## Techniques to Maximize Solar Panel Power Output

Two recent articles, Energy Harvesting With Low Power Solar Panels and Solar Battery Charger Maintains High Efficiency at Low Light, discuss how to efficiently harvest energy with low power solar panels. Both of these articles mention a concept known as maximum power, which in the context of solar panels is the ability to extract as much power as possible from the solar panel without collapsing the panel voltage. When discussing solar panels and power, terms such as Maximum Power Point Tracking (MPPT) and Maximum Power Point Control (MPPC) are often used. Let’s look into the definition and meaning of these terms in more detail.

As can be seen in Figure 1, the output current of a solar panel varies nonlinearly with the panel voltage. Under short-circuit conditions the output power is zero since the output voltage is zero. Under open-circuit conditions the output power is zero since the output current is zero. Most solar panel manufacturers will specify the panel voltage at maximum power (VMP). This voltage is typically around 70 – 80% of the panel’s open circuit voltage (VOC).In Figure 1 the maximum power is just under 140W with VMP just under 32V and IMP just under 4.5A.

Ideally, any system using a solar panel would operate that panel at its maximum power output. This is particularly true of a solar powered battery charger, where the goal, presumably, is to capture and store as much solar energy as possible in as little time as possible. Put another way, since we cannot predict the availability or intensity of solar power, we need to harness as much energy as possible while energy is available.

There are many different ways to try to operate a solar panel at its maximum power point. One of the simplest is to connect a battery to the solar panel through a diode. This technique is described here in the article Energy Harvesting With Low Power Solar Panels. It relies on matching the maximum power output voltage of the panel to the relatively narrow voltage range of the battery. When available power levels are very low (approximately less than a few tens of milliwatts), this may be the best approach.

The opposite end of the spectrum is an approach that implements a complete Maximum Power Point Tracking (MPPT) algorithm. There are a variety of MPPT algorithms, but most will have some ability to sweep the entire operating range of the solar panel to find where maximum power is produced. The LT8490 and LTC4015 are examples of integrated circuits that perform this function. The advantage of a full MPPT algorithm is that it can differentiate a local power peak from a global power maximum. In multi-cell solar panels, it is possible to have more than one power peak during partial shading conditions (see Figure 2). Typically, a full MPPT algorithm is required to find the true maximum power operating point. It does so by periodically sweeping the entire output range of the solar panel and remembering the operating conditions where maximum power was achieved. When the sweep is complete, the circuitry forces the panel to return to its maximum power point. In between these periodic sweeps, the MPPT algorithm will continuously dither the operating point to ensure that it operates at the peak.

An intermediate approach is something that Linear Technology calls Maximum Power Point Control (MPPC). This technique takes advantage of the fact that the maximum power voltage (VMP) of a solar panel does not, typically, vary much as the amount of incident light changes (see **Solar** Battery Charger Maintains High Efficiency in Low Light for more information). Therefore, a simple circuit can force the panel to operate at a fixed voltage and approximate maximum power operation. A voltage divider is used to measure the panel voltage and if the input voltage falls below the programmed level, the load on the panel is reduced until it can maintain the programmed voltage level. Products with this functionality include the LTC3105, LTC3129, LT3652(HV), LTC4000-1, and LTC4020. Note that the LT3652 and LT3652HV datasheet refer to MPPT rather than MPPC, but this is largely because Linear Technology had not come up with the MPPC terminology when the LT3652 product was released.

A final note about MPPC and the LTC3105 – the LTC3105 is a boost converter that can start up at the exceedingly low voltage of 0.25V. This makes the LTC3105 particularly well suited for boosting the output voltage of a “1S” solar panel (i.e. a solar panel whose output voltage is that of a single photovoltaic cell, even if the panel has many photovoltaic cells in parallel). With a 1S solar panel, there will be only one maximum power point – it is not possible to have multiple power peaks. In this scenario, differentiating between multiple maxima is not necessary.

In summary, many different ways of operating a **solar** panel at its maximum output operating condition exist. The panel can be connected to a battery (through a diode) whose voltage range is close to the maximum power voltage of the panel. A full MPPT algorithm, including periodic global sweeps to find the global maximum and a continuous dither to remain at that maximum (an example is the LT8490), can be used. Other products implement an input voltage regulation technique (MPPC) to operate a solar panel at a fixed operating voltage including the LTC3105, LTC3129, LT3652(HV), LTC4000-1 and the LTC4020. In the coming months, Linear Technology will introduce yet another technique for operating a solar panel at its maximum power point. Stay tuned!

### Author

Trevor Barcelo has over 15 years of experience at Linear Technology as an analog IC design engineer, design manager and product line manager. He began his career at Linear Technology’s headquarters in Milpitas, CA by designing the LTC1733 Lithium-ion battery charger. After moving to the company’s Boston Design Center, he continued designing battery chargers and USB power managers including the LTC4053, LTC4066 and LTC4089. He holds five patents related to power management. He currently defines battery charging, power management and wireless power products while managing a team of design engineers developing those products.

Trevor received an M.S. in Electrical Engineering from Stanford University and a B.A. in Physics from Harvard University.

## Product Categories

**Introduction** To **Electricity** for Solar PV **Systems**

This article will cover some of the key electrical concepts that you will need to understand if you are hoping to design your own solar PV system. The other articles on this website will assume that you have a good grasp of these topics.

## Power

Electric power is the rate at which **electricity** is transferred or used. It is measured in Watts (W).

The unit kilowatt (kW) is often used when talking about power. This is equal to one thousand Watts.

One thousand watts = 1000W = 1kW = One kilowatt

Solar panels are sold as having a specific power rating. You might buy a 250W panel, or a 300W panel for example. However, this is not the amount of power that they will always produce. Instead, it is the maximum power they can be expected to produce under standard test conditions (STC).

Standard Test Conditions (STC) Irradiance: 1000W/m2Cell temperature: 25°CAir mass: 1.5

In reality, the power produced varies depending on how much sunlight is hitting the panels, what the ambient temperature is, and several other variables. On a cool day with high irradiance, the panel may actually produce more than its “maximum” power.

## Energy

Energy is the amount of **electricity** used. It is measured in watt-hours (Wh).

As with power, it is typical to talk in terms of kilowatt-hours (kWh). Where:

One thousand watt-hours = 1000Wh = 1kWh = One kilowatt-hour

Energy can be calculated by multiplying the amount of power produced or used, by the amount of time it is being produced or used.

Energy = Power x Time

For example:Let’s consider you have three devices that each use 100W, and that you typically run these three devices for 10 hours each day. In a typical day, the total energy they will use is:

3 x 100W x 10 ÷ 1000 = 3000Wh per day (or 3kWh per day)

When sizing a grid-tied solar PV system you need to first calculate your yearly energy usage, and then design a system that will produce this amount of energy. (See our article How to Size a Grid-tie Solar PV System for more info).

In off-grid **systems**, as well as considering your usage and solar generation, you also have to correctly size your battery system so that it can store enough energy for when the sun is not shining.

## AC and DC **Electricity**

AC (short for Alternating Current) is what is provided by the grid and supplied to the electrical outlets in our homes. It is used because it is easy to step up to higher voltages which is required to transmit the **electricity** over large distances without high losses.

DC (short for Direct Current) is used in many electrical devices, such as computers and mobile phones. Since household sockets provide AC, most electronic devices need an AC to DC converter to work.

Solar Panels and batteries produce DC **electricity**. That is why in a normal household, we need to install inverters in a solar PV system to convert the DC into AC. It can then be connected to the existing AC board of the house so it can either be used by the household or exported into the **electricity** grid.

Some off-grid solar PV **systems** can be set up to only supply and use DC. This gives the advantage of not requiring an inverter, saving costs and electrical losses. It is possible to purchase many different appliances that run off DC power, such as lights, computers, fridges and freezers, fans, pumps, and mobile phone chargers.

## Voltage

Voltage is the difference in electrical potential between two points. It is measured in Volts (V) and its symbol in electrical equations and datasheets is V (or sometimes U, depending on the country).

It is the amount of potential energy available per unit charge, to move charges through a conductor. A common analogy used to help understand this concept is to think of an electric wire like water in a hose. Voltage can be thought of as the pressure of the water.

### Solar Panel Voltage

The voltage of a solar panel is not fixed, and will vary depending on the intensity of the sunlight hitting the panel. It is also heavily affected by temperature. As the temperature of the cells in a panel increase, the voltage decreases. This also causes the power output of the module to decrease. The amount that the voltage changes with each degree change in temperature is called temperature coefficient, and can be found on the solar panel datasheet.

A solar panel datasheet will give several different voltage values. The two main ones are:

Voc (at STC) – **Solar** Panel open-circuit voltage at STC. This is the voltage the solar panel can be expected to show across its terminals when it is not connected to any other device, under standard test conditions (STC). This value is used in string length calculations.

Vmpp (at STC). Solar Panel voltage at the maximum power point. The maximum voltage the panel will produce at STC when connected to an inverter with maximum power point tracking (MPPT).

### Solar Array Voltage

When solar panels are connected in series into what are called strings, their voltages are added together. When they are connected in parallel, the voltage stays the same.

The total voltage of a string must not go over the maximum voltage allowed at the input of the inverter or charge controller being used. The solar panels themselves also have a maximum system voltage that must not be exceeded. Typically the maximum voltage of the system is either 600V or 1000V (or 1500V in utility-scale **systems**). Typically residential **systems** will be 600V and in the U.S. the NEC sets this as the legal limit for dwellings with 1-2 families.

See our article on calculating solar PV string size for further information.

Note that 1000V solar panels can still be used in a 600V system. This is the maximum voltage they are designed to handle, so the 600V system will stay well below their maximum.

## Current

Current is the rate of flow of electrical charge. It is measured in amperes (A) or amps for short, and its symbol in electrical equations and datasheets is ‘I’.

When solar panels are connected to an inverter or charge controller, and are exposed to sunlight, current will flow. The higher the irradiance hitting the module, the higher the current it will produce.

When solar panels are connected in series the current does not change as more panels are added. When they are connected in parallel however, they add together. This is the opposite to voltage.

In solar (and other electrical circuits) current is important for sizing the cables and protection equipment (fuses and circuit breakers). As **electricity** is fed through cables they heat up. If more current is fed through a cable than it is designed for, perhaps due to a fault, it can get too hot and be damaged or start a fire. To prevent this, fuses or circuit breakers are used which break the circuit before the current can go above the limits of the cable.

Two important solar panel currents to be aware of are Isc and Impp.

Isc (at STC) – Short circuit current at STC. This is the amount of current that can be expected to flow when the positive and negative leads of the panel are connected together under standard test conditions. It is the maximum current that the panel can be expected to produce under STC.

Impp (at STC) – The maximum current a solar panel will produce at STC when connected to an inverter with maximum power point tracking (MPPT).

Are there any other basic solar concepts you would like to be covered here? Please let us know in the Комментарии и мнения владельцев below.

## Photovoltaic array reconfiguration under partial shading conditions for maximum power extraction via knight’s tour technique

This paper introduces a novel reconfiguration technique, called Knight’s tour to extract maximum power from photovoltaic (PV) arrays in partial shading conditions. The Knight’s tour reconfigures the PV arrays based on the Knight’s movements on the chessboard. The proposed procedure achieves the maximum power values by spreading partial shadows in all rows. Knight’s tour can be applied to a variety of PV arrays in different dimensions and sizes. Accordingly, the Knight’s tour procedure is applied to four cases in square and rectangular shapes with different dimensions and various shading conditions in each case. To make a direct comparison and present the effectiveness of the suggested procedure, the total-cross-tied connection model and conventional methods such as SuDoKu, optimal SuDoKu, improved SuDoKu, and Skyscraper puzzle are also implemented to the introduced cases. The results of the maximum power point tracking in each case are evaluated by indicators such as global maximum power point (GMPP), fill factor, mismatch loss, and efficiency. Finally, evaluations emphasize the ability and effectiveness of the Knight’s tour solution compared to other methods by achieving the GMPP values such as \(74.7\,\). \(66.6\,\). \(46.8\,\). and \(109.8\,\) for cases 1 to 4, respectively. The Knight’s tour method can be utilized as an efficient tool for the PV arrays in real-world **systems** that suffer from partial shading.

**Introduction**

The increasing energy users and the widespread expansion of renewable energy have significantly expanded the distribution of power **systems** around the world (García Márquez et al. 2018). On the other hand, rising energy consumption has led to issues such as global warming, climate change, and environmental problems (Mostafaeipour et al. 2019). In recent years, tackling these problems has posed many challenges around the world and increased the willingness to use renewable energy resources. Solar radiation is considered to be the main source of energy for the earth among renewable energy sources (Peinado Gonzalo et al. 2019; Sadeghian et al. 2021). Therefore, the direct conversion of solar radiation into electrical energy is one of the most principle technologies in the world (Owusu and Asumadu-Sarkodie 2016; Sadeghian et al. 2020). Achieving this important goal and using solar energy as electrical energy is created by a photoelectric effect technology in the photovoltaic (PV) arrays (Shahsavari and Akbari 2018). This technology has some advantages such as easy transportation and installation of solar panels, which are led to use in many places to produce electrical energy. The output power of PV can be reduced by factors such as the shade of trees, buildings, clouds, etc. (Peinado Gonzalo et al. 2020). One of the most important factors in reducing the output power is the placement of shadows in a row (Lappalainen and Valkealahti 2020). Modules under partial shading conditions (PSCs) receive less radiation than other modules (Pillai et al. 2018; Huerta Herraiz et al. 2020). Mismatch loss (ML) affects the entire PV system because shaded PV modules limit the output current of an array, which may damage PV cells or modules (Pillai and Rajasekar 2018; Hashemzadeh 2019). Solutions to deal with the aforementioned problems caused by the PSCs in the PV arrays are divided into two categories of passive and active techniques (Tabanjat et al. 2015; Pillai et al. 2018; Yousri et al. 2020). Each of these techniques has details that have been studied in various studies. The description of the passive technique can be referred to as the use of the bypass diodes and different types of inter-connections for PV modules for reducing the partially shaded losses. Figure 1 shows the series–parallel (SP), total-cross-tied (TCT), Honey-Comb (HC), and bridge-linked (BL), some of the most important and widely used PV array inter-connected schemes (Akrami and Pourhossein 2018; Dhanalakshmi and Rajasekar 2018). Recent studies demonstrate the ability and efficiency of the TCT connection model in obtaining maximum power from the PV array compared to the other inter-connection models (Picault et al. 2010).

Active techniques for PSC fall into three categories as (Satpathy and Sharma 2019): utilizing multi-tracker converters; utilizing micro converters, and; reconfiguration of PV arrays.

The multi-track converters technique tracks the maximum power point with the same shading independently for each set of the PV arrays (Dhanalakshmi and Rajasekar 2018; Pillai et al. 2018). This technique, due to the use of a large number of converters, is expensive (Sanseverino et al. 2015). Utilizing the micro converters technique is also an expensive method (Akrami and Pourhossein 2018). Finally, the PV array reconfiguration method configures the modules in the PV array by switches between them (Subramanian and Raman 2021). This method can be mainly employed for the TCT and SP inter-connection models. It is economically viable and has been able to extract high energy efficiency in the PSC of the PV array (Yang et al. 2019). Reconfiguration of the PV array eliminates the effect of mismatch losses under partial shadow conditions of the PV array in extracting maximum power (Dhimish et al. 2017; Sai Krishna and Moger 2019a). Accordingly, a novel PV array reconfiguration-based technique is presented in this study. Thus, the next section of this paper, after stating the related works, introduces the proposed method of the paper and its advantages over other related methods.

The organization of the paper in the following sections is as follows: Sect. 2 introduces the related works. The proposed Knight’s tour method is explained in Sect. 3. Section 4 introduces the performance appraisal indicators used in this paper. The simulation results are presented in Sect. 5. Finally, Sect. 6 concludes the paper.

## Related works

In general, the PV arrays are reconfigured by two categories of static and dynamic techniques (Yousri et al. 2019). In dynamic techniques, the modules are electrically configured inside the PV array to extract the maximum output power under PSCs (Vaidya and Wilson 2013; Yang et al. 2019). While static techniques refer to the physical displacement of modules and follow a fixed connection scheme in which the modules are displaced in the PV array without changing the electrical connections. Static techniques do not require any sensors or switching matrices (Rezk et al. 2019; Rezazadeh et al. 2021).

The various PV array reconfiguration designs that follow static methods are Sudoku (Rani et al. 2013), optimal Sudoku (Potnuru et al. 2015; Horoufiany and Ghandehari 2018), improved Sudoku (Sai Krishna and Moger 2019b), Zig-Zag method (Vijayalekshmy et al. 2016), Latin square method (Pachauri et al. 2018), magic square (Yadav et al. 2017), placement of shadows with distance *d* (Malathy and Ramaprabha 2018), skyscraper puzzle (Nihanth et al. 2019), and shadow puzzle (Yadav et al. 2016).

A reconfiguration solution called the power comparison technique has been proposed to extract maximum power from the PV arrays (Akrami and Pourhossein 2018). The SuDoKu technique has been introduced in reference (Rani et al. 2013) to increase the maximum output power under PSCs in the TCT PV array. In this study, the physical position of the PV modules in the TCT PV array is reconfigured based on the SuDoKu scheme. Then, the optimal SuDoKu and improved SuDoKu methods for distributing partial shadow effects in the TCT PV array have been suggested in references Potnuru et al. (2015) and Sai Krishna and Moger (2019b), respectively. It was concluded that the improved SuDoKu performed better than the SuDoKu and the optimal SuDoKu. Vijayalekshmy et al. (2016) employed decreasing partial shading losses and increasing power generation, being done by a static PV module reconfiguration technique called the Zig-Zag method. A magic square arrangement for the TCT PV array has been presented by Yadav et al. (2017) to extract maximum output power under PSC. The results obtained show the effectiveness of the magic square arrangement in reducing mismatch losses compared to other settings. Malathy and Ramaprabha (2018) employed a static reconfiguration method based on displacement distance d between adjacent panels that configure the modules of a PV array under PSC to extract maximum output power from the PV. Nihanth et al. (2019) utilized a new procedure, namely the skyscraper puzzle, employed for enhancing output power production in the PSC. Yadav et al. (2016) used two shadow distribution models in an asymmetric PV array for the TCT PV array. The compounds proposed in this paper significantly reduce the drop in the mismatch. Srinivasa Rao et al. (2014) employed the distribution of shadow effects on the PV array, performed using an interconnection scheme. The results of comparisons in this study show the superiority of the proposed scheme compared to the SP, TCT, and BL PV settings. Reducing the effects of the PSC on the power generation of TCT PV arrays has been done by Yadav et al. (2020) using a new reconfiguration scheme called odd–even configuration. Reddy and Yammani (2020) utilized a novel magic-square puzzle PV module reconfiguration technique to reduce mismatch losses under PSCs. In Prince Winston et al. (2020), novel PV array topologies are proposed to improve performance under PSCs. The method proposed in this paper is tested on seven types of array configurations by applying eight shading patterns. A static-based reconfiguration approach called the Ken-Ken puzzle has been suggested in (Palpandian et al. 2021) for reconfiguring a \(4\times 4\) TCT PV array under PSCs.

Each of the aforementioned static techniques has some advantages and disadvantages. Therefore, Sudoku has partial shade distribution, and optimal Sudoku and improved Sudoku-based methods have been proposed to overcome Sudoku’s problems. These solutions, despite having advantages such as reducing the ML and effective shadow scattering, suffer from issues such as reduced power output during effective shadow scattering. Meanwhile, the Zig-Zag method is different from other methods and has so far shown its effectiveness only for simple \(3\times 3\) PV arrays. Numerical methods such as puzzle shade and magic square have also been used to disperse the shadow. However, these methods only make the scattering of the shadow in column possible, in which case the reliability of the system is reduced.

As studies on static reconfiguration techniques were evaluated, in some other valuable studies, dynamic techniques have been employed to reconfigure the PV arrays. The bubble sorting algorithm (adaptive bank), despite having some ideal benefits, suffers from problems such as replacement constraints and difficult access to complete shadow scattering solutions (Nguyen and Lehman 2008). The results of the branched and limited algorithm showed great importance in solving the problems of the PV array reconfiguring. However, no convincing evidence has been presented for the implementation of this method for large PV power plants. Based on the radiation equation, optimization intelligence for the PV array reconfiguring is performed by hierarchical sorting based on repetition. In this method, continuous switching and complex calculation reduce the reliability of the installation method (Shams El-Dein et al. 2013). Oriented plant configuration is one of the most architectural interconnection **systems** of the PV array due to its simplicity and low cost. However, this method is associated with the frequent occurrence of short circuit faults in the array due to its unbalanced switching patterns (Velasco-Quesada et al. 2009). The rule-based Rough-Set theory concept has been presented as a reconfiguration system. However, to achieve the production of the switching matrix, it follows an inconsistent decision table, which is one of the main problems of this method (Wang and Hsu 2011). In a valuable study (Srinivasan et al. 2021), increasing the energy conversion under PSCs in a \(4\times 4\) PV array has been accomplished by introducing a new reconfiguration approach called the L-shaped propagated array configuration procedure with a novel dynamic reconfiguration algorithm. The use of clustering-based artificial neural networks (ANNs) has been published as an ideal solution to reconfiguration PV arrays and minimize power losses based on the dynamic structure (Monteiro et al. 2020). Simple structure and high accuracy are the advantages of this method, while the operation of ANNs requires a suitable database for network training (Moradzadeh et al. 2020, 2021). Sometimes prolonged training and network processing are major problems with this method (Monteiro et al. 2020). Another dynamic method of working with different loads is DC voltage using a dynamic PV array (Matam and Barry 2018). The configuration scanning algorithm that scans the array and decides how compatible components can be connected to the fixed part to extract maximum efficiency was introduced by Parlak (2014) for PV arrays reconfiguring. In using the scanning algorithm, it is necessary to calculate the short circuit of each row, which is difficult to get the short circuit current from all rows. The dynamic programming algorithm and Munkres assignment method have been suggested for reconfiguring the modules used in non-homogeneous radiation in the TCT structure to increase the output power in Sanseverino et al. (2015). In some valuable studies, various interesting optimization algorithms for the equality of solar radiation have been proposed. Storey et al. (2013) employed an optimal solution called the best–worst sorting algorithm. A random search algorithm has been used by Zabinsky (2011) for dynamic reconfiguration of the solar panels based on the principle of radiation equality. Despite the high speed of operation of this algorithm, it may provide different results in different runs depending on the inherent random mode of this method (Zabinsky 2011).

The literature review was performed for the static and dynamic techniques of reconfiguring the PV arrays. The literature indicates that the dynamic techniques reconfigures all PV modules electrically by repeating the gaining response to a specific shade, see for example (Pillai et al. 2018; Satpathy and Sharma 2019; Yousri et al. 2020). Although the electrical placement of the PV arrays can provide the highest power efficiency, those techniques are also not cost-effective due to their complexity and the need for complex sensors and circuits. On the other hand, the literature represents the cost-effectiveness and simplicity of static techniques. In addition, these techniques do not require additional peripherals such as sensors and switches. But, it can be seen that the static techniques used to reconfigure the PV modules have not been applied to large dimensions and rectangular PV arrays.

In this paper, a novel reconfiguration technique is introduced and employed to distribute shadows on the PV array and to achieve the global maximum power point (GMPP). Because one of the most important factors in reducing the output power is the PSCs in the same rows, in this paper, the losses corresponding to the PSCs and mismatch are reduced by a static method called the Knight’s tour. The suggested method, by solving the problems related to the methods reviewed in the literature and with high efficiency, causes the distributing of shadows and significantly reduces the output power losses. This method targets shadow distribution in rows to eliminate problems such as reduced system reliability and achieve a maximum output power of the PV. The proposed technique can be implemented without the need for any sensors and switches and is cost-effective compared to other electrical array reconfiguration methods in terms of economic costs and implementation. The Knight’s tour technique has been able to overcome the limitations such as high connections and complex wiring that classical reconfiguration methods suffer from. Independence of PV dimensions and type of shadow are other advantages of the Knight’s tour procedure. Thus, this method can be applied contrary to many conventional techniques to rectangular PV arrays. The remarkable point is that the Knight’s tour method, unlike other conventional methods, can be applied to the PV system in a short time, such as a few microseconds, which is a very short time against the behavior of PV. In addition, evaluation and comparison of the results showed that the proposed technique could significantly improve the performance of conventional methods such as TCT, SuDoKu, optimal SuDoKu, improved SuDoKu, and skyscraper puzzle. Despite these advantages, the proposed procedure suffers from some limitations. For example, the proposed procedure does not work very well on very small size PV arrays such as \(4\times 4\). which are mainly used in building applications. As with all reconfiguration techniques, the performance of the proposed technique is based on the I–V and P–V characteristic curves. However, various factors such as internal faults, grid-induced harmonics, and the like can have a variety of effects on characteristic curves that reduce the accuracy of the reconfiguration approach. In addition, the need for a powerful processor system and a microcontroller to run the algorithm designed on the PV system is another limitation of the technique presented in this paper.

## Knight’s tour technique

Knight’s tour is a method derived from the chess and mathematic sciences. The Knight’s tour refers to a sequel to a Knight’s movements on a chessboard, in which the Knight crosses exactly one square at a time (Alfred 2017). In expressing this method, we must keep in mind that chess and mathematics are inextricably linked. Thus, many of the characteristics of a successful mathematician or chess player such as strong pattern recognition, analytical ability, intuition, high level of creativity, spatial awareness, etc., can overlap. If the Knight finishes on a square that is one Knight’s move from the beginning square, the tour is closed. Otherwise, it is open (Conrad et al. 1994; Sandifer 2006). The problem with the Knight’s tour, which is credited to the Swiss mathematician Leonhard Euler, is the math problem of finding a Knight’s tour. In the late 1770s, Euler was able to find the first suitable solutions to this problem, and hence, it is even referred to as the Euler problem of chess and Knights (Sandifer 2006).

In this case, the route starts in the field adjacent to the starting point and is considered an open tour because the Knight cannot return directly to his starting point. Knowing the proper pattern of each of the chess pieces can play an important role in solving the problem of rearranging solar panels. Thus, the chessboard n × n is considered as a PV array with n rows and n columns. Solving the puzzle related to the Knight’s tour movement problem is one of the simple, convenient, and efficient solutions for statically reconfiguring solar panels. The goal of the puzzle is to find a sequence of moves that allows the Knight to visit each square on the chessboard exactly once.

The legal movement of a Knight is to move from a square vertically or a square horizontally and then two perpendiculars to it. Thus, the Knight can move (± 1, ± 2) or (± 2, ± 1) in the coordinates of the chessboard (Erde et al. 2012). Figures 2 and 3 show examples of closed and open Knight’s tours that have legal movement, respectively. It should be noted that our goal in this paper is not just to use closed or open tours. We can freely use any that brings us closer to the goal. Therefore, the movement of Knight’s tours with coordinates (i ± 2, j ± 3) and (i ± 3, j ± 2) on the chessboard is the goal (Singhun et al. 2019).

In this paper, considering a 10 × 10 PV array, five orientation modes of Knight movement are introduced and applied to it as follows:

- (a) The Knight moves two squares down and then three squares to the left.
- (b) The Knight moves three squares upwards and then two squares to the left.
- (c) The Knight moves two squares down and then three squares to the right.
- (d) The Knight moves three squares down and then two squares to the left.
- (e) The Knight moves three squares upwards and then two squares to the right.

The movement and placement of the Knight in each dimension are done based on the five movements mentioned above. The order of these movements will be different for a panel in different dimensions. In the continuation of this section, their movement in different dimensions and in the form of two movement patterns is mentioned.

In pattern 1, the first, second, third, fourth, fifth, sixth, seventh, eighth, and ninth movements are in the orientations a, a, b, c, d, c, e, c, and a, respectively. This algorithm can be repeated for n times.

In pattern 2, the first, second, third, fourth, fifth, sixth, seventh, eighth, and ninth movements are in the orientations d, a, b, a, d, c, e, c, and e, respectively. This algorithm can also be repeated for n times.

In general, the following points should be considered to move the Knight:

- For each loop of the Knight movement, the starting point is in row 1.
- Number 1 in each loop is \(10k-9;k\in \left\\).
- In each movement, the start and stop points cannot be in the same row.

After introducing and getting acquainted with the rules and how to move the Knight, it moves according to the first pattern. After determining the position of the starting point (number one), as shown in Fig. 4a, the position of the remaining numbers is determined based on the first pattern as follows:

- Second number: third row, sixth column (3, 6).
- Third number: fifth row, the third column (5, 3).
- Fourth number: second row, the first column (2, 1).
- Fifth number: fourth row, the fourth column (4, 4).
- Sixth number: seventh row, the second column (7, 2).
- Seventh number: ninth row, the fifth column (9, 5).
- Number eight: sixth row, the fifth column (6, 5).
- Ninth number: eighth row, the tenth column (8, 10).
- Tenth number: tenth row, the seventh column (10, 7).

After placing the numbers in the first loop and determining the direction of movement of the first loop, in the second loop, the numbers are placed, such as the first loop, according to the pattern of the first type. Thus, the first number in the first row is placed in front of the first number of the first loop (Fig. 4b). After placing the numbers in the first and second loops and determining the direction of movement of these loops, in the third loop, the numbers are placed, such as the second loop, according to the pattern of the first type. As shown in Fig. 4c, the first number in the first row is placed in front of the first number of the second loop. Thus, after placing the numbers in the first, second, and third loops in the same way as Fig. 4d–j show, the first numbers in the fourth to tenth loops are similar to the second and third loops in the first-row one by one are placed after the previous number, and the path of the Knight’s movement continues according to the pattern of the first type.

Knight’s movement was introduced according to the first pattern and tested on a 10 × 10 board. For the second pattern, the Knight’s moves according to the introduced positions. In the proposed method, it is not important whether the Knight’s tour is open or closed, but it is important to choose the right positions for the Knight’s tour to move in the correct places. Also, the purpose of using a Knight’s tour is to prevent the PV panels of one type from being in a row as it reduces the output power. Using the explained positions to move the Knight, this method can be used for 6 × 6 and 9 × 9 dimensions and dimensions in any size.

As shown in Fig. 5, the motion layout of the proposed method on a 6 × 6 PV array can be as follows:

Number 1 can be placed in the first row in each of the columns, and this is the beginning of the first loop, and this loop is repeated 5 times. The next numbers are according to the presented pattern and are in order of d, a, b, a, and b, respectively. As Fig. 6 shows for the 9 × 9 board, the number one in the first row can be placed in each of the columns, and the first loop is started. This loop is repeated eight times and the next numbers are placed according to the pattern presented in the form of e, d, a, b, b, a, d, and d, respectively.

Figure 7 shows the flowchart of the shaded modules reconfiguration in a PV array based on the proposed Knight’s tour technique. The proposed method was introduced in this section and implemented on a variety of boards in different dimensions. In the continuation of the paper, by introducing different types of PV array modeling, the suggested method will be implemented on PV arrays in different dimensions and its results will be presented.

## Performance assessment metrics

Performance appraisal of any method can be considered the most important part of the work. Thus, the efficiency and effectiveness of each method and the comparison of various methods are obtained by evaluating the results of their performance. In this paper, after applying the introduced methods on different types of PV arrays in order to reconfigure and distribute the shadow, the performance of each method is evaluated with various indicators. The GMPP, fill factor (FF), ML, and efficiency are introduced and utilized as performance evaluation indicators in this paper. Each of the aforementioned indicators is defined and calculated as follows:

GMPP is tracked and obtained by calculating the generated current in each row of the PV array.

FF is one of the defining indicators in the overall behavior of a solar cell and measures the area of a PV array module. The FF depends on the maximum power point ( \(_\) ), open-circuit voltage ( \(_\) ), and short circuit current ( \(_\) ). It can be calculated as follows:

ML is the difference between the maximum power under uniform radiation ( \(_\) ) and the GMPP under PSC ( \(_\) ). The ML can be calculated as:

Efficiency \((\eta )\) is the ratio between the maximum power point \((P_)\) and the solar energy input ( \(_\) ). The efficiency can be expressed as the following equation:

## Simulation results

The efficiency of the reconfiguration methods is obtained by influencing different models of the PV arrays. For this purpose, in this paper, the Knight’s tour method is applied for reconfiguring 4 different cases of the PV arrays. PV arrays have been considered in square and rectangular shapes and with different dimensions to better compare and express the efficiency of the proposed method. In addition, the shadows intended for all cases are in various dimensions. In each shading condition, the GMPP for the TCT, SuDoKu, optimal SuDoKu, improved SuDoKu, Skyscraper Puzzle, and Knight’s tour methods are tracked in the Simulink environment of MATLAB 2018b. The cases studied in this paper are described as follows:

Case 1: A \(9\times 9\) TCT PV array containing \(4\times 4\) shading.

Case 2: A \(9\times 9\) TCT PV array containing \(4\times 5\) shading.

Case 3: A \(8\times 7\) TCT PV array containing \(3\times 5\) shading.

Case 4: A \(8\times 14\) TCT PV array containing \(3\times 5\) shading.

It should be noted that the standard test condition specifications of the panels used in all four cases are presented in Table 1. So, under these conditions, only the size of the PV panels and the shadow conditions in each case have changed.

## Solar Panel Maximum Voltage Calculator

Just so you know, this page contains affiliate links. If you make a purchase after clicking on one, at no extra cost to you I may earn a small commission.

Use our calculator to easily find the maximum open circuit voltage of your solar array.

Note: Based on your inputs, this charge controller has a suitable maximum PV voltage for your solar array. However, it may not be the right option for your setup based on other factors such as current rating and battery bank voltage, so check that it meets all your other requirements before going with this option.

### Calculator Assumptions

- All the solar panels you input into the calculator are wired together in a single series string. If you have multiple series strings wired in parallel, I recommend using the calculator to find the max voltage for each series string. Then use the lowest max voltage as your array’s max open circuit voltage. This is because, when wiring different series strings in parallel, the voltage of the resulting array is equal to the voltage of the lowest-rated series string.
- If you don’t enter a temperature coefficient of Voc for a panel, the calculator assumes that all the panels with those specs are monocrystalline and/or polycrystalline silicon solar panels, the predominant types of solar panels on the market today.

## How to Use This Calculator

Find the technical specifications label on the back of your solar panel. For example, this is the label on the back of my Renogy 100W 12V Solar Panel.

Note: If your panel doesn’t have a label, you can usually find its technical specs in its product manual or online on its product page.

Enter the open circuit voltage (Voc). My panel’s was 22.3V.

Enter how many of this solar panel you’re wiring in series. For this example, let’s say that I have 4 of these Renogy 100W 12V Solar Panels. They’re identical panels and I’m wiring them all 4 of them in series. In this case, I’d enter 4 in the Quantity field.

Optional: Enter the panel’s temperature coefficient of Voc and select the correct unit (%/°C or mV/°C). My panel’s was.0.28%/°C. You can leave this field blank, in which case the calculator uses the appropriate voltage correction factor based on your lowest expected temperature.

If you’re wiring different solar panels together in series, click Add a Panel and repeat the above steps to add that panel’s specs and quantity. At any point you can click Remove a Panel to remove the last panel.

Enter the lowest temperature you expect your solar array to experience in daylight and select the correct unit (°F or °C). Often, people will use the lowest recorded temperature at their location. For example, I live in Atlanta and did a quick Google search to find out that the lowest recorded temperature here was.9°F (-22.8°C).

Note: If your solar panels are mounted on a vehicle, consider the various locations you plan on visiting in your vehicle when entering your lowest expected temperature.

Click Calculate Max Voltage to get your results. For the example I gave of the 4 Renogy panels, I got a maximum solar array voltage of 101.1V. When designing my solar system, I need to pick a charge controller whose max PV voltage rating is greater than this number.

## Ways to Calculate Maximum Solar Panel Voltage

Here are a couple more ways to find your max solar panel voltage besides using our calculator. Use one of these methods if you’d like to understand the math underlying the calculations.

Note: If you’d also like to calculate the power output of your solar array, check out our solar panel series and parallel calculator.

### Use Correction Factors

The National Electrical Code (NEC) provides a table of voltage correction factors for solar panels based on ambient temperature. The correction factors make it easy to calculate your maximum solar system voltage yourself.

FactorAmbient Temperature (°F)Ambient Temperature (°C)1.02 | 76 to 68 | 24 to 20 |

1.04 | 67 to 59 | 19 to 15 |

1.06 | 58 to 50 | 14 to 10 |

1.08 | 49 to 41 | 9 to 5 |

1.10 | 40 to 32 | 4 to 0 |

1.12 | 31 to 23 | -1 to.5 |

1.14 | 22 to 14 | -6 to.10 |

1.16 | 13 to 5 | -11 to.15 |

1.18 | 4 to.4 | -16 to.20 |

1.20 | -5 to.13 | -21 to.25 |

1.21 | -14 to.22 | -26 to.30 |

1.23 | -23 to.31 | -31 to.35 |

1.25 | -32 to.40 | -36 to.40 |

Note: The above table has been adapted from Table 690.7(A) from the 2023 edition of the NEC. It applies to monocrystalline and polycrystalline silicon panels, the predominant types of solar panels on the market today.

For this method, you’ll need the table along with the following numbers:

- Open circuit voltage (Voc) of each solar panel
- Number of each type of solar panel
- Lowest expected temperature

### Instructions

Find the appropriate correction factor from the above table using your lowest expected temperature.

Calculate the max open circuit voltage of each solar panel by multiplying its open circuit voltage by your correction factor.

If your panels are identical:

Max solar panel Voc = Solar panel Voc × Correction factor

If your panels are different:

Max solar panel Voc #1 = Solar panel Voc #1 × Correction factor Max **solar** panel Voc #2 = Solar panel Voc #2 × Correction factor Max solar panel Voc #3 = Solar panel Voc #3 × Correction factor. etc

Sum the max open circuit voltages of all your solar panels wired in series.

If your panels are identical:

Max solar array Voc = Max solar panel Voc × Number of panels

If your panels are different:

Max solar array Voc = Max solar panel Voc #1 Max solar panel Voc #2 Max solar panel Voc #3

Pretty easy! For once, the NEC makes life a little easier.

### Example #1: Identical Solar Panels

Let’s say these are the specs for 2 identical solar panels you’re wiring in series:

- Solar panel Voc: 19.83V
- Number of
**solar**panels wired in series: 2 - Lowest expected temperature:.10°F (-23°C)

Here’s how you’d find your max solar array voltage:

Find the appropriate correction factor using the above table. In this example, based on my lowest expected temperature of.10°F (-23°C), my correction factor is 1.2.

Multiply solar panel Voc by your correction factor.

Max solar panel Voc = 19.83V × 1.2 = 23.796

Multiply the max solar panel Voc by the number of panels wired in series.

Max solar array Voc = 23.796V × 2 = 47.592V ≈ 47.6V

In this example, the max open circuit voltage of your solar array is 47.6V.

### Example #2: Different Solar Panels

Let’s say instead that your 2 solar panels are different. They have the following open circuit voltages:

- Solar panel Voc #1: 22.6V
- Solar panel Voc #2: 21.4V
- Number of panels wired in series: 2
- Lowest expected temperature:.25°F (-32°C)

Here’s how you’d find your max solar array voltage:

Find the appropriate correction factor using the above table. In this example, based on my lowest expected temperature of.25°F (-32°C), my correction factor is 1.23.

Multiply each panel’s Voc by your correction factor.

Max solar panel Voc #1 = 22.6V × 1.23 = 27.798V Max solar panel Voc #2 = 21.4V × 1.23 = 26.322V

Sum the panels’ max open circuit voltages together.

Max solar array Voc = 27.798V 26.322V = 54.12V ≈ 54.1V

In this example, the max open circuit voltage of your solar array is 54.1V.

### Use Temperature Coefficient of Voc

For this method, you’ll need the following numbers:

- Voc of each solar panel
- Temperature coefficient of Voc of each solar panel
- Number of solar panels wired in series
- Lowest expected temperature (°C)

Note: I’ll just cover how to use this method if your temperature coefficient’s unit is %/°C, which, in my experience, is much more common than mV/°C.

### Instructions

Calculate the maximum temperature differential by subtracting 25°C from your lowest expected temperature. We use 25°C because that is the industry-standard temperature at which solar panels are rated. If using Fahrenheit, I recommend converting your lowest expected temperature to Celsius. It makes the calculations easier.

Max temp differential = Lowest expected temperature. 25°C

Calculate the maximum voltage increase percentage for each solar panel by multiplying the maximum temperature differential by the panel’s temperature coefficient of Voc. Once again, this is assuming your solar panel’s temp coefficient is given in %/°C.

If your panels are identical:

Max voltage increase percentage = Max temp differential × Temp coefficient of Voc

If your panels are different:

Max voltage increase percentage #1 = Max temp differential × Temp coefficient of Voc #1 Max voltage increase percentage #2 = Max temp differential × Temp coefficient of Voc #2 Max voltage increase percentage #3 = Max temp differential × Temp coefficient of Voc #3. etc

Calculate the maximum voltage increase of each panel by multiplying its maximum voltage increase percentage by its open circuit voltage.

If your panels are identical:

Max voltage increase = Solar panel Voc × Max voltage increase percentage

If your panels are different:

Max voltage increase #1 = Solar panel Voc #1 × Max voltage increase percentage #1 Max voltage increase #2 = Solar panel Voc #2 × Max voltage increase percentage #2 Max voltage increase #3 = Solar panel Voc #3 × Max voltage increase percentage #3. etc

Calculate the maximum open circuit voltage of each panel by summing its open circuit voltage and maximum voltage increase.

If your panels are all identical:

Max solar panel Voc = Solar panel Voc Max voltage increase

If your panels are different:

Max solar panel Voc #1 = **Solar** panel Voc #1 Max voltage increase #1 Max **solar** panel Voc #2 = **Solar** panel Voc #2 Max voltage increase #2 Max solar panel Voc #3 = Solar panel Voc #3 Max voltage increase #3. etc

Sum the max open circuit voltages of all your solar panels wired in series.

If your panels are all identical:

Max solar array Voc = Max solar panel Voc × Number of panels in series

If your panels are different:

Max solar array Voc = Max solar panel Voc #1 Max solar panel Voc #2 Max solar panel Voc #3

### Example #1: Identical Solar Panels

Let’s run through an example using the following numbers:

- Solar panel Voc: 20.2V for all panels
- Number of solar panels wired in series: 3
- Lowest expected temperature:.15°C (5°F)
- Temperature coefficient of Voc:.0.3%/°C for all panels

Subtract 25°C from your lowest expected temperature.

Max temp differential =.15°C. 25°C =.40°C

Multiply the maximum temperature differential by the temperature coefficient of Voc.

Max voltage increase percentage =.0.3%/°C ×.40°C = 12%

Multiply the solar panel open circuit voltage by the maximum voltage increase percentage.

Max voltage increase = 20.2V × 12% = 2.424V

Add the maximum voltage increase to the solar panel open circuit voltage.

Max solar panel Voc = 20.2V 2.424V = 22.624V

Multiply the maximum solar panel open circuit voltage by the number of panels wired in series.

Max solar array Voc = 22.624V × 3 = 67.872V ≈ 67.9V

In this example, the maximum open circuit voltage of your **solar** array is 67.9V.

### Example #2: Different **Solar** Panels

Let’s say you have 2 different panels with the following specs:

- Solar panel Voc #1: 19.7V
- Solar panel Voc #2: 22.1V
- Number of panels wired in series: 2
- Lowest expected temperature:.20°C (-4°F)
- Temperature coefficient of Voc #1:.0.28%/°C
- Temperature coefficient of Voc #2:.0.3%/°C

Here’s how you’d find your max Voc in this scenario:

Subtract 25°C from your lowest expected temperature.

Max temp differential =.20°C. 25°C =.45°C

Multiply the maximum temperature differential by each panels’ temperature coefficient of Voc.

Max voltage increase percentage #1 =.0.28%/°C ×.45°C = 12.6% Max voltage increase percentage #2 =.0.3%/°C ×.45°C = 13.5%

Multiply each panel’s Voc by its maximum voltage increase percentage.

Max voltage increase #1 = 19.7V × 12.6% = 2.4822V Max voltage increase #2 = 22.1V × 13.5% = 2.9835V

Add each panel’s maximum voltage increase to its Voc.

Max solar panel Voc #1 = 19.7V 2.4822V = 22.1822V Max **solar** panel Voc #2 = 22.1V 2.9835V = 25.0835V

Sum the max open circuit voltages of all the solar panels wired in series.

Max solar array Voc = 22.1822V 25.0835V = 47.2657V ≈ 47.3V

In this example, the max voltage of your solar array is 47.3V.

## How to Size a Charge Controller Using Max Solar Panel Voltage

Now that you know your maximum solar array voltage, it’s time to pick a solar charge controller.

When shopping for a charge controller, look for its maximum PV voltage (sometimes called maximum PV open circuit voltage or maximum input voltage).

Make sure your charge controller’s maximum PV voltage is higher than the maximum open circuit voltage of your solar array.

For example, let’s say you calculate your max solar array voltage to be 105V. Then a charge controller with a max PV voltage of 100V is too low. You’ll need to instead get one with a max PV voltage of, say, 150V.

## Common Mistakes When Calculating Max Solar Panel Voltage

Based on my experience.- and lots of reader emails and Комментарии и мнения владельцев.- here are the most common mistakes I see people make when trying to find their solar system’s max open circuit voltage:

- Forgetting to correct for temperature. Solar panel voltage increases as temperature drops. Often, beginners aren’t aware of this fact. (I definitely wasn’t when I first started.) As a result, they just calculate the Voc of their solar array and use that number to size their solar charge controller. That puts them at risk of frying their charge controller on cold days.
- Using maximum power voltage (Vmp or Vmpp) instead of open circuit voltage (Voc). Many panels also list a maximum power voltage (aka optimum operating voltage), denoted Vmp or Vmpp. Some people mistakenly think they should use Vmp rather than Voc in their max voltage calculations. Always use Voc.
- Using rules of thumb without understanding their limits. A couple times, I’ve seen people online give a rule of thumb for calculating max Voc.- such as add 5V to each panel’s Voc or add 20% to the array’s Voc. These can be helpful, but readers often fail to understand that these quick and dirty methods are best suited for certain temperature ranges.

Lastly, it’s important to point out that the max solar array voltage you calculate is based on your lowest expected temperature. If your array ever gets colder than that in daylight, there’s a chance it could exceed this number.