## 3 Mathematical Formulation of the CPV  
### RADTIRC as the Reference for the Model  
The RADTIRC (see Figure 1) was invented by [PERSON] et al. [44] with the aim to improve the electrical yield which could minimise the amount of costly PV material; hence reducing the cost of PV system [45]. The procedure to create the RADTIRC has been described thoroughly by [PERSON] et al. [44]. For the past few years, several analyses have been carried out to evaluate the concentrator's performance such as evaluating its geometrical concentration gain and its optical concentration gain [45], identifying the most cost-effective fabrication technique to mass-produce the concentrator [46], integration in a double-glazed window [47] and calculating its annual energy output [48]. Based on these studies, it was found that it is more desirable to utilise a shorter RADTIRC that have a larger half-acceptance angle and a lower geometrical concentration gain than the ones with a smaller half-acceptance angle and a higher geometrical concentration gain because: (i) it requires less material hence minimises the cost of material; (ii) it captures the sun energy for extended period during daytime; and (iii) the concentrator exhibits a more uniform distribution of flux, i.e., the amount of 'hot spots' diminished on the PV cell, which translates to a lower temperature, and eventually allows the RADTIRC-PV structure to function at its optimum efficiency level. Similar findings were also recorded by [PERSON] et al. [49] and [PERSON] [50] when optimising their static CPVs. The indoor experiments to characterise the RADITIRC-PV structure have been presented by [PERSON] et al. in [51]. The RADITIRC has a geometrical concentration gain of 4.91, a height of 3 cm and a square exit aperture area of 1 cm\({}^{2}\). The concentrator is designed to achieve a half-acceptance angle of \(\pm\)40\({}^{\circ}\) along its \(x\)-axis. The indoor experiments utilised a Class A solar simulator produced by Abet Technologies from the Sun 2000 model, a Keithley 2400 source meter, which are connected to a computer that has been installed with a National Instrument software. The experiments were conducted under STCs to mimic the simulation conditions, and the setup is depicted in the following Figure 2.  
The angular response of the RADITIRC-PV structure and the bare PV cell were investigated at different incident angles between \(-\)50\({}^{\circ}\) and 50\({}^{\circ}\). The angular response enables the opto-electronic gain, C\({}_{\text{opto}}\) of the RADITIRC to be determine at different angle of incidence. The opto-electronic gain of the RADITIRC at each angle of incidence is presented in Table 1. The opto-electronic gain is a ratio of short circuit current generated from RADITRC-PV to the one generated from a bare PV cell. [PERSON] and [PERSON] [52] define the opto-electronic gain as a product of optical efficiency and the geometrical concentration gain of a concentrator. The optical efficiency value determines the amount of sun rays that travel from the concentrator aperture to its exit aperture [17]. It takes into consideration the rays that are loss due to reflection, scattering and absorption. In the case of dielectric material, it also considers the rays that escape from the side profile of the concentrator. The value of optical  
Figure 1: (**a**) A small concentrating solar PV window integrating the rotationally asymmetrical dielectric totally internally reflecting concentrators (RADITIRCs); and (**b**) a single RADITIRC that has an acceptance angle of \(\pm\)40\({}^{\circ}\). Figure 2: Indoor experimental setup to evaluate the I-V and P-V characteristics of the RADITIRC-PV structure and the bare PV cell. efficiency is \(\leq\)1 [53]. [PERSON] [54] has demonstrated that the optical efficiency value is also independent of the irradiance value. Specifically for any CPV, we propose that the value of opto-electronic gain, \(\mathrm{C_{opto-e}}\) needs to be included in the mathematical model, which was obtained from experiments.