Performance of shingled solar modules under partial shading

Significant progress in the development and commercialization of electrically conductive adhesives has been made. This makes shingling a very attractive approach for solar cell interconnection. In this study, we investigate the shading tolerance of two types of solar modules based on shingle interconnection: first, the already commercialized string approach, and second, the matrix technology where solar cells are intrinsically interconnected in parallel and in series. An experimentally validated LTspice model predicts major advantages for the power output of the matrix layout under partial shading. Diagonal as well as random shading of a 1.6‐m2 solar module is examined. Power gains of up to 73.8 % for diagonal shading and up to 96.5 % for random shading are found for the matrix technology compared to the standard string approach. The key factor is an increased current extraction due to lateral current flows. Especially under minor shading, the matrix technology benefits from an increased fill factor as well. Under diagonal shading, we find the probability of parts of the matrix module being bypassed to be reduced by 40 % in comparison to the string module. In consequence, the overall risk of hotspot occurrence in matrix modules is decreased significantly.


| INTRODUCTION
In recent years, the market for solar modules significantly changed from more or less exclusively ribbon-based interconnection of fullsquare solar cells to a wide variety of cell formats and interconnection technologies which continuously increased power outputs. 1 At the same time, the worldwide energy transition requires utilizing additional surfaces for solar power generation. Activation of already existing artificial surfaces like building envelopes offers a technical potential to install around 900 GW in Germany 2 without further land consumption. Additional potential is identified for instance in the bodies of electric vehicles and in noise barriers. However, operation requirements in urban surroundings differ from those in solar power plants. Especially partial shading becomes an important issue with a huge variety of objects casting shades on solar modules any time of the day and the year. Jahn and Nasse report that in Germany, 41 % of the PV systems on buildings face shading capable of reducing the annual energy yield by over 20 % in peak operation. 3 Further studies confirm that partial shading affects around half of all installed systems. 4 Vegetation often is a source, [5][6][7] as well as objects like poles and antennas. 4,8 Work on partial shading of vehicle-integrated PV in motion emphasizes the interest for technical solutions to such challenges. 9,10 In this work, we aim to show that shingled solar modules offer a solution to partial shading losses. At the same time, they feature a highly aesthetic appearance making them especially interesting for integrated applications. In this work, we investigate the operation of shingle modules manufactured in a state-of-the-art industrial approach using strings of shingled solar cells. Additionally, we compare this with modules using the matrix technology 11 and examine the fundamental reasons for the shading tolerance of both shingling approaches. We present a simple LTspice model capable of predicting power outputs under any kind of shading conditions and validate the approach with experiments on lab-scale solar modules. Extended simulation studies on 1m Á 1:6m solar modules subjected to shading scenarios of installed and random objects complete this work.

| SHADING OF SHINGLED SOLAR CELLS
Shingle solar cells are stripe-like solar cells cut from conventional fullsquare solar cells, usually to 1/5 th or 1/6 th of their original size, for example, by thermal laser separation (TLS). 12,13 The key attribute of this technology is the interconnection by slightly overlapping neighboring solar cells and formation of a cell-to-cell bond by electrically conductive adhesive (ECA). To meet the current-voltage output of conventional solar modules, n serial-interconnected shingle solar cells form a string, from which m are combined in parallel interconnection ( Figure 1A). By shifting the solar cells from row to row by half a cell length, an additional parallel interconnection of all solar cells within each row is achieved ( Figure 1B). Half-cut shingle solar cells at the edges compensate for this lateral shift and form a uniform rectangular matrix of solar cells. Therefore, this approach is called matrix technology.

| Modeling of the solar cell IV characteristics
The two-diode equation is a widely used approach to model the solar cell behavior under illumination. Figure 2 shows the equivalent circuit including both diodes D 0 and D 1 , the current source with its photocurrent density J ph and the resistors R s and R p . Since it is sufficient precise and easier to translate into an equivalent circuit, than the commonly used Bishop model, 14 we chose to extend the characteristics into the negative bias regime by adding a reverse breakdown characteristic to D 02 following Rauschenbach. 15 In contrast to Rauschenbach, in this, work we propose to not eliminate the shunt resistance R p , but instead to assume the reverse breakdown takes F I G U R E 1 Schematic drawing of the two shingling approaches (A) string-type in which n serial-interconnected shingle solar cells form m parallel-interconnected strings. (B) in matrix-type in which each of the n serial-interconnected rows contain m parallel-interconnected solar cells by shifting cells by half their length perpendicular to the stringing direction. Half-cut shingle solar cells at the edges complete a regular rectangle F I G U R E 2 Two-diode equivalent circuit of a solar cell. A reverse breakdown characteristic according to Rauschenbach 15 extends the behavior of D 01 (highlighted by the red dashed box) into the reverse bias regime place in the junction diode D 02 . In this way R p still considers increasing currents in the negative bias regime before the breakdown occurs.
This leads to Equation 1, where J 01 , I 02 , and J Br are the saturation current densities of D 01 , D 02 , and the reverse breakdown of D 01 , respectively. The Boltzmann constant k B , the elementary charge e, the absolute temperature T, the ideality factors n 0 ¼ 1 and n 1 ¼ 2, and the series resistance R s for ohmic losses are incorporated in the exponential function of each diode term. Additionally, R p describes the shunt resistance of the solar cell. In case of the reverse breakdown D 02 , we combine the reverse breakdown factor a Br and n 1 to n Br ¼ n 1 a Br and treat n Br as a fit factor.
We use Equation 1 to fit measured data presented in Section 2.3 and incorporate the IV characteristics into the LTspice model of shingle solar modules.

| The LTspice model
The linear representation of the solar cell characteristics allows us to virtually split the shingle solar cells in two half-sized shingles as highlighted in Figure 3A. This allows the model in LTspice for both topologies to consist of 2-mn solar cells and switching between string and matrix layout is achieved by simply adjusting the surrounding network of resistors. These resistors represent the busbar metallization on the front (FS, gray) and the rear side (RS, red) of the solar cells.
Other than in a string, where currents can only flow from one solar cell in the string to the next, the matrix layout allows currents to travel through the network of busbars perpendicular to the direction of the string. Note that we refer to string also when speaking of the direc- The model includes bypass diodes (BPDs) at flexible positions along the string, described in more detail in the model validation section. BPDs are used to limit reverse bias voltages 18 Table 1.

| Lateral resistance
Besides the extended two-diode representation, the LTspice simulation requires lateral resistances R lat for the current between two adjacent solar cells via the joint. We have three different types of joint shapes to consider. Firstly, the conventional string joint, where front and rear side busbar together with the ECA form a lateral resistor.
Secondly and thirdly, the matrix joints, where either the front side or the rear side busbar, are intermitted at the transition to the neighboring, parallel-interconnected solar cell. We want to emphasize that the absolute values obtained for R lat are dependent on the busbar shape and architecture of the joint and are therefore not universally valid.
Here, we characterize eight samples per group and nine groups per ECA. Their current conduction length corresponds to half the size (78 mm) of the solar cells used in the experiments and simulations.
After stencil printing (100 μm) of three different ECA patterns (four pads of length l ECA ½2; 5;10 mm) the samples are cured at 160 C for 3 min. We compare two commercial available ECAs, one with a high specific resistance ρ of 3:7 Á 10 À3 Ωcm and one with a low specific resistance ρ of 1:9 Á 10 À4 Ωcm, which enclose the range of today's available products.

| Results and discussion
During our experiments, module temperatures vary between 38 C and 45 C with an average of 42:0 AE 1:5 ð Þ C for the matrix sample and between 45 C and 52 C with a mean of 48:2 AE 2:0 ð Þ C for the string module. In order to compensate for dT between simulation and experiment we compare the normalized power outputs P X for each scenario X by P X ¼ P XÀi =P XÀ0 with P XÀ0 being the unshaded reference in each X. We therefore determine the error on the measured power to be as big as max dT X ð Þ, which for both string and matrix interconnection is $ 7 K. With the data sheet temperature coefficient of À0:36%K À1 for the solar cell power this leads to ε P ¼ 2:52%. We underline that this error can only be an estimate since the temperature measurement is located at one position and we expect the module temperature to be inhomogeneous over its surface. Every data point is the mean of three measurements with average relative errors ε P,rel of 0.3% and 0.5 % for string and matrix, respectively. Therefore, the overall expected error is ε P,tot ¼ ε P,rel þ ε P ffi 3%. Throughout the experiments on both modules, we notice slightly increased currents Additionally, the modules feature a transparent back sheet, which allows scattered light from the chamber rear side to enter the module.
The experimental results are displayed in Figure 8. The experimental data match nicely with the simulations with an average devia- The modules considered for the following discussion consist of n rows and m columns of solar cells as shown in Figure In the matrix layout, every complete row i is a serialinterconnected generator of currents, and we therefore need to consider the row of minimal photocurrent.
with the b i j defined as the row i of minimum current generation within column j and therefore also of minimum E We introduce the notation With i ≠ b i j we get Equations 9 and-equally valid-10 If for all j applies b i j ¼ i, thus all minima in j columns are located in the same row i and Equation 6 becomes Have in mind that the proportionality between current and irradiation under the given assumptions is equal regardless of the interconnection type. Therefore, 11 shows that the matrix and string respond equally, when minimal irradiation is found in the same row i. But if only in one column b i j ≠ i, we insert Equation 10 as one summand in Equation 6 which leads to

| Fill factor
Experimental and computational data show another differing behavior of matrix and string shingled solar modules under partial shading.
Resulting from it, higher fill factors are found for the matrix module for some cases. With this in mind, we compare the impact of an identical shading for the string and the matrix interconnection. In Figure 11, the characteristics of four strings as discussed in Figure 10 and interconnected in parallel to form a string layout are shown. Within the first string, the first solar cell is subjected to a 500 Wm À2 illumination or half covered solar cell, respectively, which corresponds precisely to the shading in Figure 10. Now, in Figure 11,   Figure 13B shows the matrix layout and the random shading scenario. A sh is the overall shaded area fraction of the solar module. Since we randomly pick individual half shingle solar cells, it corresponds to an integer number N. Therefore, the area covered by shading contains an error reciprocal to the number of solar cells N used in the module ε A ¼ 1=N. In this specific case,

| Results of the diagonal shading scenario
In Figure 14, we show a map of the power gain P þ , where x-and yaxes outline the variations in α sh and w sh , respectively. P þ is calculated according to Equation 13 and is expressed in the color code. Positive Within this study, we find the power output of the matrix under diagonal shading to exceed the string module in every data-point. This is explained, as proposed before, by the ability of the matrix technology to conduct charge carriers past the shaded area rather than blocking them inside the strings. This is not possible in the string approach and leads to significant losses in I MPP or worse revers biasing parts of the module increasing the risk of hotspots. 20  We observe also minor effects on the values for string interconnection. Those can be explained by losses in lateral current conduction between the two virtual sub cells, which for example occurs when shading affects only one half of a solar cell.
We conclude that the increased current generation enabled by the matrix technology is robust against even highest resistances for lateral current conduction and the effect is unlikely to be compensated by increasing module series resistance and fill factor losses. This offers the possibility for reduction of the silver metallization from the electrical point of view. However, from the production and reliability point of view, the silver busbar is an important feature to ensure robust mechanical interconnection between the solar cells and metallization reduction needs to be pursued with care.
Variations of O sh again affirm that the difference between string and matrix is driven by the elevated current extraction and consequently a decreasing O sh links to a strong reduction of P þ,max . On the other hand, an increasing O sh increases the differences. Since the matrix concept allows us to access photocurrents that would be blocked in their respective strings, it follows that the difference in power output maximizes with the difference in irradiation and minimizes when irradiation inhomogeneity decreases.

| Results of the random shading scenario
In Figure 15 F I G U R E 1 5 Module power output plotted against the shaded area fraction A sh for random shading of solar cells. Data points represent the results for the measured R lat values. The colored area borders the range of expected power outputs for variations of e R lat ¼ 0:2; 5 ½ ÁR lat . The unshaded reference as well as the 100 % shaded case for O sh ¼ 0:8 are plotted as dot-dashed lines are partly found under lower voltages in combination with conductive bypass diodes. Therefore, we find this effect to be most relevant, for example, in case of bird droppings, which on the one hand only cause minor power losses but on the other hand typically stay on the solar module for a long time.

| CONCLUSION
In this study, we investigated the power output under partial shading for shingle modules featuring the standard string and the matrix lay- Since the main advantage of the matrix technology is an increased current extraction, we expect that such PV modules would be also beneficial in power plant applications, where multiple serial interconnected solar modules are controlled by a string inverter. However, we emphasize that such a statement requires more detailed studies and data acquisition, which is left to future work.
We did not address heating due to reverse biasing of individual solar cells. This is beyond the scope of this publication to be addressed in an adequate level of detail. However, work by Kunz et al. 20 and Clement et al. 24 discuss this in detail. They state that also in shingle solar cell modules, critical heating is caused by reverse biasing occurs. However, as discussed in Section 4.1, the matrix module reduces the risk of reverse biasing and hence solar cell heating.
We find the matrix technology particularly interesting for integrated applications such as building and vehicle integration. Huge potentials for solar power generation meet a huge variety of irregular shading conditions, making shading tolerance a very important aspect.
Above this, matrix modules fulfill other requirements like a highly aesthetic appearance without losing power due to, for example, coloring or printing patterns on the front sheet.