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Conference Paper

NREL/CP-5K00-76023

June 2020

Maximizing Yield with Improved Single-Axis 

Backtracking on Cross-Axis Slopes

Preprint

Kevin Anderson

National Renewable Energy Laboratory

Presented at the 47th IEEE Photovoltaic Specialists Conference (PVSC 47) 

June 15 - August 21, 2020
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Conference Paper

NREL/CP-5K00-76023

June 2020

Maximizing Yield with Improved Single-Axis 

Backtracking on Cross-Axis Slopes

Preprint

Kevin Anderson

National Renewable Energy Laboratory

Suggested Citation

Anderson, Kevin. 2020. Maximizing Yield with Improved Single-Axis Backtracking on Cross-Axis 

Slopes: Preprint. Golden, CO: National Renewable Energy Laboratory. NREL/CP-5K00-76023. 

https://www.nrel.gov/docs/fy20osti/76023.pdf. 
NOTICE

This work was authored by Alliance for Sustainable Energy, LLC, the manager and operator of the National 

Renewable Energy Laboratory for the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308. 

Funding provided by the U.S. Department of Energy’s Office of Energy Efficiency and Renewable Energy (EERE) 

under Solar Energy Technologies Office (SETO) Agreement Number 34348. The views expressed in the article do 

not necessarily represent the views of the DOE or the U.S. Government. The U.S. Government retains and the 

publisher, by accepting the article for publication, acknowledges that the U.S. Government retains a nonexclusive, 

paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to 

do so, for U.S. Government purposes.

This report is available at no cost from the National Renewable 

Energy Laboratory (NREL) at www.nrel.gov/publications. 

U.S. Department of Energy (DOE) reports produced after 1991 

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Cover Photos by Dennis Schroeder: (clockwise, left to right) NREL 51934, NREL 45897, NREL 42160, NREL 45891, NREL 48097, 

NREL 46526.

NREL prints on paper that contains recycled content. 
Maximizing Yield with Improved Single-Axis

Backtracking on Cross-Axis Slopes

Kevin Anderson

National Renewable Energy Laboratory

Golden, USA 

kevin.anderson@nrel.gov 

Abstract—Many tracking PV systems avoid row-torow shading by backtracking in morning and afternoon.

Because the common single-axis backtracking implementation assumes the array is in a horizontal plane,

it does not fully prevent row-to-row shading in arrays

installed on cross-axis slopes, e.g. an east-west slope for

a tracker with north-south axis alignment. This paper

presents the simulated e ̇ect of reduced self-shading

on annual production with two modifed backtracking

strategies. Depending on the array geometry and the 

steepness of the cross-axis grade, using a modifed 

backtracking strategy improved annual yield by 3-10%. 

Index Terms—single-axis tracking, backtracking, 

photovoltaic, shading, slope, grade, yield 

I. Introduction

Single-axis trackers are commonly used in large-scale 

photovoltaic installations because of their improved insolation capture compared with fxed-tilt arrays. The basic 

tracking algorithm (called “true-tracking”) minimizes the 

incidence angle between the panel normal and incoming 

beam irradiance from the sun. This approach lays modules 

fat at solar noon and tilts modules at steep angles in early 

morning and late afternoon to face towards the sun when 

it is at low elevation. At steep tilt and low sun elevation, 

inter-row shading creates cell to cell electrical mismatch 

in modules with a conventional cell layout, resulting in 

severe nonlinear power loss. To avoid this power loss, the 

tracking algorithm is modifed to lay modules fatter as the 

sun approaches the horizon to prevent row-to-row shading. 

This modifed behavior in morning and evening is called 

“backtracking” because of the reversed tracking direction 

during those times [1]. The backtracking behavior implemented in most trackers is dependent only on the array 

ground coverage ratio (GCR) and implicitly assumes the 

array is installed on fat terrain. As such, the common 

backtracking implementation fails to prevent self-shading 

in arrays built on cross-slopes where adjacent rows are 

vertically o ̇set from each other. For example, a tracker 

array with north-south axis built on a west-facing slope 

would have to backtrack more than usual in the morning 

to avoid self-shading from the height di ̇erence between 

rows. 

the e ̇ect on energy production. In this paper, the ideal 

cross-slope backtracking method is compared against the 

standard “fat backtracking” method and an “adjustedGCR” backtracking method using a photovoltaic performance model to determine the e ̇ect on annual energy 

production. Note that these three strategies focus on 

presenting the optimal angle to direct irradiance while 

avoiding self-shading in an attempt to maximize PV 

production. However, capture of direct irradiance is an 

imperfect proxy for PV production; in di ̇use-dominated 

conditions, production might be maximized by laying 

trackers fat to improve collection of downwelling di ̇use 

irradiance. Laying trackers fat in overcast conditions has 

been estimated to improve production by up to 2.5% in 

overcast climates [5]. However, the strategies considered 

here have the advantage that they are implemented purely 

in software in the tracker controllers; no additional hardware is needed to measure the di ̇use fraction of available 

irradiance or other optimization criteria. 

Section II describes the three strategies considered here. 

Section III describes the PV power and partial shading loss 

models used in this analysis. Finally, Section IV presents 

an overall production comparison of the three strategies. 

II. Backtracking Strategies

This section describes the three backtracking strategies 

considered in this analysis. The frst is the “standard” 

backtracking routine widely used in deployed tracker systems today as described in [1]. The second uses the same 

equations as the frst but with a di ̇erent GCR from the 

array’s true GCR. The third is the slope-aware backtracking routine described in [4]. 

The qualitative behavior of each method is described 

in terms of irradiance capture and self-shading on the 

“upslope” and “downslope” parts of the day, which refer to 

the halves of the day (morning or afternoon) when the sun 

is on the uphill and downhill side of the array, respectively. 

In each case, the backtracking angle B is calculated as 

the sum of the true-tracking angle T and a backtracking 

correction angle c: 

B = T + c (1) 

Previous work has compared backtracking geometries in Each of the three methods yields a value for c given the 

fat arrays and sloped arrays [2]–[4] but did not investigate required array geometry and true-tracking angle T (which 

1

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is defned identically across each of the three strategies 

because it does not consider array geometry). 

A. Standard Backtracking

As presented in [1], the standard backtracking correction 

angle is defned simply by 

cos T

cos c = − (2) GCR 

At sunrise and sunset when |T | ! 90, |c| ! 90 as 

well, resulting in B ! 0. In the case of a cross-axis slope, 

this backtracking method is conservative on the downslope 

side of the day because it remains fat even though the 

vertical o ̇set between rows would permit a steeper tilt 

without self-shading. It also fails to prevent shading on the 

upslope part of the day because the vertical o ̇set places 

the lower edge of each module below where the standard 

backtracking equation expects it to be. 

B. Adjusted-GCR Backtracking

As mentioned earlier, the adjusted-GCR method uses 

the standard backtracking correction equation but with 

a modifed GCR value. The idea is that because the 

standard backtracking approach fails on a cross-slope and 

the backtracking routine only takes one parameter (GCR), 

changing the GCR value programmed into the tracker controller might avoid some of the shading. This method has 

the advantage that it doesn’t require a frmware update 

to the tracker controllers to implement, only a change 

to the GCR value confgured in the controller interface. 

However, it’s not necessarily clear how to choose the value 

of the GCR adjustment (the method is a trade-o ̇ between 

reduced shading loss and reduced irradiance capture), 

and no matter the chosen adjustment, this method can 

only ever be an imperfect solution because the limiting 

behavior is still to lay fat at sunrise and sunset. Despite 

its limitations, it is a workaround for systems that are 

unable to use the ideal method because of limitations in 

the tracker controller. 

A brute-force sweep can be used to fnd the optimal adjusted GCR for a given system specifcation and location. 

The results of such a sweep are presented in Section IV-A. 

C. Ideal (Slope-Aware) Backtracking

The ideal backtracking method can be derived intuitively by applying a rotation of the cross-axis slope angle 

c around the tracker axis to Equation 2 to accommodate 

the cross-axis slope. Here is the equation derived in [4], 

which includes some extra modifcations to correctly handle tilted tracker axes: 

  | cos(T + c)| c = −sign T arccos (3) GCR cos c

Note that rotating the system plane not only shifts 

T by the rotation angle but also changes the horizontal 

spacing between rows and therefore the GCR as well, 

Fig. 1. Three tracking algorithms for a hypothetical array on an 

East-facing slope. Orange line: standard backtracking (GCR = 0.5); 

green line: adjusted-GCR backtracking (GCR = 0.58); blue line: 

ideal backtracking (GCR = 0.5, c = 3) 

requiring the cos c term in the denominator. Here, as in 

[4], GCR is defned according to the axis spacing in the 

horizontal plane. 

Figure 1 compares a daily rotation curve using standard fat-terrain backtracking, adjusted-GCR backtracking, and ideal backtracking for a system on an east-facing 

slope. By comparing the frst two curves to the ideal curve 

it is possible to see when shading occurs for each strategy. 

In the afternoon after the system starts to backtrack, 

the standard curve is always at a steeper tilt than the 

ideal curve, indicating continuous partial shading. The 

adjusted-GCR curve avoids much of the shading in the 

afternoon, but does eventually cross over the ideal curve 

and begin self-shading near the end of the day. However, 

this comes at the price of signifcantly worse positioning 

in the morning. Neither of the slope-naive methods is able 

to accommodate the broken morning/evening symmetry 

between shading and backtracking angle. However, the 

ideal method tilts more steeply towards the sun in the 

morning (including as soon as the sun comes above the 

horizon) while still avoiding row-to-row shading and backtracks as little as necessary in the afternoon. Note that 

for this example, the adjusted GCR was chosen somewhat 

arbitrarily by visually balancing shading avoidance with 

reduced irradiance capture. 

III. Modeling PV Power Under Partial Shade

Although shading avoidance may have other benefts for 

PV arrays (for instance, preventing cells from entering reverse bias and experiencing accelerated degradation), this 

analysis studies the e ̇ect on annual energy production. 

Using the same weather data for each simulation, the 

annual energy production using standard backtracking, 

adjusted-GCR backtracking, and ideal backtracking are 

simulated and compared. 

The modeling approach in each case is to model the 

expected power of an unshaded system and then apply a 

loss predicted from a partial shading model. The power 

generation for each tracking pattern is simulated using 

2

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the open source python package pvlib-python [6], [7]. 

The weather fle used in this simulation is the year-long 

2012 dataset recorded at the Goodwin Creek, Mississippi 

SURFRAD station [8]. SURFRAD ground station instrumentation is regularly maintained and reports high-quality 

continuous 1-minute meteorological measurements. In this 

case, we use the three irradiance measurements and the 

ambient temperature readings. 

A. Partial Shading Loss

The general task of PV modeling under partial shade 

is fairly complex. In this context we can make several 

simplifying assumptions without signifcantly restricting 

the applicability of the results. First, we assume a standard 

72-cell module with three “down-and-back” cell strings. 

We assume that row-to-row shading a ̇ects each cell string 

equally, so no consideration of bypass diodes is necessary. 

We also assume that row-to-row shading a ̇ects only direct 

irradiance, meaning all cells are exposed to the same 

di ̇use irradiance. Finally, we assume that cell temperature is uniform across the module and modeled without 

consideration of the shading. 

A high-accuracy method of modeling partial shading is 

to model IV curves for each cell separately according to 

the cell’s shading conditions and combine the cell curves 

into a module-level curve. However, this process is very 

computationally intensive. Here we use a simpler model 

that is somewhat less accurate, but dramatically faster to 

compute. By making the approximation that a cell string’s 

production is limited by its most shaded cell, we defne a 

piecewise shading model: 

(

1 − (1 − fd)fsN fs < 1/N Pnorm = (4) fd fs  1/N 

Here Pnorm is the output power of the module as a 

fraction of the power it would produce without shade, 

fd is the di ̇use fraction of in-plane irradiance, fs is the 

shaded fraction of the module, and N is the number of 

cells per column in the module. Assuming a standard 72-

cell module, N = 12. Following the assumptions that 

each string is limited by its most shaded cell and each 

string performs equally, we model the most shaded cell 

and assume the rest of the module performs equally. The 

model is defned piecewise to handle the separate cases 

of when the bottom row of cells is only partially shaded 

(fs < 1/N) and when the bottom row is fully shaded 

(fs  1/N). When the bottom row is only partially 

shaded, the power is decreased by the shaded fraction of 

the cell fsN according to the direct irradiance fraction 

1 − fd. When it is fully shaded, the power is just equal to 

the di ̇use fraction. 

To validate this model, we benchmark it against a celllevel IV model as described earlier. We use the De Soto 

single-diode model presented in [9] for generating the fve 

single-diode equation parameters at operating conditions. 

Fig. 2. Predicted power output of a partially shaded module using

the simple shading model and the full single-diode equation (SDE).

In both cases, the di ̇use fraction is set to 0.3. 

Fig. 3. Predicted power output of a partially shaded module using

the simple shading model and the full single-diode equation (SDE).

Points in the upper right have higher di ̇use fraction (fd) than points 

in the lower left. Shaded fraction (fs) is shown by color. 

We then use Bishop’s method [10] to calculate the actual 

IV curves. Both of these steps are completed using the 

implementations in pvlib-python. 

Figure 2 compares the output of a full single-diode 

model against the simple model as a function of shaded 

fraction. The simple model successfully captures the general behavior of the full single-diode model (initial steep 

drop in output power until one cell is fully shaded, then 

leveling out), although there is some error, especially 

where fs is somewhat larger than 1/N. Figure 3 shows a 

more complete comparison that sweeps across both fs and 

fd. Although the simple model does tend to overpredict 

shading loss (therefore underpredicting output power), the 

agreement is fairly good overall. We choose to use the 

simple shading model for the following analysis because 

the simple model is so much more computationally eÿcient 

than the cell-level IV curve modeling procedure and shows 

acceptably small error. 

3

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B. PV Power Modeling

For each timestamp in the irradiance dataset, the modeling process is as follows: 

1) Calculate the standard, adjusted-GCR, and ideal 

backtracking angles. 

2) Transpose the measured irradiance components into 

plane-of-array (POA) direct and di ̇use components 

(using the isotropic di ̇use sky model) for the three 

backtracking positions using pvlib-python. Irradiance losses from soiling and refection are ignored. 

3) Model cell temperature with the Sandia Array Performance Model [11] using thermal coeÿcients for 

the glass/cell/polymer, open rack confguration (a = 

−3.56, b = −0.075, T = 3). 

4) Calculate the unshaded PV power using the 

PVWatts DC model [12], assuming a temperature 

coeÿcient of 
pdc = −0.43%/C. 

5) Calculate the module shaded fractions [4]. 

6) Using the simple shading loss model described in 

Section III-A, calculate the shaded power. 

Figure 4 compares the simulated PV power for each of 

the three backtracking strategies for an East-facing slope 

on a clear-sky day. In comparison to standard backtracking, adjusted-GCR backtracking avoids the majority of 

the afternoon shading loss (Region III), although it still 

experiences some shading loss (Region IV). Additionally, 

the adjusted-GCR method slightly reduces morning production (Region I). In contrast, ideal backtracking avoids 

all afternoon shading and improves morning irradiance 

capture, though the avoided shading has much greater 

e ̇ect than the improved irradiance capture. 

IV. Results

The above modeling approaches were used for two 

analyses. The frst is to fnd the optimal GCR to use 

with the adjusted-GCR method as a function of cross-axis 

grade and true array GCR. The second compares annual 

production for the three backtracking strategies across a 

range of GCR and cross-axis grades (using the optimal 

adjusted GCR in the adjusted-GCR method). 

A. Adjusted-GCR optimization

To fnd the optimal adjusted GCR for the adjusted-GCR 

method, a brute force sweep across adjusted GCRs was 

performed. Figure 5 shows the results of this sweep applied 

for two cross-axis grades (0.05 and 0.15) for an array with 

a true GCR of 0.4. Interestingly, the annual production 

curve has local maxima on both sides of the true GCR. The 

choice to increase or decrease the adjusted GCR could be 

described as the tradeo ̇ between (1) avoiding the majority 

of upslope shading without totally sacrifcing downslope 

irradiance capture and (2) writing o ̇ the upslope part of 

the day as a total loss and optimizing for production on 

the downslope side. Another interesting result is the the 

global maximum is at an increased GCR for the mild slope 

but a decreased GCR for the steep slope. 

Fig. 4. Simulated PV power for standard (“naive”) backtracking

(blue), adjusted-GCR backtracking (orange), and ideal backtracking

(green) for a clear-sky day (GCR = 0.4, grade = 0.05). In order, the

numbered regions show (I) reduced irradiance capture by the nonideal strategies; (II) identical production during the true-tracking

part of the day; (III) avoided shading using the adjusted-GCR strategy; (IV) failed shading avoidance with the adjusted-GCR strategy. 

4

Fig. 5. Annual production using the adjusted-GCR method as a function of adjusted GCR. The unshaded power is shown for comparison.

The y-axis shows annual production normalized by the production

using the true GCR. Top: grade=0.05. Bottom: grade=0.15. 

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5

Fig. 6. Annual production heatmaps as a function of adjusted GCR

and cross-axis grade. X-axis shows true array GCR, Y-axis shows

adjusted GCR, and color shows annual production, normalized to the

annual production for that true GCR. The optimal adjusted GCR for

each true GCR is plotted as black dots. The solid line shows where

adjusted GCR equals true GCR. From top to bottom: grade equals

0.05, 0.10, 0.15, 0.20. 

Fig. 7. Annual production gain heatmaps as a function of cross-axis

grade and true GCR. Color shows the ratio of annual production between the two compared strategies for each (GCR, grade) pair. Top:

production ratio between adjusted-GCR backtracking and standard

backtracking. Bottom: production ratio between ideal backtracking

and adjusted-GCR backtracking. 

To further investigate this result, a similar sweep across 

adjusted GCRs was performed across the GCR range from 

0.2 to 0.6 and from cross-axis grades from 0.0 to 0.2. The 

optimization results for each of these sweeps are shown in 

Figure 6. As grade increases, the optimal adjusted GCR is 

more likely to be less than the true GCR. Additionally, the 

optimal GCR appears to be o ̇set a constant magnitude 

from the true GCR with the magnitude determined by the 

cross-axis grade. 

B. Annual Production by Strategy

Using the optimal adjusted GCRs calculated above, 

the annual production of the three strategies are compared across GCRs and grades in Figure 7. The adjustedGCR method performs well compared to the standard 

backtracking method, showing signifcant gains for mild 

cross-axis grades. However, the gain at steeper grades 

is much smaller. The ideal method shows a signifcant 

improvement over the adjusted-GCR method even in 

cases where the adjusted-GCR method performs well 

and especially where it performs poorly. For a “realistic” 

case (grade=0.05, GCR=0.4), the adjusted-GCR method 

shows a 4% improvement in annual production, with an 

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additional 3-4% on top (7-8% total) with ideal bactracking. For steeper slopes, the gain increases signifcantly. 

Note that calling this production improvement a “gain” 

can be misleading because these gains are just partially 

recouping a loss. Implementing these strategies in a system 

with a typical no-shading energy model that is otherwise accurate will reduce the system’s underperformance 

relative to the model, not cause it to overperform the 

model. Additionally, even with perfect shading avoidance, 

a tracking system on a cross-slope will likely underperform 

an otherwise identical system on fat terrain because of 

reduced insolation capture. 

V. Conclusions

Modifying the backtracking behavior of tracker arrays 

on cross-axis slopes can prevent signifcant production 

losses associated with row-to-row shading. For the simplifed case of constant cross-axis slope, the optimal backtracking GCR was found across a range of true array 

GCRs and cross-axis grades. Although the optimization 

was performed assuming constant slope over the array, 

the results are likely close to optimal for arrays on rolling 

terrain as well, though a more specifc analysis may be 

worthwhile for systems with nonuniform grade. Simulated 

PV production for a high-quality year-long test irradiance dataset showed an increase in annual production of 

roughly 3% by backtracking according to an optimized 

programmed GCR for a PV array with realistic geometry 

(GCR = 0.4, grade = ±0.05). This production improvement was roughly doubled by using the ideal backtracking 

method that considers both cross-axis slope and GCR. 

At steeper slopes and increased GCRs, the simulated 

production improvement increased dramatically. Because 

these algorithms do not rely on external hardware and can 

be implemented purely in software, they are expected to 

be widely applicable to single-axis tracking systems. 

VI. Acknowledgement

This work was authored by Alliance for Sustainable 

Energy, LLC, the manager and operator of the National 

Renewable Energy Laboratory for the U.S. Department of 

Energy (DOE) under Contract No. DE-AC36-08GO28308. 

Funding provided by the U.S. Department of Energy’s 

Oÿce of Energy Eÿciency and Renewable Energy (EERE) 

under Solar Energy Technologies Oÿce (SETO) Agreement Numbers 34348. The views expressed in the article 

do not necessarily represent the views of the DOE or 

the U.S. Government. The U.S. Government retains and 

the publisher, by accepting the article for publication, 

acknowledges that the U.S. Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish 

or reproduce the published form of this work, or allow 

others to do so, for U.S. Government purposes. 

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