Delft University of Technology A photovoltaic window with sun-tracking shading elements towards maximum power generation and non-glare daylighting

Vertical space bears great potential of solar energy especially for congested urban areas, where photovoltaic (PV) windows in high-rise buildings can contribute to both power generation and daylight harvest. Previous studies on sun-tracking PV windows strayed into the trade-off between tracking performance and mutual shading, failing to achieve the maximum energy generation. Here we first mathematically prove that one-degree-of-freedom (DOF) and two-DOF sun tracking are not able to gain either maximum power generation or non-glare daylighting under reasonable assumptions. Then we derive the optimum rotation angles of the variable-pivot-three-degree-of-freedom (VP-3-DOF) sun-tracking elements and demonstrate that the optimum VP-3-DOF sun tracking can achieve the aforementioned goals. Despite the strict model in this study, the same performance can be achieved by the optimum one-DOF sun tracking with extended PV slats and particular design of cell layout, requiring less complicated mechanical structures. Simulation results show that the annual energy generation and average module efficiency are improved respectively by 27.40% and 19.17% ISee the supplementary document for more information. ∗Corresponding author Email addresses: Y.Gao-1@tudelft.nl (Yuan Gao), M.Zeman@tudelft.nl (Miro Zeman), G.Q.Zhang@tudelft.nl (Guoqi Zhang) Preprint submitted to Journal of Applied Energy March 10, 2018 © 2018 Manuscript version made available under CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/


A Photovoltaic Window with Sun-Tracking Shading Elements towards Maximum Power Generation and
Non-Glare Daylighting 1. Introduction

Motivation
A photovoltaic (PV) window is a daylight-management apparatus with photovoltaic solar cells, modules, or systems embedded on, in, or around a window [1,2,3,4]. PV windows take full advantage of vertical space in congested urban areas, where available horizontal lands are scarce, and local energy consumptions are tremendous. To evaluate the equivalent horizontal area (EHA) of available vertical surfaces, we define R v/h as the ratio of the annual solar energy received on the sunward (e.g. equator-facing for temperate zones) vertical unit area to that received on the horizontal unit area, i.e., where G v,global (t) indicates the global irradiance on a sunward vertical plane; and G h,global (t) indicates the global irradiance on a horizontal plane. The integration time here is an entire year (365 days). According to reliable climate  Table 1. Considering all the urban high-rise buildings around 10 the world, vertical area holds enormous potential for the utilization of solar energy, especially the window area, which is relatively large in modern buildings. Besides the potential of power generation, PV windows also contribute to the energy balance of modern architectural environment via daylight control and heat insulation. 15

Previous studies
The nature of PV windows is to manipulate photons in order to turn incident light partially into electricity and partially into transmitted light. Most reported approaches are implemented by integrating transparent, semi-transparent, regionally transparent PV, or light-directed materials with window glazing. Re- 20 gionally transparent PV windows can be simply formed by distributing available opaque solar cells discretely onto window glasses [6,7], resulting in undesired partially-blocked view and spotted shadows. By shrinking the size of opaque solar cells [8,9,10,11] or punching small holes on the opaque surface [12], the visual effects are possibly improved, however, at the cost of complicating 25 the manufacturing process. Unlike opaque PV materials, semi-transparent solar cells reveal uniform transmittance with colored [13,14,15,16,17,18] or neutrally-colored [19,20,21,22,23,24] appearance. Since photons are selectively transmitted, semi-transparent photovoltaic (STPV) materials [25] present lower efficiency comparing with the corresponding opaque materials. To pursue 30 3 crystal clear appearance, fully transparent solar cells [26,27,28] are developed by selectively harvesting near-infrared (NIR) and ultraviolet (UV) light, leading to lower efficiency than STPV. Another approach is utilizing PV and luminescent solar concentrators (LSCs) [29,30,31,32,33], which also suffer from the low-efficiency problem. Moreover, none of the approaches mentioned above can 35 enable glare protection from direct sunlight.
To overcome the obstacles faced by passive approaches, e.g. low efficiency and sunlight glare, sun-tracking PV windows, which integrated PV materials with active window treatments (e.g. blinds, shutters, etc.), have been designed and investigated by many authors. PV blinds with one degree-of-freedom (DOF) 40 slats are mostly reported due to easy-access experimental setups. Luo et al.
conducted a comparative study of PV blinds by varying the spacing between adjacent blinds (2.5 cm, 3.5 cm, and 4.5 cm) and by varying the slat angle (30 • , 45 • , and 60 • ) [34]. However, the analysis was focusing on the thermal performance of the PV blinds, in stead of the PV power generation. Hu et al. 45 compared three types of building integrated photovoltaic (BIPV) Trombe wall system in terms of their annual performance [35,36]. Comparing with existing PV Trombe walls, the type with PV blinds showed 45% higher electricity saving.
Optimum slat angles were selected from six fixed angels (from 0 • to 75 • in 15 • steps intervals) over three seasons and time of the day. But here PV blinds were 50 integrated with walls instead of windows, failing to contribute to the daylighting of indoor environment. Hong et al. mentioned that the partial shading effect caused by the slats had a nonlinear effect on the amount of electricity generation [37]. to the solar position.

Objectives
A common misconception is that BIPV sun tracking is to orient the PV surface perpendicular to the sun rays. This misconception stems from the suntracking method commonly found in conventional PV power stations, where sun trackers (or solar trackers) are used to orient flat PV panels towards the sun in order to increase the energy collection. During daylight hours, the PV panels are kept in an optimum position perpendicular to the direction of the solar radiation [48]. Theoretical explanation of ubiquitous perpendicular-suntracking methods resides in the basic model of the global irradiance on a tilt 5 plane (G t,global ) [49], i.e., where I dir e is the direct normal (or direct beam) irradiance (DNI) of the sunlight; γ is the angle between the PV surface normal and the incident direction of the sunlight; G h,d is the diffuse horizontal irradiance; R d is the diffuse transposition factor; G t,ground is the ground-reflected irradiance. The product I dir e cos γ represents the direct irradiance on the tilt plane, i.e. G t,beam , which is a dominant component contributing more than 90% of the global irradiance in a cloudless day [50]. The other two components, diffuse (G t,d = G h,d R d ) and groundreflected irradiance, contribute a small proportion to the clear-sky G t,global , and vary with the orientation of the plane. If we ignore the variations of those two components caused by the orientation and take such components as orientationindependent constants because of their small contribution, we can conclude that the maximum G t,global is achieved when γ equals to zero, i.e. the PV surface is perpendicular to the incident sun rays. The maximum G t,global leads to the maximum incident energy per unit time, i.e. the maximum input power P in , because the direct-beam-illuminated PV area S b remains as a constant; i.e.
However, the perpendicular-sun-tracking method is not necessarily applicable to BIPV due to complicated building environment and multiple sun-tracking purposes. Comparing with conventional sun-tracking PVs, building integrated sun-tracking PVs make a profound difference because S b shrinks when shadows appear on the PV surface caused by adjacent elements. In this circumstance, the product of a maximum G t,global with a reduced S b cannot guarantee a maximum P in any more. The shadows on the PV surface not only lead to a diminished S b , but also result in PV partial shading problems, which affect the PV performance, especially the module efficiency η m . η m drops dramatically when uneven shadows are found on series-connected solar cells. PV module performs the best when no shadow casts upon it. To maximize P out at a given time, a 6 straightforward way is keeping the PV surface towards the optimal orientation, where it receives the maximum P in ; and no shadow appears on it, resulting in the maximum η m (Eq. 4). Therefore, one of the purposes of sun tracking is to preserve the maximum P out at every tracking moment, so that the PV module generates the maximum energy E, which is the integral of P out over a certain period of time t (Eq. 5).
As to BIPV, sun tracking is not only aiming at the maximum E, but also the capability to fulfill building functions. For window treatments, two main functions are daylighting and glare protection. In a nutshell, the objectives of 85 building integrated solar tracking for PV window are to receive the maximum P in , to avoid shadows on the PV surface, and to enable daylighting without glare. This work focuses on the solutions to meet these objectives.
In this work, several models were first built up for simulating the performance of PV shading elements under partial shading conditions. Those models include 90 solar irradiance and shadows on the rotated PV surface, solar cells, PV modules, equivalent irradiance, and glare. Then we investigated one-DOF, two-DOF, and three-DOF sun tracking and derived corresponding rotation angles. We summarized simulation results of four sun-tracking methods using irradiance data of Shanghai. Simulations of the optimum variable-pivot-three-degree-of-freedom 95 (VP-3-DOF) and perpendicular sun tracking were conducted using irradiance data of nine big cities around the world. Finally, optimal cell patterns of one-DOF sun tracking were discussed; and an extended application of VP-3-DOF sun tracking in horizontal windows was introduced and demonstrated.  Firstly, an equator-facing window in the sunward side of a high-rise building is defined, which is rarely shaded by surrounding objects from the sun (Fig. 1a).

Methodology
We only consider the buildings located in the temperate zone (between 23.5 • and 66.5 • for both north and south latitude) to ensure the sun stays the same side of the building during the PV-functioning hours for an entire year. Usually, the solar position is defined by the solar altitude α s and the solar azimuth A s in the horizontal coordinate system. Here, we denote the solar position by a unit vector n s (x s , y s , z s ) in corresponding Cartesian coordinate system (Fig. 1b).
Eq. 6 transforms the spherical coordinates into the Cartesian coordinates.
Analogously, the orientation of the PV surface on the shading element is denoted by the altitude α P V and the azimuth A P V of the normal of the PV surface in the horizontal coordinate system, and n P V (x n , y n , z n ) in the Cartesian coordinate system (Fig. 1d). By the aforementioned definitions, we succeed in including n P V and n s in the same three-dimensional Cartesian coordinate system (Fig.   1e). Since n P V only indicates the orientation of the PV surface instead of the exact position of the shading element, here we define the initial position of the shading element (a rectangular PV module) as a vertical plane facing equator (n P V 0 (1, 0, 0)), and let one side of the rectangle be parallel with the horizontal plane. An arbitrary position can be achieved from the initial position by a series of rotations, which is mathematically expressed as a rotation matrix, denoted as R (Fig. 1e). n P V can be derived by Based on above definitions, the following assumptions are made to simplify 115 the physical building structures and the solar radiation models. These assumptions are commonly found in similar studies [46,49], and are not restrictive as compared with the real scenario.
1. The window is an equator-facing rectangle perpendicular to the horizontal plane. The dimensions of the window and window treatments are 120 given, whose thicknesses are ignored to simplify the analyses. Window treatments are mounted interiorly behind the window glass, or within the double-glazing window. The transmittance of the outer glass is high, i.e.
the absorption and reflection of sunlight can be ignored. The PV window treatments are just able to cover the whole window area for the sake of 125 daylight control and privacy protection, i.e. the total area of PV material S P V equals to wl (Fig. 1c).
2. The shading elements in the window treatments rotate simultaneously so that they receive identical solar irradiance, which benefits the performance of series-connected mini modules. Therefore, the position of an individual 130 shading element can be obtained from one target shading element by a simple translation.
3. The total diffuse irradiance on the PV surface from the sky, ground, and interior reflection is isotropic. In other words, the surface receives identical diffuse irradiance from any direction. The ground-reflected irradiance 135 G t,ground is ignored here. We also simply take the irradiance on the shading area as the isotropic diffuse irradiance, i.e. G h,d .
According to aforementioned definitions and assumptions, we can build an isotropic solar irradiance model for the sun-tracking PV window. Since cos γ equals to n T P V · n s , where the symbol T indicates the transpose operator, referring Eqs. 2 and 7, the global irradiance on the tilt PV shading element G t,global is derived as According to Assumption 3, the irradiance on the shading area of the PV surface is G h,d . Therefore, the solar input power on a diffuse partially-shaded plane is derived as where S P V indicates the entire PV area. In this model, the solar position (n s ) of a specific date and time is predictable with the given longitude and latitude [53]; I dir e and G h,d are accessible climate data [5]; n P V 0 and S P V are constants;

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S b can be treated as a function of R for certain geometrical structures of shading elements. Therefore, an optimum R is the key solution to meet aforementioned objectives.
Notably, we consider that the shading elements are covered with lightweight thin-film PV materials. In industry, thin film PV modules contain series-145 connected solar cells formed by laser scribing technology, which makes it difficult to integrate bypass diodes. Therefore, PV modules in shadows are possible to suffer from the partial shading effects. Also, we assume the shape of solar cells is rectangular, which is the standard shape for industrial PV cells and modules, though other geometric design is possible [54].

Models of G t,global and shadows on PV shading elements
According to Eq. 9, the global irradiance on the tilt PV shading element G t,global and shadows on PV shading elements are two key models to derive the input power P in . Furthermore, shadows also affect the module efficiency η m , then consequently affect the output power P out of the PV module (Eq. 4). Here, 155 G t,global and shadows are studied under three types of sun-tracking conditions.

One-DOF sun tracking
In daily life, a most common window treatment with one DOF is a Venetian blind, which usually contains several identical rectangular slats (Fig. 2a). In terms of the model mentioned above, one DOF here refers to the rotation of the rigid PV plane around a single horizontal axis. Mathematically, we use the rotation matrix R y (θ y ) to describe such rotations (Fig. 2b), i.e., where the rotation is around y-axis; θ y equals to α P V . According to Eq. 8, G t,global can be derived as The direct-beam-illuminated PV area on the individual slat S b0 in this model is then derived as T w o-D O F su n tr ac ki ng Dual-axis sun tracking is commonly used in PV power stations since it can maximize P in by positioning PV panels perpendicular to the sunbeam [48].
In this model, two-DOF refers to free rotations of the PV shading element around two axes (Fig. 3a). To achieve free rotations around both axes, we define that shading elements are identical squares; and the centre of each square is its pivot, i.e. the cross point of two axes. According to Assumption 2, we only need to study the rotation of an individual shading element because the positions of other squares can be obtained by simple translations due to fixed pivots. Therefore, we define the centre of the target square as the origin of the Cartesian coordinates. The altitude of the target PV square α P V varies with the rotation around y-axis, denoted by the rotation matrix R y (θ y ) (see Eq. 10).
The azimuth of the target PV square A P V is changed by the rotation around z-axis, denoted by the rotation matrix R z (θ z ), i.e., The orientations of θ y and θ z are illustrated in Fig. 3b. According to Eq. 8, G t,global can be further derived as 13 It's interesting to notice that the one-DOF sun tracking can be regarded as a special case of the two-DOF sun tracking. Comparing with the one-DOF case, the PV shading elements with two DOFs produce more complicated patterns of shadows, whose area has no closed-form solution. In order to calculate S b with arbitrary θ y and θ z , a series of algorithms have been developed considering all

Three-DOF sun tracking
Based on two-DOF rotational elements, one more DOF is added to the rotation of the PV shading elements. As before, the centre of the target PV square is defined as its pivot, i.e. the cross point of the three axes. Note that the position of the pivot does change the relative positions of all squares. Thus, the centre can be used as the pivot, when we study the shadows on the target square from its surrounding neighbors. The three-DOF sun tracking can be taken as three-step rotations and mathematically defined using three rotation matrices ( Fig. 4). The first and second rotations can be mathematically denoted by the rotation matrices R y (θ y ) and R z (θ z ), which are exactly the same as those in the two-DOF model. The third rotation is denoted as R n (θ n ), which means that the target square rotates θ n around its normal n P V clockwise (viewing from the positive direction of n P V ). After the first and second rotations, n P V is derived from the initial PV orientation n P V 0 (1, 0, 0) as The third rotation matrix R n (θ n ) can be expressed as Th re e-DO F su n tra ck in g a b Figure 4: Three-DOF sun tracking and definition of rotation angles wheren P V and n * P V can be obtained by Eq. 19 and 20, i.e.
x n y n x n z n x n y n y 2 n y n z n x n z n y n z n z 2 where n P V (x n , y n , z n ) is given by Eq. 17.
The overall rotation matrix for the target square with three DOF can be expressed as Since the third rotation does not change the normal of the PV square, 175 G t,global in this three-DOF model is the same as that in the two-DOF model (see Eq 16). The aforementioned algorithms are also applicable to the calculation of S b on the three-DOF PV squares.  The two-diode model of the solar cell is used to simulate the PV power generation in certain conditions of irradiance. The equivalent circuit is shown in Fig. 5, where the output current is described as where I ph is the light-induced current. I o1 and I o2 are the reverse saturation currents of diode 1 and diode 2 respectively. V is the voltage across the solar cell electrical ports. R s and R p are the series and parallel resistances respectively. a 1 and a 2 are the quality factors (or called diode emission coefficients) of diode 1 and diode 2 respectively. V T 1,2 denotes the thermal voltage of the PV module having N s cells connected in series, defined as, where k is the Boltzmann constant (1.3806503 × 10 −23 J/K) , T is the temperature of the p-n junction, and q is the electron charge (1.60217646 × 10 −19 C).
Detailed model description can be found in [55]. The solar cell model in MAT-LAB Simulink is simplified by 5 parameters. In this study, the model is param-  To simulate the partial shading effects, the equivalent global irradiance G eq t,global of an individual solar cell is derived as where S i b is the direct-beam-illuminated area on the individual solar cell. S i P V is the total area of the individual solar cell. G eq t,global is a critical input of the partial-shading simulation. S i b can be derived by the aforementioned models of shadows under different sun-tracking methods.

Glare model
To evaluate the visual comfort under different sun-tracking methods, the Rhinoceros model of a reference room is used in this study [52]. In this model, point-in-time glare can be calculated by Grasshopper, a graphical algorithm editor tightly integrated with Rhinoceros.

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Currently, there is a number of different indices for assessing visual comfort [56]. In this study, we use Unified Glare Rating (UGR) and Discomfort Glare Probability (DGP) to evaluate the level of glare.

CIE's Unified Glare Rating (UGR) is defined as
subject to ω s ∈ [3 × 10 −4 , 10 −1 ]sr (25) where the subscript s is used for those quantities depending on the observer with a three-unit step [56].

Discomfort Glare Probability (DGP)
Discomfort Glare Probability (DGP) is is defined as where E v is the vertical eye illuminance. DGP reveals a stronge correlation with the user s response regarding glare perception [56].

One-DOF sun tracking
As discussed in the model of one-DOF PV blind, rectangular and triangular shadows are observed in the typical shading conditions. Usually, the area of triangular shadow on a long narrow slat is negligible due to its relatively small size. Therefore, Eq. 14 is simplified as In this case, according to Eq. 9, the input power P in for all slats in the PV blind is derived as I dir e lwx s + G h,d lw, 0 θ y 2 arctan z s x s ; I dir e lw(x s cos θ y + z s sin θ y ) + G h,d lw, 2 arctan We notice that P in is independent of l 0 , the length of the individual slat.
It means that the number of slats does not affect P in as long as the dimension 235 of the window is given and the triangular shadows are ignored. We also notice 19 that P in remains maximum when θ y ∈ [0, 2 arctan(z s /x s )], which means the quasi-perpendicular position (θ y = arctan(z s /x s )) where G t,global reaches the peak is not the only option for the maximum P in (see Supplementary Note 3 for detailed explanations). To better illustrate G t,global , S b , and P in in different 240 tilt positions, a set of example data is introduced (see Supplementary Note 4) to draw the semicircular color maps (Fig. 7b, c, d). Referring to Eq. 4, the maximum P out is gained with the maximum P in and η m , i.e. no shadow on the PV plane (S b = S P V ). In regard to this one-DOF PV blind, the optimum position is located where θ y equals to 0 or 2 arctan(z s /x s ).

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However, θ y = 0 means the blind stays in the closed position forever, which is not appropriate, because it turns the window into a PV wall and disables the function of daylighting. Therefore, the only feasible option of the optimum θ y is 2 arctan(z s /x s ).
Shadow simulation in a SketchUp [51] model (Fig. 8)   this optimum θ y can effectively avoid rectangular shadows from upper slats.
However, it cannot eliminate triangular shadows from window frames. Such triangular shadows are ignored when we estimate P in because of the small area.
But they cannot be ignored regarding η m due to partial shading effects of PV modules. What is worse, on the other side of the blind, incident sunlight forms 255 glare zones in the interior space. We have also tested the PV blind with vertical slats, whose optimum position (θ z = 2(π −A s )) cannot avoid triangular shadows and glare zones either (see Supplementary Note 3). Therefore, we conclude that PV window treatments with one DOF are not able to achieve the maximum P out and not able to avoid glare in the optimum position in the proposed model.

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Despite the restrictions of this model, improved design of the one-DOF PV blind will be discussed later.

Two-DOF sun tracking
As mentioned above, algorithms are developed for the calculation of two-DOF sun tracking method. By using the same data set, G t,global , S b and P in 265 21 are calculated under a full range of conditions of θ y and θ z (see Supplementary Note 5). As before, we ignore the shadows from walls and window frames at first. Apparently, G t,global hits the peak when the PV plane is perpendicular to the sunbeam (Fig. 9b). However, S b reaches the its minimum value at the very 270 same position (Fig. 9c). As their product, P in remains the maximum within a certain range, instead of a single point (Fig. 9d). This conclusion is similar to that under the one-DOF conditions (Fig. 7d). To have the maximum P out , the optimum position should be located where P in and S b climb to the peak simultaneously.  Figure 10: Two-dimensional maps of G t,global , S b , and P in are observed for G t,global , S b , and P in (Fig. 10a). Therefore, we only focus on the period nearest to the initial position, where three eligible positions are 280 found (Fig. 10b). However, such three positions are located at either θ y = 0 or θ z = 0, i.e., they are equivalent to the one-DOF sun tracking. Specifically, among the three optimum positions in the θ z -θ y coordinates (Fig. 10b), (0, 0) indicates the closed position, which is meaningless for windows as discussed before; (0, 2 arctan(z s /x s )) and (2(π − A s ), 0) represent the optimum positions of 285 the one-DOF sun tracking with horizontal axes and vertical axes respectively.
Therefore, in terms of the optimum position of sun tracking, the PV shading elements with two DOFs perform exactly the same as that with one DOF. Triangular shadows caused by walls and window frames affect the module efficiency the same way as discussed in one-DOF sun tracking. Therefore, we can draw a 290 similar conclusion that PV window treatments with two DOFs are not able to achieve the maximum P out and not able to avoid glare in the optimum position in the proposed model.

Three-DOF sun tracking
Comparing with the two-DOF rotations, the three-DOF sun tracking re- shown in Fig. 11. Therefore, it is difficult to determine the optimum positions by only visual observation. According to Eq. 21, an optimum R yzn (θ y , θ z , θ n ) corresponds to an optimum sun-tracking position, where the maximum P in and η m are observed. Therefore, theoretically, the optimum R yzn (θ y , θ z , θ n ) can 300 be derived based on the following two main conditions. First, there shall be no shadow on the target square from surrounding squares. Second, the input power P in shall stay the maximum, which is the same as that in the initial position.
where k y is an arbitrary integer. By substituting θ y into Eq. 29, θ z is derived as where k z is an arbitrary integer. From Eq. 29, we can also derive θ n , i.e.
To verify the above derivations and determine k y and k z , the same example data and algorithms are applied to calculate S b and P in as discussed previously.
Apparently, θ n does not affect G t,global at all because it does not change α P V and A P V (Fig. 11b). However, it changes the shadows on the squares, and thus influences S b (Fig. 11c). Therefore, P in varies with θ n , θ z , and θ y (Fig. 11d).
From the periodical contours of G t,global , S b , and P in , we can conclude that the solutions can fulfill the optimum conditions. The optimum position nearest to the initial position is found, where k y = 1 and k z = 0 (Fig. 12a). Therefore, the optimum rotation angles for the three-DOF sun tracking are concluded as ), x s y s cos θ y < 0; arccos( 2x 2 s − 1 cos θ y ), x s y s cos θ y 0, Besides the solutions mentioned above, we also found other solutions meeting the optimum conditions. However, those solutions share a common problem that 310 they cannot avoid the shadows from walls and window frames, even without the shadows coming from the surrounding squares (see Supplementary Note 6).
Only the solution provided by Eq. 33 describes the shadows with the same shape as that of the illuminated area through an unshaded window. Therefore, this solution is the only one capable of avoiding shadows from walls and window frames.
However, this solution for the three-DOF sun tracking still suffers from shading, when the pivots lie in the centre of the PV squares. Though the shape of shadows fulfills the requirement, the deviation of shadows caused by the fixed centres leads to interior glares and shadows on the PV squares from walls and window frames (Fig. 12b). Fortunately, a trick is found to eliminate such a deviation by changing the position of the pivot according to the solar position.
Specifically, the bottom left corner A of the target square is used as the pivot, when the solar azimuth A s is less than the azimuth of the window. Similarly, the right bottom corner B is taken as the pivot, when A s is greater than the azimuth of the window (Fig. 12c). Mathematically, to switch the pivot from the centre to the corner A or B, translations are required before and after the rotations. Let Q 0 (x q0 , y q0 , z q0 ) be an arbitrary point on the target square in the initial position, and Q(x q , y q , z q ) be the same point after the rotations. Also, we With Eq. 34, we can obtain the trajectories of the four corners of the target square. Such defined mixed rotations and translations can ensure that no shadow is on the PV squares and no glare appears inside (Fig. 12c). The perfect solution comes into effect with three-step rotations (see Eq. 33) and an 320 ingenious switch of pivots (see Eq. 34). Therefore, we name this sun-tracking method as the variable-pivot-three-DOF (VP-3-DOF) sun tracking. Here we use the phrase "3-DOF" instead of "3-axis" because it is not necessary to actually have three axes in the physical structures as long as the corners of the target square move along the trajectories. Note that the pivots only need to 325 switch one time a day when θ z = 0. The movement of the squares is continuous, as illustrated by the trajectories in Fig. 12c. Therefore, we conclude that the VP-3-DOF sun tracking is able to achieve the maximum power generation and non-glare daylighting for this model. By inputing a set of G eq t,global for each solar cell in the PV module, the simulation models generate hourly output power and module efficiency. Then the * Note that the number of solar cells on the individual shading element for illustation is not necessarily the same as the that for simulation. ** The performance of the one-DOF optimum sun tracking with slats covered by horizontal solar cells is depending on the ratio of the width (w) to the side length (l 0 ) of the slat (see Fig. 7).  sun tracking with horizontal stripes shows competitive results in aspect of E a andη m , it cannot protect glare from the sun properly. Besides, the PV performance of one-DOF optimum sun tracking with horizontal stripes depends on the ratio of the width (w) to the side length (l 0 ) of the slat, i.e. R w/l0 (Fig. 16).

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E a andη m drop dramatically with the decrease of R w/l0 , and they cannot reach the max value obtained by the optimum VP-3-DOF sun tracking. Therefore, we conclude that the optimum VP-3-DOF sun tracking is capable to gain the maximum annual energy generation and annual average efficiency, and also capable to protect glare from the sun.

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Comparing with conventional two-DOF perpendicular sun-tracking method, the proposed optimum VP-3-DOF sun tracking reveal better performance in terms of PV outputs and glare protection. In the example of Shanghai, with the VP-3-DOF sun tracking, E a is improved by 13.12%; andη m is improved by 9.39%. To draw a general conclusion, E a andη m are calculated using the 370 simulation results of the other eight cities in the world. As the average over the nine cities, E a is improved by 27.40%;η m is improved by 19.17% using our proposed optimum VP-3-DOF sun tracking (see Supplementary Note 7).    effects. In terms of the one-DOF sun tracking, triangular shadows caused by walls and window frames are inevitable. In this case, the cell layouts of vertical stripes (Fig 18a) and horizontal stripes (Fig 18c) are affected by partial shading effects. Obviously, vertical stripes suffers more since the series current is limited by the most shaded cell. To alleviate the decrease of PV module efficiency, optimal layouts are applicable if the restriction in Assumption 1 (PV area equals to wl) is relaxed. In regards to vertical stripes, we can leave the shading area blank, i.e. without covering the solar cells (Fig 18b). The length of blank area is 2l tri , where the side length of the triangular shadow l tri is derived as l tri = y s x s sin θ y l 0 .

Point-time glare
To avoid shadows, l tri shall use the maximum among all possible values. As to horizontal stripes, we can extend the width of the slats to w (Fig 18d), where w = w + 2l tri .
Theoretically, the improved layout of horizontal stripes is able to achieve the maximum power generation and non-glare daylighting with one-DOF sun track-33 ing (θ y = 2 arctan(z s /x s )). Comparing with the optimum VP-3-DOF sun tracking, the optimum one-DOF sun tracking with the improved layout of horizontal stripes achieve the same performance with simpler mechanical structures. How-385 ever, the extension of slats costs more PV material, whose area is 2l tri l for the window. In contrast, the optimum VP-3-DOF sun tracking does not rely on improved cell layout and costs less PV material to achieve the same goal. The mechanical realization of the VP-3-DOF motion is out of the scope of 390 the current study. Some recommendations to realize the VP-3-DOF motion are given as follows. Firstly, it is not necessary to have physical axes to achieve the rotation. The only requirement is to follow the trajectories provided by our mathematical model. Secondly, since it is an interior lightweight application, the use of fine translucent wires can be considered to actuate PV shading elements, 395 similar to but more sophisticated than what the normal window blinds are using.
Thirdly, electrical cables can be considered to be installed along the wires to interconnect the PV modules.
Besides vertical windows, the proposed VP-3-DOF sun tracking is also applicable to the horizontal sun roof. In terms of special scenarios, e.g. a glass 400 greenhouse, the roof area is large and the incident sunlight need to be controlled.
Comparing with the case with vertical windows (Fig. 19a), the optimum solution to the case with horizontal windows (Fig. 19b) can be derived in a similar way. Besides square PV shading elements, the rectangular PV shading elements can also apply to the VP-3-DOF sun tracking. It has been demonstrated by shadow simulations with SketchUp ( Fig. 19c & d).

Conclusions
In this paper, we have investigated the performance of the one-degree-offreedom (one-DOF), two-DOF, and three-DOF sun tracking using our proposed irradiance model. Two solutions, the optimum one-DOF sun tracking with the improved layout of horizontal stripes and optimum VP-3-DOF sun tracking, 415 enable the sun-tracking PV window to achieve the maximum power generation and non-glare daylighting at the same time. Comparing with conventional perpendicular sun tracking, the proposed sun tracking methods improve the annual energy generation by 27.40% and the annual average efficiency by 19.17% as the average over nine cities in the world. Such module-level improvements are more 420 pronounced than that triggered by new materials and process in most studies.
Comparing the two proposed solutions, the optimum one-DOF sun tracking with extended PV slats and particular cell layout requires simpler mechanical structure of rotations; while the optimum VP-3-DOF sun tracking requires less area of PV material and simpler design of cell layout.
Besides the benefits in energy generation, both solutions provide the building occupants with comfortable diffuse daylight and open exterior view. As an extended application, the optimum VP-3-DOF sun tracking for PV shading elements on horizontal glass roof of a greenhouse is capable to maximize the power generation, and also provides the crops with certain amount of diffuse daylight.

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An economic PV horticultural system can be built by applying the proposed sun-tracking method, which can increase the production of crops and reduce the energy consumption. Theoretically, the optimum variable-pivot-three-DOF suntracking method is applicable to any occasions requiring the maximum power generation and the access to the natural diffuse light.