Solar PV and Battery Sizing: Method

Solar PV and Battery Sizing

The tool sizes a photovoltaic array from the monthly solar resource, in two forms chosen by the System select. A grid-tied array is sized for a share of the load or checked at an entered power, arranged in strings, given a backup battery check and a cash flow. A stand-alone array with a battery is sized by the worksheets of the Sandia National Laboratories handbook. Both use the same site, resource, temperature and module inputs. Every method here works on monthly means, which is the level of a first sizing; an hourly simulation (the NREL PVWatts calculator and System Advisor Model do one) is the next step for clipping, shading and self-consumption. Values owned by a code, a standard or a maker (the National Electrical Code voltage table, inverter limits, module coefficients, battery ratings) are entered inputs. The default array and its costs are illustrative.

Solar resource

The resource is twelve monthly means of the daily irradiation in kWh/m², which is the same number as the peak sun hours (hours at 1,000 W/m²). It can be entered three ways. Peak sun hours in the plane of the array are used as entered; the monthly solar radiation printed by the NREL PVWatts calculator for the chosen tilt and azimuth, and National Solar Radiation Database values already at the tilt, are of this kind. Horizontal irradiation H, or a clearness index K_T, is moved to the array plane by the tool.

Each month is represented by its average day, the day whose declination is closest to the month's mean (Klein): 17 January, 16 February, 16 March, 15 April, 15 May, 11 June, 17 July, 16 August, 15 September, 15 October, 14 November and 10 December. For day n of the year the declination is Cooper's δ = 23.45° sin[360° (284 + n) / 365], the sunset hour angle is ω_s = arccos(−tan φ tan δ) at latitude φ, the day length is 2 ω_s / 15° hours, and the extraterrestrial irradiation on a horizontal surface over the day is

H₀ = (24 / π) G_sc [1 + 0.033 cos(360° n / 365)] (cos φ cos δ sin ω_s + ω_s sin φ sin δ),

with ω_s in radians and the solar constant G_sc entered (1,367 W/m² in the cited method; recent measurements give about 1,361). The clearness index is K_T = H / H₀. The diffuse part H_d is either an entered fraction of H or the monthly correlation of Erbs, Klein and Duffie: H_d / H = 1.391 − 3.560 K_T + 4.189 K_T² − 2.137 K_T³ when ω_s is at most 81.4°, and 1.311 − 3.022 K_T + 3.427 K_T² − 1.821 K_T³ when it is larger. The correlation was fitted for K_T from 0.3 to 0.8; outside that range the tool holds K_T at the limit for this step and says so.

The irradiation on the array of tilt β is the isotropic-sky sum of Liu and Jordan, H_T = (H − H_d) R_b + H_d (1 + cos β) / 2 + H ρ (1 − cos β) / 2, where ρ is the ground reflectance. R_b is the ratio of the day's beam irradiation on the array to that on the horizontal, taken outside the atmosphere as Liu and Jordan proposed. The cosine of the incidence angle on a fixed surface has the form A + B cos ω + C sin ω in the hour angle ω, so the tool integrates it exactly over the hours when the sun is above the horizon and in front of the array, for any azimuth; for an array facing the equator this reduces to the closed form of the textbooks. The isotropic sky leaves out the brighter sky around the sun and near the horizon, so it is slightly conservative for tilted arrays facing the equator. The best fixed tilt, for the year and for the poorest month, is found by repeating the calculation for every degree from 0 to 90.

The results section does not print the time-of-day geometry, but the library holds it and the test script checks it: the equation of time and declination series published by NOAA, the hour angle, zenith, azimuth, the incidence angle on a tilted surface and sunrise and sunset with the 90.833° horizon. Against the printed example of the NREL Solar Position Algorithm (17 October 2003, Golden, Colorado) this simple geometry is off by 0.35° in declination, 0.32° in zenith, 0.15° in azimuth, 0.22° in incidence and about one minute at sunrise. That is adequate for sizing and not for tracking or instrument work.

Cell temperature

The monthly mean cell temperature follows Evans' relation as used by RETScreen: T_c = T_a + C_f (219 + 832 K_T) (NOCT − 20) / 800, with the monthly mean ambient T_a, the module's nominal operating cell temperature NOCT, and C_f = 1 − 1.17 × 10⁻⁴ (s_M − β)² for a tilt away from the optimum s_M = φ − δ. When peak sun hours are entered there is no clearness index, and the bracket is replaced by an entered operating irradiance. The hot cell temperature for the string voltage and the peak power is taken at the design high ambient and 1,000 W/m², either from the NOCT definition, T_c = T_a + (NOCT − 20) G / 800, or from the Sandia model T_m = G exp(a + b v) + T_a, T_c = T_m + (G / 1000) ΔT with the coefficients of King, Boyson and Kratochvil for the mounting chosen and the wind speed v at 10 m.

Grid-tied array

The losses that are not modeled one by one are the categories of the PVWatts Version 5 manual (soiling 2, shading 3, snow 0, mismatch 2, wiring 2, connections 0.5, light-induced degradation 1.5, nameplate 1, age 0, availability 3 percent). They are multiplied, not added: L = 100 [1 − Π (1 − L_i / 100)], which gives 14.08 percent for the defaults, printed as 14 percent in the manual. The DC power follows the manual's module model, P_dc = P_dc0 (G / 1000) [1 + γ (T_c − 25)], with the power temperature coefficient γ from the data sheet (the manual uses −0.47, −0.35 and −0.20 percent per °C for standard, premium and thin-film modules). On monthly means this becomes the monthly performance ratio PR = (1 − L / 100) [1 + γ (T_c − 25) / 100] η_inv and the monthly energy E = P_dc0 × peak sun hours × days × PR.

The array power for a share f of the annual load is P = f E_load / Y₁, where Y₁ is the year's energy of 1 kW (the specific yield, kWh per kW). The module count is rounded up, and to whole strings when string sizing is on. The page prints the annual performance ratio E / (P_dc H_T,year), the capacity factor E / (8,760 P) on the DC and the AC rating, and the monthly production against the load. The inverter is taken as P_dc over the DC to AC ratio; the page compares the array's AC output at 1,000 W/m² with that rating and says whether the inverter clips. The energy lost to clipping, and the inverter's part-load efficiency curve, need hourly data and are not estimated.

String sizing

The method is the temperature-coefficient one. The module open-circuit voltage at the record low ambient is V_oc [1 + β (T_low − 25) / 100] with the data-sheet coefficient β in percent per °C, the cells being at ambient at dawn; the longest string is the inverter's maximum input voltage over it, rounded down. The maximum-power voltage at the hot cell temperature, moved by the same coefficient, sets the shortest string against the lower end of the tracker window, rounded up; the cold maximum-power voltage is checked against the upper end. The maximum circuit current of a string is 1.25 I_sc, which limits the strings on a tracker input, and conductors and overcurrent devices are rated a further 1.25 above it, 1.56 I_sc, before the conditions-of-use factors. The National Electrical Code (Article 690) allows this coefficient calculation and also gives a table of correction factors for crystalline silicon when no coefficient is available; the table is not reproduced here. Among the string lengths that pass, the tool takes the arrangement with the fewest modules that covers the requirement.

Backup battery

The energy a battery delivers to the critical loads is units × nominal energy × depth of discharge × ageing factor × discharge efficiency. Makers quote a round-trip efficiency; the tool takes the discharge leg as its square root, which splits the loss evenly between charging and discharging. The energy is compared with the critical daily energy over the outage, and the continuous power with the critical peak. The array's average daily production in its poorest month is then set against the critical daily energy, first directly and then with all of it passing through the battery, to say whether an outage can be carried on average days. Motor starting surges and the power available on a cloudy day are outside this check.

Economics

The installed cost is the cost per watt times the DC rating plus the battery; an entered percentage is deducted as the incentive (the tool gives no tax advice). In year t the production is E₁ (1 − d)^t−1 with the degradation rate d; Jordan and Kurtz found a median of 0.5 percent per year in nearly 2,000 published rates. With net metering the production up to the annual load is worth the retail rate and any surplus the export rate. With net billing the self-consumed share is worth the retail rate and the exported rest the export (avoided-cost) rate; that share depends on the hourly load and the battery and is an entered value. The yearly cash flow is the saving less operation and maintenance and one replacement. The page gives the simple payback (net cost over the year-1 cash flow), the year in which the cumulative cash flow turns positive, the net present value, the internal rate of return, and two levelized costs: the life-cycle form LCOE = [net cost + Σ costs_t / (1 + d)^t] / Σ E_t / (1 + d)^t of the NREL economic evaluation manual, and NREL's simple form (net cost × CRF + year-1 operation and maintenance) / E₁ with the capital recovery factor CRF = d (1 + d)^N / [(1 + d)^N − 1]. The two agree when production and costs are level. Financing, taxes on the savings and salvage value are not modeled.

Stand-alone systems: the Sandia worksheets

Worksheet 1 lists the loads. Each load's daily ampere-hours at the system voltage are watts × quantity × hours per day × (days per week / 7) / conversion efficiency / V_sys, with the inverter efficiency for AC loads (handbook default 0.85) and 1 for DC loads. The total is divided by the wire efficiency (0.98) and the battery efficiency (0.9) to give the corrected load. Worksheet 2 divides the corrected load by each month's peak sun hours on the array; the largest quotient is the design current and its month the design month. The handbook repeats this for tilts of latitude − 15°, latitude and latitude + 15° and takes the tilt with the smallest design current; here the tilt scan of the resource section serves that purpose.

Worksheet 3 sizes the battery: required capacity = corrected load × days of autonomy / (maximum depth of discharge × temperature derate), to which the tool adds an optional ageing derate. Batteries in series are the system voltage over the battery voltage, and in parallel the required capacity over the battery capacity. Worksheet 4 sizes the array: the design current over the module derate factor (0.9 for crystalline modules) is the array current; modules in parallel are that current over the module's maximum-power current; modules in series are the voltage required to charge the batteries, 1.20 × battery voltage × batteries in series, over the module voltage at the highest expected temperature. Worksheet 5 forms the array-to-load ratio, the derated design current times the system voltage over the daily watt-hours, which the handbook plots against the daily load to suggest a generator hybrid; that chart is not reproduced.

The worksheets say to round the parallel counts up; the handbook's designers round down for noncritical loads, so the tool offers both. Added to the worksheets: the discharge rate with every load running against an entered limit, the charge-controller rating of 1.25 times the array short-circuit current, the array size by energy balance for a maximum-power-point-tracking controller (the worksheets assume a controller that holds the array at the battery voltage), the monthly supply of the installed array against the load, and the generator running hours for an entered charger. The stand-alone default is the handbook's AC/DC residence; with the handbook's module (3.0 A, 15.9 V, 14.4 V hot) and rounding to the nearest, the page reproduces its worksheets: 72.9 and 82.7 Ah/day, 17.5 A, 709 Ah, eight 350 Ah batteries, twelve modules.

Checks

The library is tested against the handbook's residential and navigation-beacon worksheets, the PVWatts loss table, the printed example of the Solar Position Algorithm, the monthly average days and declinations tabulated by NASA, the levelized-cost examples of the NREL manual, numerical integration of the extraterrestrial irradiation and of R_b, and an independent numpy and scipy implementation on the default case and twenty seeded systems.