## Calculations

for an Off-Grid PV System

Judy LaPointe's home is on its way to becoming a finished, off-grid home.

The walls are up and the PV system is being assembled for the off-grid home described in Code Corner in HP94. This article presents most of the calculations required to design the photovoltaic (PV) system within the requirements of the National Electrical Code (NEC).

These calculations may not be all that are needed in the total design of every PV system. Local electrical codes may impose other requirements, and building codes may require calculations involving the mechanical installation. The calculations shown here are typical for a stand-alone PV system. But PV design is very system specific, and the calculations will be different for other PV systems.

The PV system detailed in this article will provide electricity for a residence located about 0.5 miles (0.8 km) from the utility grid in rural New Mexico. The PV array consists of twenty, 165 watt PV modules—3,300 watts DC at standard test conditions (STC) of 1,000 watts per square meter of irradiance and a module temperature of 25°C (77°F).

See Code Corner in HP94 for a description of the loads and system. References to the NEC are presented in brackets.

PV Source Circuit Calculations

The PV source circuits consist of the wiring from the modules to the combiner box.

Overcurrent Protection—Step 1. Overcurrent protection is required for each ungrounded conductor. The first overcurrent device is a fuse installed in series with each string of two modules. The fuse size for each of the ten PV source circuits was determined by meeting several requirements. The first requirement is to allow PV output to flow unimpeded to the charge controller. By multiplying the module short circuit current (Isc) of 5.46 amps by two adjustment factors of 1.25, we get a design current of 8.52 amps (5.46 x 1.25 x 1.25 = 8.52).

One of the 1.25 adjustment factors is due to expected and normal module current outputs above the rated value around solar noon. The other 1.25 factor is related to the NEC requirement to keep overcurrent devices and conductors from operating above 80 percent of rating (1 -f 1.25 = 0.80).

Although a 9 amp fuse is the next highest standard value above the design current of 8.52 amps and is available by special order, a 10 amp fuse is more commonly available and will meet all requirements for conductor ampacity and overcurrent protection discussed below. The above calculation determines the basic minimum fuse rating and conductor ampacity required by the NEC [690.8-9].

Module Conductors. The Sharp 165 modules we chose have #14 (2 mm2) pigtail leads and no junction box. The ampacity of a #14 USE-2 conductor in free air is 35 amps at 30°C (86°F) [310.17]. The ampacity temperature correction factor for an estimated maximum 75°C (167°F) module operating temperature is 0.41 [310.17]. See the table on page 97.

The ampacity of conductors and temperature correction factors can be found in the NEC [310.15 and Tables 310.16 (conduit installations), 310.17 (free air installations)]. The correction factor is multiplied by the conductor ampacity at 30°C (86°F) to determine the corrected ampacity at the elevated operating temperature. The temperature-corrected ampacity of the #14 conductor at 75°C is 14.35 amps. (0.41 x 35 = 14.35).

We will splice #10 (5 mm2) USE-2 conductors to the #14 pigtails. Their ampacity in free air is 55 amps at 30°C [310.17]. Some of these conductors will touch the backs of the PV modules and are therefore exposed to 75°C module operating temperatures. The temperature correction factor for an estimated maximum 75°C module operating temperature is 0.41. The temperature-corrected ampacity of the #10 conductor at 75°C is 22.55 amps (0.41 x 55 = 22.55).

Overcurrent Protection—Step 2. Ten, 10 amp fuses protect the module conductors from excess currents from the battery or from parallel strings of modules. The fuse rating is equal to the maximum module series fuse of 10 amps (marked on the back of the module), which protects the internal connections of the module. It cannot be more than this marked value. It is less than the cable ampacity of 14 amps (#14) or 23 amps (#10), and protects both conductor sizes used in the module wiring in this system.

The fuse rating is above the required rating of 8.52 amps needed to carry the current from each module.

These fuses are installed in the DC combiner boxes that combine the outputs of the ten modules in each subarray to the two circuits running to the charge controllers. RV Power Products MPPT PV charge controllers are being used with a 48 volt input and a 24 volt output.

Voltage Drop Calculations. Although voltage drop calculations are not an NEC requirement, the length of your wire runs should be a factor that you consider in system design. In our design, each PV module has a 50 inch (127 cm) length of #14 (2 mm2) conductor connected to a length of #10 (5 mm2) conductor to reach the combiner box. The maximum length (for both the positive and negative conductors) in any of the source circuits totals about 20 feet (6 m).

Just meeting code ampacity requirements may not always yield an efficient system.

Wire resistance is specified in ohms per 1,000 feet (305 m) of conductor length. [Ch. 9, Table 8]. To determine the total resistance for a wire run, the wire resistance in ohms per 1,000 feet is multiplied by the number of feet and then divided by 1,000. If we considered (for simplicity) that the entire run of cable is #14 (with a resistance of 3.14 ohms per 1,000 ft.), the resistance would be 0.0628 ohms (20 x 3.14 -1,000 = 0.0628).

Using the formula, voltage = amperage x resistance (V = I x R), we can determine the voltage drop. At a peak power current of 4.77 amps (Imp), the conductors attached to each set of two modules have a voltage drop of 0.3 volts (4.77 x 0.0628 = 0.299). To figure out the percentage of voltage drop as a result of resistance, you divide the voltage drop by the nominal system voltage and multiply by 100 percent. On a 48 volt system, this is a 0.625 percent loss on the longest circuit (0.3 -48 x 100 = 0.625).

With a portion of the circuit consisting of a #10 (5 mm2) conductor, and on circuits where the circuit length is less than 20 feet (6 m), the voltage drop and power loss (which is expressed as the same percentage because P = V x I) are even less. It is not practical to use a larger size cable at this point because the combiner box accepts no conductors larger than a #10 conductor.

PV Output Circuit Calculations

The PV output circuits include all the wiring from the combiner box to the charge controller.

Conductor Sizing. The next step in the system design is to calculate the size of the conductors between the PV combiner boxes and the DC power center. Pairs of modules are series-connected in sets of two for a 48 volt nominal output. Five strings (sets of two) of modules are paralleled in each combiner box.

The continuous output current from each of the combiner boxes (for conductor ampacity calculations) is determined by multiplying the number of paralleled strings of modules (five) by the short-circuit current (5.46 amps) of each string, and then by a current adjustment factor of 1.25 to yield an expected current of 34.125 amps (5 x 5.46 x 1.25 = 34.125) [UL Standard 1703, 690.9]. An additional 1.25 factor is then applied to get a current of 42.65 amps, and this is the current on which the conductor size and the overcurrent device must be based (34.125 x 1.25 = 42.65) [690.9].

Ambient temperatures for this system are 45°C (113°F) around the exposed portions of the metal conduits running from the combiner box to the DC power center. But the conductor ampacity tables [310.16] are based on 30°C (86°F) ambient temperatures, so we must use temperature correction factors to select a properly sized conductor.

For 90°C (194°F) insulated conductors (RHW-2 or THWN-2) in conduit, the temperature correction factor is 0.87 at an ambient temperature of 45°C (113°F) [310.16]. To determine the required ampacity for the conductor at 30°C (86°F), divide the 42.65 amps by the temperature correction factor to get 49 amps (42.65 - 0.87 = 49). Use this number to find the proper wire size on the 30°C ampacity tables in the NEC [310.16].

This ampacity value of 49 amps dictates that a #8 (8 mm2) conductor be used. We can verify our selection by working the calculation backward. A #8 conductor in conduit has a 30°C ampacity of 55 amps [310.16]. At 45°C, the ampacity is corrected to 47.9 amps (55 x 0.87 = 47.9), which exceeds the requirement of 42.65 amps.

Voltage Drop Calculations. (Not an NEC requirement) From the combiner boxes located across the driveway from the house to the DC power center, the total conductor distance (positive and negative conductors) is 300 feet (90 m). The resistance of a #8 (8 mm2) conductor is 0.778 ohms per 1,000 feet and for the 300 foot length, the resistance is 0.233 ohms (300 x 0.778 - 1,000 = 0.233) [Ch. 9, Table 8].

At the maximum power point for the PV array, the five strings of modules with a current of 4.77 amps each generate 23.85 amps when connected in parallel in the combiner box (5 x 4.77 = 23.85). The voltage drop in each of the PV output circuits is calculated by multiplying the current by the resistance, and is 5.56 volts (23.85 x 0.233 = 5.56). In a 48 volt system, this represents an 11.6 percent voltage drop, which also represents an 11.6 percent power loss (5.56 - 48 x 100 = 11.58).

Just meeting code ampacity requirements may not always yield an efficient system. A design goal (not a code requirement) was to keep the voltage drop and power loss below 2 percent. This required increasing the size of the PV output circuit conductors.

Laying out the energy conversion equipment.

Laying out the energy conversion equipment.

A 2 percent voltage drop can be translated into a drop of 0.96 volts on a 48 volt system (48 x 0.02 = 0.96). The allowable maximum conductor resistance can be calculated by dividing the maximum voltage drop (0.96 volts) by the current (23.85 amps). This yields 0.04 ohms for the entire 300 feet of conductor (0.96 - 23.85 = 0.04). The resistance per 1,000 feet would need to be 0.133 ohms (0.04 - 300 x 1,000 = 0.133) or less. This indicates that a #1/0 (5 mm2) conductor should be used, which has a resistance of 0.122 ohms per 1,000 feet [Ch. 9, Table 8].

Using this #1/0 (53 mm2) conductor with a resistance of 0.122 ohms per 1,000 feet yields a voltage drop of 1.82 percent when carrying 23.85 amps (0.122 x 300 - 1,000 x 23.85 - 48 x 100 = 1.82). A larger conductor could be used to reduce the voltage drop and power loss even further. Using a #2/0 (67 mm2) conductor, for example, would reduce the voltage drop to 1.44 percent (0.0967 x 300 -1,000 x 23.85 - 48 x 100 = 1.44). This is not a very significant decrease in the voltage drop or power loss. Also, #2/0 is larger than the terminals on some of the equipment will accept.

The one-time expense of the larger wire should be weighed against the loss in energy over the life of the system. It usually pays to install the largest conductor that can be easily connected to the devices at each end. Standalone PV energy has been estimated to cost as much as US\$2 per kilowatt-hour over the 20 to 30 year life of a system! Why go to the trouble of eliminating hidden loads and increasing the efficiency of all other loads, or choosing a more efficient inverter and charge controller when you don't address a constant (forever) loss of PV energy (and power) due to smaller than maximum (although code compliant) conductor sizes.

### Overcurrent Protection. The DC

circuit breakers used in the power center for PV output circuit overcurrent protection are rated at 100 percent duty in their listed enclosures and do not require an NEC 80 percent derating [690.8(B)(1)EX]. These circuit breakers are mounted in the power center and protect the PV output conductors from overcurrent from possible backfed current from the batteries or the inverter. These circuit breakers must be rated to carry the continuous short-circuit current of 34 amps, determined previously when making calculations for conductor sizing (5 x 5.46 x 1.25 = 34.125). The second 1.25 factor is not used in this calculation because the circuit breakers do not have to be derated to 80 percent of rating.

Circuit breakers rated as low as 35 amps could have been used. We are using circuit breakers rated at 75 amps. They were ordered when the PV modules were going to be connected

Copper Conductor Temperature Correction Factors

 Con Temp. Rating ductor Types 21-25 (70-77) 26-30 (78-86) 31-35 (87-95) 46-50 (114-122) °C & (°F) 51-55 (123-131) 56-60 (132-140) 61-70 (141-158) 71-80 (159-176) 75°C (167°F)

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