening by cracking. After dimensioning, the derived equivalent plate is converted into strut-like structures possessing equal stiffnesses with respect to the partial module size. These modules are circularly post-tensioned to form a heliostat. Numerical investigations of the modules prove their accuracy. A full solar concentration, that is, the reflected solar radiation is completely focused on the receiver, is achieved. Due to the multitude of modules within a solar field, serial production with integrated quality control is recommended.


| Heliostats for concentrated solar power plants
Concentrated solar power (CSP) is a relatively young technology with an installed power of 4.5 GW worldwide. 1 In CSP plants, the incident solar radiation is focused onto a receiver. At the receiver, the solar radiation is bundled to heat a fluid. This fluid is used to produce steam in a downstream power block, where electricity is generated through a turbine. In combination with thermal energy storage, it is an alternative to conventional power plants. Next to parabolic troughs that belong to line-like focusing systems, one of the most promising technologies for CSP plants are point-focusing solar power towers. The solar concentration hereby is achieved by a multitude of mirroring collectors, so-called heliostats. These heliostats track the sun biaxially and are arranged around a central tower where the receiver is located (Figure 1, left). Process temperatures of around 700 C are achieved. For the point-wise concentration of the solar radiation, the mirror surfaces have the shape of flat paraboloids. Hence, the efficiency is mainly dependent on the accuracy of the mirror shape.
Typically, heliostats are built up as so-called T-type structures consisting of a steel structure and glass mirror facets. The mirror elements are supported by cross beams that are mounted to a torque tube. The torque tube and the pylon form the "T" (cf. Figure 1, left).
Doing so, deformations of the cross beams and the torque tube superimpose. However, a multitude of heliostat concepts exists, for example, steel frameworks or sandwich facets. 2 They mainly aim for cost reductions since the heliostats represent about 40% of the total CSP plant investment costs. 3 Thereby, not only the supporting structure but also the mirrors, bearing materials, foundations, drives, and installation are topics of ongoing research. An overview is given by Pfahl et al. 4 One of the most advanced heliostat with respect to efficiency and costs is the so-called Stellio developed by schlaich bergermann partner (sbp). 5 The pentagonal concentrator is supported by radial cantilever arms and a central mount 6 (Figure 1, right). Thereby, the load path is reduced compared to T-type heliostats. Moreover, the steel framework cantilever arms are evenly utilized. This design is characterized by a low mass structure with a high optical accuracy, which serves as a benchmark in this contribution.

| Conceptual design of modular concrete heliostats
A first known concrete heliostat is shown, among others, in the review provided by Pfahl. 2 It is made from regular concrete and possesses thicknesses in the range of decimeters. For sun tracking, this concept requires a support and rotation axis in the center of gravity, so that a massive counter weight was necessary. However, for parabolic troughs, it could be shown that slender light-weight shells made from high-performance concrete (HPC) are an economic alternative to conventional collectors due to the low costs of concrete. 7,8 HPC is characterized by a high compressive strength of approximately >70 to 80 MPa and an increased durability compared to regular concrete. 9 The findings for parabolic shell collectors are now transferred to heliostats. This goes along with higher accuracy demands. Moreover, shell-like collectors 10 that are appropriate for parabolic troughs will now be dissolved in strut-like structures. Thereby, the statical height is increased to ensure higher stiffness. 11 The conceptual design of the HPC heliostats is orientated at the Stellio collector. A central mounted structure with main radial beams is chosen. For the bearing that is arranged in the structure's center of gravity, an inner ring is formed, where the radial beams are connected transferring the loads, that is, self-weight and wind loads, to the support. The whole heliostat also exhibits a circular shape allowing a compact field layout and minimizes the mutual shading of heliostats in the solar field. In comparison, the corner areas of rectangular concentrators might shade parts of concentrators located behind (cf. Figure 1, left). A circumferential post-tensioning is used to compress the strut structure utilizing the high compressive strength of the used concrete.
F I G U R E 1 Solar power tower pilot plant in Jülich, Germany with detail of a heliostat (left), and Stellio heliostat 6 (right) source: Left: German Aerospace Center (DLR), right: sbp sonne GmbH F I G U R E 2 Conceptual design of a modularized, post-tensioned strut-like heliostat made from concrete It minimizes or even avoids reinforcements, as discussed for axially compressed columns by Schmidt and Curbach. 12 Therefore, an outer ring is necessary to apply the tensioning forces. Secondary struts are added for the transfer of forces to the support but also to bear mirror elements or for stability purposes of the radial and outer beams. Due to the rotationally symmetric shape, the heliostat is designed by system reduction methods. 13,14 Hence, the structure can be modularized in equal partial segments which are held together by post-tensioning.
The conceptual design of the heliostat is illustrated in Figure 2 for an exemplary modular structure made from 8 modules. It exhibits a diameter of 6 m which results in a mirror area of around 30 m 2 and lies in the range of cost-optimized mirrors. 3 The heliostats will be made from HPC based on the binder Nanodur compound 5941 which has two crucial advantages for the design.
First, the Young's modulus E C is much higher than for regular concrete. Thereby, low deformations are secured. Second, also the compressive and, even more important, the tensile strength are much higher. Due to the compressive strength, the post-tensioning stresses can be sustained. The high tensile strength is necessary to ensure a non-cracked state I. This is crucial since softening by cracking would cause enlarged deformations and thus endanger a precise solar concentration. In previous designs of concrete shell collectors for parabolic troughs, the Nanodur concrete has proven to be suitable. 7,11 It exhibits good workability and has self-compacting behavior and high durability. 16 The material properties differ with respect to the used aggregates and additives. However, the Young's modulus used for all analytical derivations is set to 47.500 MPa which is a more or less a conservative lower value. The mean flexural tensile strength could be determined to f ctm,fl = 20 MPa in 3-point bending tests using prisms with dimensions of 40/40/160 (mm) according to  and DIN EN12390-5. 18 The age of the concrete prisms was 28 days with water storage of 27 days. According to Schmidt,19 it is transferred to the axial tensile strength and, based on the scattering data of the tests, the characteristic 95% quantile value is calculated to

| Geometry and accuracy demands
The shape of the reflecting surface of a heliostat is an elliptical paraboloid. For a circular structure, it corresponds to the surface of a parabola rotated around its axis of symmetry (z-axis). The shape of the parabola is defined by its focal length f, which describes the distance to the so-called focal point. In simplified terms, the focal length corresponds to the distance between the heliostat and the receiver at the central tower (cf. Figure 3). With r as the radial coordinate, the curve z(r) of the paraboloid is given by: In Figure   In a solar field, the different positions of the heliostats would require individual shapes. However, similar shapes of heliostats can be summarized into group arrangements. A coarse classification is shown in Figure 3.
For almost equal differences of height (Δh ≈ 2 cm), the mean heights of these ranges are chosen and lead to focal lengths of 36, 60, and 138 m ( Figure 3). This classification allows areas with similar focal lengths of the solar field to be covered by identical structures. However, for the design of a whole solar field, deviations to the "correct" focal lengths have to be considered since they can lead to losses in solar concentration.
The optical efficiency, that is, the solar concentration, significantly depends on an ideal paraboloidal-shaped reflecting surface. Deformations due to specific action effects, for example, self-weight and wind loads, or initial deviations, for example, due to manufacturing or assembly, cause slope deviations of the surface which are derived from the gradient of the deformations. These slope deviations (SD) are widely used to determine the optical quality by means of the rootmean-square (rms) value. For the Stellio heliostat, an SDrms value of 1.25 mrad 5 was verified, which defines the benchmark for the conceptual designs presented here. Since this value was determined at the already assembled and functional heliostat, a separation between SD from bending, that is, because of deformations of the bearing structure, and from waviness, that is, due to deformations of the mirror surface, is made. The square root of the squared sum defines the accuracy criteria used here (Equation (2)).
Assuming that both components are equally weighted, the limit value for dimensioning the bearing structure is derived to SD rms,bending = 0.88 mrad. The second design aspect is the limitation of tensile stresses, so to ensure an uncracked state I.

| Symmetry reduction
Heliostats possess a relatively low curvature based on high focal lengths. Additionally, heliostats made from concrete are mainly loaded due to self-weightsarising from the concrete structure and mirror elementsand are most stressed in the horizontal positionmeaning that they look straight upwith dominant vertical loads. Hence, it is possible to idealize the heliostats for a conceptual design as rotationally symmetric plates which are point-wise supported in the center (Figure 4, top). According to Markus and Otto, 13 the plate can be idealized as a cantilever arm with the radial coordinate r and a length set to the heliostat's radius R (Figure 4, bottom). The dimensionless coordinate ρ defines the ratio of both (Equation (3)).
The thickness of the plate or cantilever arm, respectively, exhibits a hyperbolical increase in thickness which is defined by the height h 1 at the edge and a shape factor n within a power law of ρ (Equation 4).
Increasing n results in more and more reduced heights ( Figure 5, left). For a modular segmentation of the plate, partial structures can be determined defined by the angle φ (in [rad]) (cf. Figure 4). Their (radial) moment of inertia I r can be derived to: As expected, I r also essentially depends on n. For n = 1, I r is constant over the length since rigidity losses due to the decreasing height are fully compensated by the increasing circumference of the circular partial structure u ϕ (Equation (6)).
In Figure 5 (right) the normalized moments of inertia are shown for the corresponding heights with different shape factors n. Each height and moment of inertia is normalized to its value for ρ = 0.025 since the height tends analytically to infinity for ρ = 0. For a conceptual design, n = 1.5 is chosen since it represents a good compromise between plate thickness, that is, weight, and rigidity.
F I G U R E 4 Rotationally symmetric plate with hyperbolical thickness and loads acc. to 13 (top); partial structure defined by the angle φ and corresponding equivalent statical system of a cantilever arm (bottom) Moreover, the applied loads are simplified. They comprise from a constant area load p, representing wind loads and dead loads of reflector elements and secondary concrete struts, a circular line load P r , depicting the outer ring beam for post-tensioning, and the selfweight g, defined by the thickness of the plate (cf. Figure 4). Posttensioning forces are not yet considered here. The resulting loads from self-weight G, area load P, and circular line load G r according to Figure 4 are given by: with P r = A c γ c Hereby, the area load p is assumed to 0.5 kN/m 2 , the area A c represents the outer ring stiffener's cross section, which is assumed to 0.01 m 2 (width to height ratio of 0.1 m to 0.1 m, cf. Table 2) and the concrete's bulk density yields to γ c = 25 kN/m 3 (cf.

| Derivation of stiffness-equivalent beams and partial structures
For the concrete heliostat, the developed plate segments ("piece of cake") have to be transferred into equivalent struts. Based on the thickness of the plate h, radial beam elements are derived which exhibit an equal resultant stiffness with respect to φ (cf. Equation (5)).
In   Table 2.

| Wind loads
Wind loads mainly depend on the wind flow around the structure.
Codified wind load coefficients cover wide ranges of applications and boundary conditions. Insofar, they might overestimate specific load situations. Consequently, DIN EN 1991-1-4 23 permits properly configured wind tunnel experiments to gain coefficients for individual cases. Thus, various wind tunnel tests 24 have been performed, most of them for heliostats with rectangular shapes or parabolic dish collectors. [25][26][27] Results are mainly force and moment coefficients c F or coefficient of pressure distributions c p , respectively. They differ for the spatial position of the heliostats. Load coefficients are used to design the supporting structure.
In this study, the pressure coefficients according to Gong et al. 27 are adapted and applied in the decisive horizontal position. Therefore, a conservative approach with a constant pressure distribution is applied.
The pressure coefficient is set to c p = 0.30, whereby local suction is disregarded and the resulting wind load is overestimated. Local effects at borders are not considered. with: Thereby, the air density ρ air is 1.25 kg/m 3 . In operation state, q w results to 0.02 kN/m 2 and 0.20 kN/m 2 in survival state.

| Post-tensioning
The post-tensioning is achieved using an external monostrand, stressed Hereby, μ describes the friction coefficient of 0.05 for external monostrands. 29 Assuming the sum of Θ being 2π, a mean post-tensioning force P m = 153 kN is derived for a one-sided tensioning. Based on P m , an inward, line-like deviation load u (Equation (13)) arises that holds the modules together and prestresses the radial struts. It results to ≈ 50 kN/m.

| Numerical analysis of deformations
The developed structures are loaded by the described action effects.
The analyses are performed using characteristic values in operation and survival state.
For the accuracy analysis in operation state, deformations are derived. These deformations result from an approach with reduced stiffnesses whereby stability constraints of the struts have been disregarded. Thus, it is a conservative approach. The deformations are illustrated exemplarily in Figure 9 for D45 with a focal length of f = 36 m.
Thereby, a group of heliostats near the central tower is defined (cf. Figure 3). Figure 9 shows the deformations caused by dead loads Two parameters of variations should be discussed. First, an increase of prestressing. A monostrand with fixed cross section is used here that could only be doubled, so applying two strands.
Due to cost reasons, the design is restricted to one strand. Second, a regular HPC could be used as a much cheaper material that easily withstands the occurring compressive stresses. However, it would require a larger concrete cover due to durability reasons.
The cross sections of all struts would pronouncedly increase.
Moreover, it fails in providing sufficient tensile strength capacities.
Consequently, the chosen compound turns out to be the most effective for this design case.

| Discussion of designs
The derived designs differ in accuracy, module sizes and, as a consequence, the number of modules, and weight. The main characteristics are summarized in Table 3. For every design, the accuracy demands are met. However, the highest accuracy accounts for D30, a minimum accuracy results from D45. Thereby, SD are mainly dependent on the waviness of the mirror elements and, hence, their stiffness as well as the distance between the concrete struts. Thus, the accuracy could easily be improved by stiffer mirror elements, which then entail increasing costs. The maximum weight of the heliostat structure results from D60, whichconsequently possesses the maximum weight for the segmental modules. D30 exhibits the minimum weight of the modules, but almost an equal weight as D60 for the whole structure. To cut material costs, D45 is preferred, since it reduces >10% of the mass compared to D60.
For further reductions elaborated methods like topology optimizations are recommended. 30,31 They help to identify more effective material distributions.
Due to the high amount of identical modules, serial production seems to be appropriate, especially when considering a whole solar field.
For assembly, precise joints have to be ensured since geometrical uncertainties from the production process would accumulate for the whole heliostat. Figure  This means that with increasing modules the expectable uncertainty of the heliostat also increases and leads to gaps in-between the modules.
Thereby, the post-tensioning, assembly and, consequently, operation of T A B L E 3 Characteristics of the concrete heliostats with respect to their partial structure's size • Numerical analyses reveal that deformations due to self-weight and post-tensioning almost equalize each other. Deformations which cause significant slope deviations mainly result from the waviness of the mirror elements.
• The modular structures distinguish between their number of modules, weight, and stiffness. The one with a module size defined by φ = 30 exhibits the highest stiffness and the module defined by φ = 45 the lowest weight. Thus, this last type is preferred.
• The heliostats should be fabricated in serial productions. However, increasing numbers of modules let inaccuracies superimpose that must be controlled by quality assurance.