**A Novel Module Independent Straight Line-Based Fast Maximum Power Point Tracking Algorithm for Photovoltaic Systems**

#### **Anjan Debnath, Temitayo O. Olowu, Imtiaz Parvez, Md Golam Dastgir and Arif Sarwat \***

Electrical and Computer Engineering, Florida International University, 10555 W Flagler St, Miami, FL 33174, USA; adebn001@fiu.edu (A.D.); tolow003@fiu.edu (T.O.O.); iparv001@fiu.edu (I.P.); mdast001@fiu.edu (M.G.D.)

**\*** Correspondence: asarwat@fiu.edu

Received: 17 May 2020; Accepted: 12 June 2020; Published: 22 June 2020

**Abstract:** The maximum power point tracking (MPPT) algorithm has become an integral part of many charge controllers that are used in photovoltaic (PV) systems. Most of the existing algorithms have a compromise among simplicity, tracking speed, ability to track accurately, and cost. In this work, a novel "straight-line approximation based Maximum Power Point (MPP) finding algorithm" is proposed where the intersections of two linear lines have been utilized to find the MPP, and investigated for its effectiveness in tracking maximum power points in case of rapidly changing weather conditions along with tracking speed using standard irradiance and temperature curves for validation. In comparison with a conventional Perturb and Observe (*P&O*) method, the *Proposed* method takes fewer iterations and also, it can precisely track the MPP s even in a rapidly varying weather condition with minimal deviation. The *Proposed* algorithm is also compared with *P&O* algorithm in terms of accuracy in duty cycle and efficiency. The results show that the errors in duty cycle and power extraction are much smaller than the conventional *P&O* algorithm.

**Keywords:** linear approximation; MPPT algorithm; duty cycle; global horizontal irradiance; mathematical modeling

#### **1. Introduction**

Electricity generation from renewable energy sources has consistently increased over the past decade, with the largest percentage of renewable energy sources integrated being photovoltaic (PV) systems [1–5]. There has been a consistent increase in installed PV capacity globally which has lead to a corresponding increase in power generation from PV systems as shown in Figure 1 [6].

The massive integration of PV systems has also been aided by the declining cost of PV modules and improvement in their efficiencies [7]. Other than grid-tied PV systems, the use of PV modules has been extended to various applications such as rooftop solar power supply for residential homes, mobile charging systems, wearable devices, standalone PV arrays as car parks and electric vehicle charging stations, remote weather stations, international space station, and robots for space exploration.

In contrast to PV systems, Solar Thermal Plants (STPs) allow power to be generated by concentrating solar radiation which causes a very high temperature. The heat produced is subsequently used to convert water to steam which then used to turn power turbines for electricity generation. STPs are oriented to allow maximum tracking of the sun's insolation. The plants usually consist of reflectors and receivers. The reflectors (which are typically mirrors) are used to capture and concentrate the sun's light rays onto the receivers. The heat absorbed by the receivers is used to convert water to steam in order to drive the conventional steam turbine generators. It is of uttermost importance for the reflectors used in solar thermal plants to constantly track the movement of the sun in order to ensure

maximum extraction of solar rays in order to maximize the efficiency of the plant [8–10]. In comparison with PV systems, Thermal Energy Storage (TES) technologies can be used to store energy for use at night, during severe cloud coverage, or periods with little or no sunlight. This allows STPs to be more dispatchable and achieved a higher capacity factor compared to PV. On the other hand, the decking cost of PV systems makes them cheaper than the STPs. A report by [11] suggests that STP technology might in the future help to increase the amount of PV penetration in the grid rather than being a direct competitor or alternative.

**Figure 1.** Power generation from renewable energy sources globally [6].

Due to the relatively low efficiency of PV systems compared to conventional power sources such as coal, diesel, or gas-fired plants, extraction of the maximum power per unit PV module becomes imperative. Maximum Power Point Tracking (MPPT) algorithms are developed to allow PV systems to operate at their maximum power under prevailing weather conditions [12–14]. The power generation from PV systems largely depends on the global horizontal irradiance and the atmospheric (consequently module) temperature. These parameters are stochastic in nature which means an accurate tracking system (simply called the Maximum Power Point Tracking (MPPT) system) to extract the maximum power out of the PVs is necessary. Several techniques and algorithms have been proposed for MPPT applications. These techniques and algorithms include the machine learning [15,16], hill climbing method [17–21], incremental conductance technique [22–25], Perturb and Observe [26–28], fractional open circuit voltage [29–31], fractional short circuit current [32–34], and fuzzy logic-based MPPT algorithms [35–38] amongst others. These algorithms aim to achieve high efficiency, fast-tracking speed, reduced steady-state oscillations, and reduced complexity in hardware implementation [38]. Out of these algorithms, the Perturb and Observe (P&O) algorithm, Incremental Conductance, and Fractional Open Circuit algorithm are most popular because of their simplicity. The *P&O* MPPT algorithm has dual shortcomings: enormous power loss due to large oscillations and deflection of tracking under rapid weather conditions. The large power loss can be minimized by choosing a small step size around the Maximum Power Point (MPP). However, the convergence speed has to be sacrificed [39]. The main drawbacks of Incremental and Conductance algorithms are the complex circuitry because of the derivative finding and choice of perturbation extent. The fractional open circuit based algorithm depends on the open circuit voltage of the panel, so at every change of irradiance, there is a temporary power loss due to the measurement of *VOC*. Moreover, it is not module independent since the *VOC* value has to be known prior to the application of the algorithm. Fractional short circuit current algorithm has the same issue as the fractional open circuit voltage. The algorithm is required to know the characteristics of the PV module and manufacturing specifications which makes it module dependent and less efficient. Machine learning (such as ANN) based MPPT algorithms are dependent on features

(weather data) to generate the desired duty cycle for finding the MP point and one of the most common features is irradiance which requires costly sensors to measure. Though these algorithms are fast in finding MPPs, they are not cost-effective. Moreover, they require a lot of data to train which is the most important part of those models to demonstrate good accuracy. Therefore, it can not be generalized for any module. The PV module characteristics also change with time which necessitates the need for periodic training of the ANN-based controller to accurately track the MPP.

It is obvious that selecting an MPPT algorithm for many applications is usually a trade-off between efficiency, complexity, speed, cost, and ease of hardware implementation. This paper proposes a novel MPPT algorithm where the Maximum Power Point is acquired by the intersection of two linear lines which are tangents to the power vs voltage curve of the photovoltaic module. The major contributions of this paper are the following:


MATLAB/SIMULINK environment is used for the simulation and verification of the *Proposed* algorithm. BP Solar BP SX 150S PV module is chosen in the MATLAB simulation model. The module is made of 72 multi-crystalline silicon solar cells in series and provides 150 W of nominal maximum power [40]. Table 1 shows the electrical parameters of the module:

**Table 1.** Electrical characteristics of PV Module Specifications.


The rest of this paper is structured as follows: Section 2 presents the development of *Proposed* algorithm and description of the *P&O*; Section 3 represents simulation results using the *Proposed* algorithm under slow and fast varying weather conditions, a comparison between the *Proposed* algorithm with the *P&O* algorithm and validation of the *Proposed* algorithm using the CENELEC EN50530 standard test procedures; Section 4 concludes this paper.

#### **2. Development of** *Proposed* **Algorithm and Description of the** *P&O*

Several MPPT algorithms have been proposed in literature. The P&O, which is one of the widely used MPPT algorithms is selected as a baseline to compare the performance of the *Proposed* algorithm. The detailed formulation and modeling of the *Proposed* algorithm and a brief description of the *P&O* is as presented the following subsection.

#### *2.1. Development of the Proposed Algorithm*

Figure 2, the flowchart of the *Proposed* algorithm, depicts the construction of the algorithm. In developing the algorithm, a straight line approximation technique is utilized which is based on the P-V curve shapes. There is no evidence of significantly different P-V curve shape other than an inverted tilted '*V*'. So, the *Proposed* algorithm can be employed universally for a controller coupled with any kind of P-V panel without taking the panel's preset values. The details of the flowchart are depicted in the following:


#### 2.1.1. Mathematical Model of the *Proposed* Algorithm

In the developed method, MPP can be tracked very quickly and this operation is graphically illustrated in Figure 3. To understand the *Proposed* algorithm, let us take the operating point *P*<sup>1</sup> which is at the left side of MPP. By a small perturbation of voltage, another point *P*<sup>2</sup> can be found as: Δ*P* = *P*<sup>2</sup> − *P*<sup>1</sup> > 0

Now, by straight a line approximation, we can write an equation:

$$P = m\_1 V + \mathbb{C}\_1 \tag{1}$$

where:

$$m\_1 = (P\_2 - P\_1) / (V\_2 - V\_1) \quad \text{and}$$

$$\mathbb{C}\_1 = (V\_1 P\_2 - V\_2 P\_1) / (V\_2 - V\_1)$$

Now by taking another two points in the right side of MPP where Δ *P* = *P*<sup>4</sup> − *P*<sup>3</sup> < 0 does match, another equation can be written also.

$$P = m\_2V + \mathcal{C}\_2\tag{2}$$

where:

$$m\_2 = (P\_4 - P\_3) / (V\_4 - V\_3) \quad \text{and}$$

$$C\_2 = (V\_3P\_4 - V\_4P\_3)/(V\_3 - V\_4)$$

By solving Equations (1) and (2), the intersecting point *VA* can be found as follows:

$$\begin{aligned} m\_1V + \mathcal{C}\_1 &= m\_2V + \mathcal{C}\_2 \quad \text{and} \\\\ V\_A &= (\mathcal{C}\_2 - \mathcal{C}\_1)/(m\_1 - m\_2) \end{aligned}$$

After getting *VA* which is very near to MPP, the algorithm would reach MPP very fast as depicted in Figure 3.

**Figure 2.** Flowchart of the *Proposed* algorithm.

**Figure 3.** Straight line approach to find the MPP.

2.1.2. Response of the *Proposed* Algorithm When Irradiance Increases Rapidly

Here, the reaction towards rapidly changing weather conditions has been analyzed. In Figure 4, it is shown how the normal *P&O* method fails for a rapid variation in irradiance. Starting from an operating point located at *P*<sup>1</sup> (right side of MPP) we see, after a little perturbation, the new value is *P*2. If the irradiance does not change during the perturbation, then *P*<sup>2</sup> will be lying on the same curve with *G*1, *I*1. So:

Δ*P*= *P*<sup>2</sup> − *P*<sup>1</sup> < 0 <sup>Δ</sup>*V*= *Vk* − *Vk*−<sup>1</sup> > <sup>0</sup>

**Figure 4.** Response of *Proposed* method for rapidly increasing irradiance.

Consequently, the *P&O* algorithm will drive the operating point leftward which would be the correct direction. However, if the irradiance changes rapidly during the perturbation, then *P*<sup>2</sup> should lie on the *G*<sup>2</sup> curve at corresponding *VK*. Let us call it *P* 2.

$$\begin{array}{c} \Delta P = P\_2' - P\_1 > 0 \\ \Delta V = V\_k - V\_{k-1} > 0 \end{array}$$

Accordingly, the *P&O* algorithm would drive the operating point into the right side which is the wrong direction even though it should have driven it to the left, whereas, the developed method would draw the straight line passing through these two points *P*1, *P* <sup>2</sup> and will take another two points *P* <sup>3</sup>, *P* 4 on the *I*<sup>2</sup> curve and draw the straight line as depicted in Figure 4. After calculating the intersection point, *VA* of these two straight lines, the algorithm would then drive the operating point to leftward to reach the MPP on *G*<sup>2</sup> curve.

#### *2.2. The Conventional P&O Algorithm*

In the conventional *P&O* algorithm, duty signal is perturbed till the operating point converges at the MPP. The algorithm compares the voltage and power at two consecutive samples. A small voltage perturbation is performed, if the change of power is positive, the perturbation is continued in the same direction and for a negative change in power, the perturbation is reversed in order to reach the MPP. By this approach, the whole P-V curve is checked using fixed perturbations to find the MPP. If the perturbation size is small, the algorithm takes a longer time in reaching the MPP with a high level of accuracy and vice versa. A large perturbation size could cause the algorithm to produce steady-state oscillations around the MPP [41]. Figure 5 illustrates the steps required in determining the MPP using the *P&O* approach.

**Figure 5.** Flowchart of the conventional *P&O* Algorithm.

#### **3. Simulation Results Using the** *Proposed* **Algorithm**

In this section, simulation was conducted under normal and rapid weather conditions to test the response of the *Proposed* algorithm. Then comparison was made between *P&O* and *Proposed* algorithm in terms of the total number of iterations in reaching the MPP. After that, the percentage of error in duty signal and power corresponding to the MPPs was compared. Finally, the validation of the *Proposed* algorithm was performed using standard irradiance curves.

#### *3.1. Simulation Results under Normal and Rapid Weather Conditions*

In order to investigate the responses of the algorithm, practical irradiance data was taken a for a sunny day and a cloudy day for 12 h period from 6 a.m. to 6 p.m. This Global Horizontal Irradiance (GHI) is plotted for a sunny day and cloudy day as shown in Figure 6a,b respectively in order to certify the efficacy of the *Proposed* MPPT algorithm.

Figure 7a–c, illustrate the accuracy of the *Proposed* algorithm for sunny and cloudy weather conditions. In Figure 7a,b, the module power vs module voltage graph is plotted for when irradiance was varying for a particularly sunny day and cloudy day respectively. The red dots shown in Figure 7a,b, are the exact MPPs and the green curve is the output of the developed algorithm. Itis very clear from those figures that the *Proposed* algorithm accurately identified the exact MPPs. The cloudy day irradiance shown in Figure 6a changed very rapidly; nonetheless, the *Proposed* algorithm was able to determine the MPPs (red dots) very successfully which is very clear from Figure 7b. Figure 7c shows how the module power could be varied just by varying the duty cycle which ensured that the operating point of the PV panel could be varied by changing the duty ratio.

**Figure 6.** (**a**) Sunny Day Irradiance Data from 6 a.m. to 6 p.m. (**b**) Cloudy Day Irradiance Data from 6 a.m. to 6 p.m.

**Figure 7.** (**a**) Panel voltage vs. power for a sunny day (**b**) Panel voltage vs Power for a Cloudy Day (**c**) Duty vs. Output Power.

#### *3.2. Comparison of the Proposed Algorithm with Conventional Perturb and Observe Algorithm*

Here, the number of iterations to reach the MPP was compared between the developed method and Perturb and Observe method for the following conditions:

Irradiance, *G* = 1 KW/m2 Temperature, *TaC* = 25 ◦C

Operating point, Va=5V (left side of MPP )

When the operating point lay at the left of MPP, Table 2 shows a comparison between the number of iterations used by the *Proposed* and *P&O* algorithm, where *Nop*−*I p* is the number of iterations from operating point to intersection point and *NI p*−*MPP* is the number of iterations from intersection point to MPP. The number of iterations used by the developed algorithm was much less than that of the *P&O* method. Initially, the developed method had big steps (2 V) in order to get the intersection point (32.2346 V) of two straight lines and took 15 iterations. The intersection point was close to the MPP as mentioned earlier. From MPP intersection point to the, the algorithm took 22 iterations since it went with small steps (0.1 V). The small steps after finding the intersection point not only made sure of small oscillations around MPP but also ensured high efficiency.

When the operating point lay at the right side of MPP, Table 3 data confirms again that the number of iterations of the *Proposed* algorithm was much smaller than the conventional *P&O* method.

Irradiance, *G* = 1KW/m2 Temperature, *TaC* = 25 ◦C Operating point, Va = 40 V (right side of MPP).

**Table 2.** Number of iterations comparison between developed and *P&O* algorithm when the operating point lies at left of MPP.


**Table 3.** No. of iterations comparison between developed and *P&O* when the operating point lies at right of MPP.


#### *3.3. Comparison between Proposed and Perturb and Observe Algorithm in Terms of Duty and Maximum Power Point*

Here, the irradiance and temperature were varied to analyze the performance and compare the two algorithms. It is very apparent from Table 4 that the *Proposed* method obtained the duty cycles and maximum power points much closer to the exact duty cycle and maximum power points than the Perturb and Observe method under varying weather conditions which secured better efficiency.

The comparison in terms of percentage error in duty cycle and percentage error in finding MPPs are illustrated in Figure 8a,b. It is evident in Figure 8a that the *Proposed* algorithm had a very small error in finding the correct duty cycle under varying irradiance conditions. The error was less than ±0.5% whereas the Perturb and Observe algorithm had more than ±1% error in finding the correct duty cycle. The similar scenario was observed in Figure 8b as well where the percentage error in finding the Maximum Power Point under varying irradiance conditions was less than 0.03% for the *Proposed* algorithm whereas it was more than 0.23% in the case of the Perturb and Observe method.

**Figure 8.** (**a**) Comparison of percentage of error in duty cycle under 25 ◦C between *Proposed* and conventional Perturb and Observe (P&O) algorithm (**b**) Comparison of percentage of error in Maximum Power Point under 25 ◦C between *Proposed* and conventional *P&O* algorithm.


**Table 4.** Comparison of duty cycle and maximum power between *Proposed* and *P&O* algorithm at different irradiance and temperature.

#### *3.4. Validation of the Proposed MPPT Algorithm*

In order to validate the *Proposed* algorithm, the standard test irradiance profile (according to CENELEC EN50530) was used [42,43] which is shown in Figures 9a–12a. Figures 9a–12c show the performance of the developed algorithm under irradiation ramps with very steep gradients of 0.5, 5, 20, 50 and 100 W/m2/s. From these figures, it can be seen that the quality of the MPP tracking was very good and it was accurate with all gradients. The figures show that the current always tracked the irradiance changes accurately, whereby the voltage was either lagging or leading the MPP by a very small amount. This can be explained by the fact that the current was linearly proportional to irradiance and it was almost impossible to detect the exact MPP voltage because the algorithm had to use an epsilon(stopping condition when close to MPP) value which was less than a preset value (the difference between exact *VMPP* and estimated *VMPP* which was very close to exact MPP but not equal) to get very close to exact *VMPP*. Therefore the detected *VMPP* points stayed either before or after the exact *VMPP* point, however very close.

**Figure 9.** (**a**) Very steep gradient (**b**) accuracy of the *Proposed* algorithm in finding *Vmpp* at very steep gradient (**c**) accuracy of the *Proposed* algorithm in finding *Impp* at very steep gradient.

**Figure 10.** (**a**) Irradiance with gradient of 0.5 W/m2/s (**b**) accuracy of *Proposed* algorithm in finding *VMPP* at 0.5 W/m2/s gradient (**c**) accuracy of the *Proposed* algorithm in finding *IMPP* at 0.5 W/m2/s gradient.

(**c**)

**Figure 11.** (**a**) Irradiance at a gradient of 5 W/m2/s (**b**) accuracy of the *Proposed* algorithm in finding *Vmpp* at 5 W/m2/s gradient (**c**) accuracy of the *Proposed* algorithm in finding *Impp* at 5 W/m2/s gradient.

**Figure 12.** (**a**) Irradiance at gradients of 20, 50, 100 W/m2/s (**b**) accuracy of the *Proposed* algorithm in finding *Vmpp* at gradients of 20, 50, 100 W/m2/s (**c**) accuracy of the *Proposed* algorithm in finding *Impp* at Gradients of 20, 50, 100 W/m2/s.

#### **4. Conclusions**

In this paper, a simple and accurate photovoltaic maximum power point tracking algorithm is proposed with its mathematical formulation. The *Proposed* MPPT algorithm is tested for its tracking speed, tracking accuracy, capability to rapid transition, and maximum power point efficiency and compared with the conventional hill climbing *P&O* method. The results show that the system reaches to MPP voltage in much fewer iterations and hence is very fast to converge to MPPs under fast-changing weather conditions. The duty cycle and maximum power point under various weather conditions have been analyzed also and compared with the *P&O* method. The results also show that the percentage of errors in finding duty cycles and maximum power points of the *Proposed* algorithm are much less than its counterpart ( *P&O* method). Subsequently, the developed algorithm was validated according to the CENELEC EN50530 standard which stipulates how the efficiency of the MPPT algorithm should be measured. It is also worthy of note that the *Proposed* algorithm does not require the technical specifications of the PV module as an input to the algorithm.

As a future work, the algorithm will be implemented using a micro-controller and other necessary hardware for a PV-Battery microgrid system and voltage stability will be studied for varying load conditions along with other control-algorithms such as fuzzy logic, artificial intelligence, model predictive control, etc. PV power ramp rate control also will be studied with the *Proposed* algorithm for high penetration level of PV systems as future smart grid applications.

**Author Contributions:** Conceptualization, A.D.; Funding acquisition, A.S.; Investigation, A.D.; Methodology, A.D.; Project administration, I.P.; Supervision, I.P. and A.S.; Writing–original draft, T.O.O.; Writing–review & editing, T.O.O. and M.G.D. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by National Science Foundation (NSF) under the grant number 1553494. **Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


c 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Application of Genetic Algorithm for More Efficient Multi-Layer Thickness Optimization in Solar Cells**

**Premkumar Vincent 1,†, Gwenaelle Cunha Sergio 1,†, Jaewon Jang 1, In Man Kang 1, Jaehoon Park 2, Hyeok Kim 3, Minho Lee <sup>4</sup> and Jin-Hyuk Bae 1,***<sup>∗</sup>*


Received: 9 March 2020; Accepted: 1 April 2020; Published: 4 April 2020

**Abstract:** Thin-film solar cells are predominately designed similar to a stacked structure. Optimizing the layer thicknesses in this stack structure is crucial to extract the best efficiency of the solar cell. The commonplace method used in optimization simulations, such as for optimizing the optical spacer layers' thicknesses, is the parameter sweep. Our simulation study shows that the implementation of a meta-heuristic method like the genetic algorithm results in a significantly faster and accurate search method when compared to the brute-force parameter sweep method in both single and multi-layer optimization. While other sweep methods can also outperform the brute-force method, they do not consistently exhibit 100% accuracy in the optimized results like our genetic algorithm. We have used a well-studied P3HT-based structure to test our algorithm. Our best-case scenario was observed to use 60.84% fewer simulations than the brute-force method.

**Keywords:** genetic algorithm; solar cell optimization; finite difference time domain; optical modelling

#### **1. Introduction**

Simulations of optoelectronic devices have helped to understand and design better optimized structures with efficiencies nearing the theoretical maximum. Lucio et al. analyzed the possibility of achieving the limits of c-Si solar cells through such simulations [1]. Simulations have reduced the time it takes for researchers to find optimized device structure. However, the most common way to obtain results over a large range of a parameter's values is through parameter sweep method. This brute-force method is ineffective in most cases where the user only requires the end optimized device structure. Genetic algorithm (GA) is an optimization algorithm in artificial intelligence based on Darwin's evolution and natural selection theory, in which the fittest outcome survives [2,3]. This algorithm sets an environment with a random population and a function, which is called the fitness function, that scores each individual of that population. The environment then selects individuals to become the parents of the next generation through a selection process. The next generation of individuals (children of the previous generation's parents) is obtained via a crossover method. Similar to natural mutation in genes of the offspring, the new generation's individuals can also suffer mutation in their genes. After several generations, the population converges to the individuals representing the optimal solution. Through the application of genetic algorithm, Jafar-Zanjani et al. designed a

binary-pattern reflect-array for highly efficient beam steering [4] and Tsai et al. was able to beam-shape the laser to obtain up to 90% uniformity in intensity distribution [5]. Genetic algorithm has been used by Donald et al. to improve the focusing of light propagating through a scattering medium [6] and by Wen et al. for designing highly coherent optical fiber in the mid-infrared spectral range [7]. It has also been used for designing nanostructures to improve light absorption in solar cells. Chen et al. were able to surpass the Yablonovitch Limit using genetic algorithms to design light trapping nanostructures [8]. Rogério et al. used a genetic algorithm to design surface structures on a Si solar cell to increase the short-circuit current density obtained from it [9].

In this article, we have demonstrated the optimization of an organic solar cell through the optimization of the optical spacer layers. Traditionally, finite difference time domain (FDTD) method was used to simulate the ideal short-circuit current density (*Jsc*) of the solar cell through the Lumerical, FDTD solutions software similar to our previous reported study [10]. Parameter sweep or brute-force method was used to vary the thickness of the optical spacer layers of the solar cell. At the optimized layer thicknesses, the solar cell will be observed to have the highest *Jsc* output. Although not computationally intensive for a single-layer optimization, the number of simulations expands as in Equation (1) for multi-layer optimization problems.

$$N = n\_1 \cdot n\_2 \cdot n\_3 \dots NN \tag{1}$$

where *N* is the total number of simulations and *n*1, *n*2, and *n*<sup>3</sup> are the number of simulations performed for layers 1, 2, and 3, and *NN* is the total number of layers (*NN* > 3), respectively.

To alleviate the brute-force method's limitations, we propose the use of GA. This article then aims to heuristically assert the hypothesis that GA is a more efficient approach than brute-force algorithms in tasks such as optimizing optoelectronic device structures.

#### **2. Methodology**

Figure 1 shows the device structure that was constructed in Lumerical, FDTD solutions. It consists of a 150 nm indium tin oxide (ITO) similar to our previous study [11]. The Al thickness was set to 100 nm as light gets completely reflected from the Al electrode at this thickness. Any further increase in its thickness would not affect the outcome of our optical simulation. The active layer, poly (3-hexylthiophene) (P3HT): indene-*C*<sup>60</sup> bisadduct (ICBA), was designed to be 200 nm as it exhibited good efficiency in our previous study [12]. The charge transport layers, zinc oxide (ZnO) and Molybdenum oxide (MoOx), also act as optical spacer layers and are variable quantities in our simulation. In order to qualify as an optical spacer, the layer material should have the refractive index properties to shift the light induced electric field inside the solar cell structure by varying its thickness. ZnO was already shown to be a good optical spacer in our previous research [11]. While a ITO/PEDOT:PSS/P3HT:PCBM/ZnO/Al is a more commonly used solar cell structure [13], PEDOT:PSS does not have the suitable refractive index to control the distribution of electric field for the wavelengths under consideration. MoOx was found to be a suitable hole transport layer substitute to the PEDOT:PSS by a previously reported study by Bohao et al. [14]. MoOx was also found to have good optical spacer properties in our simulations and so it was used as the other optical spacer layer.

The simulation was done in perpendicular illumination onto the solar cell as done for device simulations. This method does not simulate the real outdoor operation of the solar cell, and thus, the optimized thickness values calculated in this article would not be applicable in the view of energy yield optimization. However, our study was a comparison of the algorithms, and our conclusion should be consistent.The solar cell layers were stacked along the *y*-axis in the FDTD software. The incident light was set up as a plane wave with the spectral intensity of AM1.5G. The plane wave's propagation direction was the same axis along which the solar cell's layers were stacked. It was incident through the ITO electrode. Periodic boundaries were set along the *x*-axis and perfectly matched layers were used along the *y*-axis to set the FDTD boundary conditions. The device was meshed good enough for the results to converge. Further simulation setup details are published elsewhere [11,12]. To simulate the ideal *Jsc*, 100% internal quantum efficiency was assumed.

**Figure 1.** Solar cell device structure. The electron transport layer is ZnO, while the hole transport layer is MoOx. Both also act as optical spacer layers as they affect the distribution of light inside the device. We optimized these optical spacer layers for maximizing photon absorption inside the active layer of the solar cell.

#### *2.1. Brute Force*

Using the software's parameter sweep option, we simulated our device structure according to three sections. The first section optimized only the ZnO layer, while keeping the MoOx layer at 10 nm thickness. The second simulation section consists of optimizing only the MoOx layer, while the ZnO layer was fixed at 30 nm. Our final section optimized both optical spacer layers together. The results of the brute-force method are provided in Figure 2.

**Figure 2.** Brute-force method results: (**a**) single ZnO layer optimization, (**b**) single MoOx layer optimization, (**c**) multiple ZnO and MoOx layers optimization, (**d**) 2D data representation of (**c**) using labels pointing to the ZnO and MoOx layer thickness combinations for the ease of computation.

Using parameter sweep to simulate every possible combination is time-consuming. Figure 2 contains multiple local maxima and minima points. This makes it tedious to extrapolate the optimal thickness from fewer simulation points. Due to this, every possible simulation point is required to find the optical result. This way of obtaining the optimal result is termed as brute-force method henceforth. In order to make a 2-layer optimization problem similar to that of the single-layer optimization, we replaced the optical spacers' thickness combinations with a label in Figure 2d, effectively converting 3D data to a 2D data. The label number is given by *label number* = *max*(*ZnO thickness*) × *MoOx thickness* + *MoOx thickness* + 1.

#### *2.2. Genetic Algorithm*

To alleviate the computational and time inefficiencies of the brute-force method, we found inspiration in Darwin's natural selection theory and proposed the use of a genetic algorithm [3] for our optimization problem. Consider that there is a random population and their adaptability to the environment is given by a fitness function. In our optimization problem, the fitness function was to maximize *Jsc* output from the FDTD simulation. Each population contains several individual chromosomes, which in turn consists of an array of bits called genes. Different selection methods are used to choose certain chromosomes in each generation (see Section 2.2.1) in order to reproduce off-springs using a crossover method (see Section 2.2.2). To further mimic biology, there is also a probability that a chromosome might suffer mutation, which is provided by the mutation rate, which allows the algorithm to escape local minima in the data. In the end, the fittest members of the population prevail, meaning that the algorithm converges to the optimal solution.

A step-by-step of the works of the genetic algorithm is shown in Algorithm 1. The initial population *pop* of size *p* is randomly selected from the search space, which in our case is the maximum and minimum thickness of the optical spacer layers. The first iteration of the algorithm then starts. For each population value, the GA calls the FDTD software to simulate and extract the *Jsc* result. The fitness function (*Jsc*) is applied to all individuals in the population and ranked from the highest *Jsc* to the lowest. A selection method is then applied taking into consideration each chromosome and their respective fitness score. After selecting the parents responsible for the next generation, they reproduce to obtain the next generation, a step detailed in Algorithm 2. Please note that in our algorithm, the fittest individual in the current generation is cloned to be part of the next generation. The current generation is then updated with the next generation, with the algorithm continuing until the maximum number of generations has been reached.



As for the reproduction algorithm, detailed in Algorithm 2, the first step is to allocate an empty array for the new population. The second step is to select two parents to produce *C* children. Due to the crossover method used and mutation ratio, the same parents can reproduce different children. The new population is then updated with the new children obtained through crossover and mutation.


**Data:** *next*\_*parents*, *p*, *mutation*\_*prob* **Result:** New population *new*\_*pop* ← []; **for** *i=1 to p / 2* **do** *parent*1 ← *next*\_*parents*(i); *parent*2 ← *next*\_*parents*(len(*next*\_*parents*) - i + 1); Create *C* children from *parent*1 and *parent*2 with Crossover and Mutations; *new*\_*pop* ← *new*\_*pop* + new children

#### 2.2.1. Selection Methods

Four selection methods were examined in this work: random, tournament, roulette wheel, and breeder. For better clarity of the following explanations, the fittest individual means the individual with highest fitness score, since our problem is a maximization problem.

#### Random

This is the simplest selection method since it does not incorporate selection criteria. This method consists of randomly selecting individuals to be the next generation's parents, with no regards to the fitness function. Because of this Monte-Carlo-like approach, it can take very long for the algorithm to converge.

#### Tournament

Tournament Selection [15] samples *k* individuals with replacement from a population of *p* and applies the fitness function to those individuals in order to select the one with best fitness score, also known as the fittest individual. One can think of this method as a battle of the fittest, where *k* individuals face each other in tournament fashion to decide the fittest. The fittest individuals from each tournament round will then constitute the parents responsible for forming the next generation.

#### Roulette Wheel

Roulette-Wheel Selection (RWS) [16] is a popular way of parent selection in which individuals have a fitness-proportionate probability of being selected. In that way, if an individual is very fit, it has a higher chance of being chosen, otherwise their chance is lower.

#### Breeder

Breeder Selection [17] follows the same strategy used for breeding animals and plants, where the goal is to preserve certain desired properties from the parents in their children. This is achieved by conserving the genetic material from the fittest individuals while still giving some mutation leeway by adding a few random individuals (lucky few) to the mix of parents for the next generation.

#### 2.2.2. Crossover

Crossover is the reproduction method in genetic algorithms, and it consists of choosing the parts of each parent that will be present in their child.

#### Uniform

A uniform crossover is shown in Figure 3. In this type of crossover, each bit is chosen from one of the parents with probability of 0.5. The advantage of this method is that the same parents can form many children with a more diverse set of genes.

**Figure 3.** Uniform crossover: bitwise reproduction.

#### *k*-Point

In a *k*-point crossover, each parent is divided into *k* segments. These segments are then chosen with equal probability to compose the new chromosomes for their children. This crossover, shown in Figure 4, can be a one-point crossover (*k* = 1), or a multi-point crossover (1 < *k* < *NC*, where *NC* is the length of a chromosome). The disadvantage of the one-point crossover is that it is only able to generate two distinct children.

**Figure 4.** *k*-point crossover: blockwise reproduction.

#### 2.2.3. Mutation

This genetic operator is used to ensure genetic diversity within a group of individuals and to ensure the algorithm does not converge to a local minimum. The mutation operator works by flipping bits in a chromosome according to a mutation probability, as shown in Figure 5.

**Figure 5.** Mutation of a bit in a chromosome.

#### **3. Complexity Analysis**

In this section, we aim to further explain our algorithm's suitability with respect to its complexity. In brute force, the total simulation count increases with respect to the solar cell's layers which are thickness-optimized, as shown previously in Equation (1). In Big-O notation, which represents the upper bound for time complexity in an algorithm, the brute-force method has algorithmic complexity as in Equation (2):

$$O\_{bruteforra} = O(n^l) \tag{2}$$

where *l* is the number of layers and *n* is the number of fitness function evaluations for a layer. In other words, the complexity increases exponentially with the number of layers, and that can be very expensive as that number grows.

Our goal is to efficiently optimize layer thickness in devices composed of a single layer and multiple layers alike. Hence, we used genetic algorithm for our global optimization requirement. GA can converge to an optimal solution by evaluating less individuals than the brute-force method. However, in the classical approach, the same individual might be evaluated multiple times. To prevent this redundancy from happening, we implement GA with dynamic programming. In this approach, there is a dynamic dictionary, also called lookup table, which serves as memory to store the already evaluated individuals and their respective fitness scores. The dictionary is said to be dynamic because it may change size if the algorithm receives an individual whose fitness score has not been calculated yet. This approach is illustrated with an example in Figure 6, where in every generation, or iteration, the algorithm searches the memory for a desired individual. If this individual has already been computed, its fitness score can simply be used by the algorithm. Otherwise, the fitness function is evaluated, and the lookup table is updated.

**Figure 6.** Genetic algorithm with dynamic programming, where blue represents values that need to be added to the lookup table in order to update it and grey represents values that have already been evaluated and need only to be copied when necessary.

#### **4. Results and Discussion**

Since GA is a stochastic algorithm, we calculated the average number of simulations required by the GA and its standard deviation from 5000 repeated runs per section. The accuracy of the data discussed below are all 100%, which means that all the 5000 runs converged to the optimal solution. We have discussed three sections below: single layer—ZnO thickness optimization; single layer—MoOx thickness optimization; and multiple layers—concurrent ZnO and MoOx thickness optimization.

#### *4.1. Single Layer*

#### 4.1.1. ZnO Optical Spacer Layer

For the single ZnO optical spacer layer optimization, we fixed the MoOx layer thickness as 10 nm. We applied the brute force and the genetic algorithm to our task and solved it within the same thickness limits of 0 to 80 nm. The best *Jsc*, which is also the fitness value, was obtained when the ZnO thickness was optimized to 30 nm (as shown in Figure 2a). We have compared the total number of simulations and their respective accuracies for different selection methods. The initial population size, generations count, and mutation probability were the factors that were iterated in order to find the conditions that would use the least number of simulations to optimize the device structure. The population size was varied from 10 to 80 in increments of 10, the generation count from 10 to 100 in increments of 10, and the mutation probability from 5 to 100 in increments of 5. While the brute-force method required 81 simulations in total, the number of simulations required by the genetic algorithm was dependent on the selection method and initialization parameters used. The initialization parameters which provided the least average number of simulations from the 5000 simulations, while keeping the accuracy at 100%, is provided in Table 1.



**Figure 7.** Single ZnO layer optimization using GA with tournament selection: accuracy (%) distribution data sliced at population size of 70, generation count of 30, and mutation probability of 60%.

Figure 7 presents the accuracy distribution over different initialization parameters. It was observed that the best result was obtained while using tournament selection model with the population size of 70, generation count of 30, and mutation probability of 60%. It required 78.16 ± 1.65 simulations to reach the optimal solution. Although the GA algorithm was observed to produce only a reduction of 2.26 ± 2.04% in the number of simulations required, this was mainly due to two optimal result points in the data. Since the device with a 24 nm ZnO layer thickness exhibited a *Jsc* of 116.62 A/m2 and the one with the optimal 30 nm ZnO layer thickness exhibited a near same *Jsc* of 116.67 A/m2, the algorithm took longer to converge at the optimal structure. In a practical scenario, however, if both the above-mentioned structures were regarded as optimal, the algorithm would converge with high accuracy with much lesser number of simulations.

#### 4.1.2. MoOx Optical Spacer Layer

For the MoOx optical spacer single-layer optimization, the ZnO thickness was fixed at 30 nm. The brute-force method required 31 simulations to determine the optimized layer thickness of 8 nm (Figure 2b). For finding the best initialization parameter combination, we varied the population size from 5 to 20 in increments of 5, the generation count from 10 to 100 in increments of 10, and the mutation probability from 5 to 100 in increments of 5. The best case results for each selection model is provided in Table 2.


It was observed that while the roulette method was able to use 57.9 ± 10.45% fewer simulations to determine the optimal MoOx thickness, the other methods required nearly the same number of simulations as the brute-force method. We hypothesize that the roulette method's preferential weighing of the fittest model aided in the convergence to the optimal solution faster. Figure 8 presents the accuracy distribution for the optimum initializing parameter values.

**Figure 8.** Single MoOx layer optimization using GA with roulette selection: accuracy (%) distribution data sliced at population size of 5, generation count of 100, and mutation probability of 75%.

#### *4.2. Multi-Layer: ZnO + MoOx*

As mentioned earlier, multi-layer optimization can take a large number of simulations. The computational and time cost required to run these simulations are expensive. GA can be used to refine the optimization process to take as less simulations as required. The ZnO layer thickness was incremented by 1 nm from 0 to 80 nm, while the same thickness increment was done from 0 to 30 nm for the MoOx layer. The brute-force method used 2511 simulations to find the optimized optical spacer layer thicknesses of 24 nm ZnO and 8 nm MoOx. We converted the 3D Figure 2c to a 2D Figure 2d by applying labels for each ZnO and MoOx thickness combination. Thus, we have 2511 labels and the GA algorithm was applied to determine the label pointing to the optimal result. The population size that was varied from 500 to 1500 in increments of 500, the generation count from 10 to 100 in increments of 10, and the mutation probability from 10 to 100 in increments of 10. Table 3 presents the result from multi-layer optimization.


Figure 9 presents a 29.96 ± 1.58% reduction in the number of simulations required by the roulette selection method to obtain the optimal solution. As the complexity became higher, having more randomness in the population through a high mutation probability rate aided constructively to reduce the number of simulations required. Due to this, the roulette selection method was able to show a best average number of simulation count of 1758.77 ± 39.75 from 5000 repetitive runs.

**Figure 9.** Multi-layer optimization using GA with roulette selection: accuracy (%) distribution data sliced at population size of 1000, generation count of 90, and mutation probability of 90%.

#### *4.3. Performance Comparison: Uniform vs K-Point Crossover Methods*

The roulette method demonstrates satisfactory performance in the ZnO layer thickness optimization problem and the best performances for both MoOx and multi-layer thickness optimization problems. These results are obtained using the uniform crossover genetic operator, whereas performances with the *k*-point crossover method remain uncharted. This section aims to fill that gap and provide a performance comparison between uniform and *k*-point crossover methods. We use *k* values of 1, 2, and 4 for our *k*-point crossover method.

Table 4 shows the average number of simulations required to solve the optimization problem when using uniform versus when using *k*-point crossover. It also compares the result obtained through the various crossover methods to the chromosome's binary bit size. In the MoOx case with *k* = 1 and *bits* = 12, the chromosome is divided into segments of 6 bits. Since the maximum number of bits required to define the MoOx layer thicknesses is 5 bits, one of the parents does not pass on its chromosome to the child. Due to this, there is no genetic variation between the parents and children, essentially meaning that the children are copies of one parent, hence the solution did not converge. We also observed that in most cases using k-point crossover, the best results were obtained when the chromosome was segmented into segments of 4 bits in length. The exception to this were the MoOx case with 5-bit chromosomes and the ZnO case with 12-bit chromosomes. Uniform crossover and 4-point crossover proved better in these cases, respectively. We hypothesize that the poorer performance at other *k*-point values was linked to the amount of variation in the chromosome. Lower *k*-point values meant fewer segmentations in the parent chromosome, which in turn meant fewer genetic variations occurred, leading to a slower convergence of the result. However, at the same time, higher *k*-point value meant high genetic variation which also lead to slower convergence.


**Table 4.** Comparison results for uniform and *k*-point crossover methods.

\* Solution did not converge.

The best initialization parameters for the optimization problems discussed in this article are tabulated in Table 5.

**Table 5.** Initialization parameters for best performances.


#### **5. Conclusions**

We have demonstrated that the genetic algorithm can perform better than the conventional parameter sweep used in simulations. In our best-case scenario, it exhibited no loss in accuracy, while outperforming the brute-force method by up to 57.9% with the correct initialization parameters. In the worst-case scenario, the GA utilized the same number of simulations as the brute-force method,

demonstrating that it cannot be outperformed by brute force. We found that the best selection method was the roulette-wheel selection. For uniform crossover method, it exhibited a satisfactory performance for ZnO layer optimization and an outstanding performance for the MoOx and 2-layer optimization problems. When using k-point crossover, the roulette method was able to further decrease the number of simulations required to converge to the optimal result. In conclusion, we were able to reduce the average number of simulations required for MoOx layer optimization to 12.14 (brute force = 31), ZnO layer optimization to 70.76 (brute force = 81), and ZnO-MoOx layers optimization to 1140.06 (brute force = 2511). The GA is dependent on its initialization parameters and the selection method chosen. This article does not discuss an automated way to assign these parameters as it is not in its research scope. However, the results suggest that there is possibility for greatly refining the parameter sweep method using genetic algorithms as shown with both single and multi-layer optimization of the solar cell structure. Code and additional results are at https://github.com/gcunhase/GeneticAlgorithm-SolarCells.

**Author Contributions:** Conceptualization, P.V.; Methodology, P.V. and G.C.S.; Software, P.V., and G.C.S.; Validation, J.-H.B., J.J., I.M.K., J.P., H.K., and M.L.; Formal analysis, P.V., G.C.S., J.-H.B., J.J., H.K., and J.P.; Investigation, P.V., and G.C.S.; Resources, H.K., and J.-H.B.; Data curation, P.V., and G.C.S.; Writing—original draft preparation, P.V., and G.C.S.; Writing—review and editing, P.V., G.C.S., J.-H.B.; Visualization, P.V., and G.C.S.; Supervision, J.-H.B., and M.L.; Project administration, J.-H.B.; Funding acquisition, J.-H.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (2018R1A2B6008815), and also by the BK21 Plus project funded by the Ministry of Education, Korea (21A20131600011).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**

The following abbreviations are used in this manuscript:


#### **References**


c 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **A Side-Absorption Concentrated Module with a Di**ff**ractive Optical Element as a Spectral-Beam-Splitter for a Hybrid-Collecting Solar System**

#### **An-Chi Wei 1,2,\*, Wei-Jie Chang <sup>2</sup> and Jyh-Rou Sze <sup>3</sup>**


Received: 2 November 2019; Accepted: 27 December 2019; Published: 1 January 2020

**Abstract:** In this paper, we propose a side-absorption concentrated module with diffractive grating as a spectral-beam-splitter to divide sunlight into visible and infrared parts. The separate solar energy can be applied to different energy conversion devices or diverse applications, such as hybrid PV/T solar systems and other hybrid-collecting solar systems. Via the optimization of the geometric parameters of the diffractive grating, such as the grating period and height, the visible and the infrared bands can dominate the first and the zeroth diffraction orders, respectively. The designed grating integrated with the lens and the light-guide forms the proposed module, which is able to export visible and infrared light individually. This module is demonstrated in the form of an array consisting of seven units, successfully out-coupling the spectral-split beams by separate planar ports. Considering the whole solar spectrum, the simulated and measured module efficiencies of this module were 45.2% and 34.8%, respectively. Analyses of the efficiency loss indicated that the improvement of the module efficiency lies in the high fill-factor lens array, the high-reflectance coating, and less scattering.

**Keywords:** solar concentrator; spectral beam splitting; diffractive optical element; diffractive grating

#### **1. Introduction**

Solar technologies have drawn significant attention due to the Earth's extreme climate and energy crises, and these technologies have recently made great progress. Mostly, those solar technologies transform sunlight into electricity or thermal power. Although one of the dominating technologies for generating electricity from sunlight is photovoltaics, this technology entails significant energy loss, including thermal loss and spectral loss. To enhance or extend the energy usage of photovoltaic systems (PV), researchers have collected waste thermal energy, such as cascading photovoltaic and thermal modules [1], integrated a PV-powered air-conditioning unit with a boiler [2], reduced the PV temperature for higher efficiency by using air-based hybrid photovoltaic/thermal systems (PV/T), water-based PV/T, and refrigerant-based PV/T [3–5], and even extended the operating bands by means of multi-junction photovoltaics, spectral-beam-splitters (SBS), and so on [6–9]. Among the systems belonging to the PV/T regime, SBS has the following advantages. Photovoltaic cells are no longer used as thermal receivers, so over-heating can be avoided. Moreover, their relatively low operation temperatures have led to high efficiency. The temperature of the heat transfer fluid (HTF) of thermal modules can be unrestricted by the operating temperature of the cells, resulting in a broad range of thermal applications [8]. Because of the characteristic of spectral splitting, SBS can be applied not

only to the PV/T but also other hybrid-collecting solar systems, such as a dual-photovoltaic system with its cells operating in different energy-conversion bands [10,11]. In terms of configurations, the technique of SBS includes the following categories: dichroic filtering, liquid absorption, diffraction, and others [8,12]. Diffractive type of SBS can be sorted mainly by the alignment method of the sunlight receivers. The first type of alignment adopts a common optical axis for different receivers, such as photovoltaic cells and thermal tubes [13–15]. Thus, the shadow effect becomes unavoidable. The second kind of alignment arranges multiple receivers with different optical axes [16–19]. In this arrangement, the sunlight receivers can be photovoltaic cells with different absorption bands and lateral arrangements [12,13]. The third kind is proposed as a planar concentrator with a zig-zag optical axis, with the receivers located at different sides of the concentrator [20,21]. Herein, we propose a novel SBS configuration (of the third kind of alignment), consisting of a zig-zag optical axis, with integration of the diffractive optical element (DOE) and other planar optics to form a side-absorption concentrated module. In addition to the design principle of the module constructed by a single unit, we present and discuss a practical demonstration of seven units. Notably, because of its side-absorption structure, the system's thickness, complexities in component alignment, and wire connections can be reduced [22–24]. In brief, the proposed side-absorption concentrated module inherits the benefits of SBS, such as the improved operating efficiency of photovoltaics, more probable thermal-applications, etc., while its novel configuration facilitates a compact PV/T system with a simplified packaging process because of its side-absorption structure.

#### **2. Principles**

#### *2.1. Side-Absorption Concentrated Module for Spectral-Beam Splitting*

The proposed side-absorption concentrated module utilizes lenses, DOEs, and a light-guide as the condenser, the spectral beam-splitter and the out-coupling adapter, respectively. The structure of the proposed module is illustrated in Figure 1. Assume that the whole module is allocated on a solar tracker and that sunlight is regarded as normally incident to the module. When sunlight illuminates the positive lens, the light will be condensed and will pass through a DOE (considered a diffractive grating, herein). Then, the grating will diffract the light according to its wavelength. In this study, a structured light-guide with its top surface as the entrance and two side surfaces as two output ports will receive the diffracted waves, while the two ports export the sunlight in the visible and infrared bands individually. By means of this light-guide, the diffracted waves will be directed toward different output ports according to their spectra, resulting in the module splitting sunlight spectrally. Meanwhile, the proposed module inherits the advantages of the planar concentrator such that the output ports lie in the same horizontal plane, thereby allowing for planar outputs and saving the system volume. Afterwards, this module can be applied to a hybrid solar system with different energy-conversion mechanisms. As for the issues and challenges of the proposed module, they are discussed in Section 5.3 along with the comparisons to other techniques.

**Figure 1.** Schematic of the side-absorption concentrated module for spectral-beam splitting.

#### *2.2. Performance Evaluation*

The evaluation of the proposed side-absorption concentrated module consists of two stages. The primary stage evaluates the diffraction performance when the light passes through the grating, and the secondary stage evaluates the out-coupling performance when the light propagates through all components and couples out from the different output ports of the light-guide. Compared to the lens, the grating is more dominant during the first stage because its dispersion affects the performance of beam-splitting, and its high-order diffraction results in a loss. Accordingly, the lens is regarded as ideal to simplify the grating analyses in the primary stage. In this study, we evaluate grating by its diffractive efficiency in the visible band, the infrared band, and the whole solar spectrum. Meanwhile, the grating is designated to diffract the visible portion of sunlight toward the first-order diffraction and the infrared portion toward the zeroth-order diffraction. The efficiency of the grating in the visible band, η*vis grating*, can be then considered as the ratio of the optical power of the first-order visible output to that of the visible sunlight:

$$\eta\_{\text{greating}}^{\text{vis}} = \frac{\int\_{\lambda\_s^{\text{vis}}}^{\lambda\_{\text{r}^{\text{vis}}}^{\text{res}}} \eta\_{diff,\text{greating}}^{1}(\lambda) \cdot \mathcal{S}(\lambda) d\lambda}{\int\_{\lambda\_s^{\text{vis}}}^{\lambda\_{\text{r}^{\text{vis}}}^{\text{vis}}} \mathcal{S}(\lambda) d\lambda}, \tag{1}$$

where η<sup>1</sup> *di f f*,*grating*(λ) denotes the first-order diffraction efficiency of the grating under a source with wavelength λ, and *S* represents the solar spectrum. λ<sup>s</sup> *vis* and λ<sup>e</sup> *vis* are the starting and ending wavelengths of the considered visible spectrum, respectively. Similarly, the grating efficiency in the infrared band, η*IR grating*, relates to the optical power of the zeroth-order infrared output:

$$\eta\_{\text{greating}}^{\text{IR}} = \frac{\int\_{\lambda\_s^{\text{IR}}}^{\lambda\_\varepsilon^{\text{IR}}} \eta\_{diff,\text{greating}}^0(\lambda) \cdot S(\lambda) d\lambda}{\int\_{\lambda\_s^{\text{IR}}}^{\lambda\_\varepsilon^{\text{IR}}} S(\lambda) d\lambda},\tag{2}$$

where η<sup>0</sup> *di f f*,*grating*(λ) denotes the zeroth-order diffraction efficiency of the grating, and <sup>λ</sup><sup>s</sup> *IR* and λ<sup>e</sup> *IR* are the start and the end wavelengths of the considered infrared spectrum, respectively. Likewise, all out-coupling waves from the designated diffraction orders are counted to derive the grating efficiency within the whole solar spectrum:

$$\eta\_{\text{graiting}}^{\text{total}} = \frac{\int\_{\lambda\_s^{\text{vir}}}^{\lambda\_c^{\text{vir}}} \eta\_{diff,\text{graiting}}^1(\lambda) \cdot S(\lambda) d\lambda + \int\_{\lambda\_s^{\text{IR}}}^{\lambda\_c^{\text{IR}}} \eta\_{diff,\text{graiting}}^0(\lambda) \cdot S(\lambda) d\lambda}{\int\_{\lambda\_s^{\text{total}}}^{\lambda\_c^{\text{tail}}} S(\lambda) d\lambda}, \tag{3}$$

where λ<sup>s</sup> *total* and λ<sup>e</sup> *total* are the start and the end wavelengths of the considered solar spectrum, respectively.

At the second stage, the synergetic effects of all components, including the lens, the grating, and the light-guide, are taken into account. The module efficiency for the visible band relates to the optical power of visible light out-coupled from the visible port of the light-guide:

$$\eta\_{\text{modulue}}^{\text{ris}} = \frac{\int\_{\lambda\_s^{\text{ris}}}^{\lambda\_e^{\text{ris}}} \eta\_{\text{lens}}(\lambda) \cdot \eta\_{\text{diff,gravity}}^{\text{1}}(\lambda) \cdot \eta\_{\text{lightguide}}(\lambda) \cdot S(\lambda) d\lambda}{\int\_{\lambda\_s^{\text{ris}}}^{\lambda\_e^{\text{ris}}} S(\lambda) d\lambda} \,\tag{4}$$

where η*lens*(λ) and η*lightguide* (λ) are the wavelength-dependent transmittance of the lens and the guiding efficiency of the light-guide, respectively. Similarly, the module efficiency for the infrared band and the whole solar spectrum can be respectively expressed as follows:

$$\eta\_{\text{modulc}}^{\text{IR}} = \frac{\int\_{\lambda\_s^{\text{IR}}}^{\lambda\_e^{\text{IR}}} \eta\_{\text{lens}}(\lambda) \cdot \eta\_{\text{diff,spring}}^0(\lambda) \cdot \eta\_{\text{ligt}\text{tryindex}}(\lambda) \cdot S(\lambda) d\lambda}{\int\_{\lambda\_s^{\text{IR}}}^{\lambda\_e^{\text{IR}}} S(\lambda) d\lambda} \tag{5}$$

and

$$\eta^{\text{total}}\_{\text{modulle}} = \frac{\int\_{\frac{\lambda^{\text{int}}\_{\text{s}}}{\lambda^{\text{int}}\_{\text{s}}} \eta\_{\text{sum}}(\lambda) \cdot \eta^{1}\_{\text{diff, gravity}}(\lambda) \cdot \eta\_{\text{hydride}}(\lambda) \cdot S(\lambda) d\lambda + \int\_{\frac{\lambda^{\text{IR}}\_{\text{s}}}{\lambda^{\text{IR}}\_{\text{s}}}}^{\text{IR}} \eta\_{\text{sum}}(\lambda) \cdot \eta^{0}\_{\text{diff, gravity}}(\lambda) \cdot \eta\_{\text{hydride}}(\lambda) \cdot S(\lambda) d\lambda}{\int\_{\frac{\lambda^{\text{IR}}\_{\text{s}}}{\lambda^{\text{tail}}\_{\text{s}}}}^{\lambda^{\text{tail}}\_{\text{f}}} S(\lambda) d\lambda}. \tag{6}$$

According to Equations (1)–(6), the spectral-splitting performance of the grating and that of the entire side-absorption concentrated module can be calculated and evaluated. In the following sections, these equations will be utilized for the design and optimization of the optical components in the proposed module, as well as for evaluation of the whole module.

#### **3. Modeling**

In order to facilitate the demonstration of a single unit and an assembled array of the proposed module, we considered the commercial specifications of lenses and then designated a lens as the condenser with a diameter of 25.4 mm and a distance of 30 mm between the lens and the grating. Since a lens with a long focal length produces a thick module, and that with a short focal length brings about large spherical aberrations, a lens with an appropriate focal length was plotted, and a commercial lens (model: AC254-080-A) with a focal length of 80 mm was selected. Notably, the selected lens uses achromatic coating to reduce chromatic aberration, which can simplify the optical properties of the waves incident to the grating and facilitate the grating design.

#### *3.1. Optimization of Di*ff*ractive Grating*

There are several considerations for designing the diffractive grating in the proposed module, including the grating period, the geometric shape, the material, and the diffraction orders in use. Because of the available fabrication processes and the requirement to achieve duplication with sufficient efficiency, the shape and material of the grating were designed as blazed and polyethylene terephthalate (PET), respectively. Meanwhile, the zeroth and the first diffraction orders were utilized. According to the grating equation, the diffractive beams can be characterized by their diffractive angles. For a normally incident light source, the diffractive angle of the *m*th-order diffraction can be expressed as in [25]:

$$\theta\_m = \sin^{-1}\left(\frac{m\lambda}{d}\right),\tag{7}$$

where *d* is the grating period. By differentiation, the angular dispersion of the grating is derived as in [26]:

$$\frac{\partial \partial\_m}{\partial \lambda} = \frac{m}{d \times \cos \theta\_m}. \tag{8}$$

Equation (8) indicates that diffraction order *m* and grating period *d* are the determinative factors of angular dispersion. In the proposed module, the diffractive angle of every spectral wave influenced the overlapping of the diffractive spots on the following light-guide. Therefore, the effects of the grating period were analyzed based on Equation (7) for every wavelength. By means of the optical tool, LightTools, the diffractive spots from a broad-band source were simulated. The results show that when the grating period was less than 16 μm, the overlapping of the diffractive spots was eliminated. With consideration of the fabrication accuracy, the grating period was optimized as 15 μm.

Next, since the designated shape of the grating was blazed, the blaze angle was determined in terms of its diffraction efficiency, which is the merit-function to determine the blaze angle when the grating period is given [27]. Further, because of the wide spectral range of the light source, the diffraction efficiency was unable to be calculated accurately via scalar diffraction theory. Thus, the simulation was executed by means of the rigorous-coupled-wave-analysis (RCWA) software, Gsolver, to derive the diffraction efficiency of the blazed grate with different blaze angles. According to Equations (1)–(3), the grating efficiencies for the considered visible band (380–780 nm), infrared band (780~2520 nm), and whole solar band (280~2520 nm) were calculated, as shown in Figure 2. To maximize the grating efficiency within the whole solar spectrum, the optimized blaze angle was 3.51◦, which is equivalent to a grating height of 0.92 μm. Then, the maximum grating efficiency for the whole solar spectrum reached 63.3%, while the corresponding efficiencies for the visible and infrared spectra were 73.8% and 54.4%, respectively. Additionally, the spectral-splitting performance illustrated in Figure 3 reveals that the unused diffraction orders were greatly suppressed.

**Figure 2.** Relations between the grating efficiency and the blazed angle for different spectral bands.

**Figure 3.** Diffraction efficiency for different diffraction orders in the spectral range from 280 to 2520 nm.

#### *3.2. Design of the Module*

After the lens and the grating had been designed, the following step was to determine the parameters of the light-guide. Herein, a single-unit module was considered for the design. An isosceles v-groove was assumed to exist on the bottom side of the light-guide with reflective coating to direct the spectral-split beams toward different output ports. The ray-tracing diagram for a collimated light source impinging on the proposed module is illustrated in Figure 4. This diagram shows that, for the first-order diffraction, the spot on the light-guide shifts along the +y axis when the wavelength increases, but this y-shift phenomenon does not occur for the zeroth-order light. Accordingly, adjusting the length, vertex angle, and center of the reflective v-groove, one can direct the first-order diffractive light, mainly in the visible spectrum, toward the visible output port and allow the on-axis zeroth-order diffractive light (mainly infrared) to be reflected toward the infrared output port. In order to maintain efficient propagation, the v-groove must make the light propagating within the light-guide to fulfill the condition of total internal reflection (TIR). Then, the v-groove was assigned as an isosceles triangle with a vertex angle of 120◦ to facilitate fabrication. On the other hand, when the proposed module was constructed as an array to enlarge the solar insolation, the greater number of units resulted in longer optical paths for the rays reflected by the v-groove and a higher possibility for them to encounter other v-grooves. Subsequently, the propagation inclination of these rays changed, and the probability of rays exiting the light-guide through the entrance surface increased. In this way, more loss was induced, and the module efficiency was reduced. Thus, the geometries of the v-groove and light-guide were optimized by maximizing the module efficiency of the whole solar spectrum, as defined in Equation (6). The material of the light-guide was assigned as polymethylmethacrylate (PMMA). The simulation was performed using the software LightTools with consideration of the antireflective coating on the lens, the Fresnel loss at every interface, and the half-angle subtended by the sun, while other factors, such as the reflectance of the v-groove, were regarded as ideal. Moreover, the sizes of the grating and the light-guide in the simulation were designed according to the dimensions of the lenses, as well as the required borders and output area compatible with the receiver.

**Figure 4.** Single unit of the proposed module with the ray-tracing results for (**a**) the first-order and (**b**) the zeroth-order diffractive beams exiting through the visible and the infrared output ports, respectively.

Based on Equation (6), the module efficiency was counted from the simulated output power. Based on the results shown in Figure 5a, although the reflectance of the v-groove was assigned as unity, a longer v-groove length (as illustrated in Figure 5b) did not always contribute to higher module efficiency. The reason for this result is that an increased v-groove length enlarges the probability of rays encountering the second or successive grooves, resulting in more possible rays being reflected out of the light-guide through the entrance surface and leading to greater possible loss. Based on these simulated results, the geometric parameters of the v-groove were determined after the number of units for a module was given. The designed parameters of the v-groove and the light-guide will be presented in the next section along with those of the other components.

**Figure 5.** (**a**) Relation between the module efficiency for the whole solar spectrum, the length of the v-groove, and the number of units based on the simulation results and (**b**) structure of the v-groove.

#### **4. Demonstration and Experiment**

The proposed module was demonstrated in the form of an array comprising seven units. Each unit consisted of three components, including a commercial achromatic lens (AC254-080-A), a blaze grating made of PET, and a PMMA light-guide with a carved v-groove. Based on the aforementioned design considerations, the parameters of the components were determined under the condition of a seven-unit module, as listed in Table 1. It is worth noting that the gratings of all units can be integrated into one device to simplify fabrication, and the light-guide has similar properties. Thus, the models of these two components were constructed as monolithic. Using Equations (4)–(6), the simulated module efficiency of the seven-unit module for the whole solar spectrum was 45.2%, and the efficiency for the visible and the infrared output ports was 53.8% and 37.5%, respectively. The grating sheet was fabricated via a roll-to-roll process. Meanwhile, the structured light-guide was fabricated by ultra-precision diamond machining, and then the carved v-grooves were coated with a reflective aluminum film. Moreover, in order to assemble all the components of the module, a fixture was designed and fabricated. The assembled module is shown in Figure 6.

After the assembly of all components with the fixture, the whole module was mounted on a dual-axis solar tracker for measurement. A customized integrating sphere, a power meter (NOVA II, Ophir Optronics), a visible spectrometer (SE1020C-025-VNIR, OtO Photonics Inc.), and an infrared spectrometer (SW2830S-050-NIRA, OtO Photonics Inc.) were utilized to measure the solar irradiance along with the out-coupling power and spectra from both the output ports. According the specifications of these instruments, the guaranteed power accuracy and the spectral accuracies of the visible and infrared bands were ±3%, less than 0.4 nm, and less than 1 nm, respectively. A photo of the experimental setup is shown in Figure 7.


**Table 1.** Designed parameters of the components for each unit in the 7-unit module.

\* A commercial sample treated with achromatic coating was utilized.

**Figure 6.** Photo of the demonstrated module.

**Figure 7.** Assembled module with the measuring instruments allocated on the solar tracker.

The experiment was performed on a clear sunny day with the whole module fixed on the solar tracker. The irradiance of sunlight was measured as 606 W/m2, and the incident solar power was counted as 2149.5 mW. The measured power in the visible spectrum from the visible output port was 523.7 mW, and that in the infrared spectrum from the infrared output port was 225.28 mW. Based on Equations (4)–(5), the module efficiencies were 44.8% and 24.6% for the visible and the infrared spectra, respectively. The experimental module efficiency for the whole solar spectrum was then counted as 34.8%, according to Equation (6).

When the proposed side-absorption concentrated module is connected to different receivers that are energy conversion devices, the system efficiency will vary. Ideally, the visible and the infrared output ports will be linked to the devices with complete energy conversion in the visible and infrared bands, respectively. Because of its proper absorption band and low cost, a silicon solar cell is an economic and efficient choice to link the visible output port. For the infrared port, the successive receiver for energy conversion can be either infrared solar cells or high-efficiency thermal modules, such as a solar water heater coupled with a phase-change material [28]. The system efficiencies of several feasible configurations were analyzed, and their details are reported in the following section.

#### **5. Analyses and Discussions**

#### *5.1. System E*ffi*ciency*

As mentioned above, different energy conversion devices for the proposed module will result in different system efficiencies. For example, the visible port of the proposed module can be connected to a commercial mono-crystalline silicon solar cell, with the external quantum efficiency (EQE) coarsely calculated as 75% [29], and the infrared port can be connected to a common water heater with a thermal efficiency of 70% [30,31]. Along with the measured data of the proposed module, the system efficiency is then estimated to be 25.9%.

Preferably, high-performance devices will be utilized for the proposed module. An example of such a device is an inter-digitated back contact (IBC) silicon solar cell with a high EQE [32]. This type of potential device will preferably be coupled to the visible port of the proposed module. On the other hand, a commercial germanium cell with a high EQE is one possible choice for the infrared port [33]. According to the reported EQE data for these cells and the simulated spectral responses of the proposed module, as illustrated in Figure 8, the system efficiency is calculated as 29.5%. The proposed module using this combination of silicon and germanium cells is comparable in efficiency to other double-cell techniques (for example, advancing tandem cells made of silicon and perovskite [34,35]).

**Figure 8.** External quantum efficiency (EQE) data for the aforementioned silicon and germanium cells [32,33] and the normalized spectral responses of the visible and infrared ports in the proposed module.

#### *5.2. Loss Analyses and E*ffi*ciency Enhancement*

In order to investigate the discrepancy between the simulated and measured module efficiency, we performed loss analyses via simulation. The pertinent factors are described as follows. Unlike the original design with the lenses integrated compactly, the fixture of the fabricated module produced certain spaces for mounting the lenses, as shown in Figure 9. The modeling results show that such incompactness reduced the module efficiency. Thus, a lens array designed with a high-fill-factor is preferred. Meanwhile, the reflectance of the v-groove was not as ideal as unity. Thus, the reflectance spectrum of the aluminum-coated film, as plotted in Figure 10, is not negligible. In addition, a laser beam was utilized as a testing source to examine the surface quality of the v-groove, and scattering due to the surface roughness was inspected. Based on these factors, the module efficiency was analyzed, and the results are listed in Table 2. This analysis shows that the effects of these fabrication factors on the module efficiency for the whole solar spectrum are close (each around 3% to 4%), and their synergy dominates a 10% loss of such efficiency.

According to the above root cause analysis of the efficiency loss, improvement of module efficiency lies in the high fill-factor for the lens array, the high-reflectance coating (such as a silver coating), and the decreased scattering through the advanced polishing process. Furthermore, through modeling, we found that the Fresnel loss occurring at each refractive interface is also a dominant factor. When all Fresnel losses are eliminated, the module efficiency over the whole solar spectrum can be increased theoretically from 45.2% to 51.2%. Although producing a module without any Fresnel loss is difficult, a feasible approach for such a module would be to process two significant surfaces—the entrance

surfaces of the grating and the light-guide—with an antireflective coating. Considering this feasible antireflective-coating, the module efficiency over the whole solar spectrum is 49.3%, based on the simulation results.

**Figure 9.** Exploded drawing of the practical module.

**Figure 10.** Spectral reflectance of the aluminum-coated film.


<sup>a</sup> The primary model considered the geometry and material of each component, the antireflective coating of lens, the Fresnel loss at every refractive interface, and the half angle subtended by the sun. <sup>b</sup> The expected system efficiency was calculated on the basis of model No. (4) for an embodiment with the proposed module linking assumptively to an inter-digitated back contact (IBC) silicon solar cell and a commercial germanium cell [32,33].

#### *5.3. Techniques Comparisons*

The proposed side-absorption concentrated module is a zig-zag type of diffractive SBS. Although SBS possesses the aforementioned electrical and thermal advantages, there are different issues and challenges for different categories of SBS. The dichroic filtering category requires a dichroic filter to perform spectral beam splitting [12]; however, filters made of metal suffer from a considerable absorption loss, while dielectric filters made from multiple thin layers generally decrease performance after receiving solar insolation for a period of time. The liquid-absorption category uses liquid to absorb the thermal portion of sunlight and cool the photovoltaics [36]. However, the flow of liquid may affect the stability of the output power of the photovoltaics. The diffractive category can realize a relatively compact structure. However, the fabrication accuracy of the diffractive element is a critical factor and relates highly to the cost.

Among the available diffractive SBS techniques, the type using a common-axis possesses a simple structure but suffers from shadowing effect and its resultant loss. The type with multiple optical axes results in a compressed system by reducing the effective length of the optical axis. Nevertheless, the energy loss occurring at the spacing between its spectrally adjacent receivers is unavoidable. The type with a zig-zag axis contributes to another compact system by laterally (rather than longitudinally) allocating the electric or thermal modules. However, the longer zig-zag path may induce more loss due to the more probability of rays encountering the successive grooves. The performance of each representative diffractive-SBS technique is listed in Table 3 for comparison. This comparison shows that the performance of the proposed module is acceptable.


**Table 3.** Comparisons between the different diffractive-spectral-beam-splitters (SBS) techniques.

<sup>a</sup> Equivalent to "Optical efficiency" in some other literatures", <sup>b</sup> simulation data, <sup>c</sup> estimated results from the measured optical efficiency, <sup>d</sup> measured data.

#### **6. Conclusions**

In this study, we proposed a side-absorption concentrated module as a spectral-beam-splitter to separate the visible and infrared bands of sunlight for different energy-conversion applications. The proposed module integrates diffractive grating, lenses, and a light-guide to split sunlight according to spectral bands and to export the spectrum-split rays to individual planar ports. A design example with seven units has been provided for demonstration. In the simulation, the grating efficiency and

module efficiency of this module were counted as 63% and 45.2%, respectively, for the whole solar spectrum. Experimentally, the components of the proposed module were fabricated and assembled; the module efficiency for the whole solar spectrum was then measured as 34.8%. Using this type of module with its visible and infrared ports linked to high-EQE silicon and germanium solar cells, respectively, a system efficiency of 29.5% is expected (based on the simulation results and the released data of solar cells). Accordingly, this system is comparable in efficiency to other double-cell systems. Moreover, the details of efficiency loss were analyzed for the proposed module, while the approaches to enhance module efficiency were discussed and presented. This analysis shows that when all Fresnel losses and the pertinent factors, including fixture limitation, reflectance of Al-coated film, and scatting effect are well-controlled, a module efficiency of 51.2% is expected for the whole solar spectrum. As for the energy gain in a PV/T system, the worthiness of the hardware development, and the payback time, they relate highly to PV materials, thermal system designs, the number of units of the proposed module, the climate and weather conditions of the demonstration site, and so on. Since some of them, such as the climate conditions, generally need sufficient time to acquire, collecting the necessary data, optimizing the performance, and lowering the cost of the proposed module will be our future researches for system realization.

**Author Contributions:** Conceptualization, A.-C.W.; formal analysis, A.-C.W.; funding acquisition, J.-R.S.; investigation, W.-J.C.; methodology, J.-R.S.; project administration, A.-C.W.; resources, A.-C.W.; software, W.-J.C.; supervision, A.-C.W.; validation, W.-J.C.; writing—original draft, W.-J.C.; writing—review and editing, A.-C.W. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Ministry of Science and Technology, Taiwan, R.O.C., grant number MOST 107-2221-E-008-093 and 107-2221-E-492-026-MY3 by Academia Sinica, Taiwan, R.O.C., grant number AS-SS-108-04-02.

**Acknowledgments:** The authors thank Pi-Cheng Tung, Department of Mechanical Engineering, National Central University, Taiwan, R.O.C., and his group for the technical support on solar tracker.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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