### 8.2.6. Case #8 – WHITE_wC_00

<figure><img src="image_150.png" /><figcaption>Figure 144 Case #8 – int. irradiance correlation modelled values as a function of measured values</figcaption></figure>

<figure><img src="image_151.png" /><figcaption>Figure 145 Case #8 – modelled transmittances as a function of solar azimuth angle</figcaption></figure>

<table><caption>Table 22 Validation case #8 – results and specifications</caption><thead><tr><th colspan="4">Case specifications</th></tr></thead><tbody><tr><td>case #8</td><td>case description:</td><td>win.state:</td><td>window closed</td></tr><tr><td>case name:</td><td></td><td>slats:</td><td>white, type K80</td></tr><tr><td>WHITE_wC_00</td><td></td><td>slat-angle:</td><td>0°</td></tr><tr><td colspan="4">Transmittance τ<sub>coGSys</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [1]:</td><td></td><td>errors:</td><td>MAEτ: 0.005<br>MEτ: 0.002<br>SEτ: 0.009</td></tr><tr><td>0.105 | 0.117 | 0.123 | 0.188</td><td></td><td></td><td></td></tr><tr><td colspan="4">Internal irradiance E<sub>int model</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [W/m²]:</td><td></td><td>errors:</td><td>MAE<sub>E</sub>: 1.7 W/m²</td></tr><tr><td>2.3 | 52.6 | 74.2 | 90.0</td><td></td><td></td><td></td></tr><tr><td colspan="4">Internal irradiance correlation model vs. measurement:</td></tr><tr><td>number of points:</td><td>136</td><td></td><td></td></tr><tr><td>lin. regression through origin</td><td>y=0.9981·x</td><td>R²:</td><td>0.998</td></tr><tr><td>lin. regression</td><td>y=1.0011·x - 0.1970</td><td>R²:</td><td>0.987</td></tr><tr><td colspan="4">Measurement specifications (min | q.1 | q.3 | max):</td></tr><tr><td>sun azimuth range [°]:</td><td>117.3 | 147.1 | 221.7 | 280.1</td><td></td><td></td></tr><tr><td>sun elevation range [°]:</td><td>3.1 | 35.9 | 46.2 | 52.0</td><td></td><td></td></tr><tr><td>incidence angle range [°]:</td><td>46.0 | 51.7 | 59.8 | 100.1</td><td></td><td></td></tr><tr><td>diffuse ratio (DHI/GHI) range [1]:</td><td>0.16 | 0.25 | 0.47 | 1.00</td><td></td><td></td></tr><tr><td>meas. period:</td><td>2020.08.28 | 2020.08.30 | 2020.09.13 | 2020.09.14</td><td></td><td></td></tr></tbody></table>
### 8.2.7. Case #10 – Z-SILVER_wC_00

<figure><img src="image_152.png" /><figcaption>Figure 146 Case #10 – int. irradiance correlation modelled values as a function of measured values</figcaption></figure>

<figure><img src="image_153.png" /><figcaption>Figure 147 Case #10 – modelled transmittances as a function of solar azimuth angle</figcaption></figure>

<table><caption>Table 23 Validation case #10 – results and specifications</caption><thead><tr><th colspan="4">Case specifications</th></tr></thead><tbody><tr><td>case #10<br>case name:<br>Z-SILVER_wC_00</td><td>case description:</td><td>win.state:<br>slats:<br>slat-angle:</td><td>window closed<br>silver, type VZ90<br>0°</td></tr><tr><td colspan="4">Transmittance τ<sub>co,sys</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [1]:<br>0.161 | 0.163 | 0.164 | 0.165</td><td>errors:</td><td>MAEτ:<br>MEτ:<br>SEτ:</td><td>0.012<br>0.010<br>0.008</td></tr><tr><td colspan="4">Internal irradiance E<sub>int.model</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [W/m<sup>2</sup>]:<br>3.7 | 4.4 | 5.6 | 10.6</td><td>errors:</td><td>MAE<sub>E</sub>:</td><td>0.3 W/m<sup>2</sup></td></tr><tr><td colspan="4">Internal irradiance correlation model vs. measurement:</td></tr><tr><td>lin. regression through origin<br>number of points: 6</td><td>y=0.9572·x</td><td>R<sup>2</sup>:</td><td>0.998</td></tr><tr><td>lin. regression<br>number of points: 6</td><td>y=0.9858·x - 0.1873</td><td>R<sup>2</sup>:</td><td>0.990</td></tr><tr><td colspan="4">Measurement specifications (min | q.1 | q.3 | max):</td></tr><tr><td>sun azimuth range [°]:<br>sun elevation range [°]:<br>incidence angle range [°]:<br>diffuse ratio (DHI/GHI) range [1]:<br>meas. period:</td><td>123.1 | 125.9 | 217.5 | 226.6<br>25.6 | 27.3 | 32.0 | 35.1<br>49.5 | 53.3 | 58.6 | 60.5<br>1.00 | 1.00 | 1.00 | 1.00<br>2020.09.25 | 2020.09.25 | 2020.09.25 | 2020.09.25</td><td></td><td></td></tr></tbody></table>
### 8.2.8. Result overview and discussion

When assessing the comparison results, it must be considered that the virtual model measurements are not compared against an ideal standard but against a complex empirical measurement depending on many variables. While the modelled results can be considered noise-free, the empirical measurement contains significant noise. For this reason, the (signed) mean errors $ME_{\tau}$, indicating systematic deviation, are considered most relevant to assess the quality of the model, provided the statistical errors are in an acceptable range. The latter can be assessed by the standard deviation of the errors $SE_{\tau}$, as well as the regression analysis, provided that sufficient data points are available.

<figure><img src="image_154.png" /><figcaption>Figure 148 Mean error of transmittance $ME_{\tau}$ (bars) and standard deviation of errors $SE_{\tau}$ (red lines) for all test cases</figcaption></figure>

<figure><img src="image_155.png" /><figcaption>Figure 149 Mean absolute error of transmittance $MAE_{\tau}$ for all test cases</figcaption></figure>
Figure 148 and Figure 149 provide an overview of the error measures in graphical form. Note that the mean absolute error $MAE_τ$ covers both statistical errors as well as systematic deviations. Additionally, the values are provided in Table 24. Note that, based on their definition, the stated errors are absolute errors. Hence, like their base quantity transmittance, they are related to the incidence irradiance.

<table><caption>Table 24 Overview of determined transmittance errors</caption><thead><tr><th>group</th><th>case no</th><th>case name</th><th>meas. count</th><th>ME<sub>τ</sub></th><th>SE<sub>τ</sub></th><th>MAE<sub>τ</sub></th></tr></thead><tbody><tr><td rowspan="3">unsh. w.c.</td><td>1</td><td>UNSHADED_SOUTH</td><td>3347</td><td>1,3%</td><td>4,0%</td><td>2,4%</td></tr><tr><td>2</td><td>UNSHADED_WEST</td><td>6091</td><td>0,3%</td><td>4,0%</td><td>3,0%</td></tr><tr><td colspan="2">group mean values:</td><td>4719,0</td><td>0,8%</td><td>4,0%</td><td>2,7%</td></tr><tr><td rowspan="11">window closed, shaded</td><td>3</td><td>ANTHRA_wC_00</td><td>206</td><td>0,3%</td><td>1,6%</td><td>1,0%</td></tr><tr><td>4</td><td>ANTHRA_wC_45</td><td>58</td><td>0,1%</td><td>0,0%</td><td>0,1%</td></tr><tr><td>5</td><td>ANTHRA_wC_m45</td><td>42</td><td>1,2%</td><td>1,7%</td><td>1,7%</td></tr><tr><td>6</td><td>SILVER_wC_00</td><td>78</td><td>-0,6%</td><td>1,4%</td><td>1,1%</td></tr><tr><td>7</td><td>SILVER_wC_m45</td><td>64</td><td>1,2%</td><td>2,2%</td><td>2,2%</td></tr><tr><td>8</td><td>WHITE_wC_00</td><td>136</td><td>0,2%</td><td>0,9%</td><td>0,5%</td></tr><tr><td>9</td><td>WHITE_wC_53</td><td>22</td><td>1,7%</td><td>0,5%</td><td>1,7%</td></tr><tr><td>10</td><td>Z-SILVER_wC_00</td><td>6</td><td>1,0%</td><td>0,8%</td><td>1,2%</td></tr><tr><td>11</td><td>Z-SILVER_wC_58</td><td>40</td><td>-0,5%</td><td>0,3%</td><td>0,6%</td></tr><tr><td colspan="2">group mean values:</td><td>72,4</td><td>0,5%</td><td>1,1%</td><td>1,1%</td></tr><tr><td rowspan="16">window open, shaded</td><td>12</td><td>ANTHRA_wO_00</td><td>123</td><td>0,2%</td><td>1,3%</td><td>1,0%</td></tr><tr><td>13</td><td>ANTHRA_wO_45</td><td>50</td><td>1,1%</td><td>0,7%</td><td>1,1%</td></tr><tr><td>14</td><td>ANTHRA_wO_m45</td><td>44</td><td>-2,3%</td><td>3,0%</td><td>3,3%</td></tr><tr><td>15</td><td>SILVER_wO_45</td><td>27</td><td>-3,1%</td><td>1,0%</td><td>3,1%</td></tr><tr><td>16</td><td>SILVER_wO_m21</td><td>23</td><td>1,0%</td><td>2,7%</td><td>2,2%</td></tr><tr><td>17</td><td>SILVER_wO_m45</td><td>27</td><td>1,2%</td><td>3,5%</td><td>2,4%</td></tr><tr><td>18</td><td>WHITE_wO_00</td><td>37</td><td>-2,2%</td><td>1,8%</td><td>2,4%</td></tr><tr><td>19</td><td>WHITE_wO_45</td><td>24</td><td>2,0%</td><td>1,1%</td><td>2,0%</td></tr><tr><td>20</td><td>WHITE_wO_m45</td><td>24</td><td>-6,1%</td><td>4,5%</td><td>6,7%</td></tr><tr><td>21</td><td>Z-SILVER_wO_00</td><td>15</td><td>-0,9%</td><td>3,5%</td><td>3,0%</td></tr><tr><td>22</td><td>Z-SILVER_wO_45</td><td>2</td><td>-1,3%</td><td>0,1%</td><td>1,3%</td></tr><tr><td>23</td><td>Z-SILVER_wO_58</td><td>13</td><td>-1,2%</td><td>0,4%</td><td>1,2%</td></tr><tr><td colspan="2">group mean values:</td><td>34,1</td><td>-1,0%</td><td>2,0%</td><td>2,5%</td></tr></tbody></table>

### 8.2.8.1. Unshaded cases

It has to be noted that the evaluations for the unshaded case are not trivial, as they deviate significantly from the simple on-plane irradiance. As can be seen in Figure 135 and Figure 138, the determined transmittance values for the two unshaded cases vary significantly in the range of approx. 20% to 45%. This is a result of the strong angular dependence that the transmittance of the tripled-glazed window exhibits. Beyond that, the shading of the reveal limits the amount of diffuse irradiance reaching the measurement area (located at the centre of the glazing). If only diffuse radiation reaches the window, either because direct sunlight originates from the opposite directions or due to cloud cover, the modelled transmittance converges to a constant value of around 34%. This represents the
average value for all directions of the visible sky and ground segments. The convergence to a constant value for diffuse irradiance can also be observed in the following shadowed cases.

The two unshaded test cases generally showed the highest level of variance. This can partly be attributed to the generally higher level of transmittance but also seems to indicate that reflections significantly impair the measurement. If no shading is present, specular reflections in the environment can directly reach the vertically oriented sensor. Considering the related, relatively high standard deviations of errors (4% each), the mean errors of determined transmittances were low, with only 1.3% for the south-facing case and 0.3% for the west-facing case.

### 8.2.8.2. Shaded cases with closed window

The results of these practically most relevant cases show good agreement of the modelled and measured transmittances. Most errors range well below 2%, with standard deviations in the same range. Only one case (#9) indicates a systematic error, as the mean error exceeds the standard deviation. However, only 22 measurements were available for the related case. The determined transmittances again showed a strong variation depending on the slat angle, sun angle and sky condition. The resulting transmittances range from approximately 1% to 20%. The lowest values are achieved for the anthracite slats in a low-diffuse sky condition, whereas test cases with horizontal slats in diffuse conditions show the highest transmittance.

### 8.2.8.3. Shaded cases with an open window

The measurements taken with open windows generally exhibit the highest errors and variances. Approximately half of the measurements show systematic deviations in the sense that the mean error exceeds the related standard deviation. Mainly two reasons could be the potential cause for the errors. Firstly, opening the window negatively impacted the internal scattering situation, as the black curtain behind the *PyroScanner* had to be partially retracted. An additional blanket was then used to cover the window sash, which was opened at an angle slightly greater than 90 degrees. The now visible parts of the window frame additionally contained some metallic parts that were left uncovered. Beyond the scattering, the open window also allowed external air to reach the measurement equipment leading to thermal gradients. Additional and more significant thermal gradients arose as the internal pyranometer and electronic equipment were exposed to the strongest direct solar irradiance in these cases. Accordingly, the highest deviations were observed in case #20 with a negative slat angle. In this shading configuration, the internal sensor was exposed to up to 900 W/m² direct normal irradiance with a strong temporal (spatial) fluctuation behind the tilted slats.

### 8.2.8.4. Albedo and ground reflected irradiance

The evaluation of the measurements showed that the combination of the bright metallic roof on the neighbouring building and the relatively dark floor is not captured appropriately by the idealised constant albedo approach. Performing adaptations of the albedo values used for the evaluation significantly improved the results regarding correlation and absolute error (not shown here). Nonetheless, for the purpose of validation, three constant albedo values (see section 8.1.11) were used for all evaluations performed here. The observed parameter sensitivity regarding the ground-reflected irradiance demonstrates the need for more detailed modelling of the surrounding environment, as discussed in section 6.2.4. Implementing such a refined approach that considers the geometry and reflectance values in the environment is potentially the most efficient method to increase the general accuracy of calculations for solar-induced energy components.
### 8.2.8.5. Conclusion PyroScanner measurements

Though the empirical measurements proved to be challenging, as they are influenced by many external factors, the comparison of the empirical measurement results with the model measurement results is satisfactory. The mean transmittance values that were determined in the measurement for each case ranged from approximately 1% (ANTHRA\_wC\_45) to approximately 75% (WHITE\_wO\_m45) depending on the material, slat type, slat angle and presence of the window. In addition, variations of the transmittance of 10 to 20 % and beyond (in terms of “absolute” transmittance) were recorded for the individual test cases. These variations depend on the actual irradiance profiles, mainly determined by cloud cover and solar angles.

The *RadiCal* method is able to predict the transmittances values that vary significantly depending on the shading type and configuration, as well as on the irradiance profile, mostly with accuracies in the range of 1 % to 2% in terms of “absolute” transmittance (see Table 24). Nevertheless, some cases show systematic deviations that would require further investigation. Potential sources for the errors can either be linked to the empirical measurements, e.g. in the form of thermal gradients (as “window-open” measurements generally exhibited higher errors). They could, however, also be caused by reflections in the environment or testing facility that were not covered by the model. The simplification implied by the constant albedo model generally restricts the achievable (see section 8.2.8.4). Further, it must be considered that the detailed spectral and angular information, as required to apply the RadiCal method to its full extent, were not available, neither for the glazing nor for the materials of the shading devices. The inversion method developed in the PhD (sections 4.12 and 4.13) were applied to derive the required parameters of available data. Hence, the *PyroScanner* validation measurements prove that the method can be applied with limited data to predict the real-world performance of shaded or unshaded glazings with reasonable accuracy. It has already been demonstrated that the implemented optical models are able to provide high accuracy if the detailed data required to model the surface is available (see section 4.9.3).
# 9. Application case

In order to demonstrate the feasibility and potential of the method, a typical application case is presented in this chapter. The dynamic energy balance model of a triple-glazed IGU fenestration with Venetian blinds is established based on a detailed geometric model. The energy model is evaluated with a high temporal resolution, and the results are used to determine key performance figures for different boundary conditions. The calculation was presented and published at the *ibpsa BauSim 2022 conference* (Rüdisser, 2022).

## 9.1. Target object

The window model corresponds to the actual window used for the validation measurements of the method (see chapter 8). The model comprising the window-reveal, window-frame, triple-glazing and Venetian blinds represents a commonly used building detail of modern constructions. Despite appearing simple at first glance, the detail is, in fact, very complex from an optical point of view. The incoming direct and diffuse radiation is first scattered at the surfaces of the blinds, where a fraction of the irradi-

ance is reflected towards the glazing. The glazing consists of three glass panes, the first and the last of which are coated with a low-E-film on the side facing towards the inner glass pane. The entire configuration, consisting of the blinds, two coated glass panes and one uncoated glass pane, gives rise to a series of interreflections among the different layers. Finally, a significant fraction of the irradiance is absorbed in the four layers, while a part is transmitted into the interior. In order to accurately determine the total transmitted power (or energy), a detailed knowledge of these five energy flows is essential, as not only the directly transmitted irradiance but also the secondary heat flows will contribute significantly (see also section 3.1). Therefore, the model is well suited to demonstrate the advantages of the *RadiCal* method, which allows for accurately determining transmittance and absorptance for any parts of the target objects. The window's specifications can be found in Table 25; a *RadiCal* rendering of the window is presented in Figure 116 (section 7.11).

<table><caption>Table 25 Specifications of fenestration model</caption><tr><td>geometry:</td><td>reveal: 1.214 x 1.450 m<br>glazing: 1.024 x 1.229 m<br>blind cover: 1.210 x 0.192 m<br>depths (relative to glazing front):<br>cover: 0.152 m, reveal: 0.105 m</td></tr><tr><td>shading:</td><td>Venetian blinds, 8 cm, rounded, white<br>PE-coated aluminium, mass of all slats:<br>2.68 kg</td></tr><tr><td>glazing:</td><td>Triple IGU 4/18/4/18/4, low-E coating<br>(3%) on layer 2 and 5, t<sub>e</sub>=0.47, q<sub>i</sub>=0.06,<br>g=0.53, U<sub>g</sub>=0.53 W/m<sup>2</sup>K, mass:<br>3x13.2 kg</td></tr></table>
## 9.2. Model and workflow

A representation of the entire workflow and model is presented in Figure 150. In the first step to creating the window’s energy model, all SIOPs relevant to the energy balance are generated. In the analysed case, the absorption-SIOPs for the three window panes, the SIOP for the solar direct transmission to the interior and the absorption SIOP of the blinds are required. The latter is used to model the blind’s surface temperature, which is considered in the longwave radiative exchange with the first glazing surface. As mentioned, all SIOPs can be generated within a single raytracing scan of the object (see section 7.10).

<figure><img src="image_156.png" /><figcaption>Figure 150 Overview of model and work-flow: top: 3D model of the target object, centre: determined SIOPs, bottom: dynamic thermal simulation model overview</figcaption></figure>

The fenestrations' dynamic thermal simulation model is established in the next step. In order to have full control over the simulation parameters and avoid any third-party-related issues, the author has implemented a simple nodal dynamic thermal simulation platform that is used for this application.
It allows setting up models based on thermal nodes and heat transfers based on a library containing a variety of classes. The simulation platform features thermal mass and heat input modelling. The latter is, of course, essential to model the solar inputs to the specific nodes. The dynamic simulation platform has been validated against other validated tools based on simple reference cases.

The simulation model for the presented application case can consider the following boundary conditions: air temperature, sky and ground temperature (for longwave radiative heat exchange), wind speed (for convective heat exchange), and interior temperature. The model comprises seven variable thermal nodes. The first one represents the blind temperature, whereas the other six represent the internal and external surfaces of the three glass panes. While the solar inputs and view factors are based on detailed three-dimensional raytracing data, the energy model is set up in one dimension, i.e. one temperature is assigned to each layer. Due to the relatively high thermal conductivity of the glass panes, this approach is reasonable and commonly applied. However, the same approach can be used to create a spatially refined model, either by vertical stratification or two-dimensional segmentation.

All radiative, conductive and convective heat exchanges were modelled according to the specifications provided by the standards EN 673 (EN 673, 2011) and ISO 6946 (EN ISO 6946, 2015). In order to determine the heat transfers of the IGU accurately, all relevant parameters, i.e. surface temperatures, gas temperatures and gas properties, were calculated dynamically for each time step. This leads to a significant variation in the heat flows within the IGU and, consequently, in the thermal transmittance of the entire glazing. In order to be able to model transient effects, thermal inertia is considered for each node based on the component's relevant mass and specific heat capacity. An overview of the simulation model with all relevant nodes and heat transfers is depicted at the bottom of Figure 150.

The SIOPs are evaluated using the Perez irradiance model as specified in section 6.2.3. The generic Perez incident profiles are created with irradiance data provided by weather data time series. As outlined in section 4.5, the local solar position is calculated using the SG2 (Blanc *et al.*, 2012) algorithm in the implemented module (section 4.5.2). Subsequently, the Perez irradiance components are determined based on the measured irradiance data (see section 6.2.5). Finally, the SIOPs can be evaluated to provide the power inputs to the specific nodes with high temporal resolution.

The derived solar power input time series and additional time-series data relevant to the air body, i.e. air temperature and wind speed, are used to update the model's boundary conditions on each time step. Finally, the equation system is solved on every time step, and the resulting node temperatures are used to derive all relevant heat flows. The entire time series is processed twice, with all solar inputs set to zero in the first run. The resulting heat flows contained in this time series that excludes any solar inputs are further used as a reference to identify heat flows resulting from solar gains. This approach, therefore, follows the principles expressed in equation (2).
## 9.3. Evaluation with temporally highly resolved data

<figure><img src="image_157.png" /><figcaption>Figure 151 Dynamic thermal simulation results of IGU window shaded with slats at 0°, based on measured irradiance data with high temporal resolution (1min)</figcaption></figure>

Typical climate reference data provides time series with hourly time intervals. To be able to perform dynamic simulations with sub-hourly intervals, the data is typically interpolated. However, this interpolated data usually does not reflect the true dynamic behaviour of the environmental parameters. This is especially true for irradiance parameters, where, e.g. broken cloud cover leads to characteristic time profiles with high gradients. In order to capture the reaction of the glazing in this environment, measured data of the AEE INTEC weather station (AEE INTEC, 2023) with a high temporal resolution of 1 min is used for the simulation. Figure 151 shows the related simulation results. The simulated case assumes a shading of the window with a horizontal (0°) slat position from December 22^th^ to December 25^th^. It can be seen how the individual panes specifically absorb the incoming radiation. In the analysed period, 11.1% of the total incident irradiance is absorbed by pane 1, 3.2% by pane 2 and 3.8% by pane 3. The absorption of the inner two panes is significantly lower than that of the first pane, as the absorption in the glass and coating of the outer pane means that already spectrally filtered light will reach the inner panes. The centre pane shows less absorption than the inner pane, as only the latter has a low-E coating applied.

The 18.1% of the incident irradiance absorbed by all three panes effectively leads to a solar-related (secondary) internal heat flux of 8.9%. It can be attributed directly to the absorbed radiation and the resulting increase of the related heat transfers. Correlation analysis reveals that the effective time lag between the incident solar radiation and the resulting internal heat flow due to the thermal masses involved is as long as 90 minutes on average. This represents an interesting finding of the analysis performed with high temporal resolution.
9.4. Evaluation of energetic performance

In order to evaluate the total energetic performance of the single window element, two specific loca-
tions with very different irradiance profiles were selected as examples: Hamburg in Germany and
Sevilla in Spain. Further, two orientations (south and west) and two different shading states were
considered: unshaded and fully shaded with slats in a static horizontal position (0°). EPW files
(Lawrie *et al.*, 2019) representing typical meteorological years can be imported directly. Data of
climate.OneBuilding.org (Crawley *et al.*, 2022) was used to model the local climate for the chosen
locations. The reference years *DEU_HH_Hamburg-Fuhlsbuttel_101470 TRY2010_03_Jahr* and
*ESP_AN_Sevilla.083900_TMYx* were used to reflect the climate of Hamburg and Sevilla. For the
sake of comparability and simplicity, a constant indoor temperature of 22 °C was assumed for the
entire year. The annual simulation runs were processed with 1-minute time steps. The total calcula-
tion time for the whole year was less than one second on a single CPU. The simulation model's high
performance is ideal for carrying out parametric studies or control-strategy optimisations efficiently.

<figure><img src="image_158.png" /><figcaption>*Figure 152 Total heat-balance for the IGU unit calculated from energy flows*
*determined with dynamic thermal simulation based on the RadiCal method*
*(total values represented as bars, components represented as curves)*</figcaption></figure>

Figure 152 shows the results of the simulation runs displayed as biweekly average values for the total power values transmitted through the window. Due to the high insulation effect of the triple glazing, all simulation cases show positive annual totals. However, a significant seasonal variation can be seen. While the west-oriented window in Hamburg shows total energy losses between November and March, the south-facing window in Sevilla clearly shows the highest total gains, with minima occurring in the summer months. The significantly different seasonal pattern results from the lower latitude and clearer sky conditions compared to Hamburg. The strong angular dependence of shading-, projection- and reflection-effects result in remarkably low solar gains for the unshaded south-oriented window in Sevilla during the spring and summer months.
## 9.5. Determination of effective performance parameters

For further in-depth analysis, the simulation results are related to the incident irradiance to calculate relative transmittance values. By dividing the total transmitted power and the power of the solar-induced secondary heat-flows by the incident solar power, the coefficients *solar direct transmittance* $τ$, the *secondary internal heat factor* $q_i$ and their sum, representing the *total solar energy transmittance* or *g-value* can be determined. The coefficients are denoted with the subscript *sys* (system) to distinguish them from the corresponding coefficients as defined in the standard EN 410 (EN 410, 2011). In order to determine the incident power in this “system approach”, a reference area must be defined. This could be either the area of the entire reveal, the area of the window (glazing and frame) or only the externally visible glazing area. Depending on the application and objective of the study, either approach may be appropriate. For the present analysis, the reference provided by the glazing area seems most suitable for comparison with the standard calculations. As pointed out in section 2.6, the standard values are calculated solely for the case of normal incidence and do not contain any shading or reflection of the window reveal. In contrast, the coefficients calculated here consider all relevant “real-world effects”. Therefore, they show a significant deviation from the standard values. In order to illustrate the magnitude of this deviation, average values for the coefficients were computed for the entire reference years on a daily basis.

<figure><img src="image_159.png" /><figcaption>Figure 153 Daily averages for the entire IGU
for total solar energy transmittance (g), solar direct transmittance ($τ_e$) and secondary internal heat factor ($q_i$) computed of energy flows determined in dynamic thermal simulation</figcaption></figure>

The results of this inversion approach to derive the effective performance parameters of the glazings using the simulation results are depicted in Figure 153. It can be seen that the solar direct transmittance $τ$, and consequently the total solar energy transmittance $g$, range significantly below the standard values of 0.47 and 0.53, respectively (see Table 25). This reduction was expected and can be attributed to two factors: first, high proportions of diffuse radiation cause a noise-like variation, which can be seen especially well for the Hamburg location. Second, higher incidence angles significantly reduce the transmittance, which can be seen well in the case ‘Sevilla/south/unshaded’, for example. Note that, unlike in the heat balances presented in Fig. 152, the geometric projection effect
(cosine-law) is already compensated since the external irradiance on the reference plane is used to calculate coefficients. Hence, the observed incidence-angle-dependent reduction of the transmittance is solely a consequence of reveal-shading, increased reflectivity of the glazing surfaces and higher absorption within the window panes.

Interestingly, in contrast to solar direct transmittance, the secondary internal heat factor shows only a limited variance and ranges relatively close to the normal incidence value calculated according to the standard (0.06). A comparison of the polar plot representation of the absorption *SIOPs* of glass panes vs. the transmittance *SIOP* for the entire glazing (not depicted here) reveals that relative to transmittance absorption increases with growing incidence angles. The longer optical paths and higher number of internal reflections within the glass panes cause this significantly higher absorption at higher angles. Therefore, the secondary heat flux is less sensitive to the angle of incidence, as the increased reflectivity that occurs at higher angles is partly compensated by the increased absorption.
# 10. Discussion

## 10.1. State-of-the-art methods and novel method

The state-of-the-art analysis, part of objective 1 and covered in Chapter 3, showed that the currently applied methods have various limitations. The methods proposed by the current standards are based on highly simplified models that can lead to significant inaccuracies. The main issues of the models used in the standards are their restriction to near-normal incidence and inadequate modelling of diffuse and direct radiation components (see sections 3.4 and 3.5). The methods were originally designed for performance benchmarking of building components under lab conditions. They should not be applied for real-world irradiance profiles, as required for building performance simulation. The application case presented in this work (see Figure 153) demonstrates that a constant total solar energy transmittance coefficient (“g-value”) approach is not applicable, as g-values determined by inversion vary in the range of several tens of per cent depending on the solar angles and irradiance profile.

More advanced methods that consider the angular profile of the incidence light mostly rely on a combined approach. The properties of the glazings are determined based on empirical data and advanced models by applying the methods implemented in LBNL Window Software (Curcija et al., 2018). The combination of the glazing with the complex geometry of shading elements or surrounding parts is modelled with raytracing tools, particularly Radiance (Radiance Community, 2022). The resulting bidirectional scattering distribution data (BSDF) described in the form of Klems matrices (Klems, 1994a) is then further processed to directional solar heat gain coefficients (DSHGC) or directly evaluated in building performance simulation tools. The DSHGC method requires the assumption of thermal properties and boundary conditions that have to be anticipated for the application they are later applied to, which leads to inaccuracies, see e.g. (Wermke, 2021). Hence, a strict separation of the optical model and thermal domain should be followed (see section 2.5).

However, beyond this issue, the BSDF/Klems approach shows some systematic limitations that arise from its definition. The limitations are related to simplifications regarding the spectral, spatial, angular and polarisation properties of the transmitted or reflected light beam. The impact of these limitations becomes more significant if a system involving two or more BSDF layers is modelled. This is a typical case, as the commonly used raytracing tools do not allow accurate modelling of glazings, while LBNL Window does not allow the modelling of detailed 3D shading layers or other relevant three-dimensional geometry. Hence, both methods are commonly applied to form a combined system of two or more BSDF layers.

The RadiCal method proposed here allows accurate, microscopic modelling of coated and uncoated transparent layers as part of a complex, detailed three-dimensional structure. This allows a unified approach as all relevant optical processes can be considered based on a single, comprehensive three-dimensional model. The optical modelling of the glazings is potentially more accurate than the LBNL Window method, as the angularly and spectrally resolved model considers the entire global radiation spectrum and interference or polarisation effects on material boundaries are intrinsic parts of the model. This determines the angular and spectral properties of the scattering on the modelled surface, as well as the outcome of any further scattering events. However, the achievable accuracy relies on the quality of the available optical data for the modelled surfaces.

The RadiCal method proposes a strict separation of the thermal and optical domains. The newly introduced solar incidence operator (SIOP) provides an interface to pass the optical data to thermal
simulations. The definition of the *SIOP* allows for an accurate modelling of transmitted and absorbed powers or energies. As the output of the *SIOP* is a scalar value, it cannot readily be used for lighting applications. For this purpose, the *RadiCal* raytracer can still be used to generate BSDF information based on a single comprehensive and detailed three-dimensional optical model.

## 10.2. Physical optics integration

Objective 2 of the PhD was to develop a raytracer that incorporates models of physical optics. The application of models based on fundamental laws of optics and electrodynamics allows for achieving more accurate results with a reduced demand for input parameters compared to data-driven models. In addition, more fundamental, physically based models potentially allow easier adaptation to new materials, new material combinations or future requirements.

The implementation of physical optics is achieved by applying more complex models for scattering events, while the propagation of light is still modelled based on geometrical optics, i.e. linear propagation of discrete rays. Modelling light propagation based on the principles of physical optics would require solving Maxwell's equations with high spatial resolution, which is virtually impossible for large objects with complex geometry. Deviating from classical geometrical optics, the polarisation state of the ray of light is tracked, and the attenuation in a medium is considered during light propagation.

In order to model scattering events occurring on material interfaces, the complex-valued Fresnel equations are solved. The solution of Fresnel's equations is embedded in Müller matrices and Stokes vector algebra to be able to consider and track the ray's polarisation state. Similar approaches are usually only applied on a microscopic scale, e.g. for ellipsometry in material science or on a macroscopic scale in astrophysics, but have not yet been applied in the field of building science. By applying this method, many important spectral and angular dependencies of reflection, absorption and refraction arise naturally. The accuracy of the approach relies, however, on the availability of spectrally resolved material data. If data containing the required information is unavailable, two inversion methods that allow deriving this data of available measurement data are proposed and implemented in the method. Alternatively, the application of estimated or generic constants in fallback cases allows for a gradual downgrading of the optical models.

A matrix method for modelling one or more layers of thin-films applied to the surface is part of the implementation. The feature is required for modelling thin coatings that are applied on surfaces to alter their optical behaviour by exploiting interference effects. It is an essential part of the method as it is required to model coated glasses. Most modern glazings have spectrally selective coatings applied to one or more of its glass panes (e.g. low-E or solar control coatings).

Polarisation of light is considered throughout all models. This is not a supplemental feature of the method, but an integral part, as solving Fresnel equations in the general form requires considering the state of polarisation. Any energetic impact of the occurring polarisations is, therefore, implicitly contained in the model. However, its quantitative significance has to be specifically analysed and demonstrated in future studies.

In order to implement the physical optics models, Monte-Carlo methods have to be applied. Finding analytic solutions to the models is impossible as this would require solving a complex set of nested multidimensional integrals. However, the stochastic Monte-Carlo approach allows for an efficient solution, considering all spectral, spatial, directional and polarisation dependencies.

Of the more commonly occurring and energetically relevant optical processes, only diffraction could not be integrated into the current raytracing model. As demonstrated in section 4.3.5, diffraction
usually plays a minor role, with the important exception of microstructured materials, particularly
textile materials. To still be able to model such surfaces, the scattering behaviour resulting from
diffraction must be modelled on a macroscopic scale, e.g. by deriving models based on measured
BSDF data.

10.3. Required material data and inversion approaches

Specific material data is required to apply the physical-optics-based modelling to its full potential. This is, in particular, the spectral complex-valued index of refraction (cRI) that combines information on the refractive index of the material and its absorption properties. While this information is used in many physical optics applications and material sciences, it is often not readily available for building components. Generally, it is possible to determine the complex-valued index of refraction in direct measurements by applying spectral ellipsometry. Nevertheless, the necessary measurement equipment is not very common. Therefore, measured cRI information can rarely be found for the relevant global radiation spectrum.

In order to address this issue, two inversion approaches have been developed in the course of the
PhD (sections 4.12 and 4.13). The methods are able to derive the cRI functions of available meas-
urement data providing spectral information. Such data is generally determined using UV-Vis-NIR
spectrometers that, unlike ellipsometers, are quite common. Spectrometers are used to determine the
absorption, reflection or transmittance of materials with high spectral resolution. The measured
spectra are used to derive cRI functions in the proposed inversion methods. In essence, this is achieved
by mimicking the measurement principle in virtual measurements. The unknown cRI function is
parameterized and iteratively optimised until the measured reflectance, absorption or transmittance
spectra match the empirically determined spectra.

In the test cases covered in this work (sections 4.11.3 and 4.12.3), the methods have proven to be
able to fit the provided spectra with good precision. Moreover, the values determined in the numerical
process are physically plausible and exhibit resemblance with available references. Nonetheless, the
inversion will generally not have unique solutions. They will be able to reproduce the provided re-
flectance, absorption and/or transmittance spectra for near-normal incidence with high precision.
But this does not imply that the modelled values for oblique angles correctly match the true behav-
iour of the material.

The performed comparisons with known directional dependencies (sections 4.11.3 and 4.12.3) and
the appearance of the materials in the renderings indicate that the angular behaviour is plausible.
However, in order to achieve a higher level of accuracy, measurement data for oblique angles is
required. The inversion results for uncoated and opaque materials will generally show a good agree-
ment with the true behaviour, as the physical processes are less complex and the degrees of freedom
in the optimisation are limited. However, the inversion results for coated materials should be treated
cautiously, as the involved thin-film interference effects are linked to a spectrally and angularly
selective behaviour. Beyond that, the required degrees of freedom in the optimisation are considerably
higher, as the material parameters for all model layers must be determined.

The discussed issues regarding the uncertainty of the material parameters must not be seen as a flaw
of the *RadiCal* method but as a requirement arising from the potentially more accurate modelling.
Simpler scattering models can be applied if the required data remains unavailable and the use of
estimated values is inappropriate. The implemented fallback principle always allows the currently
applied models to be degraded based on fewer, single or constant material parameters.
## 10.4. Results of validation steps

Since the underlying theory and implementation of the *RadiCal* method is multifaceted and complex, a high priority was given to validation steps and test cases. The performed tests should prove in a cumulative way that the applied models and equations, their implementation and their mostly numerical solutions are correct. The following validation and testing steps are contained in the work:

1. Sampling of the global radiation spectrum

2. Solar position algorithm

3. Import and processing of spectral refractive index data

4. Solution of complex-valued Fresnel equations

5. Implementation of Stokes-Vector algebra

6. Müller-matrix calculus

7. Microfacet roughness model

8. Lambert-diffuse subsurface scattering

9. Inversion to determine spectral complex-valued refractive index function

10. Inversion to determine spectral subsurface reflectance function

11. Validation of Perez irradiance module

12. Distribution of directions on Fibonacci hemisphere

13. Validation of *SIOP* evaluation vs. standard model

14. Validation of *SIOP* evaluation vs. BPS software

15. Validation of complex models via backward raytracing/rendering

16. Full-system *PyroScanner* empirical validation

Of these steps, steps 6 and 16 are considered of particular importance. Both tests cumulatively include many other tests, cover large parts of the entire work and are performed against empirical and practical data. The results of the Müller-matrix applications (6.) show that the modelling of optical scattering is able to reproduce empirical data with high precision on a microscopic scale. The accuracy is, of course, determined by the quality of the available material data that can be considered high for these tests. In contrast, the full-system validation (16.) demonstrates that the method can be applied at a large scale, involving multiple materials and objects, complex geometry and material data with limited quality. Beyond that, the full-system validation represents a typical application for which the method was designed. Both validation steps showed satisfactory results. The results of the Müller-matrix applications exceeded expectations, while the full-system validation showed some inaccuracies. Presumably, most of the issues are linked to the quality of the empirical data, as measuring transmittance values under natural conditions is challenging. Nevertheless, the significantly varying transmittance values for different shading configurations and environmental conditions were still predicted with satisfactory accuracy in the range of a few per cent and below. The results indicate that applying a more complex model for the ground-reflected irradiance is essential to achieve higher accuracy. The current definition of the *SIOP* allows the integration of such an approach straightforwardly (see 6.2.4).
## 10.5. Limitations

The implementation of the raytracer is performed based on object-oriented programming (OOP) models. The inheritance and polymorphism principles of OOP allow for efficiently replacing, adapting or enhancing the scattering models for new materials or surfaces. Currently, apart from a few auxiliary generic classes, only one scattering class (TLSISroughPol, see section 7.9) is used to model materials. While the integrated features allow modelling a wide range of opaque or transparent, coated or uncoated and rough or flat surfaces, it will still be necessary to add new models for modelling practically relevant surfaces realistically.

The statistical approaches used to model the surface's roughness as well as the implemented diffuse subsurface reflection model are based on the assumption of randomly distributed surface orientations or scattering particles. Consequently, the implemented models are unable to correctly describe scattering processes on surfaces that exhibit significant regular structures on the microscopic or nanoscopic scale.

The methods to derive new scattering classes can essentially be divided into three types, which can be referred to as white-, grey- and black-box approaches. The white box approach can be applied to model micro-structured surfaces, like coarse textiles or perforated metals. In this approach, light scattering is explicitly modelled based on a microscopic three-dimensional model of the surface. Apart from the three-dimensional model, these models generally require limited empirical data to calibrate their scattering properties to real-world materials. However, the validity of the approach is limited to geometric structures of larger scales, as diffraction becomes more significant when the relevant apertures are small.

To model materials dominated by smaller-scale structures, appropriate statistical models can be applied to model the distribution and orientation of scattering events. Again, measured data can be used to calibrate the model parameters. Finally, in the black-box approach, scattering classes can be derived based on measured data only. The information from angular measurement samples (BSDF) can be used to model the related scattering events directly. However, polarisation information is generally disregarded in this approach, as it is usually not contained in BSDF measurements. Beyond the need for additional scattering classes, the demand for suitable material data can pose an issue. The topic is discussed in detail above (section 10.3).

Another limitation is not related to the optical modelling but to the *SIOP* operator, which is used to pass the raytracing information to other applications. According to its definition, the output of the operator is limited to one dimension. Therefore, the concept cannot readily be used to replace the currently applied Klems matrices. The directionally resolved output information is a fundamental requirement for most lighting applications. It is, however, not relevant for the energetic simulations this work focuses on. However, since the *RadiCal* method intrinsically covers the visible spectrum of light, it may be well worthwhile to consider its application in this field. The *RadiCal* raytracer can be used to generate Klems matrices, allowing its immediate use for lighting applications. While this approach has the advantage of being compatible with the existing methods, it would be desirable to exploit the advantages of the *SIOP* definition in the lighting field as well. One potential approach involves generating a series of *SIOPs* covering the hemisphere for the outgoing light with the desired resolution. However, this would still correspond to a discrete distribution on the output side. A more complex solution would involve the development of a more complex, multidimensional representation still based on spherical harmonics. E.g., the resulting function values of the current approach could provide coefficients for another spherical harmonics expansion describing the directional distribution of the outgoing light.
## 10.6. Integration into current frameworks

A fundamental goal, not of this work but of potential subsequent developments, is providing the method to a broad group of potential users in research and industry. The immediate path for achieving this is to provide the method in the form of a tool that allows the generation of the Klems matrices in the commonly used format. This allows the immediate application of the *RadiCal* method for both lighting and thermal simulation. For the latter case, however, the application of the *SIOP* interface, as described in this work, is highly recommended, as it offers the discussed advantages. The implementation required to evaluate *SIOPs* is concise and can be provided in multiple programming languages. An integration for Modelica (Dymola) simulations is already available and applied in a research project.

## 10.7. Discussion with respect to research questions

In the following, the research questions raised in section 1.2 are addressed in brief.

### 1. How can the accuracy of modelling solar-induced energy flows in building science be enhanced?

In the present work, the limitations of current models were identified and discussed, and suitable, more complex models were proposed, implemented and tested. Further, a new operator and workflow to integrate the optical simulation results into building performance simulation were developed and implemented. Finally, the validity and potential of the proposed method were demonstrated in validation cases and an exemplary application case.

#### 1.1. Which optical processes, beyond simple geometrical optics, are relevant, and how can they be considered? (e.g. thin-film interference, polarisation, surface roughness)

A more general approach that is intrinsically able to cover significant optical effects was identified. Essentially, it is based on solving Fresnel's equations in their complex-valued form, considering the wavelength of light, its polarisation state, as well as electrodynamic properties of the materials (see section 10.2). Additional models are integrated in the approach to consider surface roughness and subsurface scattering of materials.

#### 1.2. Is it possible to create a physical-optics-based raytracer that can be used to efficiently evaluate the solar properties of relevant building components, in particular for combinations of glazings and shading devices?

A full implementation of the raytracer was performed in the course of the PhD. The raytracer applies principles of physical optics and relies on Monte-Carlo simulation methods (see Chapter 7). Its capabilities for accurate modelling of coated (and uncoated) glazings allow unified modelling of shaded windows, façade elements or technical applications.

#### 1.3. Can the raytracer be designed in a way that allows a seamless transition or downgrading to existing models?

For the case of missing material data, the implementation provides several fallback cases that allow a gradual downgrading of the model. Commonly applied constants or spectrally integrated values can be applied in these generic models.
1.4. If and how can the highly accurate raytracing be deployed to practical application in dynamic building performance simulation?

A novel method, referred to as SIOP – solar incidence operator, was developed for this purpose (see Chapter 6). Its feasibility and potential are demonstrated in application and validation cases in the present work. The operator allows efficient deployment of the detailed data gathered in the raytracing scans. It, therefore, enables dynamic thermal simulation tools to perform solar-related calculations with raytracing accuracy at almost no computational cost.
# 11. Conclusion

## 11.1. Summary

In order to improve the modelling of solar-related energy flows for building applications, a novel method has been developed, implemented and tested. First, the potential inaccuracies and limitations of the currently available and applied models and methods were analysed. Currently applied approaches to model solar gains through transparent surfaces of buildings mostly rely on combinations of at least two methods. Limitations are related to the individual methods but also arise from combining the methods. In order to overcome these restrictions, a unified approach that applies models of physical optics based on a single, detailed 3D CAD model was developed. Energetically significant optical processes were identified, and the related advanced optical models, primarily used in material science and astronomy, were adopted for their use in the building science context. A Monte Carlo raytracing method has been developed to incorporate the implemented optical models. While the propagation of light is still modelled based on the principles of geometrical optics, the scattering events, i.e. reflection, absorption and transmission, are modelled based on the relevant electromagnetic models. In order to do this, light's polarisation state and spectral distribution must be considered in all modelled processes. Further, a matrix-based thin-film method allows modelling interference effects on surfaces covered with one or multiple thin layers of different materials. The feature is essential to the method, as modern glazings rely on such spectrally selective coatings. In addition to the physical optics approach, which fundamentally relies on solving Fresnel's equations in their complex-valued form, a statistical roughness model and a generic subsurface-scattering model had to be added to allow for modelling of practically relevant surfaces. All optical models were implemented by applying the concepts of object-oriented programming. This allows straightforward future adaptation or extension of the models for specific surfaces or refined modelling.

In the applied physical optics models, the materials are characterised by their electrodynamic properties in the form of the spectrally resolved complex-valued index of refraction (cRI). Although the underlying physical quantities forming the cRI are accessible to measurement, they are often not readily available. To address this issue, two approaches allowing the application of the method if the required material data is initially unavailable were implemented. Firstly, two inversion methods were developed that allow deriving cRI functions of measured reflection, transmission or absorption spectra. Additionally, a fallback approach has been implemented if the required optical properties remain unavailable. It allows a gradual downgrading of the optical models to generic versions relying only on a single or a few parameters.

The combination of in-depth optical modelling of different materials and consideration of detailed geometry constitutes a complex and multifaceted model that cannot be evaluated in analytical closed-form. Nonetheless, it can still be solved efficiently by means of stochastic approaches. For this reason, a raytracer fundamentally based on Monte Carlo methods was developed, implemented and tested in the course of the PhD.

In addition to the novel method proposed for optical modelling, a new interface for deploying the detailed raytracing data to building performance simulation was developed. The method is called *SIOP* (solar incidence operator) and is based on spherical harmonics, known from quantum physics. The *SIOP* concept allows providing the detailed optical data gathered in the raytracing process to thermal simulation in an efficient and comprehensive manner. Hence, *SIOPs* effectively enable sim-
ulation tools to perform solar calculations with raytracing accuracy, but virtually at no computa-
tional costs. SIOPs are not only used to model solar gains resulting from irradiance directly trans-
mitted through transparent surfaces but also provide information on the power absorbed in any
relevant component of the façade or window. The combined information allows accurate dynamic
simulation of all relevant temperatures and energy flows within a building component for any given
boundary conditions. Thus, all temperature-related or thermal-mass-related dependencies can be
modelled accurately.

Many validation and test cases are contained throughout the work. The tests are performed at different levels of the model to prove and demonstrate the validity, not only of the models and equations but also of their algorithmic implementation. In an essential validation, advanced virtual optical measurements, e.g. ellipsometry, are compared against available empirical data. Further, the evaluation of the SIOP based on the Perez diffuse sky model is validated against the standard calculation performed with a building performance simulation tool. At the highest level, the entire RadiCal workflow is applied to perform virtual measurements of a triple-glazed window equipped with four different shading devices. Real-world measurements performed over several months with varying solar incidence profiles are compared with virtual measurements performed on the modelled window. A dedicated measurement device and method were developed in the course of the PhD specifically for this purpose. This was necessary in order to be able to measure the spatially highly inhomogeneous irradiance field behind the shaded window under real-world conditions. Considering the generally higher error level linked to such measurements, the comparison shows a good level of agreement. The deviation of the calculated from the measured effective transmittances for 22 different shading configurations with variable irradiance profiles is in the range of a few per cent and below. Finally, the application of the method is demonstrated in a typical use case. The thermal performance of a triple-glazed window is accurately determined for different locations and orientations.

The established novel method can be used to address a wide range of research and industrial appli-
cations, reaching beyond the standard task of building performance simulations. Its full potential
must be explored in further studies.

11.2. Key findings

The work performed in the PhD demonstrates that more accurate modelling of solar-induced energy flows in building science is possible if optical processes are modelled on a more fundamental level, and a strict separation of optical modelling and thermal simulation is maintained. In detail, the most significant findings are:

11.2.1. Physical optics approach

In order to accurately determine solar-related energy flows, a detailed and spectrally resolved mod-
elling of all relevant optical processes is required. The optical model needs to consider detailed three-
dimensional geometry and the materials' optical properties object. Hence, a raytracing process is
required. Currently used raytracers are data-driven, as the optical properties of surfaces are mostly
modelled based on empirical BSDF data. For building science applications, it is of essential im-
portance that comprehensive and accurate models are available for the transparent surface, particu-
larly for uncoated and coated glasses. Accurate modelling of such surfaces requires the consideration
of the fundamental nature of light, i.e. its electrodynamic character. This can be performed most
efficiently by solving Fresnel's equations in complex-valued form. Compared to currently applied
methods, the approach relies on a different type of material specification. This information is often
not readily available, which can hamper the application of the method. However, the use of physical
optics models leads to significant advantages as many significant dependencies are intrinsically con-
tained and do not require additional assumptions or models. These are, in particular, the angular
and spectral dependencies of scattering processes, i.e. reflection, transmittance/refraction and ab-
sorption. Since the models are based on fundamental physical principles, they generally require less
input data than data-driven models. Their application potentially leads to plausible and realistic
approximations for the case that limited material data is available or generic values are used.

11.2.2. Modelling of coatings

Accurate modelling of coatings applied on surfaces is a challenging but essential task, as they are an
integral component of modern glazings (e.g. low-E or solar protection layers). Interference effects in
coatings are responsible for intended and unintended spectrally and directionally selective properties
of these coatings. In order to model the related optical properties efficiently, it is necessary to apply
an extended Fresnel approach that is able to consider thin-films.

11.2.3. Additionally required models

In order to model practically relevant surfaces, it is essential to additionally consider the surface's
roughness as well as subsurface scattering processes. Two generic models are currently part of the
method. The integrated parameters allow their adaptation for modelling a wide range of surfaces.
However, it will be necessary to develop additional models to consider specific surfaces, e.g. micro-
or nanostructured surfaces and metallic paints.

11.2.4. Consideration of polarisation

Polarisation is a fundamental, yet often neglected, property of light. Polarisation generally occurs on every scattering event at oblique incidence angles. In order to fully apply the Fresnel approach, the consideration of polarisation is a requirement. For the intended application, the concepts of the four-dimensional Stokes vectors and Müller matrices are well suited, as they allow efficient and straightforward integration of polarisation into the raytracing process and scattering models.

11.2.5. Inversion methods to determine optical properties

As the optical properties of materials are commonly not available in the required form, two inversion
methods were implemented to derive them from available data. Although the methods rely on a
stochastic optimisation of models with a relatively high degree of freedom, the results are consistent
and indicate a plausible physical behaviour. The ambiguity of the results depends on the amount of
input information provided. Spectral measurements at non-normal directions a rarely performed but
would significantly increase the quality of the inversion results. If only near-normal measurements
are available for the inversions, the results have to be treated cautiously. However, they potentially
lead to an increased quality of the overall simulation results, as plausible spectral, angular and
polarisation dependencies are intrinsically contained in the models.

11.2.6. Separation of optical and thermal domain

The concept of directional solar heat gain coefficients (DSHGC) is regarded as problematic, and its
application should be questioned. The derivation of DSHGC values requires anticipation of results
and boundary conditions of the dynamic thermal simulations that they will be applied to in later
simulations. This constitutes a circular reference. From a fundamental point of view, it can be stated
that the energy flows induced by solar radiation have significant thermal impacts, while temperatures
or temperature gradients generally have little to no impact on the related solar processes. Based on
this unilateral interaction, it is appropriate to separate the two domains entirely. The optical evalu-
ations must be solved first, independent of any thermal considerations. The results of the optical
model characterising only transmitted and absorbed energy flows induced by solar radiation should
then be deployed to dynamic thermal simulation in an appropriate form.

11.2.7. SIOPs – representation of optical properties

The currently applied form to provide the results of the optical calculations in the form of 145
discrete values cannot satisfactorily cover the entire hemisphere. A smooth and continuous angular
representation of the results is favourable as the temporal and spatial integrations and differentia-
tions required in the evaluation processes also result in smooth and continuous functions. This in-
creases both the accuracy and performance of the related dynamic thermal simulations.

A method that provides smooth and continuous functions on the hemisphere is proposed in this
work. The representation is based on spherical harmonics expansion and allows a flexible and efficient
way to store, deploy and evaluate the results of the optical calculations.

11.2.8. Variability and dependencies of solar-induced energy flows

The performed full-system validation cases and application examples demonstrate the remarkable
variability of solar-induced energy flows, thus demonstrating the need for accurate modelling. Even
for the supposedly trivial case of an unshaded triple-glazed window, the optical and solar total trans-
mittance values generally exhibit diurnal and seasonal variations in the double-digit percentage
range. The most significant dependencies are those on the solar angles and on the sky condition.
However, the results also demonstrate that a more refined modelling of the local environment is
necessary to increase the accuracy of solar calculations further.

11.2.9. Usefulness of rendering

While initially not an objective of the thesis, rendering via backward raytracing turned out to be a
substantial component of the method. Creating visual representations of the target objects allows for
validating and demonstrating the quality of light modelling and the level of detail the method
achieves. The quality of the backward raytracing used for the renderings represents the quality of
the forward raytracing applied for the energetic calculations. Creating a single rendering requires
evaluating billions of scattering events covering an extensive range of spectral and directional cases.
Hence, the creation of renderings is a valuable validation step regarding the models' accuracy, the
implementation's functionality and the correctness of material assignments.
## 11.3. Further directions

The methods and tools developed in this PhD open up a wide field for further research and application. An essential and immediate goal is to engage the research community in this process. For this purpose, providing access to the method through transparent, easy-to-use and well-documented tools is a primary objective for the near future.

Furthermore, the following tasks are planned for the foreseeable future:

* *Intermodel comparisons*

As this work focuses primarily on empirical validation, it will be essential to perform comparative studies in which the results of the *RadiCal* method are compared against commonly used calculation methods. The comparisons should involve building performance simulation and also focus on the related heating and cooling energy deviations.

* *Derivation of daylighting parameters*

Essential daylighting parameters, like luminance, daylight factor or glare measures, can be derived from the already implemented *RadiCal* simulation approach if the irradiance is weighted according to the luminous efficiency function and suitable detector surfaces are considered in the model.

* *Studies on the modelling of coated glazings*

In order to analyse and improve the accuracy and efficiency of the inversion method proposed in section 4.12, further research is necessary. Primarily angularly resolved measured data is required for this purpose.

* *Studies of polarisation-related effects*

The consideration of polarisation is a key distinguishing feature of the novel method and an intrinsic part of all calculations. In order to meet the already perceptible interest (and scepticism), it is important to transparently demonstrate light polarisation's qualitative and quantitative significance for various applications.

* *Development/implementation of a refined model for reflected radiance*

Theoretical considerations and the results of the empirical measurements demonstrated the importance of detailed modelling for radiation reflected in the environment. A new feature will allow detailed modelling of this environment based on 3D models or photogrammetric information.

* *New models for microstructured surfaces*

The object-oriented approach allows the seamless integration of new scattering models into the raytracer. New models can be derived by developing statistical models for the microscopic scattering events based on measured BSDF information. The models created in such a process also contain polarisation information, allowing a full application of the *RadiCal* method. Textile surfaces and metallic paints should be the first new integrations.

* *Volume scattering models*

Currently, only attenuation is considered during propagation in media. Additionally, the redirection, i.e. diffusion of light during its propagation in certain media, e.g. specific glasses and liquids, should be considered. While relevant only for special applications, the integration of volume scattering is easy to accomplish and will, therefore, be performed in the foreseeable future.
List of figures

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  <tbody>
    <tr>
      <td>Figure 1 Graphical overview of the thesis structure</td>
      <td>4</td>
    </tr>
    <tr>
      <td>Figure 2 Overview of implemented modules and addressed objectives</td>
      <td>6</td>
    </tr>
    <tr>
      <td>Figure 3 RadiCal application workflow</td>
      <td>10</td>
    </tr>
    <tr>
      <td>Figure 4 Comparison of the workflow for modelling a shaded triple-glazed window</td>
      <td>12</td>
    </tr>
    <tr>
      <td>Figure 5 Components relevant for the total solar transmittance</td>
      <td>14</td>
    </tr>
    <tr>
      <td>Figure 6 Klems matrix representation of LBNL BSDF Viewer</td>
      <td>16</td>
    </tr>
    <tr>
      <td>Figure 7 Thermal model for the two additional heat gain components</td>
      <td>19</td>
    </tr>
    <tr>
      <td>Figure 8 Workflow for modelling with LBNL Window and Radiance/genBSFD</td>
      <td>20</td>
    </tr>
    <tr>
      <td>Figure 9 Separation into disturbed and undisturbed components according to ISO 15099</td>
      <td>26</td>
    </tr>
    <tr>
      <td>Figure 10 Timeline of essential persons influencing the current model and understanding of light</td>
      <td>29</td>
    </tr>
    <tr>
      <td>Figure 11 Overview of the global radiation spectrum and relevant spectral ranges</td>
      <td>31</td>
    </tr>
    <tr>
      <td>Figure 12 Principle of superposition; "collision" of two wavelets with different wavelengths</td>
      <td>32</td>
    </tr>
    <tr>
      <td>Figure 13 Wave packets in space and time domain and corresponding frequency spectra</td>
      <td>34</td>
    </tr>
    <tr>
      <td>Figure 14 Different polarization states resulting from different phase shifts</td>
      <td>36</td>
    </tr>
    <tr>
      <td>Figure 15 Polarisation caused by reflection and refraction</td>
      <td>37</td>
    </tr>
    <tr>
      <td>Figure 16 Angular Fresnel reflection/transmission coefficients</td>
      <td>38</td>
    </tr>
    <tr>
      <td>Figure 17 Photo of polarised reflection on window glass</td>
      <td>39</td>
    </tr>
    <tr>
      <td>Figure 18 Effective scattering regimes by particle size and wavelength figure based on (Petty, 2006)</td>
      <td>40</td>
    </tr>
    <tr>
      <td>Figure 19 Specular and diffuse reflection on car paint</td>
      <td>41</td>
    </tr>
    <tr>
      <td>Figure 20 Huygens-Fresnel principle</td>
      <td>43</td>
    </tr>
    <tr>
      <td>Figure 21 Diffraction explained by Huygens-Fresnel principle</td>
      <td>44</td>
    </tr>
    <tr>
      <td>Figure 22 Intensity profile of Fraunhofer diffraction model</td>
      <td>44</td>
    </tr>
    <tr>
      <td>Figure 23 Tangent of half intensity diffraction angle θh for various slit sizes and wavelengths</td>
      <td>45</td>
    </tr>
    <tr>
      <td>Figure 24 Intensity profile resulting from diffraction at edge of semi-infinite screen at 1 m distance</td>
      <td>45</td>
    </tr>
    <tr>
      <td>Figure 25 Illumination pattern on the floor</td>
      <td>46</td>
    </tr>
    <tr>
      <td>Figure 26 Examples for true diffraction</td>
      <td>47</td>
    </tr>
    <tr>
      <td>Figure 27 Attenuation in medium according to Beer-Lambert's law</td>
      <td>48</td>
    </tr>
    <tr>
      <td>Figure 28 Coherent vs. non-coherent processes</td>
      <td>49</td>
    </tr>
    <tr>
      <td>Figure 29 Components of the ASTM G173 reference spectrum and black-body radiation at 5778 K</td>
      <td>51</td>
    </tr>
    <tr>
      <td>Figure 30 Inversion of CDF to provide random wavelengths following the desired spectrum</td>
      <td>52</td>
    </tr>
    <tr>
      <td>Figure 31 Validation result for global radiation sampling module</td>
      <td>54</td>
    </tr>
    <tr>
      <td>Figure 32 Overview of refractive index functions in the global radiation spectral range</td>
      <td>61</td>
    </tr>
    <tr>
      <td>Figure 33 Thin-film assembly</td>
      <td>64</td>
    </tr>
    <tr>
      <td>Figure 34 Angular reflection of material/air interfaces for three wavelengths and two polarisation states</td>
      <td>66</td>
    </tr>
    <tr>
      <td>Figure 35 Reflection and transmission on material interface considering polarization</td>
      <td>70</td>
    </tr>
    <tr>
      <td>Figure 36 "Example 1" of Tetsu Anan working paper (Anan, 2018)</td>
      <td>73</td>
    </tr>
    <tr>
      <td>Figure 37 Angular dependence of Müller matrix elements</td>
      <td>75</td>
    </tr>
    <tr>
      <td>Figure 38 Sciamachy scan mirror reflectance as a function of wavelength</td>
      <td>76</td>
    </tr>
    <tr>
      <td>Figure 39 Ellipsometric angles (Ψ, Δ) as function of SiO<sub>2</sub> film thickness</td>
      <td>77</td>
    </tr>
    <tr>
      <td>Figure 40 Müller matrix ellipsometer</td>
      <td>78</td>
    </tr>
    <tr>
      <td>Figure 41 Ellipsometric angles as function of angle</td>
      <td>79</td>
    </tr>
    <tr>
      <td>Figure 42 Microfacet and macrosurface normal vectors</td>
      <td>80</td>
    </tr>
    <tr>
      <td>Figure 43 BSDF functions resulting of implemented microfacet roughness model for float glass surface with different roughness</td>
      <td>84</td>
    </tr>
    <tr>
      <td>Figure 44 BSDF functions resulting of implemented microfacet roughness model for float glass surface with roughness 0.3 and different incidence angles [red: 80°, violet: 45°, grey: 10°]</td>
      <td>85</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>Figure 45 BSDF functions resulting of implemented microfacet roughness model for silver glass surface</td>
      <td>85</td>
    </tr>
    <tr>
      <td>Figure 46 BSDF functions resulting of implemented microfacet roughness model for silver surface with roughness 0.3 and different incidence angles</td>
      <td>86</td>
    </tr>
    <tr>
      <td>Figure 47 BSDF function silver surface with roughness 1.0 and incidence angle 45°</td>
      <td>86</td>
    </tr>
    <tr>
      <td>Figure 48 Spectral dir.-hemi. reflectance for silver and float glass surface as a function of surface roughness α</td>
      <td>87</td>
    </tr>
    <tr>
      <td>Figure 49 Various surface roughnesses rendered with RadiCal algorithm</td>
      <td>88</td>
    </tr>
    <tr>
      <td>Figure 50 Reflection types currently covered by the RadiCal implementation</td>
      <td>89</td>
    </tr>
    <tr>
      <td>Figure 51 Frequency/sample density of randomly generated Lambert directions as a function of z-coordinates</td>
      <td>92</td>
    </tr>
    <tr>
      <td>Figure 52 Frequency/sample density of randomly generated Lambert directions as a function of angle to surface normal</td>
      <td>92</td>
    </tr>
    <tr>
      <td>Figure 53 Applied "thick-slab" model</td>
      <td>93</td>
    </tr>
    <tr>
      <td>Figure 54 Applied parametric models for uncoated (left)</td>
      <td>96</td>
    </tr>
    <tr>
      <td>Figure 55 Determined complex-valued refractive index function of uncoated glass pane (Planibel Clearlite)</td>
      <td>99</td>
    </tr>
    <tr>
      <td>Figure 56 Empirical complex-valued refractive index function of clear-glass determined by Rubin</td>
      <td>99</td>
    </tr>
    <tr>
      <td>Figure 57 Uncoated window pane - measured T/R-spectra vs. model results using the derived cRI</td>
      <td>100</td>
    </tr>
    <tr>
      <td>Figure 58 Uncoated window pane - angular dep. of transmittance and reflectance based on derived cRI</td>
      <td>100</td>
    </tr>
    <tr>
      <td>Figure 59 Determined cRI functions for the layers of the two-layer substitute model for the low-E coating (iplus Top 1.1)</td>
      <td>101</td>
    </tr>
    <tr>
      <td>Figure 60 Measured cRI function of 20 nm-thick silver film deposited on SiO2 substrate</td>
      <td>101</td>
    </tr>
    <tr>
      <td>Figure 61 Low-E coated window pane - measured T/Rf/Rb-spectra vs. model results</td>
      <td>102</td>
    </tr>
    <tr>
      <td>Figure 62 Low-E coated window pane - angular dep. of transm., refl. and absorption based on derived cRI</td>
      <td>102</td>
    </tr>
    <tr>
      <td>Figure 63 Determined complex-valued refractive index function of ETFE-membrane</td>
      <td>103</td>
    </tr>
    <tr>
      <td>Figure 64 Measured index of refraction and absorption coefficients of ETFE membranes (blue lines)</td>
      <td>103</td>
    </tr>
    <tr>
      <td>Figure 65 ETFE membrane - measured T/R-spectra vs. model results using the derived cRI</td>
      <td>104</td>
    </tr>
    <tr>
      <td>Figure 66 ETFE-membrane - angular dependence of transmittance and reflectance based on derived cRI</td>
      <td>104</td>
    </tr>
    <tr>
      <td>Figure 67 Reflectance data as provided by the manufacturer</td>
      <td>107</td>
    </tr>
    <tr>
      <td>Figure 68 Subsurface reflectance functions determined in optimisation process</td>
      <td>108</td>
    </tr>
    <tr>
      <td>Figure 69 Convergence of MC simulations</td>
      <td>113</td>
    </tr>
    <tr>
      <td>Figure 70 simulated annealing progression of cost function</td>
      <td>114</td>
    </tr>
    <tr>
      <td>Figure 71 Fibonacci pattern on the succulent plant Epithelantha</td>
      <td>117</td>
    </tr>
    <tr>
      <td>Figure 72 Fibonacci sphere generation ε offset applied for optimised distribution</td>
      <td>118</td>
    </tr>
    <tr>
      <td>Figure 73 Results of the optimisation process to determine the offset value ε0</td>
      <td>119</td>
    </tr>
    <tr>
      <td>Figure 74 2D polar representation of angular segmentations of the hemisphere</td>
      <td>119</td>
    </tr>
    <tr>
      <td>Figure 75 3D view of angular segmentations of the hemisphere</td>
      <td>120</td>
    </tr>
    <tr>
      <td>Figure 76 Comparison of relative solid-angle sizes</td>
      <td>120</td>
    </tr>
    <tr>
      <td>Figure 77 polar plot of the first 12 real-valued spherical harmonic functions (j=0..11)</td>
      <td>126</td>
    </tr>
    <tr>
      <td>Figure 78 SIOP polar plot representation and reference system</td>
      <td>128</td>
    </tr>
    <tr>
      <td>Figure 79 3D model of target used</td>
      <td>132</td>
    </tr>
    <tr>
      <td>Figure 80 Results of raytracing - sampled data for total transmitted power</td>
      <td>132</td>
    </tr>
    <tr>
      <td>Figure 81 Polar plot representation of the sampled data (right) vs. the fitted SIOP (left); both blended (center)</td>
      <td>133</td>
    </tr>
    <tr>
      <td>Figure 82 Polar plot representation of generated SIOPs for shaded window</td>
      <td>133</td>
    </tr>
    <tr>
      <td>Figure 83 Lambert-SIOP - sphere shaped SIOP representing absorption by a Lambertian plane</td>
      <td>136</td>
    </tr>
    <tr>
      <td>Figure 84 The five irradiance components considered by the Perez model</td>
      <td>137</td>
    </tr>
    <tr>
      <td>Figure 85 Relevant solid-angles as function of tilt-angle (0=vertical plane)</td>
      <td>141</td>
    </tr>
    <tr>
      <td>Figure 86 Cosine-SIOP averages as function of tilt-angle</td>
      <td>141</td>
    </tr>
    <tr>
      <td>Figure 87 Products of solid-angles and SIOP averages as function of tilt-angle</td>
      <td>141</td>
    </tr>
    <tr>
      <td>Figure 88 Perez evaluation of SIOP orientation in world-coordinate for different tilt angles</td>
      <td>142</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>Figure 89 projection of radiance patches on located and oriented SIOP</td>
      <td>144</td>
    </tr>
    <tr>
      <td>Figure 90 Perez clearness interpolation</td>
      <td>148</td>
    </tr>
    <tr>
      <td>Figure 91 Comparison of Perez diffuse sky irradiance</td>
      <td>150</td>
    </tr>
    <tr>
      <td>Figure 92 Comparison of Perez diffuse sky irradiance. effect of interpolation feature</td>
      <td>150</td>
    </tr>
    <tr>
      <td>Figure 93 Comparison SIOP evaluation vs. IDA ICE (Perez model) clear sky; 14.06.2021</td>
      <td>151</td>
    </tr>
    <tr>
      <td>Figure 94 Comparison SIOP evaluation vs. IDA ICE (Perez model) mostly overcast; 17.07.2021</td>
      <td>152</td>
    </tr>
    <tr>
      <td>Figure 95 Triangulated cube with normal vectors</td>
      <td>156</td>
    </tr>
    <tr>
      <td>Figure 96 Vector definitions contained in the TRCface class</td>
      <td>158</td>
    </tr>
    <tr>
      <td>Figure 97 Collision detection to determine intersection point closest to ray origin (P0) in direction d</td>
      <td>159</td>
    </tr>
    <tr>
      <td>Figure 98 Octree partition of imported CAD model to increase efficiency of collision detection</td>
      <td>160</td>
    </tr>
    <tr>
      <td>Figure 99 Octree hierarchy</td>
      <td>161</td>
    </tr>
    <tr>
      <td>Figure 100 Principle of ray relocation at periodic planes</td>
      <td>163</td>
    </tr>
    <tr>
      <td>Figure 101 Smooth shading / linear interpolation of surface normal vectors</td>
      <td>164</td>
    </tr>
    <tr>
      <td>Figure 102 Smooth shading example - partially silver coated ETFE membrane cushion</td>
      <td>164</td>
    </tr>
    <tr>
      <td>Figure 103 Overview of main components of TMaterialInterface</td>
      <td>165</td>
    </tr>
    <tr>
      <td>Figure 104 Typical scattering on an opaque, slightly rough painted surface (α = 0.1, n = 1.5, ρss = 0.8)</td>
      <td>166</td>
    </tr>
    <tr>
      <td>Figure 105 Process flow diagram of TLSISrouhPol showing the processing of a scattering event</td>
      <td>169</td>
    </tr>
    <tr>
      <td>Figure 106 Schematic overview of the directional scanning process</td>
      <td>172</td>
    </tr>
    <tr>
      <td>Figure 107 Appropriate geometries for caster objects</td>
      <td>174</td>
    </tr>
    <tr>
      <td>Figure 108 Uniform sampling of quad face</td>
      <td>175</td>
    </tr>
    <tr>
      <td>Figure 109 Uniform sampling of triangular face</td>
      <td>175</td>
    </tr>
    <tr>
      <td>Figure 110 Schematic overview of typical measurement configuration</td>
      <td>178</td>
    </tr>
    <tr>
      <td>Figure 111 Typical convergence during raytracing</td>
      <td>180</td>
    </tr>
    <tr>
      <td>Figure 112 Typical full-panoramic HDRI environment</td>
      <td>181</td>
    </tr>
    <tr>
      <td>Figure 113 Working principle of backward raytracing (idealised, environment at infinite distance)</td>
      <td>181</td>
    </tr>
    <tr>
      <td>Figure 114 Process flow diagram of the RadiCal backward raytracing/rendering algorithm</td>
      <td>182</td>
    </tr>
    <tr>
      <td>Figure 115 Glass/metal demo rendering using RadiCal raytracer</td>
      <td>183</td>
    </tr>
    <tr>
      <td>Figure 116 RadiCal-rendering of validation window model</td>
      <td>183</td>
    </tr>
    <tr>
      <td>Figure 117 Demo rendering "octopus" using RadiCal raytracing algorithm</td>
      <td>184</td>
    </tr>
    <tr>
      <td>Figure 118 Aerial view of measurement location</td>
      <td>186</td>
    </tr>
    <tr>
      <td>Figure 119 Weather station with sun-tracker SOLYS2</td>
      <td>187</td>
    </tr>
    <tr>
      <td>Figure 120 top: south panorama view; bottom-left: west panorama view; bottom-right: lab roof</td>
      <td>187</td>
    </tr>
    <tr>
      <td>Figure 121 left: PyroScanner indoor pyranometer unit on moveable platform;</td>
      <td>188</td>
    </tr>
    <tr>
      <td>Figure 122 PyroScanner data and control flow diagram</td>
      <td>189</td>
    </tr>
    <tr>
      <td>Figure 123 Web interface of PyroScanner to remotely control and monitor measurements</td>
      <td>189</td>
    </tr>
    <tr>
      <td>Figure 124 colocation calibration of internal pyranometer</td>
      <td>190</td>
    </tr>
    <tr>
      <td>Figure 125 Calibration curve used for evaluation</td>
      <td>190</td>
    </tr>
    <tr>
      <td>Figure 126 Potential sources for internally scattered light</td>
      <td>191</td>
    </tr>
    <tr>
      <td>Figure 127 Photogrammetric determination of slat angle using a 3D-printed tool</td>
      <td>192</td>
    </tr>
    <tr>
      <td>Figure 128 Averaging method applied to consider periodic length</td>
      <td>193</td>
    </tr>
    <tr>
      <td>Figure 129 Location of model- and PyroScanner-measurement regions</td>
      <td>194</td>
    </tr>
    <tr>
      <td>Figure 130 Sampling with exponential response</td>
      <td>195</td>
    </tr>
    <tr>
      <td>Figure 131 3D model (sliced)</td>
      <td>196</td>
    </tr>
    <tr>
      <td>Figure 132 3D models demonstrating the applied level of detail of slat types used</td>
      <td>202</td>
    </tr>
    <tr>
      <td>Figure 133 Overview of the four slat specimens used</td>
      <td>202</td>
    </tr>
    <tr>
      <td>Figure 134 Case #1 - int. irradiance correlation</td>
      <td>204</td>
    </tr>
    <tr>
      <td>Figure 135 Case #1 - modelled transmittances</td>
      <td>204</td>
    </tr>
    <tr>
      <td>Figure 136 Case #1 - modelled and measured internal irradiance values as time series</td>
      <td>205</td>
    </tr>
    <tr>
      <td>Figure 137 Case #2 - int. irradiance correlation</td>
      <td>206</td>
    </tr>
    <tr>
      <td>Figure 138 Case #2 - modelled transmittances</td>
      <td>206</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>Figure 139 Case #2 -modelled and measured internal irradiance values as time series</td>
      <td>207</td>
    </tr>
    <tr>
      <td>Figure 140 Case #3 - int. irradiance correlation</td>
      <td>208</td>
    </tr>
    <tr>
      <td>Figure 141 Case #3 - modelled transmittances</td>
      <td>208</td>
    </tr>
    <tr>
      <td>Figure 142 Case #6 - int. irradiance correlation</td>
      <td>209</td>
    </tr>
    <tr>
      <td>Figure 143 Case #6 - modelled transmittances</td>
      <td>209</td>
    </tr>
    <tr>
      <td>Figure 144 Case #8 - int. irradiance correlation</td>
      <td>210</td>
    </tr>
    <tr>
      <td>Figure 145 Case #8 - modelled transmittances</td>
      <td>210</td>
    </tr>
    <tr>
      <td>Figure 146 Case #10 - int. irradiance correlation</td>
      <td>211</td>
    </tr>
    <tr>
      <td>Figure 147 Case #10 - modelled transmittances</td>
      <td>211</td>
    </tr>
    <tr>
      <td>Figure 148 Mean error of transmittance ME&tau; (bars) and standard deviation of errors SE&tau; (red lines)</td>
      <td>212</td>
    </tr>
    <tr>
      <td>Figure 149 Mean absolute error of transmittance MAE&tau; for all test cases</td>
      <td>212</td>
    </tr>
    <tr>
      <td>Figure 150 Overview of model and work-flow: top: 3D model of the target object</td>
      <td>217</td>
    </tr>
    <tr>
      <td>Figure 151 Dynamic thermal simulation results of IGU window shaded with slats at 0&deg;</td>
      <td>219</td>
    </tr>
    <tr>
      <td>Figure 152 Total heat-balance for the IGU unit calculated from energy flows</td>
      <td>220</td>
    </tr>
    <tr>
      <td>Figure 153 Daily averages for the entire IGU</td>
      <td>221</td>
    </tr>
  </tbody>
</table>
# List of tables

<table><tr><td>Table 1 Overview of coherence time and corresponding coherence lengths of different light sources</td><td>34</td></tr><tr><td>Table 2 Statistical analysis of global radiation reference spectrum ASTM G-173</td><td>51</td></tr><tr><td>Table 3 Input parameters for the validation example</td><td>56</td></tr><tr><td>Table 4 Comparison of solar position: SG2 RadiCal implementation vs. SPA algorithm</td><td>57</td></tr><tr><td>Table 5 Currently implemented descendant classes of TRIbaseclass</td><td>60</td></tr><tr><td>Table 6 Stokes vectors for basic polarisation states</td><td>67</td></tr><tr><td>Table 7 Müller matrices of specimen 1E – measured data vs. RadiCal calculation results</td><td>78</td></tr><tr><td>Table 8 Spectral directional-hemispheric reflectance values for test cases</td><td>87</td></tr><tr><td>Table 9 Analysis of optimisation results for generic coated surface models</td><td>109</td></tr><tr><td>Table 10 Exemplary list of potential SIOP target quantities and relevant units</td><td>124</td></tr><tr><td>Table 11 Key figures of generated SIOPs</td><td>132</td></tr><tr><td>Table 12 Error statistics, comparison of Perez diffuse sky irradiance RadiCal vs. IDA ICE</td><td>149</td></tr><tr><td>Table 13 Error statistics, comparison SIOP evaluation for plane vs. IDA ICE for irradiance on plane</td><td>152</td></tr><tr><td>Table 14 Currently implemented descendant classes of TLightSurfaceInteractionStokes</td><td>168</td></tr><tr><td>Table 15 Key hardware components of the PyroScanner</td><td>188</td></tr><tr><td>Table 16 Material assignments used for validation model</td><td>196</td></tr><tr><td>Table 17 Overview of test cases</td><td>203</td></tr><tr><td>Table 18 Validation case #1 – results and specifications</td><td>205</td></tr><tr><td>Table 19 Validation case #2 – results and specifications</td><td>206</td></tr><tr><td>Table 20 Validation case #3 – results and specifications</td><td>208</td></tr><tr><td>Table 21 Validation case #6 – results and specifications</td><td>209</td></tr><tr><td>Table 22 Validation case #8 – results and specifications</td><td>210</td></tr><tr><td>Table 23 Validation case #10 – results and specifications</td><td>211</td></tr><tr><td>Table 24 Overview of determined transmittance errors</td><td>213</td></tr><tr><td>Table 25 Specifications of fenestration model</td><td>216</td></tr></table>
References

Abramowitz, M., Stegun, I.A. and Romer, R.H. (1988) ‘Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables’, *American Journal of Physics* [Preprint]. Available at: https://doi.org/10.1119/1.15378.

AEE INTEC (2023) AEE - Institute for Sustainable Technologies. Available at: https://www.aee-intec.at/technisches-labor-demonstration-b24 (Accessed: 1 January 2023).

Anan, T. (2018) *Derivation of Mueller matrix from Zemax Definition of Stokes Q, U, & V*. Available at: https://www.kwasan.kyoto-u.ac.jp/~anan/shed/temporary/scip/mm_zemax_ta_20180310.pdf.

Ashikhmin, M. and Shirley, P. (2011) ‘An Anisotropic Phong BRDF Model’, *Graphics Tools---The jgt Editors’ Choice*, (1), pp. 303–311. Available at: https://doi.org/10.1201/b10628-37.

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# Appendix A: Full-system validation results

## A1. Case #1 – UNSHADED_SOUTH

<figure><img src="image_160.png" /><figcaption>Figure A1 - Case #1 – int. irradiance correlation modelled values as a function of measured values</figcaption></figure>

<figure><img src="image_161.png" /><figcaption>Figure A2 - Case #1 – modelled transmittances as a function of solar azimuth angle</figcaption></figure>

<table><caption>Table A1 Validation case #1 – results and specifications</caption><thead><tr><th colspan="4">Case specifications</th></tr></thead><tbody><tr><td>case #1<br>case name:<br>UNSHADED_SOUTH</td><td>case description:</td><td>win.state: window closed<br>slats: -<br>slat-angle: -°</td><td></td></tr><tr><td>Transmittance τ<sub>cog,sys</sub><br>range of determined values (min | q.1 | q.3 | max) [1]:<br>0.175 | 0.307 | 0.356 | 0.359</td><td>errors:</td><td>MAEτ: 0.024<br>MEτ: 0.013<br>SEτ: 0.040</td><td></td></tr><tr><td>Internal irradiance E<sub>int,mode</sub><br>range of determined values (min | q.1 | q.3 | max) [W/m²]:<br>2.4 | 16.7 | 46.2 | 169.5</td><td>errors:</td><td>MAE<sub>E</sub>: 1.8 W/m²</td><td></td></tr><tr><td>Internal irradiance correlation model vs. measurement:<br>number of points: 3347<br>lin. regression through origin y=0.9785·x<br>lin. regression y=0.9738·x + 0.4422</td><td>R²:</td><td>0.999<br>0.998</td><td></td></tr><tr><td>Measurement specifications (min | q.1 | q.3 | max):<br>sun azimuth range [°]: 56.7 | 84.7 | 279.6 | 302.3<br>sun elevation range [°]: 0.6 | 15.3 | 37.3 | 65.8<br>incidence angle range [°]: 64.9 | 84.8 | 106.2 | 123.3<br>diffuse ratio (DHI/GHI) range [1]: 0.08 | 0.21 | 0.91 | 1.00<br>meas. period: 2021.07.02 | 2021.07.05 | 2021.07.10 | 2021.07.12</td><td></td><td></td><td></td></tr></tbody></table>
## A2. Case #2 – UNSHADED_WEST

<figure><img src="image_162.png" /><figcaption>Figure A3 Case #2 – int. irradiance correlation modelled values as a function of measured values</figcaption></figure>

<figure><img src="image_163.png" /><figcaption>Figure A4 Case #2 – modelled transmittances as a function of solar azimuth angle</figcaption></figure>

<table><caption>Table A2 Validation case #2 – results and specifications</caption><thead><tr><th colspan="4">Case specifications</th></tr></thead><tbody><tr><td>case #2<br>case name:<br>UNSHADED_WEST</td><td>case description:</td><td>win.state: window closed<br>slats: 0<br>slat-angle: -°</td><td></td></tr><tr><td>Transmittance τ<sub>cog,sys</sub></td><td>range of determined values (min | q.1 | q.3 | max) [1]:<br>0.208 | 0.339 | 0.373 | 0.450</td><td>errors: MAEτ: 0.030<br>MEτ: 0.003<br>SEτ: 0.040</td><td></td></tr><tr><td>Internal irradiance E<sub>int,mode</sub></td><td>range of determined values (min | q.1 | q.3 | max) [W/m²]:<br>2.5 | 24.1 | 173.2 | 397.0</td><td>errors: MAE<sub>E</sub>: 6.1 W/m²</td><td></td></tr><tr><td>Internal irradiance correlation model vs. measurement:</td><td>number of points: 6091<br>lin. regression through origin y=1.0105·x<br>lin. regression y=1.0119·x - 0.3049</td><td>R²: 0.996<br>R²: 0.992</td><td></td></tr><tr><td>Measurement specifications (min | q.1 | q.3 | max):</td><td>sun azimuth range [°]: 59.4 | 104.8 | 258.8 | 303.1<br>sun elevation range [°]: -1.0 | 22.4 | 52.1 | 64.6<br>incidence angle range [°]: 16.8 | 38.8 | 138.4 | 162.8<br>diffuse ratio (DHI/GHI) range [1]: 0.12 | 0.36 | 1.00 | 1.00<br>meas. period: 2021.07.12 | 2021.07.17 | 2021.08.01 | 2021.08.05</td><td></td><td></td></tr></tbody></table>
## A3. Case #3 – ANTHRA_wC_00

<figure><img src="image_164.png" /><figcaption>Figure A5 Case #3 – int. irradiance correlation modelled values as a function of measured values</figcaption></figure>

<figure><img src="image_165.png" /><figcaption>Figure A6 Case #3 – modelled transmittances as a function of solar azimuth angle</figcaption></figure>

<table><caption>Table A3 Validation case #3 – results and specifications</caption><thead><tr><th colspan="4">Case specifications</th></tr></thead><tbody><tr><td><strong>case #3</strong><br>case name:<br><strong>ANTHRA_wC_00</strong></td><td>case description:</td><td>win.state:<br>slats:<br>slat-angle:</td><td><strong>window closed</strong><br><strong>anthracite, type K80</strong><br><strong>0°</strong></td></tr><tr><td colspan="4"><strong>Transmittance τ<sub>cog,sys</sub></strong></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [1]:<br><strong>0.029 | 0.041 | 0.120 | 0.207</strong></td><td>errors:</td><td>MAEτ: <strong>0.010</strong><br>MEτ: <strong>0.003</strong><br>SEτ: <strong>0,016</strong></td><td></td></tr><tr><td colspan="4"><strong>Internal irradiance E<sub>int,mode</sub></strong></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [W/m<sup>2</sup>]:<br><strong>2.8 | 17.3 | 25.7 | 39.3</strong></td><td>errors:</td><td>MAE<sub>E</sub>: <strong>2.1 W/m<sup>2</sup></strong></td><td></td></tr><tr><td colspan="4"><strong>Internal irradiance correlation model vs. measurement:</strong></td></tr><tr><td>lin. regression through origin</td><td>number of points: <strong>206</strong><br>y=0.9741·x</td><td>R<sup>2</sup>: <strong>0.985</strong></td><td></td></tr><tr><td>lin. regression</td><td>y=0.9184·x + 1.3131</td><td>R<sup>2</sup>: <strong>0.870</strong></td><td></td></tr><tr><td colspan="4"><strong>Measurement specifications (min | q.1 | q.3 | max):</strong></td></tr><tr><td>sun azimuth range [°]:</td><td colspan="3"><strong>85.3 | 153.4 | 223.5 | 289.8</strong></td></tr><tr><td>sun elevation range [°]:</td><td colspan="3"><strong>3.6 | 45.8 | 58.3 | 61.5</strong></td></tr><tr><td>incidence angle range [°]:</td><td colspan="3"><strong>53.4 | 60.5 | 65.6 | 109.8</strong></td></tr><tr><td>diffuse ratio (DHI/GHI) range [1]:</td><td colspan="3"><strong>0.11 | 0.17 | 0.96 | 1.00</strong></td></tr><tr><td>meas. period:</td><td colspan="3"><strong>2020.07.27 | 2020.08.01 | 2020.08.23 | 2020.08.25</strong></td></tr></tbody></table>
## A4. Case #4 – ANTHRA_wC_45

<figure><img src="image_166.png" /><figcaption>Figure A7 Case #4 – int. irradiance correlation modelled values as a function of measured values</figcaption></figure>

<figure><img src="image_167.png" /><figcaption>Figure A8 Case #4 – modelled transmittances as a function of solar azimuth angle</figcaption></figure>

<table><caption>Table A4 Validation case #4 – results and specifications</caption><thead><tr><th colspan="4">Case specifications</th></tr></thead><tbody><tr><td>case #4<br>case name:<br>ANTHRA_wC_45</td><td>case description:</td><td>win.state: window closed<br>slats: anthracite, type K80<br>slat-angle: 45°</td><td></td></tr><tr><td colspan="4">Transmittance τ<sub>cog.sys</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [1]:<br>0.011 | 0.012 | 0.013 | 0.014</td><td>errors:</td><td>MAEτ: 0.001<br>MEτ: 0.001<br>SEτ: 0.0005</td><td></td></tr><tr><td colspan="4">Internal irradiance E<sub>int.model</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [W/m²]:<br>6.2 | 7.4 | 8.1 | 8.6</td><td>errors:</td><td>MAE<sub>E</sub>: 0.8 W/m²</td><td></td></tr><tr><td colspan="4">Internal irradiance correlation model vs. measurement:</td></tr><tr><td>number of points: 58<br>lin. regression through origin y=0.8902·x<br>lin. regression y=0.7100·x + 1.3940</td><td>R²:</td><td>0.999<br>0.866</td><td></td></tr><tr><td colspan="4">Measurement specifications (min | q.1 | q.3 | max):</td></tr><tr><td>sun azimuth range [°]: 120.6 | 144.3 | 194.7 | 239.4<br>sun elevation range [°]: 40.0 | 46.3 | 54.3 | 55.1<br>incidence angle range [°]: 54.4 | 55.2 | 61.9 | 67.2<br>diffuse ratio (DHI/GHI) range [1]: 0.08 | 0.10 | 0.13 | 0.18<br>meas. period: 2020.08.20 | 2020.08.20 | 2020.08.22 | 2020.08.22</td><td></td><td></td><td></td></tr></tbody></table>
## A5. Case #5 – ANTHRA_wC_m45

<figure><img src="image_168.png" /><figcaption>Figure A9 Case #5 – int. irradiance correlation modelled values as a function of measured values</figcaption></figure>

<figure><img src="image_169.png" /><figcaption>Figure A10 Case #5 – modelled transmittances as a function of solar azimuth angle</figcaption></figure>

<table><caption>Table A5 Validation case #5 – results and specifications</caption><thead><tr><th colspan="4">Case specifications</th></tr></thead><tbody><tr><td>case #5<br>case name:<br>ANTHRA_wC_m45</td><td>case description:</td><td>win.state: window closed<br>slats: anthracite, type K80<br>slat-angle: -45°</td><td></td></tr><tr><td colspan="4">Transmittance τ<sub>cog.sys</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [1]:<br>0.139 | 0.161 | 0.169 | 0.173</td><td>errors:</td><td>MAEτ: 0.017<br>MEτ: 0.012<br>SEτ: 0.017</td><td></td></tr><tr><td colspan="4">Internal irradiance E<sub>int.model</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [W/m²]:<br>4.0 | 35.2 | 101.1 | 130.1</td><td>errors:</td><td>MAE<sub>E</sub>: 6.2 W/m²</td><td></td></tr><tr><td colspan="4">Internal irradiance correlation model vs. measurement:</td></tr><tr><td>number of points: 42<br>lin. regression through origin<br>lin. regression</td><td>y=0.9530·x<br>y=0.9789·x - 2.2878</td><td>R²: 0.990<br>R²: 0.966</td><td></td></tr><tr><td colspan="4">Measurement specifications (min | q.1 | q.3 | max):</td></tr><tr><td>sun azimuth range [°]:</td><td>122.9 | 147.4 | 208.5 | 279.5</td><td></td><td></td></tr><tr><td>sun elevation range [°]:</td><td>10.5 | 47.0 | 56.0 | 57.0</td><td></td><td></td></tr><tr><td>incidence angle range [°]:</td><td>56.4 | 57.2 | 64.7 | 99.4</td><td></td><td></td></tr><tr><td>diffuse ratio (DHI/GHI) range [1]:</td><td>0.35 | 0.53 | 0.98 | 1.00</td><td></td><td></td></tr><tr><td>meas. period:</td><td>2020.08.05 | 2020.08.14 | 2020.08.16 | 2020.08.16</td><td></td><td></td></tr></tbody></table>
## A6. Case #6 – SILVER_wC_00

<figure><img src="image_170.png" /><figcaption>Figure A11 Case #6 – int. irradiance correlation modelled values as a function of measured values</figcaption></figure>

<figure><img src="image_171.png" /><figcaption>Figure A12 Case #6 – modelled transmittances as a function of solar azimuth angle</figcaption></figure>

<table><caption>Table A6 Validation case #6 – results and specifications</caption><thead><tr><th colspan="4">Case specifications</th></tr></thead><tbody><tr><td>case #6<br>case name:<br>SILVER_wC_00</td><td>case description:</td><td>win.state: window closed<br>slats: silver, type K80<br>slat-angle: 0°</td><td></td></tr><tr><td colspan="4">Transmittance τ<sub>cog.sys</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [1]:</td><td></td><td>errors:</td><td>MAEτ: 0.011<br>MEτ: -0.006<br>SEτ: 0,014</td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [W/m²]:</td><td></td><td>errors:</td><td>MAE<sub>E</sub>: 1.9 W/m²</td></tr><tr><td colspan="4">Internal irradiance E<sub>int.model</sub></td></tr><tr><td>number of points: 78<br>lin. regression through origin y=1.0433·x<br>lin. regression y=1.0123·x + 1.0775</td><td></td><td>R²: 0.996<br>R²: 0.981</td><td></td></tr><tr><td colspan="4">Internal irradiance correlation model vs. measurement:</td></tr><tr><td>sun azimuth range [°]: 86.5 | 181.6 | 272.9 | 298.6<br>sun elevation range [°]: 4.9 | 28.5 | 63.4 | 65.7<br>incidence angle range [°]: 62.4 | 65.7 | 93.1 | 118.5<br>diffuse ratio (DHI/GHI) range [1]: 0.09 | 0.14 | 0.70 | 1.00<br>meas. period: 2020.07.03 | 2020.07.04 | 2020.07.06 | 2020.07.27</td><td></td><td></td><td></td></tr></tbody></table>
## A7. Case #7 – SILVER_wC_m45

<figure><img src="image_172.png" /><figcaption>Figure A13 Case #7 – int. irradiance correlation modelled values as a function of measured values</figcaption></figure>

<figure><img src="image_173.png" /><figcaption>Figure A14 Case #7 – modelled transmittances as a function of solar azimuth angle</figcaption></figure>

<table><caption>Table A7 Validation case #7 – results and specifications</caption><thead><tr><th colspan="5">Case specifications</th></tr></thead><tbody><tr><td>case #7<br>case name:<br>SILVER_wC_m45</td><td>case description:</td><td>win.state: window closed<br>slats: silver, type K80<br>slat-angle: -45°</td><td></td><td></td></tr><tr><td colspan="5">Transmittance τ<sub>cog,sys</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [1]:</td><td></td><td>errors:</td><td>MAEτ: 0.022<br>MEτ: 0.012<br>SEτ: 0.022</td><td></td></tr><tr><td>0.077 | 0.153 | 0.199 | 0.210</td><td></td><td></td><td></td><td></td></tr><tr><td colspan="5">Internal irradiance E<sub>int.model</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [W/m²]:</td><td></td><td>errors:</td><td>MAE<sub>E</sub>: 4.2 W/m²</td><td></td></tr><tr><td>6.9 | 14.4 | 49.6 | 112.2</td><td></td><td></td><td></td><td></td></tr><tr><td colspan="5">Internal irradiance correlation model vs. measurement:</td></tr><tr><td>number of points: 64</td><td></td><td></td><td></td><td></td></tr><tr><td>lin. regression through origin y=0.9080·x</td><td></td><td>R²: 0.991</td><td></td><td></td></tr><tr><td>lin. regression y=0.8829·x + 1.5012</td><td></td><td>R²: 0.978</td><td></td><td></td></tr><tr><td colspan="5">Measurement specifications (min | q.1 | q.3 | max):</td></tr><tr><td>sun azimuth range [°]:</td><td>97.3 | 185.4 | 273.5 | 285.0</td><td></td><td></td><td></td></tr><tr><td>sun elevation range [°]:</td><td>14.4 | 25.9 | 61.8 | 63.8</td><td></td><td></td><td></td></tr><tr><td>incidence angle range [°]:</td><td>62.6 | 64.0 | 93.2 | 104.6</td><td></td><td></td><td></td></tr><tr><td>diffuse ratio (DHI/GHI) range [1]:</td><td>0.21 | 0.37 | 0.99 | 1.00</td><td></td><td></td><td></td></tr><tr><td>meas. period:</td><td>2020.07.16 | 2020.07.18 | 2020.07.21 | 2020.07.24</td><td></td><td></td><td></td></tr></tbody></table>
## A8. Case #8 – WHITE_wC_00

<figure><img src="image_174.png" /><figcaption>Figure A15 Case #8 – int. irradiance correlation modelled values as a function of measured values</figcaption></figure>

<figure><img src="image_175.png" /><figcaption>Figure A16 Case #8 – modelled transmittances as a function of solar azimuth angle</figcaption></figure>

<table><caption>Table A8 Validation case #8 – results and specifications</caption><thead><tr><th colspan="4">Case specifications</th></tr></thead><tbody><tr><td>case #8<br>case name:<br>WHITE_wC_00</td><td>case description:</td><td>win.state: window closed<br>slats: white, type K80<br>slat-angle: 0°</td><td></td></tr><tr><td colspan="4">Transmittance τ<sub>cog.sys</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [1]:</td><td></td><td>errors:</td><td>MAEτ: 0.005<br>MEτ: 0.002<br>SEτ: 0.009</td></tr><tr><td>0.105 | 0.117 | 0.123 | 0.188</td><td></td><td></td><td></td></tr><tr><td colspan="4">Internal irradiance E<sub>int.model</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [W/m²]:</td><td></td><td>errors:</td><td>MAE<sub>E</sub>: 1.7 W/m²</td></tr><tr><td>2.3 | 52.6 | 74.2 | 90.0</td><td></td><td></td><td></td></tr><tr><td colspan="4">Internal irradiance correlation model vs. measurement:</td></tr><tr><td>number of points: 136</td><td></td><td></td><td></td></tr><tr><td>lin. regression through origin y=0.9981x</td><td></td><td>R²: 0.998</td><td></td></tr><tr><td>lin. regression y=1.0011x - 0.1970</td><td></td><td>R²: 0.987</td><td></td></tr><tr><td colspan="4">Measurement specifications (min | q.1 | q.3 | max):</td></tr><tr><td>sun azimuth range [°]:</td><td>117.3 | 147.1 | 221.7 | 280.1</td><td></td><td></td></tr><tr><td>sun elevation range [°]:</td><td>3.1 | 35.9 | 46.2 | 52.0</td><td></td><td></td></tr><tr><td>incidence angle range [°]:</td><td>46.0 | 51.7 | 59.8 | 100.1</td><td></td><td></td></tr><tr><td>diffuse ratio (DHI/GHI) range [1]:</td><td>0.16 | 0.25 | 0.47 | 1.00</td><td></td><td></td></tr><tr><td>meas. period:</td><td>2020.08.28 | 2020.08.30 | 2020.09.13 | 2020.09.14</td><td></td><td></td></tr></tbody></table>
## A9. Case #9 – WHITE_wC_53

<figure><img src="image_176.png" /><figcaption>Figure A17 Case #9 – int. irradiance correlation modelled values as a function of measured values</figcaption></figure>

<figure><img src="image_177.png" /><figcaption>Figure A18 Case #9 – modelled transmittances as a function of solar azimuth angle</figcaption></figure>

<table><caption>Table A9 Validation case #9 – results and specifications</caption><thead><tr><th colspan="5">Case specifications</th></tr></thead><tbody><tr><td>case #9</td><td>case description:</td><td>win.state:</td><td>window closed</td><td></td></tr><tr><td>case name:</td><td></td><td>slats:</td><td>white, type K80</td><td></td></tr><tr><td>WHITE_wC_53</td><td></td><td>slat-angle:</td><td>53°</td><td></td></tr><tr><td colspan="5">Transmittance τ<sub>cog,sys</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [1]:</td><td></td><td>errors:</td><td>MAEτ: 0.017</td><td></td></tr><tr><td>0.035 | 0.036 | 0.037 | 0.039</td><td></td><td>MEτ: 0.017</td><td></td><td></td></tr><tr><td></td><td></td><td>SEτ: 0.005</td><td></td><td></td></tr><tr><td colspan="5">Internal irradiance E<sub>int.model</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [W/m<sup>2</sup>]:</td><td></td><td>errors:</td><td>MAE<sub>E</sub>: 9.5 W/m<sup>2</sup></td><td></td></tr><tr><td>5.1 | 29.3 | 36.3 | 37.2</td><td></td><td></td><td></td><td></td></tr><tr><td colspan="5">Internal irradiance correlation model vs. measurement:</td></tr><tr><td>number of points:</td><td>22</td><td></td><td></td><td></td></tr><tr><td>lin. regression through origin</td><td>y=0.7058·x</td><td>R<sup>2</sup>: 0.999</td><td></td><td></td></tr><tr><td>lin. regression</td><td>y=0.7668·x - 2.0392</td><td>R<sup>2</sup>: 0.993</td><td></td><td></td></tr><tr><td colspan="5">Measurement specifications (min | q.1 | q.3 | max):</td></tr><tr><td>sun azimuth range [°]:</td><td>158.3 | 179.2 | 225.6 | 264.4</td><td></td><td></td><td></td></tr><tr><td>sun elevation range [°]:</td><td>15.1 | 36.1 | 44.9 | 45.7</td><td></td><td></td><td></td></tr><tr><td>incidence angle range [°]:</td><td>45.7 | 46.5 | 55.6 | 84.5</td><td></td><td></td><td></td></tr><tr><td>diffuse ratio (DHI/GHI) range [1]:</td><td>0.14 | 0.15 | 0.19 | 0.64</td><td></td><td></td><td></td></tr><tr><td>meas. period:</td><td>2020.09.03 | 2020.09.15 | 2020.09.15 | 2020.09.15</td><td></td><td></td><td></td></tr></tbody></table>
## A10. Case #10 – Z-SILVER_wC_00

<figure><img src="image_178.png" /><figcaption>Figure A19 Case #10 – int. irradiance correlation modelled values as a function of measured values</figcaption></figure>

<figure><img src="image_179.png" /><figcaption>Figure A20 Case #10 – modelled transmittances as a function of solar azimuth angle</figcaption></figure>

<table><caption>Table A10 Validation case #10 – results and specifications</caption><thead><tr><th colspan="4">Case specifications</th></tr></thead><tbody><tr><td>case #10<br>case name:<br>Z-SILVER_wC_00</td><td>case description:</td><td>win.state: window closed<br>slats: silver, type VZ90<br>slat-angle: 0°</td><td></td></tr><tr><td colspan="4">Transmittance τ<sub>cog.sys</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [1]:<br>0.161 | 0.163 | 0.164 | 0.165</td><td>errors:</td><td>MAEτ: 0.012<br>MEτ: 0.010<br>SEτ: 0.008</td><td></td></tr><tr><td colspan="4">Internal irradiance E<sub>int.model</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [W/m²]:<br>3.7 | 4.4 | 5.6 | 10.6</td><td>errors:</td><td>MAE<sub>E</sub>: 0.3 W/m²</td><td></td></tr><tr><td colspan="4">Internal irradiance correlation model vs. measurement:</td></tr><tr><td>number of points: 6<br>lin. regression through origin<br>lin. regression</td><td>y=0.9572x<br>y=0.9858x - 0.1873</td><td>R²: 0.998<br>R²: 0.990</td><td></td></tr><tr><td colspan="4">Measurement specifications (min | q.1 | q.3 | max):</td></tr><tr><td>sun azimuth range [°]:</td><td>123.1 | 125.9 | 217.5 | 226.6</td><td></td><td></td></tr><tr><td>sun elevation range [°]:</td><td>25.6 | 27.3 | 32.0 | 35.1</td><td></td><td></td></tr><tr><td>incidence angle range [°]:</td><td>49.5 | 53.3 | 58.6 | 60.5</td><td></td><td></td></tr><tr><td>diffuse ratio (DHI/GHI) range [1]:</td><td>1.00 | 1.00 | 1.00 | 1.00</td><td></td><td></td></tr><tr><td>meas. period:</td><td>2020.09.25 | 2020.09.25 | 2020.09.25 | 2020.09.25</td><td></td><td></td></tr></tbody></table>
## A11. Case #11 – Z-SILVER_wC_58

<figure><img src="image_180.png" /><figcaption>Figure A21 Case #11 – int. irradiance correlation modelled values as a function of measured values</figcaption></figure>

<figure><img src="image_181.png" /><figcaption>Figure A22 Case #11 – modelled transmittances as a function of solar azimuth angle</figcaption></figure>

<table><caption>Table A11 Validation case #11 – results and specifications</caption><thead><tr><th colspan="4">Case specifications</th></tr></thead><tbody><tr><td>case #11<br>case name:<br>Z-SILVER_wC_58</td><td>case description:</td><td>win.state: window closed<br>slats: silver, type VZ90<br>slat-angle: 58°</td><td></td></tr><tr><td colspan="4">Transmittance τ<sub>cog.sys</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [1]:</td><td></td><td>errors:</td><td>MAEτ: 0.006<br>MEτ: -0.005<br>SEτ: 0.003</td></tr><tr><td>0.016 | 0.017 | 0.021 | 0.022</td><td></td><td></td><td></td></tr><tr><td colspan="4">Internal irradiance E<sub>int.model</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [W/m²]:</td><td></td><td>errors:</td><td>MAE<sub>E</sub>: 2.9 W/m²</td></tr><tr><td>2.4 | 4.5 | 7.3 | 11.0</td><td></td><td></td><td></td></tr><tr><td colspan="4">Internal irradiance correlation model vs. measurement:</td></tr><tr><td>number of points: 40</td><td></td><td></td><td></td></tr><tr><td>lin. regression through origin y=1.5075·x</td><td></td><td>R²: 0.989</td><td></td></tr><tr><td>lin. regression y=1.7453·x - 1.6243</td><td></td><td>R²: 0.954</td><td></td></tr><tr><td colspan="4">Measurement specifications (min | q.1 | q.3 | max):</td></tr><tr><td>sun azimuth range [°]:</td><td>116.3 | 129.6 | 201.9 | 243.1</td><td></td><td></td></tr><tr><td>sun elevation range [°]:</td><td>18.6 | 26.2 | 35.9 | 40.8</td><td></td><td></td></tr><tr><td>incidence angle range [°]:</td><td>41.3 | 46.1 | 57.5 | 65.2</td><td></td><td></td></tr><tr><td>diffuse ratio (DHI/GHI) range [1]:</td><td>0.11 | 0.15 | 0.78 | 1.00</td><td></td><td></td></tr><tr><td>meas. period:</td><td>2020.09.27 | 2020.09.27 | 2020.09.30 | 2020.09.30</td><td></td><td></td></tr></tbody></table>
## A12. Case #12 – ANTHRA_wO_00

<figure><img src="image_182.png" /><figcaption>Figure A23 Case #12 – int. irradiance correlation modelled values as a function of measured values</figcaption></figure>

<figure><img src="image_183.png" /><figcaption>Figure A24 Case #12 – modelled transmittances as a function of solar azimuth angle</figcaption></figure>

<table><caption>Table A12 Validation case #12 – results and specifications</caption><thead><tr><th colspan="4">Case specifications</th></tr></thead><tbody><tr><td>case #12<br>case name:<br>ANTHRA_wO_00</td><td>case description:</td><td>win.state: window closed<br>slats: anthracite, type K80<br>slat-angle: 0°</td><td></td></tr><tr><td colspan="4">Transmittance τ<sub>cog.sys</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [1]:<br>0.068 | 0.099 | 0.125 | 0.375</td><td>errors:</td><td>MAEτ: 0.010<br>MEτ: 0.002<br>SEτ: 0.013</td><td></td></tr><tr><td colspan="4">Internal irradiance E<sub>int.model</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [W/m²]:<br>30.7 | 52.2 | 65.9 | 85.4</td><td>errors:</td><td>MAE<sub>E</sub>: 4.9 W/m²</td><td></td></tr><tr><td colspan="4">Internal irradiance correlation model vs. measurement:</td></tr><tr><td>number of points: 123<br>lin. regression through origin y=0.9690·x<br>lin. regression y=0.7245·x + 14.8686</td><td>R²:</td><td>0.990<br>0.698</td><td></td></tr><tr><td colspan="4">Measurement specifications (min | q.1 | q.3 | max):</td></tr><tr><td>sun azimuth range [°]: 92.2 | 147.9 | 200.5 | 235.9<br>sun elevation range [°]: 27.5 | 52.3 | 58.2 | 61.7<br>incidence angle range [°]: 53.1 | 59.3 | 64.0 | 88.1<br>diffuse ratio (DHI/GHI) range [1]: 0.10 | 0.17 | 0.27 | 1.00<br>meas. period: 2020.07.28 | 2020.07.30 | 2020.08.12 | 2020.08.26</td><td></td><td></td><td></td></tr></tbody></table>
## A13. Case #13 – ANTHRA_wO_45

<figure><img src="image_184.png" /><figcaption>Figure A25 Case #13 – int. irradiance correlation modelled values as a function of measured values</figcaption></figure>

<figure><img src="image_185.png" /><figcaption>Figure A26 Case #13 – modelled transmittances as a function of solar azimuth angle</figcaption></figure>

<table><caption>Table A13 Validation case #13 – results and specifications</caption><thead><tr><th colspan="4">Case specifications</th></tr></thead><tbody><tr><td><strong>case #13</strong><br/><strong>case name:</strong><br/><strong>ANTHRA_wO_45</strong></td><td>case description:</td><td>win.state: window closed<br/>slats: anthracite, type K80<br/>slat-angle: 45°</td><td></td></tr><tr><td colspan="4"><strong>Transmittance τ<sub>cog.sys</sub></strong></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [1]:</td><td></td><td>errors:</td><td>MAEτ: 0.011<br/>MEτ: 0.011<br/>SEτ: 0.007</td></tr><tr><td>0.034 | 0.038 | 0.050 | 0.054</td><td></td><td></td><td></td></tr><tr><td colspan="4"><strong>Internal irradiance E<sub>int.model</sub></strong></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [W/m²]:</td><td></td><td>errors:</td><td>MAE<sub>E</sub>: 3.9 W/m²</td></tr><tr><td>3.5 | 14.8 | 27.8 | 33.3</td><td></td><td></td><td></td></tr><tr><td colspan="4"><strong>Internal irradiance correlation model vs. measurement:</strong></td></tr><tr><td>number of points: 50</td><td></td><td></td><td></td></tr><tr><td>lin. regression through origin y=0.8214·x</td><td></td><td>R²: 0.995</td><td></td></tr><tr><td>lin. regression y=0.8437·x - 0.5467</td><td></td><td>R²: 0.967</td><td></td></tr><tr><td colspan="4"><strong>Measurement specifications (min | q.1 | q.3 | max):</strong></td></tr><tr><td>sun azimuth range [°]:</td><td>123.8 | 146.9 | 218.8 | 236.2</td><td></td><td></td></tr><tr><td>sun elevation range [°]:</td><td>42.6 | 47.9 | 54.5 | 56.1</td><td></td><td></td></tr><tr><td>incidence angle range [°]:</td><td>55.4 | 56.6 | 62.2 | 66.2</td><td></td><td></td></tr><tr><td>diffuse ratio (DHI/GHI) range [1]:</td><td>0.17 | 0.33 | 0.97 | 1.00</td><td></td><td></td></tr><tr><td>meas. period:</td><td>2020.08.17 | 2020.08.17 | 2020.08.19 | 2020.08.19</td><td></td><td></td></tr></tbody></table>
## A14. Case #14 – ANTHRA_wO_m45

<figure><img src="image_186.png" /><figcaption>Figure A27 Case #14 – int. irradiance correlation modelled values as a function of measured values</figcaption></figure>

<figure><img src="image_187.png" /><figcaption>Figure A28 Case #14 – modelled transmittances as a function of solar azimuth angle</figcaption></figure>

<table><caption>Table A14 Validation case #14 – results and specifications</caption><thead><tr><th colspan="4">Case specifications</th></tr></thead><tbody><tr><td>case #14<br>case name:<br>ANTHRA_wO_m45</td><td>case description:</td><td>win.state: window closed<br>slats: anthracite, type K80<br>slat-angle: -45°</td><td></td></tr><tr><td colspan="4">Transmittance τ<sub>cog.sys</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [1]:<br>0.367 | 0.448 | 0.585 | 0.607</td><td>errors:</td><td>MAEτ: 0.033<br>MEτ: -0.023<br>SEτ: 0.030</td><td></td></tr><tr><td colspan="4">Internal irradiance E<sub>int.model</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [W/m²]:<br>40.8 | 200.0 | 310.0 | 389.4</td><td>errors:</td><td>MAE<sub>E</sub>: 16.8 W/m²</td><td></td></tr><tr><td colspan="4">Internal irradiance correlation model vs. measurement:</td></tr><tr><td>number of points: 44<br>lin. regression through origin<br>lin. regression</td><td>y=1.0585·x<br>y=1.1113·x - 14.4652</td><td>R²: 0.998<br>R²: 0.983</td><td></td></tr><tr><td colspan="4">Measurement specifications (min | q.1 | q.3 | max):</td></tr><tr><td>sun azimuth range [°]:</td><td>118.8 | 139.4 | 201.6 | 236.7</td><td></td><td></td></tr><tr><td>sun elevation range [°]:</td><td>36.4 | 46.8 | 54.4 | 57.3</td><td></td><td></td></tr><tr><td>incidence angle range [°]:</td><td>52.7 | 57.4 | 62.8 | 67.2</td><td></td><td></td></tr><tr><td>diffuse ratio (DHI/GHI) range [1]:</td><td>0.12 | 0.17 | 0.49 | 1.00</td><td></td><td></td></tr><tr><td>meas. period:</td><td>2020.08.05 | 2020.08.13 | 2020.08.27 | 2020.08.27</td><td></td><td></td></tr></tbody></table>
## A15. Case #15 – SILVER_wO_45

<figure><img src="image_188.png" /><figcaption>Figure A29 Case #15 – int. irradiance correlation modelled values as a function of measured values</figcaption></figure>

<figure><img src="image_189.png" /><figcaption>Figure A30 Case #15 – modelled transmittances as a function of solar azimuth angle</figcaption></figure>

<table><caption>Table A15 Validation case #15 – results and specifications</caption><thead><tr><th colspan="4">Case specifications</th></tr></thead><tbody><tr><td>case #15<br>case name:<br>SILVER_wO_45</td><td>case description:</td><td>win.state: window closed<br>slats: silver, type K80<br>slat-angle: 0°</td><td></td></tr><tr><td colspan="4">Transmittance τ<sub>cog.sys</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [1]:<br>0.083 | 0.085 | 0.093 | 0.168</td><td>errors:</td><td>MAEτ: 0.031<br>MEτ: -0.031<br>SEτ: 0.010</td><td></td></tr><tr><td colspan="4">Internal irradiance E<sub>int.model</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [W/m²]:<br>6.9 | 27.8 | 34.0 | 36.3</td><td>errors:</td><td>MAE<sub>E</sub>: 14.6 W/m²</td><td></td></tr><tr><td colspan="4">Internal irradiance correlation model vs. measurement:</td></tr><tr><td>number of points: 27<br>lin. regression through origin<br>lin. regression</td><td>y=1.4498·x<br>y=0.4850·x + 30.0150</td><td>R²: 0.979<br>R²: 0.448</td><td></td></tr><tr><td colspan="4">Measurement specifications (min | q.1 | q.3 | max):</td></tr><tr><td>sun azimuth range [°]:</td><td>96.9 | 156.1 | 200.1 | 223.7</td><td></td><td></td></tr><tr><td>sun elevation range [°]:</td><td>35.6 | 60.4 | 63.3 | 64.5</td><td></td><td></td></tr><tr><td>incidence angle range [°]:</td><td>63.4 | 64.0 | 65.8 | 84.4</td><td></td><td></td></tr><tr><td>diffuse ratio (DHI/GHI) range [1]:</td><td>0.10 | 0.17 | 0.36 | 0.90</td><td></td><td></td></tr><tr><td>meas. period:</td><td>2020.07.09 | 2020.07.17 | 2020.07.20 | 2020.07.20</td><td></td><td></td></tr></tbody></table>
## A16. Case #16 – SILVER_wO_m21

<figure><img src="image_190.png" /><figcaption>Figure A31 Case #16 – int. irradiance correlation modelled values as a function of measured values</figcaption></figure>

<figure><img src="image_191.png" /><figcaption>Figure A32 Case #16 – modelled transmittances as a function of solar azimuth angle</figcaption></figure>

<table><caption>Table A16 Validation case #16 – results and specifications</caption><thead><tr><th colspan="4">Case specifications</th></tr></thead><tbody><tr><td>case #16</td><td>case description:</td><td>win.state:</td><td>window closed</td></tr><tr><td>case name:</td><td></td><td>slats:</td><td>silver, type K80</td></tr><tr><td>SILVER_wO_m21</td><td></td><td>slat-angle:</td><td>-21°</td></tr><tr><td colspan="4">Transmittance τ<sub>cog.sys</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [1]:</td><td></td><td>errors:</td><td>MAEτ: 0.022<br>MEτ: 0.010<br>SEτ: 0.027</td></tr><tr><td>0.186 | 0.207 | 0.363 | 0.504</td><td></td><td></td><td></td></tr><tr><td colspan="4">Internal irradiance E<sub>int.model</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [W/m<sup>2</sup>]:</td><td></td><td>errors:</td><td>MAE<sub>E</sub>: 7.9 W/m<sup>2</sup></td></tr><tr><td>47.6 | 93.8 | 136.1 | 147.7</td><td></td><td></td><td></td></tr><tr><td colspan="4">Internal irradiance correlation model vs. measurement:</td></tr><tr><td>number of points:</td><td>23</td><td></td><td></td></tr><tr><td>lin. regression through origin</td><td>y=0.9458·x</td><td>R<sup>2</sup>: 0.995</td><td></td></tr><tr><td>lin. regression</td><td>y=0.8400·x + 12.5163</td><td>R<sup>2</sup>: 0.921</td><td></td></tr><tr><td colspan="4">Measurement specifications (min | q.1 | q.3 | max):</td></tr><tr><td>sun azimuth range [°]:</td><td>92.5 | 155.0 | 194.6 | 218.9</td><td></td><td></td></tr><tr><td>sun elevation range [°]:</td><td>32.4 | 61.2 | 64.2 | 64.8</td><td></td><td></td></tr><tr><td>incidence angle range [°]:</td><td>64.5 | 64.8 | 66.2 | 87.9</td><td></td><td></td></tr><tr><td>diffuse ratio (DHI/GHI) range [1]:</td><td>0.10 | 0.20 | 0.47 | 1.00</td><td></td><td></td></tr><tr><td>meas. period:</td><td>2020.07.10 | 2020.07.13 | 2020.07.14 | 2020.07.14</td><td></td><td></td></tr></tbody></table>
## A17. Case #17 – SILVER_wO_m45

<figure><img src="image_192.png" /><figcaption>Figure A33 Case #17 – int. irradiance correlation modelled values as a function of measured values</figcaption></figure>

<figure><img src="image_193.png" /><figcaption>Figure A34 Case #17 – modelled transmittances as a function of solar azimuth angle</figcaption></figure>

<table><caption>Table A17 Validation case #17 – results and specifications</caption><thead><tr><th colspan="4">Case specifications</th></tr></thead><tbody><tr><td><strong>case #17</strong><br><strong>case name:</strong><br><strong>SILVER_wO_m45</strong></td><td>case description:</td><td>win.state: window closed<br>slats: silver, type K80<br>slat-angle: -45°</td><td></td></tr><tr><td colspan="4"><strong>Transmittance τ<sub>cog.sys</sub></strong></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [1]:<br>0.134 | 0.332 | 0.372 | 0.475</td><td>errors:</td><td>MAEτ: 0.024<br>MEτ: 0.012<br>SEτ: 0.035</td><td></td></tr><tr><td colspan="4"><strong>Internal irradiance E<sub>int.model</sub></strong></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [W/m²]:<br>29.4 | 134.0 | 180.9 | 200.7</td><td>errors:</td><td>MAE<sub>E</sub>: 7.7 W/m²</td><td></td></tr><tr><td colspan="4"><strong>Internal irradiance correlation model vs. measurement:</strong></td></tr><tr><td>number of points: 27<br>lin. regression through origin y=0.9958·x<br>lin. regression y=1.0641·x - 11.4049</td><td>R²:</td><td>0.996<br>0.971</td><td></td></tr><tr><td colspan="4"><strong>Measurement specifications (min | q.1 | q.3 | max):</strong></td></tr><tr><td>sun azimuth range [°]:</td><td>93.7 | 153.0 | 195.3 | 218.8</td><td></td><td></td></tr><tr><td>sun elevation range [°]:</td><td>31.8 | 60.6 | 63.8 | 64.3</td><td></td><td></td></tr><tr><td>incidence angle range [°]:</td><td>64.1 | 64.4 | 66.0 | 86.8</td><td></td><td></td></tr><tr><td>diffuse ratio (DHI/GHI) range [1]:</td><td>0.13 | 0.18 | 0.63 | 1.00</td><td></td><td></td></tr><tr><td>meas. period:</td><td>2020.07.15 | 2020.07.15 | 2020.07.16 | 2020.07.21</td><td></td><td></td></tr></tbody></table>
## A18. Case #18 – WHITE_wO_00

<figure><img src="image_194.png" /><figcaption>Figure A35 Case #18 – int. irradiance correlation modelled values as a function of measured values</figcaption></figure>

<figure><img src="image_195.png" /><figcaption>Figure A36 Case #18 – modelled transmittances as a function of solar azimuth angle</figcaption></figure>

<table><caption>Table A18 Validation case #18 – results and specifications</caption><tr><td colspan="2"><strong>Case specifications</strong></td><td></td><td></td><td></td></tr><tr><td><strong>case #18</strong><br/><strong>case name:</strong><br/><strong>WHITE_wO_00</strong></td><td>case description:</td><td>win.state:</td><td>window closed</td><td></td></tr><tr><td></td><td></td><td>slats:</td><td>white, type K80</td><td></td></tr><tr><td></td><td></td><td>slat-angle:</td><td>0°</td><td></td></tr><tr><td colspan="2"><strong>Transmittance τ<sub>cog.sys</sub></strong></td><td></td><td></td><td></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [1]:</td><td></td><td>errors:</td><td>MAEτ: 0.024</td><td></td></tr><tr><td>0.266 | 0.272 | 0.316 | 0.398</td><td></td><td>MEτ: -0.022</td><td></td><td></td></tr><tr><td></td><td></td><td>SEτ: 0.018</td><td></td><td></td></tr><tr><td colspan="2"><strong>Internal irradiance E<sub>int.model</sub></strong></td><td></td><td></td><td></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [W/m²]:</td><td></td><td>errors:</td><td>MAE<sub>E</sub>: 14.4 W/m²</td><td></td></tr><tr><td>63.2 | 134.3 | 182.7 | 231.5</td><td></td><td></td><td></td><td></td></tr><tr><td colspan="2"><strong>Internal irradiance correlation model vs. measurement:</strong></td><td></td><td></td><td></td></tr><tr><td>number of points:</td><td>37</td><td></td><td></td><td></td></tr><tr><td>lin. regression through origin</td><td>y=1.0729·x</td><td>R²:</td><td>0.995</td><td></td></tr><tr><td>lin. regression</td><td>y=0.9452·x + 21.5458</td><td>R²:</td><td>0.923</td><td></td></tr><tr><td colspan="2"><strong>Measurement specifications (min | q.1 | q.3 | max):</strong></td><td></td><td></td><td></td></tr><tr><td>sun azimuth range [°]:</td><td>122.2 | 143.2 | 218.3 | 236.8</td><td></td><td></td><td></td></tr><tr><td>sun elevation range [°]:</td><td>31.7 | 39.1 | 47.5 | 50.1</td><td></td><td></td><td></td></tr><tr><td>incidence angle range [°]:</td><td>43.0 | 50.7 | 59.9 | 63.6</td><td></td><td></td><td></td></tr><tr><td>diffuse ratio (DHI/GHI) range [1]:</td><td>0.13 | 0.19 | 0.38 | 0.92</td><td></td><td></td><td></td></tr><tr><td>meas. period:</td><td>2020.08.28 | 2020.09.03 | 2020.09.11 | 2020.09.22</td><td></td><td></td><td></td></tr></table>
## A19. Case #19 – WHITE_wO_45

<figure><img src="image_196.png" /><figcaption>Figure A37 Case #19 – int. irradiance correlation modelled values as a function of measured values</figcaption></figure>

<figure><img src="image_197.png" /><figcaption>Figure A38 Case #19 – modelled transmittances as a function of solar azimuth angle</figcaption></figure>

<table><caption>Table A19 Validation case #19 – results and specifications</caption><tr><td colspan="2"><strong>Case specifications</strong></td><td></td><td></td><td></td></tr><tr><td><strong>case #19</strong><br/><strong>case name:</strong><br/><strong>WHITE_wO_45</strong></td><td>case description:</td><td>win.state:</td><td>window closed</td><td></td></tr><tr><td></td><td></td><td>slats:</td><td>white, type K80</td><td></td></tr><tr><td></td><td></td><td>slat-angle:</td><td>45°</td><td></td></tr><tr><td colspan="2"><strong>Transmittance τ<sub>cog.sys</sub></strong></td><td></td><td></td><td></td></tr><tr><td></td><td>range of determined values (min | q.1 | q.3 | max) [1]:</td><td>errors:</td><td>MAEτ: 0.020</td><td></td></tr><tr><td></td><td>0.118 | 0.121 | 0.125 | 0.129</td><td></td><td>MEτ: 0.020</td><td></td></tr><tr><td></td><td></td><td></td><td>SEτ: 0.011</td><td></td></tr><tr><td colspan="2"><strong>Internal irradiance E<sub>int.model</sub></strong></td><td></td><td></td><td></td></tr><tr><td></td><td>range of determined values (min | q.1 | q.3 | max) [W/m²]:</td><td>errors:</td><td>MAE<sub>E</sub>: 9.9 W/m²</td><td></td></tr><tr><td></td><td>46.3 | 61.7 | 93.9 | 101.5</td><td></td><td></td><td></td></tr><tr><td colspan="2"><strong>Internal irradiance correlation model vs. measurement:</strong></td><td></td><td></td><td></td></tr><tr><td></td><td>number of points: 24</td><td></td><td></td><td></td></tr><tr><td>lin. regression through origin</td><td>y=0.8800·x</td><td>R²: 0.993</td><td></td><td></td></tr><tr><td>lin. regression</td><td>y=1.0468·x - 13.4919</td><td>R²: 0.929</td><td></td><td></td></tr><tr><td colspan="2"><strong>Measurement specifications (min | q.1 | q.3 | max):</strong></td><td></td><td></td><td></td></tr><tr><td></td><td>sun azimuth range [°]: 189.2 | 206.2 | 231.8 | 244.0</td><td></td><td></td><td></td></tr><tr><td></td><td>sun elevation range [°]: 21.7 | 29.7 | 39.6 | 42.5</td><td></td><td></td><td></td></tr><tr><td></td><td>incidence angle range [°]: 43.0 | 46.3 | 57.5 | 66.0</td><td></td><td></td><td></td></tr><tr><td></td><td>diffuse ratio (DHI/GHI) range [1]: 0.24 | 0.31 | 0.51 | 0.61</td><td></td><td></td><td></td></tr><tr><td></td><td>meas. period: 2020.09.22 | 2020.09.22 | 2020.09.23 | 2020.09.23</td><td></td><td></td><td></td></tr></table>
## A20. Case #20 – WHITE_wO_m45

<figure><img src="image_198.png" /><figcaption>Figure A39 Case #20 – int. irradiance correlation modelled values as a function of measured values</figcaption></figure>

<figure><img src="image_199.png" /><figcaption>Figure A40 Case #20 – modelled transmittances as a function of solar azimuth angle</figcaption></figure>

<table><caption>Table A20 Validation case #20 – results and specifications</caption><thead><tr><th colspan="4">Case specifications</th></tr></thead><tbody><tr><td>case #20<br>case name:<br>WHITE_wO_m45</td><td>case description:</td><td>win.state: window closed<br>slats: white, type K80<br>slat-angle: -45°</td><td></td></tr><tr><td colspan="4">Transmittance τ<sub>cog.sys</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [1]:</td><td></td><td>errors:</td><td>MAEτ: 0.067<br>MEτ: -0.061<br>SEτ: 0.045</td></tr><tr><td>0.643 | 0.715 | 0.746 | 0.762</td><td></td><td></td><td></td></tr><tr><td colspan="4">Internal irradiance E<sub>int.model</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [W/m²]:</td><td></td><td>errors:</td><td>MAE<sub>E</sub>: 33.9 W/m²</td></tr><tr><td>200.3 | 286.0 | 394.5 | 518.3</td><td></td><td></td><td></td></tr><tr><td colspan="4">Internal irradiance correlation model vs. measurement:</td></tr><tr><td>number of points: 24</td><td></td><td></td><td></td></tr><tr><td>lin. regression through origin y=1.0835·x</td><td></td><td>R²: 0.996</td><td></td></tr><tr><td>lin. regression y=0.9822·x + 37.2080</td><td></td><td>R²: 0.933</td><td></td></tr><tr><td colspan="4">Measurement specifications (min | q.1 | q.3 | max):</td></tr><tr><td>sun azimuth range [°]:</td><td>124.8 | 143.8 | 227.5 | 239.3</td><td></td><td></td></tr><tr><td>sun elevation range [°]:</td><td>31.1 | 36.1 | 42.0 | 47.5</td><td></td><td></td></tr><tr><td>incidence angle range [°]:</td><td>47.6 | 53.1 | 59.4 | 64.1</td><td></td><td></td></tr><tr><td>diffuse ratio (DHI/GHI) range [1]:</td><td>0.12 | 0.15 | 0.47 | 0.93</td><td></td><td></td></tr><tr><td>meas. period:</td><td>2020.09.09 | 2020.09.10 | 2020.09.10 | 2020.09.10</td><td></td><td></td></tr></tbody></table>
## A21. Case #21 – Z-SILVER_wO_00

<figure><img src="image_200.png" /><figcaption>Figure A41 Case #21 – int. irradiance correlation modelled values as a function of measured values</figcaption></figure>

<figure><img src="image_201.png" /><figcaption>Figure A42 Case #21 – modelled transmittances as a function of solar azimuth angle</figcaption></figure>

<table><caption>Table A21 Validation case #21 – results and specifications</caption><thead><tr><th colspan="5">Case specifications</th></tr></thead><tbody><tr><td>case #21<br>case name:<br>Z-SILVER_wO_00</td><td>case description:</td><td>win.state:<br>slats:<br>slat-angle:</td><td>window closed<br>silver, type VZ90<br>0°</td><td></td></tr><tr><td colspan="5">Transmittance τ<sub>cog.sys</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [1]:<br>0.272 | 0.278 | 0.325 | 0.356</td><td>errors:</td><td>MAEτ:<br>MEτ:<br>SEτ:</td><td>0.030<br>-0.009<br>0,035</td><td></td></tr><tr><td colspan="5">Internal irradiance E<sub>int.model</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [W/m²]:<br>54.9 | 91.4 | 177.6 | 211.8</td><td>errors:</td><td>MAE<sub>E</sub>:</td><td>14.1 W/m²</td><td></td></tr><tr><td colspan="5">Internal irradiance correlation model vs. measurement:</td></tr><tr><td>number of points:<br>lin. regression through origin</td><td>15<br>y=1.0513·x</td><td>R²:</td><td>0.991<br>0.931</td><td></td></tr><tr><td>lin. regression</td><td>y=1.0707·x - 2.9988</td><td>R²:</td><td></td><td></td></tr><tr><td colspan="5">Measurement specifications (min | q.1 | q.3 | max):</td></tr><tr><td>sun azimuth range [°]:<br>sun elevation range [°]:<br>incidence angle range [°]:<br>diffuse ratio (DHI/GHI) range [1]:<br>meas. period:</td><td>202.4 | 211.4 | 234.9 | 245.5<br>20.1 | 27.1 | 36.1 | 38.7<br>42.8 | 46.4 | 59.3 | 67.1<br>0.13 | 0.16 | 0.77 | 0.96<br>2020.09.24 | 2020.09.24 | 2020.09.30 | 2020.09.30</td><td></td><td></td><td></td></tr></tbody></table>
## A22. Case #22 – Z-SILVER_wO_45

<figure><img src="image_202.png" /><figcaption>Figure A43 Case #22 – int. irradiance correlation modelled values as a function of measured values</figcaption></figure>

<figure><img src="image_203.png" /><figcaption>Figure A44 Case #22 – modelled transmittances as a function of solar azimuth angle</figcaption></figure>

<table><caption>Table A22 Validation case #22 – results and specifications</caption><tbody><tr><td colspan="4"><strong>Case specifications</strong></td></tr><tr><td><strong>case #22</strong><br/><strong>case name:</strong><br/><strong>Z-SILVER_wO_45</strong></td><td><strong>case description:</strong></td><td><strong>win.state:</strong><br/><strong>slats:</strong><br/><strong>slat-angle:</strong></td><td><strong>window closed</strong><br/><strong>silver, type VZ90</strong><br/><strong>45°</strong></td></tr><tr><td colspan="4"><strong>Transmittance τ<sub>cog.sys</sub></strong></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [1]:<br/><strong>0.078 | 0.078 | 0.080 | 0.080</strong></td><td><strong>errors:</strong></td><td><strong>MAEτ:</strong><br/><strong>MEτ:</strong><br/><strong>SEτ:</strong></td><td><strong>0.013</strong><br/><strong>-0.013</strong><br/><strong>0,001</strong></td></tr><tr><td colspan="4"><strong>Internal irradiance E<sub>int.model</sub></strong></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [W/m²]:<br/><strong>35.2 | 35.2 | 43.3 | 43.3</strong></td><td><strong>errors:</strong></td><td><strong>MAE<sub>E</sub>:</strong></td><td><strong>7.4 W/m²</strong></td></tr><tr><td colspan="4"><strong>Internal irradiance correlation model vs. measurement:</strong></td></tr><tr><td>number of points: <strong>2</strong></td><td></td><td></td><td></td></tr><tr><td>lin. regression through origin<br/><strong>y=1.1866·x</strong></td><td><strong>R²:</strong></td><td><strong>1.000</strong></td><td></td></tr><tr><td>lin. regression<br/><strong>y=0.9991·x + 7.4329</strong></td><td><strong>R²:</strong></td><td><strong>1.000</strong></td><td></td></tr><tr><td colspan="4"><strong>Measurement specifications (min | q.1 | q.3 | max):</strong></td></tr><tr><td>sun azimuth range [°]:<br/><strong>133.8 | 133.8 | 136.8 | 136.8</strong></td><td></td><td></td><td></td></tr><tr><td>sun elevation range [°]:<br/><strong>28.8 | 28.8 | 30.3 | 30.3</strong></td><td></td><td></td><td></td></tr><tr><td>incidence angle range [°]:<br/><strong>51.0 | 51.0 | 52.7 | 52.7</strong></td><td></td><td></td><td></td></tr><tr><td>diffuse ratio (DHI/GHI) range [1]:<br/><strong>0.24 | 0.24 | 0.29 | 0.29</strong></td><td></td><td></td><td></td></tr><tr><td>meas. period:<br/><strong>2020.10.01 | 2020.10.01 | 2020.10.01 | 2020.10.01</strong></td><td></td><td></td><td></td></tr></tbody></table>
## A23. Case #23 – Z-SILVER_wO_58

<figure><img src="image_204.png" /><figcaption>Figure A45 Case #23 – int. irradiance correlation modelled values as a function of measured values</figcaption></figure>

<figure><img src="image_205.png" /><figcaption>Figure A46 Case #23 – modelled transmittances as a function of solar azimuth angle</figcaption></figure>

<table><caption>Table A23 Validation case #23 – results and specifications</caption><thead><tr><th colspan="4">Case specifications</th></tr></thead><tbody><tr><td>case #23<br>case name:<br>Z-SILVER_wO_58</td><td>case description:</td><td>win.state: window closed<br>slats: silver, type VZ90<br>slat-angle: 58°</td><td></td></tr><tr><td colspan="4">Transmittance τ<sub>cog.sys</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [1]:<br>0.047 | 0.047 | 0.056 | 0.058</td><td>errors:</td><td>MAEτ: 0.012<br>MEτ: -0.012<br>SEτ: 0.004</td><td></td></tr><tr><td colspan="4">Internal irradiance E<sub>int.model</sub></td></tr><tr><td>range of determined values (min | q.1 | q.3 | max) [W/m²]:<br>3.1 | 7.0 | 30.2 | 32.7</td><td>errors:</td><td>MAE<sub>E</sub>: 4.5 W/m²</td><td></td></tr><tr><td colspan="4">Internal irradiance correlation model vs. measurement:</td></tr><tr><td>number of points: 13<br>lin. regression through origin<br>lin. regression</td><td>y=1.2324·x<br>y=1.2017·x + 0.7648</td><td>R²: 0.997<br>R²: 0.991</td><td></td></tr><tr><td colspan="4">Measurement specifications (min | q.1 | q.3 | max):</td></tr><tr><td>sun azimuth range [°]:</td><td>126.6 | 143.0 | 187.2 | 199.5</td><td></td><td></td></tr><tr><td>sun elevation range [°]:</td><td>25.9 | 33.6 | 39.7 | 39.8</td><td></td><td></td></tr><tr><td>incidence angle range [°]:</td><td>39.9 | 40.0 | 48.5 | 57.5</td><td></td><td></td></tr><tr><td>diffuse ratio (DHI/GHI) range [1]:</td><td>0.11 | 0.12 | 0.99 | 1.00</td><td></td><td></td></tr><tr><td>meas. period:</td><td>2020.09.29 | 2020.09.29 | 2020.09.30 | 2020.09.30</td><td></td><td></td></tr></tbody></table>