A typical module will have 36 cells connected in series, each cell typically having the parameters:
Voc = 600 mV (25°C) FF = 75%
Vmp = 475 mV (25°C) Vmp = 430 mV (45°C) Imp/Isc = 0.95
Each cell can be reasonably accurately represented by the equation
where V is the terminal voltage, I is the current, IL is the light-generated current, n is the ideality factor (taken to be 1.3), Rs is the series resistance, q is the charge on an electron (1.6 x 10-9 C), к is Boltzmann’s constant (1.38 x 10-3 J/K), T is absolute temperature (typically 318 K for field operation), and I0 is the dark saturation current, given by
= 2.17 x10~7 x IL (at 45°C)
where Voc is the open circuit voltage, which is typically 600 mV at 25°C for commercial solar cells, but falls to about 555 mV at 45°C.
For commercial cells, Rs is designed to be approximately inversely proportional to the rated short circuit current, so that percentage power loss in Rs is approximately constant with cell size (about 2.5%); that is
where Isc is the short circuit current under 1 kW/m2.
To allow for variations in light intensity, let
Il = L x Isc (H.11)
where L is the factor representing the light intensity such that L = 1 corresponds to 1 kW/m2 and L = 0.5 corresponds to 500 W/m2. We can now rewrite Eqn. (H.8) as
for T = 318 K.
For a number of cells interconnected in series, the voltage at any current I from Eqn. (H.12) should simply be multiplied by the number of series-connected cells.
The next step is to generate the five current-voltage curves from Eqn. (H.12) that correspond to the five light intensities (i. e. five values for L) from Fig. H.2. These are shown in normalised form in Fig. H.3, with corresponding tables of normalised values being given in Table H.1. The currents on the vertical axis and in the normalised tables will be explained later. Each voltage on the horizontal axis is multiplied by the factor m, which is the number of nominally 12 V modules connected in series in each string.
Figure H.3. Current-voltage characteristic curves for a typical commercial PV system at the light intensities given by Fig. H.2. The curves were generated using Eqn. (H.12).
Table H.1. Normalised values for the currents and voltages corresponding to the five curves in Fig. H.3.
curve 1 curve 2 curve 3 curve 4 curve 5
Modules specifically tailored to give the exact desired voltage at maximum power point can of course be designed and constructed by using a different number of series – connected cells. This, however, will cost a premium price, making it uneconomical to use specially-designed modules unless the quantities required are enormous.
Considering the standard modules at 45°C:
• one single module (36 cells) should give Vmp = 15.5 V
• two in series (72 cells) should give Vmp = 31.0 V
• three in series (108 cells) should give Vmp = 46.5 V
• four in series (144 cells) should give Vmp = 62.0 V.
As can be seen from Fig. H.3, the maximum power point voltage changes very little with variations in light intensity.
We therefore choose an appropriate number of series-connected modules to give us a maximum power point voltage (at 45°C) as close as possible to the voltage at which the subsystem attains maximum operating efficiency (allowing for a 2% voltage drop along the length of the wiring).
The final part of the design procedure is the determination of the current-generating capacity required for the solar panels. An iterative approach is simplest and well suited to solving through the use of computers. An initial choice of array size (current rating) is made by following the guidelines of steps (4) and (5) in Section H.2, above, where it is suggested that we want the subsystem to operate at its maximum efficiency when the light intensity (Lmp) is given by
Lmp = 0.80IM (H.13)
Therefore, we want a rated maximum power point current (Imp) at 1 kW/m2 insolation of
where Im is the motor current at maximum subsystem efficiency.
Once the required Imp value is determined, we must appropriately upgrade the rating to ensure the solar panels will in fact produce the current Imp. The contributing reasons for necessitating this upgrade in rating include:
• dust on the surface (Halcrow & Partners, 1981; Hammond et al., 1997)
• insolation levels possibly lower than anticipated
• tolerances in solar panel outputs
• degradation of solar panels.
These factors can be compensated for by appropriately over-sizing the solar array design. Typically, we should allow for 6% loss owing to dust, depending on location, 10% for degradation of the solar panels (unless manufacturers give guarantees to the contrary) and 10% for combined tolerances in solar panel outputs and insolation levels. A combined reduction of 26% from rated value translates to an over-sizing requirement of 35% (1/0.74). We call this a derating factor DR, which in this case is 0.74. In other words, we select solar panels with a rated maximum power point
current 1.35 times greater than that calculated as being necessary. To convert this to a short circuit current rating, simply divide by 0.95.
We are now in a position to refer back to Fig. H.3 and consider the vertical axis. If Isc is the rated short circuit current, as specified by the manufacturer, then we must:
1. Use Fig. H.2 to make adjustment for the light intensity being different from
1 kW/m2 (= 100 mW/cm2), which means multiply by (0.99 x I / 100), for the top curve, where I is given by Eqn. (H.1).
2. Multiply by the derating factor DR (which is suggested to be 0.74). Consequently, for the top curve, we get a short circuit current of:
0.99 x 0.74 x I / 100 x Isc
(0.0073 x I) x ISc
where I is in mW/cm2.
Having now selected the array size, the vertical and horizontal axes of Fig. H.3 can be specified, and the load line (with efficiency weightings) superimposed. This facilitates, in conjunction with Fig. H.2, the calculation of pumped volumes of water and system average efficiencies throughout both sunny and cloudy days and for the whole design period. If this procedure is being carried out by a computer, it becomes a relatively simple task to try different values for the current to iteratively tune into the system design for maximum overall efficiency and preferred pumping regime.
It should be noted that a degree of conservatism has automatically been built into the design back in steps (4) and (5) when we defined the number of pumping hours (E) in terms of the number of equivalent hours of sunshine with light intensity (Isa), but then subsequently used the light intensity of 0.80Isa as the one for which the maximum subsystem efficiency would correspond. This conservatism was necessary for two reasons. Firstly, it allows for reduced subsystem performance (efficiency) when operating under light intensities greater or less than the design value of 0.80Isa. Secondly, it compensates for subsystem performance below that expected, which could result from:
• discrepancies between manufacturers’ curves and those measured in practice
• degradation of subsystem components
• unexpected changes in static head etc.
The net result of the conservatism is that the design, on paper, should result in 1020% more water being pumped than is required in the specification. This figure may be increased or decreased as required, although any modifications should probably be done in consultation with the consumer.