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Solar Photovoltaic (PV) technology is the fastest growing green energy technologies. Since year 2002, the worldwide production volume of the PV modules is increasing at a rate of 100% every 2 years . This growing trend in the adoption of the PV technology is sustained by the continual improvement in manufacturing technologies and less expensive raw materials which leads to, more efficient and lower cost PV solutions in the market.
The cost effectiveness of a PV system largely depends on its ability to meet the varying electrical load demands under varying environmental factors. Thus, accurate PV module modeling and simulating the PV system's performance subjected to varying environmental factors are critical aspects of sizing the PV system.
Various PV models are available in the literature [2-6]. These models are generally simplifications to the dual-diode model. While these model simplifications would ease system sizing calculation effort, over-simplification would result in an over- or under-sized PV system. In both cases, the PV system is not a cost effective solution.
The performance of a PV module varies significantly with temperature and solar irradiation level. The performance of PV modules connected in series and parallel configurations; PV array, gets more complicated when subjected to partial shading [7-9]. Partial shading occurs when the entire PV array is not under uniform insolation. When this occurs, the output power of the PV array would demonstrate multiple peaks. Thus, reducing efficiency of the PV system as the Maximum Power Point Tracking (MPPT) technique might not be able to distinguish the global maximum power point among the multiple peaks .
The objectives of this report are thus to study the issues as discussed in the aforementioned discussions:
To evaluate and compare the effectiveness of the PV models in modeling the I-V characteristics with varying Fill-factor (FF) as well as shading effects
To study the effects of partial shading on the I-V and P-V characteristics of a PV array
PV Cell Modeling
Operation of PV Cell
A PV cell works on the principle of photoelectric effect, converting the light energy captured from the solar irradiation into electrical energy. As a semiconductor p-n junction, the PV cell has a built-in electric field. Thus, when the PV cell is connected to an external load, the built-in voltage of the PV cell will function as the electromotive force to drive the free electrons around the circuit when under solar irradiation. The operation of the PV cell is illustrated in Figure 1.
Figure 1: Operation of PV Cell
Models of PV Cell
There are generally 2 equivalent lump circuit representations for PV cells, namely the dual-diode and single-diode model. The dual-diode model (Figure 2) has 2 diodes included in the equivalent lump circuit. The purposes of these 2 diodes are to represent the electrons and holes recombination at the surfaces and the bulk regions, and the recombination at the junction respectively. The former dominates during high insolation level while the latter dominates at lower insolation level .
Figure 2: Dual-diode Model
Unlike the dual-diode model, the single-diode model (Figure 3) only has a diode in the equivalent lump circuit. This model assumes a constant ideality factor generally varies between 1 and 5. As the discussed in the dual-diode model, the ideality factor is a function of voltage across the diodes in the equivalent lump circuit. Therefore, in the single-diode model, recombination in the junction is assumed negligible and electron-hole pair recombination at the surface dominates. In other words, the single-diode model might not be able to provide an accurate representation of the of the PV cell characteristic at low voltage or insolation conditions. Thus, it would be deemed a less accurate model as compared to the dual-diode model.
Figure 3: Single-diode Model
Apart from the diodes, the equivalent lump circuit of the PV cell is also made up of a series and parallel resistance. The series resistance represents parasitic resistances due to cell-interconnect bus-bars, cell metallization and cell solder bonds, etc. The parallel resistance represents the lump resistances that arise due to impurities and defects in the cell that results in high-conductivity paths across the solar cell p-n junction. Increasing the series resistance or decreasing the shunt resistance results in a smaller peak power and fill-factor (FF) .
The single-diode in Figure 3 could be further simplified by assuming the parallel resistance is sufficiently large and could thus be neglected as shown in Figure 4.
Figure 4: Single-diode Model without Parallel Resistance
The above mentioned 3 PV cell models will be used in the subsequent sections to study the accuracy of the models in modeling PV cells of varying fill factor and under shading effects.
Types of PV Cells
The PV cells available in the market generally fall into one of the following four PV types :
Mono-crystalline silicon (also known as single-crystal silicon)
Poly-crystalline silicon (also known as multi-crystalline silicon)
Amorphous silicon (also known as thin-film silicon)
Mono-crystalline PV cells are most commonly used in solar applications. These cells are usually blue or black in colour as shown in Figure 5. Poly-crystalline PV cells are also one of the popular options in solar applications. Unlike the mono-crystalline cells, the surface of polycrystalline cells shows a random pattern of crystal borders (Figure 6).
Figure 5: Mono-crystalline PV Cell 
Figure 6: Poly-crystalline PV Cell 
Amorphous PV cells are non-crystalline silicon. Thus, they do not have a distinct crystal structure as the previous two types of PV cells mentioned previously. The manufacturing process of the amorphous silicon units involves depositing thin layers of vaporized silicon in a vacuum onto a support of glass, plastic, or metal. Amorphous silicon cells are produced in a variety of colours (Figure 7) and are not suitable for residential applications due to their low generation density.
Figure 7: Amorphous PV Cell 
Hybrid PV cells (Figure 8) is a type of PV cell that utilizes more than one PV technology to achieve superior performance even at elevated temperature. Hybridization allows the PV cell to exploit the advantages of both organic and inorganic semiconductors.
Figure 8: Hybrid PV Cell 
Figure 9: Manufacturing Process of Crystalline & Amorphous PV Cells
The cost of the different types of PV cells differs as a result of the difference in their manufacturing process. As shown in Figure 9, the mono-crystalline PV cell undergoes the longest manufacturing process as compared to poly-crystalline and amorphous. Hence, the cost of mono-crystalline is significantly higher than that of poly-crystalline and amorphous PV cells since it consumes more raw material and energy during the manufacturing process. Similarly, hybrid type PV cell would be the most expensive PV cell since it takes more effort to produce.
Each step taken in the manufacturing process strives to purify the raw material further and to eliminate defects in the PV cells. Thus, mono-crystalline would have higher efficiency as compared to poly-crystalline and amorphous PV cells. Among the PV cell types, Hybrid PV cells have the greatest efficiency, generally more than 18%. The efficiencies of mono-crystalline, poly-crystalline and amorphous PV cells were shown in Table 1 .
Table 1: Comparison of PV Technology Efficiencies
Study of PV Cell Modeling for Varying Fill-Factor & under Shading Effects
In this section, the effectiveness of the three PV cell models previously discussed was evaluated on a qualitative basis. The PV cell models were evaluated upon the closeness-of-fit between the I-V characteristic curves of the measured I-V data against the data simulated by each of the models, on the basis of varying FF and shading effects. The PV cell models were studied along with four PV cells, each corresponding to a different type of PV technology.
The parameters necessary for each of the model were derived by curve-fitting the model equations to actual or measured I-V characteristic curve obtained from the PV cell datasheet (refer to Appendix). The curve-fitting algorithm used in this process was the Levenberg-Marquardt algorithm [17-18]. The mapping functions and parameters used in the curve-fitting algorithm for the corresponding PV cell model were as follows:
Single-Diode Model without Parallel Resistance:
Table 2 shows the extracted parameters of each model for the corresponding type of PV cell.
Comparing PV Cell Modeling across varying Fill Factor PV Cells
The FF of a PV cell serves as an index to the overall behavior of the cell and is defined as follows:
Based on the above equation, it could be concluded that the FF will be highly dependent on the efficiency of the cell. Efficiency of a PV cell varies across the PV cell type; mono-crystalline, poly-crystalline, amorphous and hybrid. Thus, the FF of the PV cell would likewise vary across the PV cell type. The FF of each of the PV cell used in this study of the effectiveness in cell modeling across PV cell type was shown in Table 3.
Mono-Crystalline (Suntech - STP180S)
Single-Diode Model w/o Parallel Resistance
Poly-crystalline (Sharp - ND-176UC1)
Single-Diode Model w/o Parallel Resistance
Amorphous (BSC BS-52)
Single-Diode Model w/o Parallel Resistance
Hybrid (Sanyo HIT Power205)
Single-Diode Model w/o Parallel Resistance
Table 2: Extracted PV Cell Model Parameters
Voc / V
Isc / A
Pmax / W
Mono-Crystalline (Suntech - STP180S)
Poly-crystalline (Sharp - ND-176UC1)
Amorphous (BSC BS-52)
Hybrid (Sanyo HIT Power205)
Table 3: Fill-factor of Different Types of PV Cells/Module
In this study, the model parameters in Table 2 were inserted into the PV cell models developed in the Simulink environment. Figure 10, 11 and 12 are the Simulink model of the PV cell single-diode without parallel resistance model, single-diode model and dual-diode model respectively. The model representations developed in the Simulink environment were the implementation of the model equations (1), (2) and (3) described in Section 3. The I-V characteristic curves of the models for each type of PV cell against the measured data were demonstrated in Figure 13, 14, 15 and 16 for the mono-crystalline, poly-crystalline, amorphous and hybrid type cells respectively.
Figure 10: Simulink Model representation of the Single-diode without Parallel Resistance Model
Figure 11: Simulink Model representation of the Single-diode Model
Figure 12: Simulink Model representation of Dual-diode Model
Figure 13: I-V Characteristic Curves for Mono-Crystalline PV Module (Insolation = 1000W/m2)
Figure 14: I-V Characteristic Curves for Poly-Crystalline PV Module(Insolation = 1000W/m2)
Figure 15: I-V Characteristic Curves for Amorphous PV Module (Insolation = 1000W/m2)
Figure 16: I-V Characteristic Curves for Hybrid PV Module (Insolation = 1000W/m2)
From Figure 13, 14 and 15, it could be observed that dual-diode model could effectively model the I-V characteristic curves of the mono-crystalline, poly-crystalline and amorphous type PV cell/module. The effectiveness of the single-diode models in modeling the I-V characteristic curves of the PV cells improved along with the FF. The simulated I-V characteristic curves of both single-diode models perfectly matched the actual I-V characteristic curve of the hybrid PV cell, which has the highest FF.
On the other, the as the FF decreases, the I-V characteristic curves of the single-diode models deviates from the measured data of the PV cell. In the case of the amorphous PV cell, the single-diode models were not able to match the I-V characteristic curve of the measured data. Unlike the crystalline PV cells, amorphous type PV cell undergo less purification during the manufacturing process. This results in more defects present in the PV cell which contributes to significant recombination within the junction of the PV cell. Thus, a dual-diode model is necessary to model amorphous type PV cell.
Comparing PV Cell Modeling of PV Cells under Shading
In this section, the insolation level on the PV cell models were varied to study the effectiveness of the models in representing the I-V characteristics under shading effects. PV cells subjected to shading effects are equivalent to reduction in insolation level. The adjusted insolation for the mono-crystalline, poly-crystalline, amorphous and hybrid type PV cells were 600W/m2, 600W/m2, 400W/m2 and 400W/m2 respectively. These insolation levels correspond to the lowest insolation I-V characteristic curves available from the data sheets.
Figure 17, 18, 19 and 20 shows the I-V characteristic curves of PV cell models along with the measured data of the four types of cells.
Figure 17: I-V Characteristic Curves for Mono-Crystalline PV Module (Insolation = 600W/m2)
Figure 18: I-V Characteristic Curves for Poly-Crystalline PV Module (Insolation = 600W/m2)
Figure 19: I-V Characteristic Curves for Amorphous PV Module (Insolation = 400W/m2)
Figure 20: I-V Characteristic Curves for Hybrid PV Module (Insolation = 400W/m2)
The I-V characteristic curves depicted in Figure 17 and 18 suggest that the effectiveness of the PV models (single-diode models and dual-diode models) were not significantly affected by the reduction in insolation level. Relative to the mono-crystalline PV cell, the deviation from the measured data is more significant for the poly-crystalline case.
Deviations from the measured data were significant in both cases of the amorphous and hybrid PV cells. It was concluded in the section 3.1, that only dual-diode model could effectively model the I-V characteristic of an amorphous PV cell. Therefore, the closeness of fit demonstrated by the single-diode models in the amorphous PV cell comparison shall be irrelevant.
The closeness of fit demonstrated in the case of mono- and poly-crystalline would suggest that the same would apply to the hybrid type PV cell since it has the largest FF among the four types of PV cells. The results in Figure 20 prove otherwise. A closer examination revealed that the hybrid type PV cell contains a composite of crystalline and amorphous materials. Since in the amorphous case (Figure 19), significant deviation could be observed on the simulated dual-diode model I-V characteristic curve, the deviation from the measured data in the hybrid PV cell could be justified.
It could be concluded that for PV cells that contains amorphous type material, the model parameters would vary significantly as the insolation level varies. Hence, when modeling amorphous type PV cells, model parameters should be varied according to variation in insolation level.
Study of Shading Effects on PV Array
The shading effects on PV arrays with solely series-connected PV modules and parallel-connected modules were well understood . During shading, PV array with parallel-connected modules generally results in less power loss than that with series-connected modules. On the other hand, PV array with parallel-connected modules would carry much larger current than the series-connected counterpart. Therefore, PV arrays are typically composed of series-connected modules, also known as PV strings, connected in parallel to deliver the desired power at practically utilizable voltage.
Effects of shading on series-parallel connected PV modules array are generally complicated to study due to the PV array configuration and unpredictable shading pattern. In order to facilitate the investigation of shading effects on the PV array, the subject matter is limited to a 4x4 PV array; four modules in a string and 4 strings in parallel. Two shading patterns were employed in this study; a static and a dynamic shading pattern. The former depicts the scenario where part of the PV array was partially shielded from full insolation as the sun is not located perpendicular to the plane of the PV array. The latter refers to the scenario when the PV array was partially shielded from full insolation by moving obstacles such as passing clouds. In the course of this study, modules under the effects of shading would be exposed to an insolation level of 200W/m2, while full insolation level refers to 1000W/m2.
PV Array and Shading Pattern Modeling
A graphical user interface (GUI) was developed in Simulink environment to facilitate the study of the shading effects on the PV array. As shown in Figure 23, by selecting the modules under shading, different shading patterns were developed.
Figure 23: GUI for Shading Pattern Development
The Simulink model for the PV array is shown in Figure 24. In this model, four PV module strings were connected in parallel and simulated with a voltage ramp input to extract the I-V characteristic curve of the PV array under shading effect. Within each of the strings; Simulink Block 'String1', 'String2', 'String3' and 'String4', four modules were connected in series. Figure 25 depicts the Simulink blocks embedded within the PV string Simulink blocks of Figure 24. The implementation of the Simulink block 'Subsystem1' of Figure 25 is shown in Figure 26, where four PV modules were connected in series.
Figure 24: PV Array - 4 Strings in Parallel Configuration
Figure 25: PV String Module
Figure 26: PV String - 4 PV Modules in Series Configuration
Effects of Static Shading
The effects of static shading on the PV array were demonstrated in Figure 27, 28, 30 and 31. It was assumed that shading only affects the modules along the edges of the PV array. Figure 27 and 28 are the I-V and P-V characteristic curves of the PV array when the number of modules under shading in the string with 'Module 1', 'Module 2', 'Module 3' and 'Module 4', increased from no modules under shading to all modules in the string subjected to shading.
Figure 27: I-V Characteristic Curves of String with Modules under Shading
Figure 28: P-V Characteristic Curve of String with Modules under Shading
Multiple peaks observed in the P-V characteristic curves (Figure 28) were due to modules in the string irradiated with different insolation level. In addition, the existence of a dominant peak could be observed. The dominant peak corresponds to the global maximum power. As the number of shaded module in the string was increased, the dominant peak remained unchanged while the other peak diminished and approached a single peak when all modules in the string were shaded.
The behavior of the P-V characteristic curve could be explained by the I-V characteristic curve of the PV array. Since the other 3 strings in the PV array were not affected by shading, their I-V characteristic curves remained unchanged as the shading patterns were modified. Thus, the difference in I-V characteristic curve was a result of the change in the I-V characteristic curve of the string with modules subjected to shading effects.
The PV string's I-V characteristic curve was the summation of the individual I-V characteristic curves of the modules within the string, performed along the voltage axis. A module under shading would have a lower insolation level as compared to a module not subjected to shading. The decreased in insolation would cause the current to decrease in a linear relationship while the voltage decreased logarithmically. As a result, the I-V characteristic curve of the shaded module would approximate that of the module not subjected to shading, with the current axis being scaled down. This was demonstrated in Figure 29.
Figure 29: I-V Characteristic Curves of Modules with & without Shading Effects
The overall I-V characteristic curve of the PV module was obtained by summing the individual I-V characteristic curve of the modules along the voltage axis. Since modules in the string were subjected to two different levels of insolation, 1000 W/m2 and 200 W/m2, two 'knees' would be observed in the PV string's I-V characteristic curve (Figure 27).
Figure 30: Summation of Modules' I-V Characteristic Curves
Figure 30 depicts the summation of the modules' I-V characteristic curve to arrive at the PV string's I-V characteristic curve obtained in Figure 27. When the current through the string exceeds 0.2Isc, the shaded module will not be conducting and its bypass diode would be conducting, resulting in zero voltage across the module. However, shaded module begun to conduct when the current falls below 0.2Isc. Therefore, two 'knees' could be observed in the PV string's I-V characteristic curve; corresponding to the case when the shaded module was not conducting and the case when it was conducting. Since the shaded module could only conduct when the current through the PV string was less than the shaded module's short circuit current, the 'knee' on the PV string's I-V characteristic curve due to the conducting shaded module would remain at the same current and voltage operating point as the number of modules subjected to shading increases. Thus, explained the unperturbed dominant peak (Figure 28) as the number of shaded modules in the string was increased.
Contrary to the abovementioned case, the value of the dominant peak decreased as the number of strings with only one shaded module was increased. The shaded modules in these shading patterns were 'Module 1', 'Module 5', 'Module 9' and 'Module 13'. The results on this shading pattern on the PV array were demonstrated in Figure 31 and 32. It could also be observed that as the number of strings with only one singly shaded module was increased, the multiple peaks became more pronounced, which was also contrary to the case of the previous shading pattern.
Figure 31: I-V Characteristic Curves of Strings with 1 Module under Shading
Figure 32: P-V Characteristic Curves of Strings with 1 Module under Shading
The results of the 2 shading patterns on the PV array have an important implication on the orientation of the PV array. As discussed, the dominant peak power was not affected by the number of shaded modules in a string. Thus, the PV array should be orientated so that if shading would occur along its edges, only adjacent modules of the array that are connected in series rather than those where modules adjacent to each other were in parallel would be shaded. In addition, if all the modules along the edge of the array were shaded, the shaded string would not result in multiple peaks. The proposed orientation of the PV array was demonstrated in Figure 33.
Figure 33: Optimal PV Array Orientation
Effects of Dynamic Shading
The dynamic shading pattern utilized in this section of study replicates a passing cloud, causing shading on the PV array in diagonal fashion. The cloud was assumed to cause shading on 'Module 1' when it begun to move across the PV array. As the cloud progressed, 'Module 1', 'Module 2', 'Module 5' and 'Module 6' were shaded. The last shading pattern involved ''Module 1', 'Module 2', 'Module 3', 'Module 5', 'Module 6', 'Module 7' and 'Module 9', 'Module 10' and 'Module 11'. The I-V and P-V characteristic curves of the PV array as a result of each of these shading patterns were illustrated in Figure 34 and 5 respectively.
Figure 34: I-V Characteristic of PV Array under Dynamic Shading
Figure 35: P-V Characteristic of PV Array under Dynamic Shading
As the number of shaded modules increased, multiple peaks became more prominent and changed in the deliverable peak power were also significant. In addition, the open-circuit voltage of the PV array decreased significantly when nine modules were shaded.
Two problems were encountered during the course of the project. The first problem encountered was in extracting the data parameters for the dual-diode models using the Levenberg-Marquardt algorithm while the next problem was the simulation of the PV models in the Simulink environment.
The curve fitting algorithm for the dual-diode model parameter extraction worked well for the amorphous PV cell. However, in the case of the mono-, poly-crystalline and hybrid PV cells, negative reverse saturation currents were obtained at the end of the curve-fitting process. This problem was resolved in the case of mono- and poly-crystalline by omitting the series and parallel resistance in the curve-fitting process. Thus, more constraints were imposed in the curve-fitting process, thereby, forcing it to arrive at positive values for the reverse saturation currents.
In the case of the hybrid PV cell, increasing or decreasing the number of constraints in the curve-fitting function did not result in positive values for the reverse saturation currents. Thus, the I-V characteristic curves of the dual-diode model for the hybrid PV cell were omitted in the study.
PV Cell Simulation in Simulink
The PV cells that were modeled in the Simulink environment utilized the 'Algebraic Constraint' block to solve the non-linear PV cell diode models equations for the PV voltage. The 'Algebraic Constraint' block solved the non-linear PV cell diode model equations through numerical methods. Thus, the initial guess of the PV voltage would be critical as to whether it could converge to a solution.
Run-time errors were often encountered as the 'Algebraic Constraint' block could not converge to a solution. The problem persisted, despite having the initial guess of the PV voltage varied. The problem was resolved by utilizing the discrete solver and smaller step-size for the simulation.
Three PV cell models were discussed in this report. The single-diode models and dual-diode models could model mono-crystalline, poly-crystalline and hybrid type PV cell effectively. On the other hand, only the dual-diode model could effectively model the I-V characteristic curve of the amorphous type PV cell. It was also discovered that the model parameters of amorphous type material PV cell would vary significantly during shading or changes in insolation level.
Shading effects on the PV array were also investigated. It was concluded that if shading along the PV array edge was inevitable, the PV array shall be orientated so that the PV strings were parallel to the portion of the array edge under shading.
The present work has neglected the effects of temperature on the PV cell or array during shading. As variation of temperature would have significant impact on the I-V characteristic of the PVs, future work could incorporate the effects of shading with temperature variations.
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