Two-Layer Ring Truss-Based Space Solar Power Station

Abstract

A space solar power station (SSPS) has become a huge potential candidate to provide abundant and clean electrical energy for terrestrial users by collecting and converting solar power in space. In this paper, an innovative two-layer ring truss-based SSPS is proposed. It consists of the top layer concentrator-based spherical one-time reflection region, the bottom layer space radiator using symmetric or asymmetric cable networks, a ring truss for a supporting structure, a photoelectric conversion system, and transmitting antennas. The construction strategies including the triangular facets modularity of top layer concentrator, area requirement of bottom layer space radiator, two-segment optimization design of generatrix of photoelectric conversion system, and aperture derivation of transmitting antenna are carried out. Then, the performance analysis mainly including the modularization theory error calculation, energy collection and distribution, and thermal characteristics in orbit of this proposed SSPS is presented. Finally, the system parameters are estimated and summarized for a better sense of the proposed SSPS. The results indicate that 100% energy collection can be achieved for an ideal concentrator, and 80% with a modular division layer of 6 and tracking error of no more than 2°. The results demonstrate the feasibility of the proposed SSPS concept and can provide a reference for future space energy harvesting and space exploration projects.

1. Introduction

In recent years, space exploration has become a hot spot with the development of space technology, and space power generation technology such as a space solar power station (SSPS) has gradually become possible. SSPS is a large-scale energy system that collects and converts solar energy to electric power in space and then transmits the electric power to earth wirelessly. It is a promising solution to the energy crisis by using clean, abundant, and sustainable space energy [1,2].

Since initially being proposed by Dr. Peter Glaser [3], many typical concepts of SSPS have been proposed around the world, which can mainly be divided into two kinds, namely the non-concentrating concept and the sunlight-concentrating concept. Typical non-concentrating SSPS include NASA’s reference model [4], Tethered-SSPS [5], Sail-Tower SSPS [6], and multi-rotary joints SSPS [7]. By the use of a concentrator to avoid the large-scale photovoltaic cell array, which can greatly improve the power mass ratio of the system, the sunlight-concentrating concept has attracted more interest than the non-concentrating concept. The integrated symmetrical concentrator (ISC) architecture [8], Sun-Tower [9], SPS via arbitrarily large phased array (ALPHA) [10], and SSPS via Orb-shape Membrane energy gathering array (OMEGA) [11] are typical representatives of this concept. Additionally, the ε-near-zero (ENZ) metamaterial-based SSPS [12], fiber-optic bundle-based SSPS [13], and secondary reflection-based SSPS-OMEGA [14] proposed in recent years all belong to the sunlight-concentrating concept. Among them, the SPS-ALPHA concept and SSPS-OMEGA concept attract more attention.

The SPS-ALPHA adopts a sandwich structure that integrates the three subsystems of photoelectric conversion, power management, and transmitting antenna for energy conversion and transmission. This concept avoids the need for high power conductive rotary joint and a long transmission electric cable, thereby improving reliability during electric power transmission. However, it brings the severe thermal problem in sandwich structure [15,16,17], as well as very complicated attitude control and the problem of light leakage [18,19], for there are thousands of individual thin-film reflectors needed to be adjusted to keep sun tracking in real-time. The SSPS-OMEGA concept uses a spherical concentrator to collect solar energy, and a photoelectric conversion system moves inside the concentrator at a uniform speed in one dimension. The control system of this concept is greatly simplified and the problem of light leakage is avoided [20]. Meanwhile, the separation among photoelectric conversion, power management, and transmitting antenna can greatly relieve the heat dissipation pressure of the whole system. Nevertheless, there are still two main problems with this concept: one is the wear and breakdown of an ultra-high power brush, resulting in low reliability in the process of energy transmission. The other is the technical bottleneck of concentrating film, simultaneously satisfying semi-transparency and semi-reflection of sunlight [21].

To overcome the problems mentioned above and make SSPS more reasonable and feasible, a new SSPS concept based on a two-layer ring truss is proposed in this paper. This proposed concept is expected to have high energy collection efficiency, good energy distribution characteristic, low system mass, and well-performing thermal control. For this aiming, the construction strategy, performance analysis, and system parameter estimation of the proposed concept are followed.

2. The Two-Layer Ring Truss-Based SSPS

The design of a two-layer ring truss-based SSPS, as shown in Figure 1, is mainly composed of four parts: concentrator, photoelectric conversion system, transmitting antenna, and space radiator. By the use of the linear concentrating characteristic of the sphere, the parallel incident sunlight is concentrated on the photoelectric conversion system located in [0.5, 1] R region (R is the radius of the spherical concentrator). Then, solar energy is converted into direct current (DC) through the photoelectric effect of the PV cell, and the DC is transmitted to the transmitting antennas at each end through the electric cable and non-contact rotary joint. The power device in the transmitting antenna converts electrical energy into radio frequency, which is finally transmitted by wireless microwave to the rectenna on the ground.

The entire system is based on a ring truss deployable structure for its advantages of large aperture, light weight, high packing efficiency, and high accuracy [22,23,24]. Unlike the traditional ring truss reflector antenna [25,26], which only has a metal mesh surface attached to the backside of the front net, a two-layer ring truss structure is proposed in this innovative SSPS design. It mainly consists of a deployable ring truss, tension ties, two cable networks (front net and rear net), and a two-layer thin-film surface. The ring truss is connected to cable networks and supports the cable networks once deployed. Tension ties connect the nodes of the two cable networks and assembly imposes tensile forces between the front net and the rear net to pretension the cable networks structure.

The two-layer thin-film surface is composed of two groups of triangular facets, one group forms the concentrator with a front net, called the top layer, and the other group forms the space radiator with a rear net, called the bottom layer, and both groups are shaped through the pretension of cable networks structure. The triangular facets at the top and bottom layers are made of thin-film materials with high reflectivity and high emissivity, respectively, for solar energy collection and waste heat radiation. The top layer concentrator concentrated several times solar energy on a photoelectric conversion system, which may lead to a serious thermal problem. Therefore, the bottom layer space radiator is used to dissipate the waste heat, transferred from the photoelectric conversion system through the heat transfer channel, to the cold space by heat radiation. The heat transfer channel adopts a multi-functional integrated design to provide support for the photoelectric conversion system while transferring heat out. In addition, the shape of the bottom layer satisfying the thermal control condition can be obtained through a form-finding design based on ensuring the accuracy of the top layer [27,28,29].

During in-orbit operation, the concentrator is tracked to the sun in real-time by an attitude controller distributed on the ring truss. The photoelectric conversion system is fixed together with the concentrator without relative movement. To compensate for the loss of self-blocking, an additional disc-shaped PV cell array is added at the top of the photoelectric conversion system to receive direct incident sunlight. The DC power is transmitted to the transmitting antenna through a non-contact rotary joint, shown in Figure 1, which mainly consists of the fixed plate connected to the ring truss, the rotary track connected with the electric cable, and a pickup system connected to the transmitting antenna. The rotary track transfers the DC to the pickup system through magnetic coupling. The waste heat generated by the photoelectric conversion system is transferred to the space radiator through an efficient heat transfer channel. The aperture of the space radiator shall not exceed that of the concentrator, to ensure that the radiator is always in the shadow of the concentrator and always towards the cold space.

3. Construction Strategy

3.1. Top Layer Concentrator-Based Spherical One-Time Reflection Region

Inspired by the SSPS-OMEGA concept, the spherical one-time reflection region is selected as the concentrator of this concept, and solar energy is collected by the use of the spherical linear concentrating characteristic. The size and shape of the triangular facet in the top layer directly determine the surface accuracy, and also directly affect the energy collection performance of the concentrator. Therefore, the concentrator modular construction strategy is discussed firstly.

3.1.1. Linear Concentrating Characteristic of the Sphere

Figure 2 illustrates the light propagation process under spherical reflection in the XOY coordinate plane. The parametric equation of the spherical surface on the coordinate plane XOY can be expressed as:

$$\{\begin{array}{l}x=R\cos \alpha \\ y=R\sin \alpha \end{array}$$

(1)

where α is the residual angle of the incident angle β.

Based on the Fresnel’s reflection law [30]

$${\mathit{r}}_{ref}={\mathit{r}}_{inc}-2\left({\mathit{r}}_{ref}\cdot \mathit{n}\right)\mathit{n}$$

(2)

where r_ref and r_inc are the unit vectors of the reflected sunlight and incident sunlight, respectively. n is the unit normal vectors of the concentrator.

The equation of reflected sunlight can be derived assuming the parallel incident sunlight can be described as x = Rcos α:

$$x\cos 2\alpha +y\sin 2\alpha -R\cos \alpha =0$$

(3)

Equation (3) exhibits some concentrating characteristics of spherical concentrator that all incident sunlight will be converged within the region of [R/2, R] of Y-axis, which is described as linear concentrating mode. In addition, when the residual angle of incident sunlight belongs to the interval [π/6, π/2], it will be concentrated on Y-axis after one-time reflection, while the residual angle of incident sunlight belongs to the interval [0, π/6), it will be concentrated on Y-axis after multiple-time reflection. Thus, the region of [π/6, π/2] is named a one-time reflection region.

After multiple reflections, the mathematical description of the light equation becomes quite difficult, which is not conducive to the design of the photoelectric conversion system in the following chapters. At the same time, with the increase in the number of reflections, the incident energy loss is serious, resulting in a decrease in the value of energy collection using this region. Therefore, in this paper, the spherical one-time reflection region is used as the concentrator to collect energy.

Moreover, Equation (3) indicates the reflected sunlight is a line family with a single parameter of residual angle α, so the envelope curve of reflected sunlight could be obtained by establishing the equation

$$\{\begin{array}{l}F\left(x,y,\alpha \right)=0\\ \frac{\partial F}{\partial \alpha }=0\end{array}$$

(4)

where $F=y\sin 2\alpha +x\cos 2\alpha -R\cos \alpha$.

Then, the equation of the envelope curve can be derived as:

$$\{\begin{array}{l}x=R{\cos }^3\alpha \\ y=\frac{R}{2}\sin \alpha \left(1+2{\cos }^2\alpha \right)\end{array}$$

(5)

Equation (5) shows another important concentrating characteristic that all incident sunlight will be reflected between the envelope curve and spherical parametric curve. It is significant for the construction of the photoelectric conversion system with high solar energy collection performance.

3.1.2. Spherical Concentrator Modularity

Compared with the equal latitude and longitude grid division, the spherical polygonal grid division has the advantages of better isometric property and structural stability [31]. Therefore, the spherical one-time reflection concentrator of this SSPS concept is constructed by spherical polygonal grid division, and the regular dodecahedron is used to generate a spherical grid.

As shown in Figure 3a, the center O₁ of a regular pentagon surface in the regular dodecahedron is firstly obtained, and the straight line OO₁ is made and intersects the sphere at point p₁, connecting p₁ with other vertices of the regular pentagon to form five isosceles triangles, as shown in Figure 3a. Then, the triangle is projected to the sphere with radius length Op₁ to form spherical isosceles triangles, as shown in Figure 3b. All three sides of the spherical triangle are geodesic arcs. In this way, the spherical triangle grid can be divided first, and then the coordinate of the entire spherical grid can be obtained by rotation and symmetry operation on the basis of the coordinate of the spherical triangle grid.

As shown in Figure 3b, in order to further simplify the calculation procedure, $\stackrel{⌢}{\Delta p_1{p}_2{p}_3}$ is placed symmetrically on both sides of coordinate plane XOZ, and the coordinates of grid points of the spherical triangle $\stackrel{⌢}{\Delta p_1{p}_2{p}_3}$ can be obtained by mirror mapping the coordinates of grid points of $\stackrel{⌢}{\Delta p_1{p}_4{p}_3}$ about plane XOZ.

Firstly, the arc $\stackrel{⌢}{p_1{p}_4}$ is evenly divided into n_c equal parts (n_c is the number of modular layers), and the arc $\stackrel{⌢}{p_1{p}_3}$ is evenly divided into n_c equal parts by a set of declination circles passing through the point a_i, and each partition point is denoted as b_i. Then, meridian circles are set through b_i, intersecting the arc $\stackrel{⌢}{p_4{p}_3}$ at point N, and intersecting the arc $\stackrel{⌢}{a_i{b}_i}$ at point M. By calculating the azimuth α and elevation β corresponding to point M, the coordinates of M can be obtained, thus the coordinates of grid points in a spherical triangle $\stackrel{⌢}{\Delta p_1{p}_4{p}_3}$ can be obtained by traversing n_c equally divided intersections. Finally, the triangular facets are constructed by connecting adjacent points.

According to the radius of the sphere and the corresponding azimuth and elevation, the coordinates of the grid points in the spherical triangle $\stackrel{⌢}{\Delta p_1{p}_4{p}_3}$ can be expressed as:

$$\{\begin{array}{l}x=R\cdot \sin {\alpha }_{i,j}\cdot \cos {\phi }_{i,j}\\ y=R\cdot \sin {\alpha }_{i,j}\cdot \sin {\phi }_{i,j}\\ z=R\cdot \cos {\alpha }_{i,j}\end{array}\left(i,j=1,2,3,\cdots ,{\mathit{n}}_c\right)$$

(6)

Finally, the modular division of the spherical triangle $\stackrel{⌢}{\Delta p_1{p}_2{p}_3}$ is shown in Figure 4a. Retaining the one-time reflection region of the spherical grid, the modularity of the top layer concentrator is obtained, as shown in Figure 4b.

3.2. Bottom Layer Space Radiator

After the construction of the top layer, the bottom layer space radiator could be constructed using symmetric or asymmetric cable networks [29], as shown in Figure 5. The grid of the bottom layer is consistent with the top layer, and the grid points are one to one.

For the symmetric cable networks, the height of the bottom layer is equal to the top layer, and the height of the ring truss after deployment is at least twice as high as the bottom layer. As for asymmetric cable networks, the height of the bottom layer can be optimized and adjusted without changing the height and surface accuracy of the top layer. So the area of the space radiator is maximum when using symmetric cable networks, which can be expressed as

$$S_r=2\pi R{h}_{bottom}$$

(7)

where h_bottom is the height of the bottom layer.

When using asymmetric cable networks, the area of the space radiator can be expressed as

$$S_r=\pi \left(\frac{3R^2}{4}+h_{bottom}^2\right)$$

(8)

The area of the space radiator depends on the thermal load of the photoelectric conversion system, which can be expressed as [32]

$$Q=\sigma \epsilon S_r\left(1+X_{sr}\right)\left(T_{sr}^4-T_{\infty }^4\right)$$

(9)

where σ is the Stefan-Boltzmann constant, 5.67 × 10⁻⁸ W/(m²·K⁴), ε is the emissivity. X_sr is the angle factor of the concave of the bottom layer toward space. T_sr is the average temperature of the space radiator, T_∞ is the environment temperature, 4 K.

When operating in orbit, the heat load can be estimated

$$Q=I_0{A}_{aper}{\eta }_c\left({\alpha }_{pv}-{\eta }_{pv}\right)$$

(10)

where I₀ is the solar constant, and I₀ = 1367 W/m² in geostationary earth orbit (GEO). A_aper is the area of aperture of the concentrator. η_c is the collection efficiency of the concentrator. α_pv and η_pv are absorptivity and efficiency of PV cell array, respectively.

3.3. Configuration of the Photoelectric Conversion System

3.3.1. Generatrix Description

An ideal spherical concentrator with a one-time reflection region is used to design the configuration of the photoelectric conversion system, and a coordinate system for the generatrix of the photoelectric conversion system in a two-dimensional plane is illustrated in Figure 6, where the envelope curve is contained. According to Equation (5), the envelope curve will intersect the Y-axis at (0, R/2). The curve AB represents the generatrix of the photoelectric conversion system. To ensure more solar energy is received, the setting of endpoint A and endpoint B is critical.

Suppose an incident ray s₁ intersecting the X-axis at point D intersects the envelope curve at point B, then all incident sunlight from the OD region will be blocked, so the abscissa value of point B determines the self-blocking by the photoelectric conversion system. Another incident light s₂ at the edge of the concentrator with an incident angle of π/3 will intersect the Y-axis at point (0, R), and its reflection intersects the reflection of s₁ at A point. As known from the concentrating characteristic of the concentrator, all incident sunlight from the DE region will be concentrated on the curve AB.

Furthermore, to ensure a more uniform energy distribution, point C is selected to carry out a two-segment design for the generatrix. An incident sunlight s₃ intersecting the X-axis at point F with incident angel π/4 is selected out for its reflected light is a horizontal line, which is conducive to find out the intersection. The reflection of s₃ intersects the reflection of s₁ at point C, and all incoming energy from DF and FE regions will be concentrated on curves AC and CB, respectively. The energy density on the curve CB is obviously higher than that on the curve AC, so it is necessary to perform a two-segment design for curve AB.

Considering the convenience of the design, the Bézier curve based on the Bézier polynomial parametrization is introduced. Bézier curve is a parameterized curve controlled by control points with good continuity, conductibility, convex hull property, and endpoint property, which is very convenient for generatrix design [33]. A Bézier curve of order n can be defined as:

$$C\left(u\right)={\displaystyle \sum _{i=0}^{\mathit{n}}{B}_{i,n}\left(u\right)P_i},0\le u\le 1$$

(11)

where $B_{i,n}\left(u\right)$ is Bernstein polynomial, Pi is the control point.

$$B_{i,n}\left(u\right)=\frac{\mathit{n}!}{i!\left(\mathit{n}-i\right)!}{u}^i{\left(1-u\right)}^{\mathit{n}-i}i=0,1,\dots ,\mathit{n}$$

(12)

Additionally, the derivative of any point on the Bézier curve can be expressed as

$$C^{\prime }\left(u\right)=\mathit{n}{\displaystyle \sum _{i=0}^{n-1}{B}_{i,n-1}\left(u\right)\left(P_{i+1}-P_i\right)}$$

(13)

3.3.2. Optimization Design

In order to design the generatrix of the photoelectric conversion system based on energy collection and distribution, the Monte Carlo ray-tracing method (MCRT) is adopted [21]. By using the MCRT method, the position and direction of each sunlight considering sunlight cone angel could be described as

$$\{\begin{array}{l}{r}_c=R_c\sqrt{R_r}\\ {\mathit{\theta }}_c=2\pi R_{\mathit{\theta }}\end{array}$$

(14)

$$\{\begin{array}{l}\gamma =\arctan \sqrt{R_{\gamma }}\tan {\gamma }_{\max }\\ \beta =2\pi R_{\beta }\end{array}$$

(15)

where R_r, R_θ, R_γ, and R_β are probability distributions of radius r_c, incident angel θ_c, conical angle γ, and circumferential angle β, respectively, which are independent and evenly distributed in the interval [0, 1]. R_c is the radius of the concentrator aperture, and α_max is the maximum cone angle.

Assuming that the sunlight incident from the plane y equals R, the parameter equation of incident sunlight can be established as

$$\left[\begin{array}{l}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}{R}_c\sqrt{R_r}\cos \left(2\pi R_{\mathit{\theta }}\right)\\ R\\ R_c\sqrt{R_r}\sin \left(2\pi R_{\mathit{\theta }}\right)\end{array}\right]+\left[\begin{array}{c}-sin\gamma \cos \beta \\ -\cos \gamma \\ -\sin \gamma sin\beta \end{array}\right]t$$

(16)

Additionally, the energy carried by single sunlight can be defined as

$$e=\frac{I_0{A}_{aper}}{N_s}$$

(17)

where N_s is the number of sampled sunlight that irradiates to the aperture.

Then, the energy collected by the photoelectric conversion system can be expressed as

$$\mathit{E}={\displaystyle \sum _{j=1}^{{\mathit{N}}_{\mathit{p}}}{\mathit{N}}_{\mathit{s},\mathit{j}}\cdot e\cdot \phi \left(\mathit{\theta }\right)}$$

(18)

where N_p represents the vector of the segmented numbers of curves AC and CB, E represents the vector of energy collected by curves AC and CB. N_s,j is the number of sunlight falling on the j_th subsection. φ(θ) represents the angular response of photovoltaic cell at incident angle θ, which is considered as

$$\phi \left(\mathit{\theta }\right)=\{\begin{array}{ll}1& 0\le \mathit{\theta }\le {\mathit{\theta }}_{\mathit{m}}\\ 0& {\mathit{\theta }}_{\mathit{m}}<\mathit{\theta }\le \frac{\pi }{2}\end{array}$$

(19)

where θ_m is the maximum responsible angle vector and is set to be 4π/9 in this paper.

Then, the energy distribution can be expressed as

$$\mathit{\xi }=\sqrt{{\displaystyle \sum _{j=1}^{{\mathit{N}}_{\mathit{p}}}{\left({\mathit{E}}_{\mathit{d},\mathit{j}}-\overline{{\mathit{E}}_{\mathit{d}}}\right)}^2}/\left({\mathit{N}}_{\mathit{p}}-\mathit{I}\right)}$$

(20)

where I is the unit vector, and

$$\begin{array}{l}{\mathit{E}}_{\mathit{d},\mathit{j}}={\mathit{N}}_{\mathit{s},\mathit{j}}\cdot e\cdot \phi \left(\mathit{\theta }\right)\\ \overline{{\mathit{E}}_{\mathit{d}}}=\frac{{\displaystyle \sum _{j=1}^{{\mathit{N}}_{\mathit{p}}}{\mathit{E}}_{\mathit{d},\mathit{j}}}}{N_p}\end{array}$$

(21)

The photoelectric conversion system is expected to receive energy as much as possible, and the energy distribution is as uniform as possible. Thus, the objective function can be expressed as

$$\mathbf{\Phi }=-{\omega }_1\frac{E}{I_0{A}_{aper}}+{\omega }_2\xi$$

(22)

Then, the mathematical optimization model can be established as

$$\begin{array}{ll}\mathrm{find} & P\\ \min & \mathbf{\Phi }=-{\omega }_1\frac{E}{I_0{A}_{aper}}+{\omega }_2\xi \\ \mathrm{s}.\mathrm{t} & f\left(x_c\right)|{}_{AC}=f\left(x_c\right)|{}_{CB}\\ & f^{\prime }\left(x_c\right)|{}_{AC}=f^{\prime }\left(x_c\right)|{}_{CB}\\ & {\eta }_s\le \left[{\eta }_s\right]\\ & P\in \mathbf{U}\end{array}$$

(23)

where P (P₀, P₁, …, P_n) = [x₁, x₂, …, x_n; y₁, y₂, …, y_n]^T, ω₁ and ω₂ are weighting factors for energy collection and distribution, respectively. The first two constraint functions indicate that curves AC and CB satisfy geometric continuity and derivative continuity at point C. η_s is the self-blocking of the photoelectric conversion system, and [η_s] is a pre-given value. U is the feasible region of the coordinates of the control points.

3.3.3. Optimization Results

In order to avoid the impact of the initial value on the optimization procedure and the difficulties caused by the sensitivity solution, Particle Swarm Optimization (PSO) is used to obtain an optimal solution based on the mathematical optimization model established above. To reduce the calculation scale, a concentrator model with R equal to 0.5 m is used for the optimization procedure. Generations are set to 600 and the size of the population is set to 20. The order of curves CB and AC are 2 and 3, respectively, and the number of subsections of curves CB and AC is 3 and 6, respectively. The number of sampled sunlight is set to 10⁵ after sensitivity analysis. The pre-given value [η_s] is set to 0.05, and the weighting factors are set to ω₁ = ω₂ = 0.5.

Then, the iterative convergence curve of the objective function is shown in Figure 7, and the optimal generatrix of the photoelectric conversion system and its 3D model are illustrated in Figure 8, the energy distribution is shown in Figure 9, and detailed parameters are listed in

Table 1. Detailed parameters of optimal design.

Parameters	First Segment (Curve AC)	Second Segment (Curve CB)
Energy collection E/(I0 × Aaper)	35%	65%
Energy distribution ξ	0.878	0.291
Maximum incident angel θ	6.80°	77.61°
Minimum incident angel θ	0.48°	0.01°
Control point P/m	(0.0837,0.4517) (0.0419,0.3953) (0.0419,0.3537) (0.0419,0.3536)	(0.0419,0.3536) (0.0540,0.3416) (0.1,0.3415)

.

As seen in Table 1, the designed photoelectric conversion system exhibits an excellent energy collection performance with 100% solar energy collection; meanwhile, the well-performed energy distribution is also obtained in both segments. The incident angle of reflected on the photovoltaic cell array is as low as 0.01°, and the maximum does not exceed 78°, which fully meets the design requirements. The coordinates of the control point are also listed.

3.4. Microwave Transmitting Antenna

After the DC power is transmitted to the transmitting antenna, the energy received by the rectenna on the ground depends on the beam collection efficiency (BCE) between the transmitting antenna and rectenna, which can be obtained according to the radio wave theory [34]

$${\eta }_{BCE}=\frac{P_r}{P_t}=1-e^{-{\tau }^2}$$

(24)

$$\tau =\frac{\sqrt{A_t{A}_r}}{\lambda L}$$

(25)

where P_r and P_t are transmitted power by transmitting antenna and received power by rectenna, respectively. A_t, A_r and λ are the aperture area of the transmitting antenna, aperture area of the rectenna, and wave length, respectively. L is the distance between two antennas, which should be satisfied the Fresnel region condition to ensure efficient transmission [35].

$$L<\frac{2D_t^2}{\lambda }$$

(26)

where D_t is the diameter of the transmitting antenna.

Moreover, carefully considering atmospheric attenuation, the microwave frequency of 5.8 GHz is chosen in this concept for it is within the microwave radio windows of the atmosphere. The efficiency of microwave transmission keeps high at this frequency even in bad weather.

Furthermore, the size of the transmitting antenna could be quite large when the SSPS is working in orbit, so a step amplitude taper to reduce the manufacturing difficulty and complexity with little loss of efficiency is adopted for facilitating modular manufacturing and assembly. For the stepped taper, the whole transmitting antenna is divided into several rings, and identical amplifiers are used in the same ring, and more details could be seen in our previous work [36].

4. Performance Analysis

4.1. Modularization Theory Error of Concentrator

The concentrator-based spherical one-time reflection region is modularly constructed with a series of triangular facets, so there must be a theoretical error, which reflects the surface accuracy of facets approaching the sphere.

As shown in Figure 10, ΔH₁H₂H₃ is any divided triangular facet, and its projection onto XOY plane is also illustrated. Point G is any point on the spherical triangular surface.

Suppose the equation of the triangular ΔH₁H₂H₃ plane is expressed as $z=ax+by+c$, then the parameters a, b, and c can be obtained from the coordinates of the vertices H₁, H₂, and H₃. Moreover, the equation of the normal line passing point G can be expressed as

$$\frac{x-x_G}{x_G}=\frac{y-y_G}{y_G}=\frac{z-z_G}{z_G}$$

(27)

Then, the coordinates of the intersection G’ of the normal line with the triangular ΔH₁H₂H₃ plane can be obtained, and the distance between points G and G’ is

$$d=\sqrt{{\left(x_G-x_{G^{\prime }}\right)}^2+{\left(y_G-y_{G^{\prime }}\right)}^2+{\left(z_G-z_{G^{\prime }}\right)}^2}$$

(28)

Thus, the modularization theory error of the concentrator can be obtained by

$$\mathit{\delta }=\sqrt{{\displaystyle \sum _{i=1}^{NUM}\mathsf{\Omega }}/{\displaystyle \sum _{i=1}^{NUM}S}}$$

(29)

where NUM is the number of triangular facets, and

$$\begin{array}{l}\mathsf{\Omega }={\mathsf{\Omega }}_l+{\mathsf{\Omega }}_r={\displaystyle {\int }_{x_{H_3}}^{x_{H_1}}{\displaystyle {\int }_{k_{H_1^{\prime }{H}_3^{\prime }}x+c_{H_1^{\prime }{H}_3^{\prime }}}^{k_{H_3^{\prime }{H}_2^{\prime }}x+c_{H_3^{\prime }{H}_2^{\prime }}}{d}^2}dy}dx+{\displaystyle {\int }_{x_{{}_{H_1}}}^{x_{H_2}}{\displaystyle {\int }_{k_{H_1^{\prime }{H}_2^{\prime }}x+c_{H_1^{\prime }{H}_2^{\prime }}}^{k_{H_3^{\prime }{H}_2^{\prime }}x+c_{H_3^{\prime }{H}_2^{\prime }}}{d}^2}dy}dx\\ S=S_l+S_r={\displaystyle {\int }_{x_{H_3}}^{x_{{}_{H_1}}}{\displaystyle {\int }_{k_{H_1^{\prime }{H}_3^{\prime }}x+c_{H_1^{\prime }{H}_3^{\prime }}}^{k_{H_3^{\prime }{H}_2^{\prime }}x+c_{H_3^{\prime }{H}_2^{\prime }}}}dy}dx+{\displaystyle {\int }_{x_{H_1}}^{x_{{}_{H_2}}}{\displaystyle {\int }_{k_{H_1^{\prime }{H}_2^{\prime }}x+c_{H_1^{\prime }{H}_2^{\prime }}}^{k_{H_3^{\prime }{H}_2^{\prime }}x+c_{H_3^{\prime }{H}_2^{\prime }}}}dy}dx\end{array}$$

(30)

where $k_{H_1^{\prime }{H}_3^{\prime }}$, $k_{H_1^{\prime }{H}_2^{\prime }}$ and $k_{H_3^{\prime }{H}_2^{\prime }}$ are slopes of lines $H_1^{\prime }{H}_3^{\prime }$, $H_1^{\prime }{H}_2^{\prime }$ and $H_3^{\prime }{H}_2^{\prime }$, respectively. $c_{H_1^{\prime }{H}_3^{\prime }}$, $c_{H_1^{\prime }{H}_2^{\prime }}$ and $c_{H_3^{\prime }{H}_2^{\prime }}$ are the intercepts of lines $H_1^{\prime }{H}_3^{\prime }$, $H_1^{\prime }{H}_2^{\prime }$ and $H_3^{\prime }{H}_2^{\prime }$, respectively. $x_{H_1}$, $x_{H_2}$ and $x_{H_3}$ are abscissas of points H₁, H₂, and H₃, respectively.

Based on Equations (27)–(30), the modularization theory errors and maximum side lengths of the concentrator with radius R equal to 0.5 m at different division layers are listed in

Table 2. The values of theory error and maximum side length.

Division Layers	δ/mm	Maximum Side Length/mm	Triangular Facet Number
6	0.819	66.8	485
9	0.364	44.6	1145
12	0.204	33.5	2040
15	0.131	26.8	5230

.

It can be seen that with the increase of the number of division layers, the theoretical error and the maximum side length decrease correspondingly, indicating more triangular facets are divided and a more accurate surface approximation is obtained. Referring to the design requirements for theoretical errors of cable-mesh reflector antennas [22], which is usually less than 1 mm. Therefore, the concentrator with modular division layers of 6 is selected as the subsequent analysis model.

4.2. Solar Energy Collection and Distribution

4.2.1. Modularization

After modularization, the concentrator changes from an ideal sphere to a series of triangular facets, which may result in energy loss as shown in Figure 11, as well as change in energy distribution. It may not achieve well-performed energy collection and energy distribution as listed in Table 1.

When analyzing the energy collection and distribution after spherical modularization, the unit normal vector n of point G in Equation (2) will no longer be (x_G/R, y_G/R, z_G/R), and changes

$$n=\overrightarrow{OG}\to n^{\prime }=\overrightarrow{H_1{H}_2}\times \overrightarrow{H_1{H}_3}$$

(31)

By using the MCRT method and the same number of sampled sunlight in Section 3.3, the energy distribution of the photoelectric conversion system is shown in Figure 12, and the detailed values of energy collection and distribution are listed in

Table 3. Detailed values of energy collection and distribution.

Parameters	First Segment (Curve AC)	Second Segment (Curve CB)
Energy collection E/(I0 × Aaper)	33%	49%
Energy distribution ξ	1.195	0.145

.

A comparative analysis of the data in Table 1 and Table 3 shows that the energy collection drops from 100% to 82% with a loss of 18% after modularization, including 2% for the first segment and 16% for the second segment. The loss of the second segment is much larger than that of the first segment. In addition, modularization improves the energy distribution of the second segment, while making it worse of the first segment.

4.2.2. Tracking Accuracy

The two-layer ring truss-based SSPS is oriented toward the sun through the electric propulsion distributed on the ring truss, and the tracking accuracy is another important factor affecting the performance of energy collection and distribution, as shown in Figure 13.

When there exists a tracking error, the unit vector of incident sunlight in Equation (16) changes

$$r_{inc}=\left[\begin{array}{c}-\sin \gamma \cos \beta \\ -\cos \gamma \\ -\sin \gamma \sin \beta \end{array}\right]\to {r^{\prime }}_{inc}=\left[\begin{array}{c}-\sin \gamma \cos \beta \\ -\cos \gamma \\ -\sin \gamma \sin \beta \end{array}\right]+{\delta }_{te}$$

(32)

where δ_te is the vector of tracking error, which can be expressed as

$${\delta }_{te}=\left[\begin{array}{c}{\mathit{\delta }}_x\\ 0\\ {\mathit{\delta }}_z\end{array}\right]=\left[\begin{array}{c}-\cos {\mathit{\theta }}_x\\ 0\\ -\cos {\mathit{\theta }}_z\end{array}\right]$$

(33)

where θ_x and θ_z are tracking angel deviations, respectively.

Figure 14 illustrates the energy collection of the photoelectric conversion system with different tracking errors. Figure 15 shows the variation curves of energy collection and distribution of different segments with tracking error.

As shown in Figure 14, due to modularity, energy collection does not change the same in both directions (θ_x = 0 and θ_z = 0), but it shows a consistent trend, that is, the energy collection decreases with the increase of the tracking error. When the tracking error is less than 2°, the energy collection of the concentrator is greater than 80%. When the tracking error is more than 5°, the energy collection is less than 74%. As seen in Figure 15, the tracking error has a greater effect on the energy collection of the second segment and a little effect on that of the second segment in both directions. However, the increase in tracking error does not result in a dramatic change in the energy distribution in both segments and directions, and even for the first segment, the energy distribution tends to get better with the increase in tracking error.

4.3. Orbital Thermal Characteristics

After completing the above system construction and energy performance analysis, the orbital thermal analysis of the system is carried out to obtain the temperature distribution of the system and evaluate the thermal control performance of the bottom layer space radiator on the photoelectric conversion system.

A 3D thermal analysis model is established with the finite-element method software FEMAP, as shown in Figure 16a, and the thermo-physical properties of this model are listed in

Table 4. Thermo-physical properties of the model.

Components	Density (kg·m−3)	Specific Heat (J·kg−1·K−1)	Heat Conductivity (W·m−1·K−1)	Emissivity	Absorptivity
Top concentrator	1400	1090	3.2	0.05	0.05
Bottom radiator	1400	880	237	0.93	0.93
Photoelectric conversion system	1910	23.1	81	0.84	0.92
Rear net	1685	1140	5300	---	---
Heat transfer channel	1800	800	3500	---	---

. The heat transfer channel, like the rear net, is made of graphene materials with high heat conductivity [37], and its configuration is shown in Figure 16b. The ring truss and tension ties mainly play a supporting role and have little effect on temperature, so they are ignored in this model. In addition, the transmitting antenna can be thought of as a separate subsystem, with only electrical connections to other components and few thermal connections, so it is also not considered in this thermal analysis model. In order to preliminarily analyze the thermal characteristics of the proposed system in orbit, the bottom layer space radiator adopts symmetric cable networks structure and no tracking error for the whole system.

The main device of the photoelectric conversion system is a thin-film gallium arsenide (GaAs) PV cell with a conversion efficiency of 40% [38]. Then, the temperature distribution of the two-layer ring truss-based SSPS is shown in Figure 17.

Clearly seen in Figure 17a, the maximum temperature is around 130 °C and located on the photoelectric conversion system, in detail, on the second segment. Although the disc-shaped PV cell array only receives direct sunlight, it is affected by the radiative heat transfer of the photoelectric conversion system, making its average temperature reaches 123 °C, as shown in Figure 17b. The temperature of the photoelectric conversion system, as shown in Figure 17c, performs well below 130 °C and shows a temperature distribution of higher temperature on the second segment and lower temperature on the first segment, partly because of the higher energy collection and partly because of the longer distance from the space radiator. Figure 17d shows a good temperature distribution of the bottom layer space radiator with the difference between the maximum and minimum temperature being approximately 7 °C.

The results indicate that the bottom layer radiator exhibits excellent thermal control performance, and can effectively control the temperature of the photoelectric conversion system and ensure that it works within a reasonable temperature range. It should also be noted that the difference between the temperature of the photoelectric conversion system and the temperature of the bottom layer space radiator is still quite large, indicating that a more efficient heat transfer channel is urgently needed and critical for achieving better thermal control.

5. System Parameter Estimation

Table 5. Preliminarily efficiencies and power level for proposed SSPS.

	Efficiency (%)	Power (GW)
Energy collection efficiency	80	10.88
Photoelectric conversion efficiency	40	8.7
DC-to-RF efficiency	80	3.48
Beam collection efficiency	90	2.78
RF-to-DC efficiency	80	2.5
DC power on the ground		2
Total efficiency	18.4

provides the preliminary efficiencies and power level for the proposed SSPS. The system parameters of the 2 GW-level two-layer ring truss-based SSPS are summarized in

Table 6. The system parameters of the 2 GW-level two-layer ring truss-based SSPS.

Parameters	Value
DC power on the ground	2 GW
System efficiency	18.4%
Frequency of the microwave beam	5.8 GHz
Orbit	GEO
Diameter of the concentrator	1600 m
Diameter of transmitting antenna	1000 m
Diameter of rectenna	4000 m

, with a preliminarily estimated weight listed in

Table 7. Preliminary system weight estimation.

	Specification	Mass (MT)
Top layer	Area specific mass goal is 0.05 kg/m2, including the thin-film reflector and front net.	~100
Bottom layer	Area specific mass goal is 0.05 kg/m2, including the thin-film radiator and rear net.	~100
Ring truss and tension ties	Area specific mass goal is 0.3 kg/m2, supporting and tension system.	~610
Photoelectric conversion system	Thin-film GaAs, efficiency goal is 40% and power/mass ratio goal is 4000 W/kg, including carbon fiber skeleton frame.	~870
Power management and distribution	High-temperature superconducting cables, and the inductive power transfer system.	~100
Transmitting antenna	Area specific mass goal is 4 kg/m2 [39], including microwave circuit substrate, power amplifier, and support structure.	~6300
Thermal management	Heat transfer channel	~50
Attitude controller	Electric propulsion	~300
Total		~8430

. It must be noted that the proposed SSPS may be built around 2050 and the parameters are estimated based on future advances in relevant science and technology.

6. Conclusions

A two-layer ring truss-based SSPS is proposed and investigated in this paper at a conceptual study level. This concept has some advantages, which are summarized below.

• High energy collection efficiency
  For the ideal concentrator, a perfect 100% energy collection can be achieved. For the modular division layers of 6 and the tracking error of no more than 2°, 80% energy collection is guaranteed.
• Good energy distribution
  For the ideal concentrator, the indexes of energy distribution on two segments of the photoelectric conversion system are only 0.878 and 0.291, respectively. For the modular division layers of 6, their values are 1.195 and 0.145, respectively.
• Low system mass
  Compared with existed SSPS concept [10,11,12,13], lower system mass is achieved, which means a higher power mass ratio.
• Well-performing thermal control
  With the bottom layer space radiator, the temperature of the photoelectric conversion system can be effectively controlled l below 130 °C.

However, it must be realized that the current research remains at the initial conceptual stage, many problems still need to be investigated in future work, such as:

• Ultra-large structure space deployment
  The entire system is based on a ring truss deployable structure, which can be directly deployed for an MW-level SSPS. However, it is still difficult for a GW-level SSPS, which needs to be studied deeply.
• High precision attitude control.
• The effects of ecliptic angle.