# Modelling, parameter estimation and assessment of partial shading conditions of photovoltaic modules

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## Abstract

This paper proposes a method for assessing the effect that different features of partial shading conditions (PSC) may have on the operation of a photovoltaic (PV) system. Simulation studies, based on an experimentally validated model of a PV system, are used to assess the influence of PSC. Three classifications of PSC are defined based on the timescale of their influence on the irradiance experienced by the PV module and the relative location of the voltage at which the global maximum power point occurs is assessed. Sample case studies are presented to illustrate the application of the proposed PSC assessment method. The results have implications for the design of future maximum power point tracking methods.

## Keywords

Photovoltaic Modelling Partial shading Maximum power point## 1 Introduction

Photovoltaic (PV) cells represent a renewable energy source that has highly non-linear characteristics and operation which depends greatly on the environmental conditions. Understanding how the operation of such cells varies under complex and changing environmental conditions is an important step in validating the performance of these systems. PV cells have an optimal operation point which is generally tracked or estimated to ensure efficient operation. Determining this optimum point becomes considerably more complex under non-uniform environmental conditions.

Non-uniform environmental conditions further complicate modelling of the current-voltage (I-V) and power-voltage (P-V) characteristics of PV systems. If multiple PV modules are connected together and one produces less power than the others, this module will limit the power produced by the other modules and may experience hot spot formation or cell damage unless bypass diodes are installed across the modules [1, 2]. Bypass diodes enable the current to flow through an alternative path and skip modules that cannot contribute to the power production. However, the integration of bypass diodes further complicates the I-V and P-V characteristics as multiple peaks are now observed and the power production may be reduced [3, 4, 5]. Non-uniform conditions arising across a module could occur due to shading of part of the module from buildings, trees or other obstacles in the environment, physical damage to the cell, cell ageing over time, or due to the passage of clouds over the module [2, 6, 7]. Especially in the case of cloud movement across the modules, these non-uniform environmental conditions may change very rapidly leading to a considerable reduction in the power extracted from the system.

Studies completed in the literature to assess the effects of partial shading conditions (PSC) in PV systems take on many different forms. In some studies the position of the PV modules and obstacles in the environment with respect to the position of the sun at the time of interest are used to map shadows onto the panels [8, 9, 10, 11, 12, 13, 14]. Other approaches involve using artificial shading conditions to explore the impact of PSC on the PV characteristics [15, 16, 17]. In mapping the shadow onto the modules, some authors [8, 14] consider the effects of direct and diffuse irradiance. Defining obstacles by linear functions and a transmission factor for non-opaque objects is suggested by [10]. The 3D mapping of shadows can be achieved using the sun’s position and simplified ray tracking with the irradiance on the cell being the averaged shaded and non-shaded irradiance [12]. A PV systems installers guide [11] recommends that close objects provide more direct shading, and therefore a more significant reduction in irradiance, than objects that are far away. The light generated current can be expressed as a function of the shaded area, photo-current density and shadow transmittance to develop PSC [15]. Shading is represented by a shading strength and shading percentage (representing what portion of the system is shaded) by [16], and as a simple percentage by [17]. The shading impacts can also be assessed by considering how the configuration of modules in a system can reduce the effect of shading from obstacles [18] or due to cloud transients [19]. A recent study into the effect of PSC on PV modules has defined five statements which define typical behaviours of PSC [20]. These statements lead to two hypotheses for predicting the number of local maximum power points (MPPs). Other recent work has combined a sky map with a sensitivity map for PV modules that have reflectors to estimate incident irradiance composed of reflected light, direct light and scattered light [21]. These results show good prediction of incident irradiance, however the method has greater complexity than the method adopted in this paper.

The shading analysis presented in this paper is unique as the primary intention of the study is to explore how the relative location of the global maximum power point (GMPP) voltage varies when the shading conditions change. To achieve this objective while minimising computational load, some essential assumptions have been made including using the global direct irradiance rather than separating direct and diffuse irradiance in this preliminary study. Three types of PSC are defined. These are constant, static and transient PSC. Constant PSC represents a reduction in irradiance that will always be present, such as due to cell ageing or damage. Static PSC represents shading which changes slowly with time, such as the shade from an object in the environment which may also have a shading strength associated with it. Transient PSC is much quicker and represents the irradiance at a particular point based on the time of day and cloud cover. Essentially transient PSC can be considered as the instantaneous irradiance reaching the Earth’s surface which may vary very rapidly based on cloud cover. As a residential scale system is considered in this paper, the geographical area of the system is small such that transient PSC (or the incident irradiance from the sun) is assumed constant across all modules in the system at any point in time. The irradiance on each module is determined by considering the constant shading factor, static shading factor and transient irradiance.

The classification of shading types considered in this paper is more comprehensive than that presented by [22] as it also considers the changing irradiance due to cloud cover and time of day as a type of shading phenomenon. The approach of using 1 min solar data and matrices to provide a shading factor for each module to model the different shading on modules has been considered by [23] however, in their paper the main shading effects considered relate more so to transient PSC defined above. The purpose of the study presented in this paper is to enable the effects of constant, static and transient PSC on the location of the GMPP voltage to be isolated and quantified. This has direct implications for the design of future global maximum power point tracking (GMPPT) methods.

- 1)
Defining three classifications of PSC and utilising these to isolate the individual impacts of shading on a PV system.

- 2)
Development of a method and case studies for assessing the effects of constant, static and transient PSC on the location of the GMPP voltage.

- 3)
Identification of the implications this study has for GMPPT.

Section 2 presents a method for assessing the impact of the three PSC types defined above on the relative location of the GMPP based on an experimentally validated PV module simulation model. In Section 3, three sample case studies are presented to show how the PSC assessment method can be used with simulated obstacles and constant cell ageing factor related partial shading on the PV modules. Finally, Section 4 discusses the key observations from the case studies and possible implications for MPPT and Section 5 presents the paper conclusions.

## 2 Modelling and assessment of partial shading conditions

Parameters of BP380 PV module

Parameter | Value |
---|---|

\(V_{oc}\) | 22.10 V |

\(I_{sc}\) | 4.80 A |

\(V_{mpp}\) | 17.60 V |

\(I_{mpp}\) | 4.55 A |

\(P_{mpp}\) | 80.10 W |

\(N_{s}\) | 36 |

In this paper residential scale PV systems are of interest and are of sufficiently small size that the transient shading factor at any point in time can be assumed constant across all the modules in the system. Transient PSC is the shading due to changing irradiance level and is represented in this study through 1 min irradiance data from the Australian Bureau of Meteorology (BOM) [27]. This idea of transient PSC being modelled by 1 min data is supported by [28], who determined that shading periods caused by clouds have an average duration of 60 s, but may range from 4 s up to 1.5 h. Constant PSC can be modelled by applying a constant shading factor to the module. Static partial shading is modelled by placing a simulated obstacle in the environment and modelling the shadow from the object based on the position of the sun in the sky.

*shaded-cells*represent the number of shaded cells.

This paper adopts a simple approach for estimating which cells within the module are shaded by the obstacle. A simple approach is deemed appropriate as the key objective of the study is to develop a preliminary set of observations related to how different shading phenomena affect the relative location of the GMPP with respect to its voltage. The number of cells that are shaded by the object are found by considering the object orientation in the environment and calculating the shadow tip position for the time and day of the year. Objects define their own coordinate basis, such that each object is located at coordinates (0, 0) on the north–east axis, and the distance to the closest corner of the PV module is defined. In this coordinate system it is possible to determine when the cells of each module lie within half of the object width of the shadow centre line (defined by joining the object origin to the shadow tip) indicating that they are most likely shaded. For simplicity, the width of the shadow is assumed equal to the object width. Complex shadow shapes are not considered in this analysis and the global direct irradiance is applied without separating the impacts of direct and diffuse irradiance. These simplifications are deemed appropriate as the primary goal of the study is to investigate the movement of the relative voltage of the GMPP as authentic shading conditions change.

The series of equations given below provides a process for evaluating the shadow tip location [9, 29, 30], where \(\phi\) denotes latitude; \(\delta\) denotes declination angle; \(\alpha\) denotes elevation angle; \(\omega\) denotes longitude angle; *LT* denotes local time; *N* denotes day of the year; \(\Delta T_{GMT}\) denotes difference of local time from GMT in hours (10 h in Tasmania).

*LSM*is:

*EoT*can be given by:

*B*is given by:

*TC*is:

*LST*:

*HRA*can be found from:

*Azimuth*gives the direction of the sun. And

*Azi*is a temporary value as the actual azimuth needs to be calculated slightly different depending on

*HRA*.

*H*is the height of the object in the environment.

*x*and

*y*coordinates can be calculated from the azimuth and elevation angle for each minute at the location of interest using (12).

Once the shadow path of an obstacle is known it can be used to determine the static shading factor on each module in the system using (1). This shading factor is then multiplied by the constant and transient shading factors to establish an estimate of the equivalent irradiance on each module. This can then be used with the eight series-connected PV modules model to develop a P–V curve trace for the system showing the impact of PSC at that particular instant in time. This process is repeated across a sample of shading and irradiance conditions to monitor how these changes impact on the relative location of the GMPP in terms of voltage. These sample conditions are based on modelling objects in the system at a particular location on a particular date and time, and utilising BOM 1 min known irradiance data that corresponds with that same time, date and location.

*P*

_{gmpp}is the power at the global maximum power point and

*V*

_{gmpp}is the voltage at the global maximum power point. This procedure is generic and relies on the equations defined in this section. In the case studies presented in Sect. 3, 1 min solar irradiance data has been used, so each characteristic is drawn on a 1 min basis.

## 3 Partial shading case studies

Four case studies on the eight series-connected modules simulation model from Sect. 2 are explored in this section. These case studies represent a subset of the extensive additional case studies completed and highlight the key findings from the other cases. Other case studies have been omitted due to space requirements. In the first case study there is only one obstacle located in the environment and in the second case study there are two obstacles in the environment. Case 3 involves a constant PSC being applied to the system to represent cell ageing and damage. Case 4 incorporates a shading strength into the calculation of shading factor for two obstacles in the environment. The simulation model enables the location of the voltage of the GMPP to be monitored and key observations on the transitions of the GMPP location under constant, static and transient PSC can be described for each case study.

### 3.1 Case 1

Transitions for Case 1

Transition | Average | Minimum | Maximum | Standard deviation | Count |
---|---|---|---|---|---|

One MPP transition | 15.55 | 0 | 18 | 2.12 | 3453 |

Transition of two MPPs | 0.07 | 0 | 5 | 0.41 | 15 |

MPP locations for Case 1

Location | Average | Minimum | Maximum | Standard deviation | Count |
---|---|---|---|---|---|

MPP6 | 36.40 | 0 | 60 | 10.50 | 8083 |

MPP7 | 29.64 | 0 | 41 | 6.62 | 6580 |

MPP8 | 4.47 | 2 | 16 | 1.97 | 992 |

From these tables, it can be seen that it is most likely that the GMPP voltage will remain around the same MPP region (77.85%) or transition to an adjacent MPP region (22.06%) between each minute of the study. In a very small number of cases (0.1%) a transition to a MPP region further away was observed. It can also be seen that during the shading time, the GMPP was most likely in the region of MPP6 (51.63%).

### 3.2 Case 2

Transitions for Case 2

Transition | Average | Minimum | Maximum | Standard deviation | Count |
---|---|---|---|---|---|

One MPP transition | 12.61 | 2 | 26 | 4.67 | 3316 |

Transition of two MPPs | 4.30 | 0 | 11 | 2.63 | 1130 |

Transition of three MPPs | 3.26 | 0 | 8 | 2.43 | 858 |

Transition of four MPPs | 0.60 | 0 | 4 | 0.92 | 124 |

Transition of five MPPs | 0.01 | 0 | 1 | 0.10 | 1 |

MPP locations for Case 2

Location | Average | Minimum | Maximum | Standard deviation | Count |
---|---|---|---|---|---|

MPP2 | 0.01 | 0 | 1 | 0.10 | 1 |

MPP4 | 4.62 | 0 | 62 | 9.06 | 966 |

MPP5 | 36.30 | 0 | 90 | 22.49 | 9546 |

MPP6 | 44.60 | 0 | 87 | 20.41 | 11730 |

MPP7 | 72.50 | 1 | 146 | 23.14 | 19068 |

MPP8 | 103.37 | 7 | 227 | 61.00 | 27187 |

### 3.3 Case 3

### 3.4 Case 4

Shading strengths of 0.2, 0.5, and 0.7 are applied to the shading scenario of Case 2 on 23 May 2010 to assess if this factor has any influence on the relative location of the GMPP voltage.

These results show that the shading factor has no influence on the time of shading as the path of the obstacle across the modules is fixed. The shading factor however works with static shading to change the movement of the GMPP and reduces the overall power available as the shading factor increases. As the shading strength increases, the shading becomes more severe leading to a greater number of transitions between adjacent MPP locations.

## 4 Key observations

- 1)
Power available at the GMPP is most affected by the transient PSC (the irradiance).

- 2)
With constant PSC small local peaks occur in the PV characteristics however the GMPP voltage is not influenced significantly.

- 3)
The most significant factor in moving the relative location of the GMPP with respect to voltage is static partial shading, resulting from the movement of shadows from obstacles in the environment.

- 4)
Shading strength influences the shading factor caused by static shading but only when the static obstacle shadow maps onto the modules.

- 5)
As shading strength increases, the likelihood of the GMPP voltage moving to an adjacent region increases based on the corresponding more significant change in shading factor in the modules of the system.

- 6)
The GMPP is more likely to remain around the same MPP region with respect to voltage or move to an adjacent MPP region, rather than undergoing a transition across several MPP regions.

The method proposed in this paper has also been applied with 1 s irradiance data available from NREL [31] demonstrating similar results. These results are not shown in this paper due to space restrictions, but are supported by the fact that shading periods caused by clouds have an average duration of 60 s [28] and the relative location of the GMPP in terms of voltage is more sensitive to the relatively static PSC arising from obstacle placement in the environment which will not substantially change from 1 s to the next.

These key observations have a direct implication for the future development of GMPPT methods. In particular, these results suggest that when trying to locate a new maxima it may be possible to search within a much smaller neighbourhood of the previous operating point. This is indicated by the representative results included in this paper and supported by the other case studies and sets of data omitted due to space, which show that in the presence of shading from an obstacle, the GMPP relative location is most likely to remain at the same point or transition to an adjacent MPP location. This significantly reduces the search range when a change in conditions is detected and provides a mechanism for monitoring adjacent peaks [2] as a method of detecting a change in the static shading situation [32]. The results also show that when small scale residential PV systems are considered, static partial shading caused by objects in the environment will have the most significant impact on the relative location of the GMPP while the transient PSC will have the greatest effect on the power available. By isolating the effects of the different types of PSC, the effects that each of these will have on a particular PV system can be estimated before the system is installed. By modelling the movement of the shadow of obstacles across the modules, it is possible to select an optimal configuration of the modules to minimise PSC losses.

## 5 Conclusion

In this paper a PSC assessment strategy has been proposed and explored through several case studies. These case studies show that the shading from objects in the environment has the greatest effect on the relative location of the GMPP voltage when compared with constant cell degradation and rapidly varying irradiance. The results have also shown that typically when the GMPP voltage moves to another region due to a change in environmental conditions, this is most commonly to an adjacent position. This has implications for the design of GMPPT strategies as it provides guidance on a suitable searching range of the previous operating point when attempting to locate a new GMPP.

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