Solar Row Spacing, Shading and Backtracking

Q: How is the minimum row spacing of solar panels calculated?

With the table width L, its tilt β and the solar elevation α measured in the plane perpendicular to the rows: p = L·(cos β + sin β / tan α). It is the condition for the top edge of one row not to cast a shadow on the next at that instant. The elevation used is a design decision; the formula only turns that decision into meters.

Q: What is backtracking and when does it act?

It is the strategy of a single-axis tracker that, with the sun low, rotates the row back from the angle that points at the sun so that one row does not shade its neighbor. On flat terrain the backtracking angle depends only on the GCR and the ideal tracking angle. It does not act in the central hours because there is no shadow to avoid.

Q: Can a shadow with a small area cut string power a lot?

Yes. A module is divided into blocks of cells protected by a bypass diode, and a shadow reaching part of one block is enough for that diode to conduct and the block to stop contributing. The loss grows in steps and is not proportional to the shaded area. That is why the design is compared against an electrical shading simulation and not just the area.

Short answer

The minimum row spacing comes from a single triangle: the table width, its tilt and the sun elevation measured in the plane perpendicular to the rows, at the hour the project decides to protect. With a tracker, backtracking applies the same condition at every instant and sets the row rotation from the GCR. What weighs most on energy is not the shaded area but how many bypass-diode cell blocks are reached, because each block that conducts through its diode is lost entirely.

Short answer

Row spacing sets how much power fits on the land and how much shading is accepted. It is calculated with the table width, its tilt and the sun elevation in the plane perpendicular to the rows, and verified with an hourly simulation. On a tracker, backtracking applies the same condition instant by instant. And because a small shadow can knock out whole blocks of cells, the energy loss is almost never proportional to the shaded area.

This guide goes deeper into the geometry and the electrical effect. The concepts of pitch, GCR, bifacial modules and tilt in Colombia are in PV field design: pitch, GCR and shading.

The starting quantities

Quantity	Symbol	How it is defined
Table width	L	Dimension of the table in its plane, in the direction of the row spacing
Pitch	p	Horizontal distance between axes or matching edges of adjacent rows
GCR	L / p	Ratio of module area to the land area they occupy
Tilt	β	Angle of the table relative to horizontal
Solar elevation	α	Angle of the sun above the horizon

The solar declination, which sets the height of the sun at noon, varies through the year between approximately +23.45° and −23.45°. At solar noon, the elevation is 90° minus the difference between the site latitude and the declination. For a site at 6° north latitude, on December 21 (declination close to −23.45°) that elevation is about 60.6°: high, but the sun rises and sets much lower, and it is in those hours that the shading is decided.

Row spacing with a fixed structure

For rows facing the equator, the triangle formed by the tilted table and the sun ray gives the minimum spacing without shading:

p = L · (cos β + sin β / tan α)

Here α is the sun elevation projected in the plane perpendicular to the rows. When the sun is not in that plane, the projected elevation must be used and not the real elevation; otherwise the shadow is under- or overestimated depending on the azimuth. The formula is equivalent to the condition L·cos(β − ω) ≤ p·cos ω, with ω the ray angle from the vertical (ω = 90° − α), which is the same one backtracking uses.

Illustrative numerical example (assumed values, not from a project): table of L = 4.4 m, β = 15° and a design elevation α = 30° in the plane perpendicular to the rows.

p = 4.4 × (cos 15° + sin 15° / tan 30°) = 4.4 × (0.966 + 0.448) = 6.22 m
GCR = 4.4 / 6.22 = 0.71

With a pitch smaller than the one calculated, at that sun elevation the back row is shaded along its lower edge. Exactly how much depends on the real geometry of the field, and that fraction does not equal the same fraction of energy loss, as shown below.

Which elevation to protect is not set by a formula. It depends on the price of land, the energy lost and the project’s criterion, and is resolved by comparing annual production with several pitches.

Trackers and backtracking

On a horizontal single-axis tracker over flat terrain, the ideal tracking angle ω points the row at the sun. With the sun low, at that angle each row shades the next. The no-shading condition is cos(θ − ω) = cos ω / GCR, where θ is the actual row rotation. From it comes the backtracking angle (Anderson and Mikofski, equation 14, as implemented in the pvlib library):

θ = ω − sign(ω) · arccos(|cos ω| / GCR)

Backtracking exists only when cos ω < GCR. In the central hours, when cos ω ≥ GCR, there is no shadow to avoid and the row points directly at the sun.

Illustrative numerical example: GCR = 0.40.

Backtracking starts when cos ω = 0.40, that is, at ω = 66.4°.
With ω = 70°: arccos(cos 70° / 0.40) = arccos(0.855) = 31.2°, and the actual rotation is θ = 70° − 31.2° = 38.8°.

A low GCR makes backtracking start later and last less, but leaves less power per hectare. On terrain with a slope across the rows, the slope-aware equations must be used; ignoring it leaves shadows the controller believes it has avoided. General tracker behavior is covered in Solar trackers: types and operation.

What shading does to a string

A crystalline silicon module is divided into blocks of cells, and each block is protected by a bypass diode. Modules with three diodes have three blocks, and half-cut cell modules have six. When a shadow reaches a block, its diode conducts and the string keeps delivering the current of the other blocks, but without that block.

Two consequences:

The loss is not proportional. The number of affected blocks is obtained by rounding up the product of the shaded fraction and the blocks in the direction of the shadow. A module with three blocks in the direction of the shadow and 12% of its height shaded loses an entire block, a third of its cells, and not 12%.
Module orientation matters. If the blocks are arranged along the direction the shadow advances, only the ones reached are lost. If each block runs parallel to the shadow edge, the shadow reaches all of them at once. That is why half-cut modules are preferred in portrait, according to the example by Martínez-Moreno et al. collected in pvlib.

That is why the electrical effect is evaluated in simulation and not deduced from the area: two designs with the same shaded fraction can lose very different amounts of energy depending on the blocks they reach.

How it is verified in design

Calculate the hourly solar elevation for the site for the design day and for several days of the year with a validated solar position algorithm.
Obtain the shaded fraction of each row per hour, with the real geometry of the plane (tilt, slope, structure heights).
Convert that fraction into shaded blocks according to the module and its orientation, and simulate the electrical loss. The uniform shading model of Deline et al. (2013), implemented in NREL’s SAM, offers an approximation with a typical error of 1% against full I-V curve simulations.
Compare annual energy with several pitches and decide with the cost of land, cable and structure.
Give electrical engineering the string limits: which modules are shaded first and which go together.

Common mistakes

Using the real solar elevation instead of the one projected in the plane perpendicular to the rows.
Taking the shaded area as a measure of the loss, without counting blocks and diodes.
Calculating the pitch on flat terrain and building on sloped terrain.
Enabling backtracking without checking that the controller has the correct GCR and slope.
Mixing in one string modules shaded at different hours.
Accepting a design elevation without comparing it against annual production.

What to ask for or verify

The design elevation or hour assumption, written in the calculation report.
The GCR, pitch and slope used, by zone of the field.
The hourly shading simulation with electrical loss, not just geometric.
The backtracking parameters loaded in the controller, including the GCR.
The module orientation and the number of bypass diode blocks from the data sheet (how to read it).
The grouping into strings against the shading (string sizing).
At commissioning, the I-V curve of the string with a known shadow and thermography during shading hours, which show the modules with the diode conducting.

We help with the design and verification of your plant

Our team does the primary engineering of solar plants, with string grouping according to shading, and the testing and commissioning of solar plants, in Colombia or elsewhere on request. For a first power estimate you can use the free PV sizing tool. Tell us about your project.

References

Anderson, K. and Mikofski, M., Slope-Aware Backtracking for Single-Axis Trackers, NREL/TP-5K00-76626, 2020. DOI: 10.2172/1660126.
Lorenzo, E., Narvarte, L. and Muñoz, J., “Tracking and back-tracking”, Progress in Photovoltaics 19(6), 747–753, 2011. DOI: 10.1002/pip.1085.
Deline, C., Dobos, A., Janzou, S., Meydbray, J. and Donovan, M., “A simplified model of uniform shading in large photovoltaic arrays”, Solar Energy 96, 274–282, 2013. DOI: 10.1016/j.solener.2013.07.008.
Martínez-Moreno, F., Muñoz, J. and Lorenzo, E., “Experimental model to estimate shading losses on PV arrays”, Solar Energy Materials and Solar Cells 94(12), 2298–2303, 2010. DOI: 10.1016/j.solmat.2010.07.029.
pvlib python documentation (tracking.singleaxis, shading.shaded_fraction1d, Martínez model example).

Frequently asked questions

How is the minimum row spacing of solar panels calculated?