## 4 Results and Discussion
### Case Study 1: Comparison with Non-Optimized Literature Work  
To verify the algorithmic design and optimization processes, the work of [PERSON] et al. [5] was selected for a case study. It compared many SOE shapes, among which a conventional non-optimized spherical dome SOE was included, based on an equal acceptance angle, \(\alpha_{90\%}\), of 1.0\({}^{\circ}\) (the incidence angle at which the optical efficiency is 90% of the maximum value). The same 625 cm\({}^{2}\) POE was used for all. The dome SOE had a geometrical concentration, \(CG\), of 257\(\times\) and an f-number of 1.5, which results in a CAP value of 0.28 (\(CAP=\sqrt{CG}\,\sin\alpha_{90\%}\)). For this work's POE algorithm, an equivalent area of 625 cm\({}^{2}\) and an f-number of 1.5 was used, with a design incidence angle of 1.0\({}^{\circ}\) and a groove width of 3 mm with PMMA lens material. The algorithm returned the prismatic characteristics in Figure 5a, indicating the prism angles, \(\theta_{Liptimum}\) and groove thicknesses, \(h_{i}\). The groove width selection was based on a sensitivity analysis that included a range of 1-5 mm values. By feeding the POE design values and applying the SOE optimization algorithm at a 1% threshold, \(T_{s}\), with a TIR prevention scheme from the SOE base, 0.2 mm of radius increment, \(r_{inc}\), and PMMA SOE material, the following concentrator parameters were obtained:  
1. Optimum SOE radius (\(r\)): 13.6 mm;
2. Optimum recommended receiver radius (\(FS_{opt}\)): 6.65 mm;
3. Optimum SOE-receiver spacing (\(SR_{opt}\)): 5.0 mm. Then, the concentrator was simulated with Monte-Carlo ray tracing (MCRT) via COMSOL Multiphysics(r) version 5.3. Ray independence analysis was carried out for 10\({}^{3}\)-10\({}^{6}\) rays, and a total of  
Figure 5: For Case 1: (**a**) prism angles (dashed blue line) and groove thickness (solid orange line) of the optimum Fresnel lens POE and (**b**) the concentrator’s optical efficiency (POE + SOE) at different incidence angles. \(10^{4}\) rays were found sufficient for this simulation to save computational time. An 850 W/m\({}^{2}\) solar flux was shone on the lens, the same as that used in literature, at a range of incidence angles from 0.0\({}^{\circ}\) to 2.0\({}^{\circ}\). The corresponding optical efficiencies, \(\eta_{o}\), are plotted in Figure 5b, from which the acceptance angle, \(\alpha_{90\%}\), was found as 1.15\({}^{\circ}\). With the optimized receiver radius of 6.65 mm, the concentrator's _CG_ was 449.6\(\times\), which resulted in a CAP of 0.426--about 52% higher than that by [PERSON] et al. [5]. The 2D ray trajectories and the focal irradiance distribution at the acceptance angle are illustrated in Figure 6a,b, respectively. Figure 6a shows all solar rays being refracted through the optimized SOE without TIR. The focal irradiance distribution, depicted by Figure 6b, peaks at 221 suns (1 sun = 1000 W/m\({}^{2}\)) with poor homogeneity. [PERSON] et al. [5] mentioned that the irradiance homogeneity by a spherical dome SOE will be poor and can be improved by an enhanced POE design but at the expense of a reduced CAP. However, as discussed later in this section changing the SOE-receiver spacing for the same POE can also fine-tune the irradiance and produce better homogeneity. A simulative study was carried out to verify the analytical optimization. SOE parameters were iterated one at a time while fixing the other two, and the effects of changing each parameter were tested with COMSOL(r). The resulting optical efficiencies are depicted against the iterated parameters in Figure 7a-c.  
Figure 6: For Case 1: (**a**) ray trajectories through the optimum SOE parameters of this work at the acceptance angle of 1.15\({}^{\circ}\) and (**b**) the receiver’s irradiance distribution in W/m\({}^{2}\) at the acceptance angle of 1.15\({}^{\circ}\).