## Comparing Tilt Angles intro page: tilt angle

This page is part of a series on tilt angles. An intro is at tilt angle. The other pages are:
load tilts,
optimal tilt,
tilt deviation, and comparing tilts.

On optimal tilt we compare several different methods for finding the optimal tilt. On this page we present a couple tables comparing the optimal annual tilt angles found by the methods discussed on that page.

The tilt angles I investigated came from the common "tilt-at-latitude for best year-round performance" tilt angle formula, a more sophisticated latitude-based formula (called here the "Macslab formula") and the site-specific (and thus more accurate than the latitude-based formulas) PV Watts calculator.

I also used the PV Watts calculator to assess how much solar energy solar panels with each tilt angle could expect to gather in an average year in the various cities. See optimal tilt for the background on the formulas and PV Watts. This footnote (1) contains a link to the "Macslab Formula" and this footnote (2) contains a link to the PV Watts calculator.

The "Macslab Formula" comes with many stipulations (solar panels at sea level, between latitudes 25° and 50°, etc) and I followed all but one in the first table. However, in the other tables, I ignore more stipulations. The second table on this page includes cities far above sea level and on a separate page I also have tables with cities that are much more extreme latitudes (high and low latitudes).

All the solar panels were pointed due South (there is a refresher explaining why that's a good idea in the Northern hemisphere at the bottom of this page.

##### General Findings

The tables on this page seemed to confirm the claim that - all other things being equal - cloudier places do better with shallower tilts (see optimal tilt). They also made me wonder if - all other things being equal - higher elevations above sea level encouraged steeper tilts.

The data in the tables on high and low latitudes also seemed to be in line with the cloudier=shallower rule.

#### Comparing Formulas - Macslab's Rules

The "Macslab formulas" (by Charles Landau (1)) assume the following conditions):
(1) You can point your solar panels due South in the Northern Hemisphere or due North in the Southern Hemisphere.
(2) You are at sea level.
(3) There are no clouds, trees, haze or etc. blocking the sunlight.
(4) Your latitude is between 25° and 50.°

The following table comes close to meeting all but one of these conditions. The solar panels are all pointed due South, the cities are all at low altitudes and at latitudes between 25° North to 50° North. But, "weather happens" and these cities are not immune to clouds, haze, pollution, etc.

The request that our solar panels should face due South in the Northern Hemisphere is of course not peculiar to the Macslab Formula. (An explanation is at the bottom of the page.

 City (lat°) ML Tilt MT kWh/ m2/ day Cen. PVW Tilt PWT kWh/ m2/ day %MT/ PWT Lat. Tilt kWh/ m2/ day %LT/ PWT elev. (met,ft) Miami (25.8°N) 22.7° 5.26 24.5° 5.26 93% 5.26 105% 2m, 7ft Cairo (30.1°N) 26° 5.68 24.5° 5.68 106% 5.66 123% 74m, 243ft LA (33.9°N) 28.9° 5.63 31.5° 5.63 92% 5.63 108% 32m, 105ft Memphis (35.1°N) 29.8° 5.19 30.5° 5.19 98% 5.18 116% 87m, 285ft Philadelphia (39.9°N) 33.4° 4.58 34.5° 4.58 97% 4.57 116% 9m, 30ft NYC (40.8°N) 34.1° 4.58 34° 4.58 100% 4.56 120% 57m, 187ft Boston (42.2°N) 35.2° 4.62 37° 4.63 105% 4.61 114% 5m, 16ft Portland (45.6°N) 37.8° 4.01 33° 4.02 115% 3.95 138% 12m, 39ft Seattle (47.4°N) 39.1° 3.82 34° 3.83 115% 3.76 139% 122m, 400ft Paris (48.7°N) 40.1° 3.25 31.5° 3.27 127% 3.17 155% 96m, 315ft The cities countries are named and the table abbreviations are explained here: 3.

This table compares the optimal annual
tilt angle according to the Macslab formula (ML Tilt) with the best tilt angle I found using the PV Watts Calculator (Cen. PVW Tilt). It assumes that the tilt angles are fixed permanently in place.

(Macslab link: 1. PV Watts link and method for finding best tilt angle with PV Watts: 2.)

I also included the average daily solar radiation falling on surfaces tilted with each of those two angles (... kWh/m2/day) and "at-latitude" (Lat. ...). All the solar radiation numbers were found using PV Watts (1)

Finally, I compared how much the tilt suggestions varied (%MT/PWT = "macslab tilt/PV Watts tilt expressed as a percentage"; LT = "latitude tilt").

A more complete explanation of the abbreviations as well as the locations of the cities is here: 3.

As you can see, the Macslab tilt was usually less than 10% off of the PV Watts tilt (my "true optimal annual tilt angle") and with two exceptions was closer to the PV Watts tilt than the "at latitude" tilt was (the two formulas did equally well for LA and in Miami tilting "at latitude" was actually a little better).

Also, the solar radiation collected with the Macslab tilt was always very close to or equal to the solar radiation gathered by the PV Watts tilt.

Excepting Miami, unless the city had a lot of diffuse radiation (indicated by having about 4 or less kWh/m2 of solar radiation on the average day) the PV Watts tilt was about 10 to 20% shallower than "at-latitude" and within 7% (usually less) on either side of the Macslab annual tilt angle.

If the city had a lot of diffuse radiation (indicated by having about 4 or less kWh/m2 of solar radiation on the average day), the PV Watts tilt was about 40 to 35% shallower than "at-latitude" and 15 to 30% shallower than Macslab.

This is in keeping with our discussion on optimal tilt - where we explain why places with a lot of diffuse radiation need shallower tilts than the tilts determined by latitude-based tilt angle formulas.

### Tilt Steeper at higher altitudes?

So, what happens if we ignore another of the Macslab assumptions? The table below includes high altitude cities. However, the cities are still at latitudes between 25° North to 50° North and the tilts still all assume your solar panels are pointed due South.

[The city's countries and the table's abbreviations are explained here: 4.]

I have a working-hypothesis that - all other things being equal - higher elevations mean that the optimal annual tilt is steeper but that this can easily be overruled by cloudiness. I'm not sure if this is true.

As we show on air mass, being high above sea level means that less solar radiation is lost to air mass. Also on that page, we explain that the effect of air mass is most felt when the sun is lowest in the sky.

When the sun is as low as 45° above the horizon, you still only get about 5% more insolation at a mile high (1.6 km) than at sea level. When the sun is 30° above the horizon, being a mile high (1.6 km) gains you about 7% more. When the sun is between 15° and 20° above the horizon, the milers (1.6 kilometerers) get about 10% more solar radiation and when the sun is about 10° above the horizon, being so high gives you about 15% more. (see the table on air mass)

At 40° North the sun is about 45° above the horizon at solar noon in early spring and late fall. It is always below 30° for most of the winter and its solar noon (the sun's high point each day is solar noon) low-point is about 26° above the horizon meaning that most of the day in the winter it is quite a bit below 30°.

Putting this together with the fact that a steeper tilt favors winter over summer (see optimal tilt), it seems reasonable that higher altitudes should favor steeper tilt angles (the higher altitude means air mass matters less and air mass is most important in winter; since steeper tilts favor winter over summer, it seems reasonable that tilts should get steeper when the winter insolation has been increased more than the summer insolation has been increased).

 City (lat°) ML Tilt MT kWh/ m2/ day Cen. PVW Tilt PWT kWh/ m2/ day %MT/ PWT Lat. Tilt kWh/ m2/ day %LT/ PWT elev. (met,ft) Kathmandu (27.7°N) 24.2° 5.61 29° 5.62 83% 5.61 96% 1337m, 4386ft New Delhi (28.6°N) 24.8° 6.00 29.5° 6.02 84% 6.02 97% 216m, 709ft El Paso (31.8°N) 27.3° 6.52 31° 6.53 88.1% 6.53 103% 1194m, 3917ft Tuscon (32.1°N) 27.5° 6.58 31.5° 6.59 87% 6.59 102% 779m, 2556ft Atlanta (33.6°N) 28.6° 5.19 30° 5.19 95% 5.18 112% 315m, 1033ft Albuquerque (35°N) 29.7° 6.47 33.5° 6.48 88.7% 6.48 104% 1619m, 5312 ft Kashi (39.5°N) 32.8° 5.09 35° 5.09 93.8% 5.08 129% 1291m, 4236ft Boulder (40°N) 33.5° 5.55 38° 5.56 88% 5.56 105% 1634m, 5361ft Madrid (40.5°N) 33.9° 5.08 33° 5.08 103% 5.04 123% 582m, 1909ft Cleveland (41.4°N) 34.6° 4.19 30° 4.20 115% 4.14 138% 245m, 804ft Buffalo (42.9°N) 35.7° 4.07 30.5° 4.08 117% 4.02 141% 215m, 705ft Missoula (46.9°N) 38.7° 4.37 34.5° 4.38 112% 4.31 136% 972m, 3189ft The cities countries are named and the table abbreviations are explained here: 4.

[The city's countries and the table's abbreviations are explained here: 4.]

I came up with my hypothesis by looking at the data in the table to the left and only later thinking about air mass. However, in trying to explain why the table supports my hypothesis, I ran into some difficulties.

I will just content myself with noting that at least for the very high elevation places I checked, my hypothesis doesn't seem to be disproven.

The very high elevation (over 3,000 ft), clearly sunny places (over 5.5 kWh/m2 of sunlight per day) did indeed have steeper optimal annual tilts than sea level cities at similar latitudes had.

Also, for places at elevations above 3,000 ft (m) with about 5 or less kWh/m2 of sunlight a day (Missoula and Kashi), the PV Watts tilts was around 30 - 35% shallower than the "at-latitude-tilt" and the relationship to the Macslab tilt was divided and seemed to be related to how sunny the city was: the PV Watts tilt in Kashi (radiation about 5 kWh/m2) was about 10% steeper than the Macslab tilt but the PV Watts tilt in Missoula (radiation about 4.4 kWh/m2) was about 12% shallower than Macslab.

Other considerations made me less sure of the hypothesis but I decided that the discussion of my confusions would not be of general interest.

It is, however, pleasant to see that fairly low elevation (around 225m/740ft) Cleveland and Buffalo further confirmed our findings that you need shallower tilts in cloudier places.

###### High and Low Latitudes

To see similar tables but for cities with latitudes outside of the 25° - 50° range, go to high and low latitudes.

#### Point Due South in Northern Hemisphere

At latitudes further to the North of the equator than the Tropic of Cancer (currently about 23.5° North), the sun stays in the Southern part of the sky year-round and is due South at solar noon (middle of the day - the daily high-point for the sun). At latitudes between the equator and the Tropic of Cancer, the sun spends most of the year in the Southern part of the sky (only at the equator is the sun evenly divided between the Northern and Southern parts of the sky).

So, in the Northern Hemisphere, all other things being equal (ie: shade from trees, clouds, etc don't interfere), you gather the most sun if your tilted solar panels are pointed due South (as in "true South", not "magnetic South").

The opposite situation is found in the Southern Hemisphere and so solar energy systems in the Southern Hemisphere generally tilt due North.

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#### Footnotes

1. The Macslab Formula is available online at optsolar.
The Macslab tilt angle recommendations are as follows:
Winter: (.89 * latitude°) + 24°
Summer: (.92 * latitude°) - 24.3°
Spring/Fall: (.98 * latitude°) - 2.3°
Annual: (.76 * latitude°) + 3.1°

The Macslab solar seasons are:
Winter: October 7 to March 5
Spring: March 5 to April 18
Summer: April 18 to August 24
Fall: August 24 to October 7

2. There are two versions of the PV Watts Calculator. A portal to both of them is currently here: About PV Watts. The NREL of the US government is responsible for the calculator. In making the tables and finding optimal tilt angles, I used the first version of the calculator.
To find the optimal tilt angle using PV Watts, I used trial and error to find all the tilt angles that provided the most annual solar radiation and then took the midpoint of those values as the "optimal tilt angle." As an example, in Tuscon, Arizona, no permanently fixed tilt gathered more than 5.68 kWh/m2 in a given year and the tilt angles that gathered this much solar energy ranged from 22° through 27° and so the "optimal annual tilt angle" was taken to be [22° + ((27° - 22°)/2)] = 24.5°.

3. In this footnote we first list the nations and states (or - for countries besides the US - whatever the equivalent to "states" is) that house each of the cities on the table. Then we explain the abbreviations. All of the solar radiation insolation data as well as the PV Watts optimal annual tilt angle (Cen. PVW Tilt) is from the PV Watts Calculator (Version 1) [see footnote #2].
The cities are:
Miami, Florida, USA
Cairo, Cairo Governorate, Egypt
Los Angeles, California, USA (LA on the table)
Memphis, Tennessee, USA
Philadelphia, Pennsylvania, USA
New York City, New York, USA (NYC on the table)
Boston, Massachussetts, USA
Portland, Oregon, USA
Seattle, Washington, USA
Paris, Īle-de-France, France
The abbreviations in the table are explained as follows:
City (lat°) = City (latitude in degrees)
MT kWh/m2/day = how much solar radiation in kilowatt-hours was gathered per square meter per day (in an average year) with the tilt angle chosen using the Macslab annual optimal tilt formula. [(.76 * latitude°) + 3.1° (see footnote #1)]
Cen. PVW Tilt = the "center PV Watts tilt" - it is the optimal annual tilt angle I found using the PV Watts calculator (my method is explained in the second part of footnote #2).
PWT kWh/m2/day = how much solar radiation in kilowatt-hours was gathered per square meter per day (in an average year) with the PV Watts optimal annual tilt (the Cen. PVW Tilt).
%MT/PWT compares the Macslab optimal annual tilt with the PV Watts optimal annual tilt (the Cen. PVW Tilt). It was found this way: %MT/PWT = [((Macslab Tilt Angle)/(PV Watts Tilt Angle)) * 100%]. For example, for Cairo, the Macslab Tilt Angle was 26° and the PV Watts Tilt was 24.5°, so %MT/PWT = [(26°/24.5°) * 100%] = 106%, meaning the Macslab tilt was a little steeper than the PV Watts tilt.
Lat. Tilt kWh/m2/day = how much solar radiation in kilowatt-hours was gathered per square meter per day (in an average year) by tilting "at-latitude" (tilting, for example, at 30.1° in Cairo because Cairo's latitude is 30.1° North).
%LT/PWT compares the "at-latitude" tilt with the PV Watts Tilt. It was found this way: %LT/PWT = [(("At-Latitude" Tilt Angle)/(PV Watts Tilt Angle)) * 100%]. For example, for Cairo (30.1° North), the "at-latitude" tilt is 30.1° and the PV Watts tilt is 24.5°, so %LT/PWT = [(30.1°/24.5°) * 100%] = 123%, meaning the "at-latitude" tilt is considerably steeper than the PV Watts tilt.
elev. (met,ft) = "elevation (meters, feet)" - the elevation (aka: "altitude") above sea level is given, first in meters and then in feet.

4. The abbreviations for this table are all given in footnote #3. In this footnote we list the nations and states (or - for countries besides the US - whatever the equivalent to "states" is) that house each of the cities on the table. All of the solar radiation insolation data is from the PV Watts Calculator (Version 1) [see optimal tilt - particularly footnote #1].
The cities are:
Miami, Florida, USA
New Delhi, Delhi, India
Cairo, Cairo Governorate, Egypt
Tuscon, Arizona, USA
Atlanta, Georgia, USA
Los Angeles, California, USA (LA on the table)
Memphis, Tennessee, USA
Philadelphia, Pennsylvania, USA
New York City, New York, USA (NYC on the table)
Boulder, Colorado, USA
Madrid, Madrid, Spain
Cleveland, Ohio, USA
Boston, Massachussetts, USA
Buffalo, New York, USA
Portland, Oregon, USA
Missoula, Montana, USA
Seattle, Washington, USA
Paris, Īle-de-France, France
City (lat°)

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