Solar Power Math Problems

Math problems are everywhere when solar power is concerned, not only for reducing power losses, but also for safety considerations. Article 690 of the National Electrical Code (NEC) requires deratings and correction factors based on the type of wiring involved and the locations the wires are installed. There's also a default "just because it's solar" calculation with a lot of good reasons related to the NEC's goal of life safety. I installed my system using the 2003 edition, which was the current edition at the time.

The first math problem deals with solar panel output. When planning your solar panel installation, technical details for stuff Pmax, Vpm, Ipm, VOC, and Isc are needed for a safe design. Here are the specs for the solar panels I used:

Solarex (now BP)
Millenia MST-43MV
Pmax 180 watts 43 watts
Vpm 54.0 volts 72.0 volts
Ipm 3.33 amps 0.6 amps
Voc 66.4 volts 98 volts
Isc 3.65 amps 0.8 amps

The NEC requires a 'weather correction factor' to determine the highest possible voltage. Solar panels produce less power the hotter they get; they produce more power the colder they get. The NEC has a table (Table 690.7, Voltage Correction Factors for Crystalline and Multicrystalline Silicon Modules) that gives the correction factor, based on the coldest possible operating temperatures expected. From the 2003 version of the NEC:
[NEC 690.7]

  • -11C thru -20C (13F thru -4F): 1.17
  • -1C thru -10C (31F thru 14F): 1.13

And from the 2008 version.

NEC Table 690-7, 2008 National Electric Code

My area's coldest daytime winter temperature is usually above 20F so my weather correction factor would be 1.13, but I've seen 30 mph winds on bright sunny winter days, so I'll use the higher number to allow for windchill.

Each group of panels is a 'PV Source Circuit'. My panels are wired in parallel (two per circuit); the math runs like this:
[NEC 690.8(A)(1)]

  • the sum of all the Isc numbers in the circuit, multiplied by the weather correction factor equals the 'PV Source Circuit Current'
  • 3.65A + 3.65A = 7.3A times 1.17 = 8.541A

The PV Source Circuit Current is then multiplied "because it's solar" by 1.25. 8.541A times 1.25 = 10.67A. This is the maximum possible current that each circuit could possibly produce. If I selected only a 10 amp fuse for this circuit, there is a strong possibility I'd have to replace fuses pretty often in the winter time, especially on bright sunny days with lots of snow reflecting even more light onto the solar panels.
[NEC 690.8(B)(1)]

Now that I've determined how much current might be produced, I need to select the correct wire size. I'm using type USE-2 cable from the solar panels to the combiner box where the circuit breakers are located. USE-2 cable is UL listed for outdoor use in hot areas (90C) and is also sunlight resistant. The temperature derating of USE-2 in 141-158F is 0.58
[NEC 310.16]

  • Ampacity of USE-2 cable, 10AWG: 40 amps
    40 amps times 0.58 = 23.2 amps
  • Ampacity of USE-2 cable, 12AWG: 30 amps
    30 amps times 0.58 = 17.4 amps
  • Ampacity of USE-2 cable, 14AWG: 25 amps
    25 amps times 0.58 = 14.5 amps

The wire size has to be able to handle 125% of the derated PV Source Circuit Current (10.67A), so 10.67A times 1.25 = 13.3A. Our wire has to be thick enough to handle 13.3 amps, so either of these sizes would meet the electrical code.

Temperature derating for multiple cables. There is an additional factor to be aware of if these wires are running through conduit. Based on the number of current carrying conductors (positive conductors), the wire is derated according to the following: [NEC 310.15(B)(2)(A)]

  • 4-6 conductors: 80%
  • 7-9 conductors: 70%
  • 10-20 conductors: 50%

If these 10 these circuits are running through conduit, then the rating for 10AWG (40A > 23.2A) is reduced yet again. 23.2 times 0.5 = 11.6 amps. I can either run 9 circuits in conduit and run 1 circuit free (allowed with USE-2 cable) and derate the circuits in conduit to 70% (16.24A), or divide the runs, with 5 circuits per conduit and derated to 80% (18.56A).

Something else to consider is resistance. Thinner wires have more resistance than thicker wires which reduces the amount of power available at the end. And, the lower the voltage, the greater the power loss.

  • DC Resistance of 14AWG wire: 2.5 ohms/1000ft
  • DC Resistance of 12AWG wire: 1.6 ohms/1000ft
  • DC Resistance of 10AWG wire: 1.1 ohms/1000ft

I have fairly short wiring runs (less than 50 feet). The following table shows the calculated voltage drop (loss) for a 50' circuit, at various DC voltages, with a 10 amp load. The voltage drops even further on longer runs. At 12 volts, a 500' circuit loses so much, that it's only 2.4 volts at the opposite end!

  12 VDC 24 VDC 48 VDC 96 VDC
2 AWG 11.84 V
  @ 10 amps
23.84 V
  @ 10 amps
47.84 V9
  @ 10 amps
95.84 V
  @ 10 amps
10 AWG 10.9 V 22.9 V 46.9 V 94.9 V
12 AWG 10.4 V 22.4 V 46.4 V 94.4 V

Once the PV Source Circuits are at the circuit breakers, they are combined to form the PV Output Circuits. The PV combiner box can combine 12 PV source circuits into 1 PV output circuit, or split those same 12 PV source circuits into 2 PV output circuits. After factoring in the math (and based on the charge controller limitations), our 10 PV source circuits are combined into 2 PV output circuits:
[NEC 690.8(A)(2)]

  • PV Source Circuit Current times number of circuits times 1.25 (twice) equals PV Output Circuit Current.
  • 7.3A times 10 circuits = 73.0A times 1.25 = 91.25 times 1.25 = 114.06 (we'll round up to 115A).
  • 7.3A times 5 circuits = 43.8A times 1.25 = 54.75 times 1.25 = 57.03A (we'll round up to 60A).

The charge controllers (Outback MX-60) are rated for continuous duty at 60 amps and 125 volts DC. In deciding the system voltages, we had to take this limitation in account. Again, more math:
[NEC 690.7]

  • sum of the maximum voltages (Voc) of panels wired in series, times the weather correction factor.
  • 66.4 + 66.4 = 132.8 volts, times 1.13 = 150 volts, which is way over the 125 volt limit.

If I need higher voltages in the future, I may be able to rewire the panels and mix them with 24V panels. Assuming the 24 volt panels have a maximum voltage of 44.2 volts (like the BP 3160 solar panels): 66.4 + 44.2 = 110.6, times 1.13 = 124.978, which is right at the 125 volt charge controller limit. Of course, I would also have to keep the source circuit currents in mind as well.

From this point (the combiner box) to the DC equipment inside the house, everything is calculated for 60 Amps.

I'm planning on adding more solar panels on the southeast facing portion of the roof, which will start collecting power a couple of hours earlier than the southwest facing panels. The expansion will require another combiner box, probably with 2 more PV Output circuits, so the conduit will be sized accordingly.

THHN/THWN wire is rated for 70&#176C and is suitable for running in conduit. The first set of solar panels is two circuits. There's room on the roof for even more solar panels, which might be an additional 2 circuits in the future. We know that we'll eventually have at least 4 circuits in the conduit, and that the conduit will be warm (but not as warm as the wires at the solar panels). [Table 310.16]

  • THWN wire is derated as: Rating times 0.88 for (96-104&#176F ambient temperature), times 80% (4 conductors in conduit)
  • 3AWG is rated as 100A times 0.88 = 88A times 0.8 = 70.4A
  • 2AWG is rated as 115A times 0.88 = 101.2A times 0.8 = 80.96A

We can use 3 AWG wire, but 2 AWG provides less power loss (and is usually readily available and in stock at most do-it-yourself places).

An equipment ground wire is also required, and its size is based on the size of the largest breaker (60A), BUT if the wiring on the PV Output Circuits has been oversized (like ours), then the equipment ground wire also has to be oversized to the size of the PV Output Circuit wires.
[NEC 690.45], [NEC 250.122]

Eventually, there will be 4 PV Output circuits plus the equipment ground wire running in conduit from the roof. We're using 2" conduit which has room for a total of 12 wires (if they're all 2AWG).

When wires are first installed in conduit, you're allowed 40% fill based on the diameter of all the wires involved. The number of wires you're installing and the 40% fill ratio determines the minimum size of conduit allowed, and just one extra wire could mean having to install larger diameter conduit (which starts getting expensive pretty fast). There's a provision in the NEC that can help save money, although it's not very pretty: If the equipment ground wire is 6 AWG or larger, the ground wire is allowed to be attached to the outside of the conduit. [NEC 250.64]

There are many types of conduit, but not all are approved for use in the outdoors where rain and sun are present. Rigid metal conduit (RMC) and Intermediate metal conduit (IMC) are approved. Liquidtight is approved if it's sunlight resistant. Schedule 40 PVC conduit is also approved if it's rated sunlight resistant, but I've still seen it deform in normal summer temperatures. Electrical metallic tubing (EMT) is not approved for outdoors where exposed to weather, and Schedule 80 PVC conduit is not approved for outdoors where exposed to sunlight.

Where multiple wires are installed in conduit, the cross section of the wires is only allowed to fill up to 40% of the cross section of the conduit. The cross section of #2 AWG THWN wire is 0.1158 square inches. The cross section of nine wires is 1.0422 square inches. The conduit fill tables in Chapter 9 of the NEC specifies that 1.5" RMC allows up to 0.829 square inches, and 2" RMC allows up to 1.363 square inches.

If we were concerned about exceeding conduit fill, there's a provision in the NEC that allows us to run the equipment ground wire attached to the outside of the conduit, IF the equipment ground wire is 6 AWG or larger. But remember, if the equipment ground wire is 6 AWG or smaller, it MUST be green (larger can be marked with green tape, etc.)
[NEC 250.64]