In a previous chapter (see Chapter 3), it has been shown that the characteristic I-of a solar cell can be expressed with sufficient accuracy by

where IL, I0,RS and RP are the photogenerated current, the dark current, the series resistance and the parallel resistance, respectively. The voltage Vt equals mkT/e (we recall that for m = 1,Vt « 25 mV at 300 K). This expression gives an adequate representation of the intrinsic behaviour of a typical c-Si solar cell. Nonetheless, it cannot be used directly to obtain the required predictions, because some parameters IL and IO in particular, cannot be established from the usually available information, often restricted to the values of and which are always included on the manufacturers’ data sheets.

This difficulty is effectively overcome when the following assumptions, which are generally valid for c-Si PV cells and modules, are made:

• The effect of parallel resistance is negligible.

• The photogenerated current and the short-circuit current are equal.

• exp((V + IRs)/Vt) » 1.

This allows Equation (22.56) to be written as

which, with I = 0, leads to the following expression for the open-circuit voltage:

Voc = vHi)

where

This is a very useful expression. As we shall see, the values of all the parameters on the right- hand side are easily obtained allowing immediate application of the expression. An inconsistency arises in the sense that I(V = 0) = ISC. Nevertheless, in all solar cells of practical use, we find that VOC » IRS ^ I(V = 0) « ISC, which therefore makes this objection irrelevant. The expression can be inconvenient to use in the sense that I is implicit (it appears on both sides of the equation), theoretically making it necessary to solve the equation iteratively. However, for voltages close to the maximum-power point, a reasonably accurate solution can be obtained with only one iteration by setting IM = 0.9 x ISC in the second term.

The calculation of the maximum power can, in principle, be carried out by considering that the power is given by the product P = VI. The values of IM and VM, defining the maximum power operation point, can be obtained from the usual condition for a maximum, dP/dV = 0. However, the implicit nature of the resulting expression makes it very cumbersome to use. It would be better to look for a simpler method, based on the existing relationship between the fill factor and the open – circuit voltage. According to M. A. Green [51] an empirical expression describing this relation is

where

and voc = VOC/Vt and rs = RS/(VOC/ISC) are defined as the normalised voltage and the normalised resistance, respectively. It is interesting to note that the series resistance at STC can be deduced from the manufacturer’s data by the expression

The values of VM and IM are in turn given by [52]

—— = 1——— In a — rs(l — a~b) and—!_ _ a~b (22.64)

voc voc isc

where:

a

a = voc + 1-2 vocrs and ft = ———— (22.65)

1 + a

This set of expressions is valid for voc > 15 and rs < 0.4. The typical accuracy is better than 1%. Their application to a photovoltaic generator is immediate, if all their cells are supposed to be identical, and if the voltage drops in the conductors connecting the modules are negligible.

Now, for the prediction of the I – V curve of a PV generator operating on arbitrary conditions of irradiance and temperature, a good balance between simplicity and exactness is obtained through the following additional assumptions.

• The short-circuit current of a solar cell depends exclusively and linearly on the irradiance. That is,

I *

hc(G) = – J^Geff (22.66)

where Geff is the ‘effective’ irradiance. This concept must take into consideration the optical effects related to solar angle of incidence, as described in Section 22.7.

• The open-circuit voltage of a module depends exclusively on the temperature of the solar cells Tc. The voltage decreases linearly with increasing temperature. Hence,

dVOC

Voc(Tc) = Voc + ^c-?;*)^ (22.67)

where the voltage temperature coefficient, dVOC/dTc is negative. The measurement of this parameter use to be included in PV modules characterisation standards [53] and the corresponding value must, in principle, be also included on the manufacturer’s data sheets. For crystalline silicon cells, dVOC/dTc is typically -2.3 mV per0C and per cell.

• The series resistance is a property of the solar cells, unaffected by the operating conditions.

• The operating temperature of the solar cell above ambient is roughly proportional to the incident irradiance. That is,

Tc — Ta + QGeff

where the constant Ct has the value:

The values of NOCT for modules currently on the market varies from about 42 to 46 °C, implying a value of Ct between 0.027 and 0.032 °C/(W/m2). When NOCT is unknown, it is reasonable to approximate Ct — 0.030 °C/(W/m2). This NOCT value corresponds to mounting schemes allowing the free air convection in both sides of the PV modules, which cannot be the case on roof-mounted arrays, that restrict some of the airflow. Then, it has been shown [54] that the NOCT increases by about 17 °C if some kind of back ventilation is still allowed, and up to 35 °C if the modules are mounted directly on a highly insulated roof.

Example: To illustrate the use and the usefulness of the equations of the preceding sections, we shall analyse the electrical behaviour of a PV generator rated at 1780 W (STC), made up of 40 modules, arranged 10 in series x 4 in parallel. The conditions of operation are Geff — 700 Wm-2 and Ta — 34 °C. It is known that the modules have the following characteristics under STC: Z|C — 3A, VOC — 19.8 V and PM — 44.5 W. Further, it is known that each module consists of 33 cells connected in series and that NOCT — 43 °C.

The calculations consist of the following steps:

1. Determination of the characteristic parameters of the cells that make up the generator under STC (Equations 22.61-22.65):

33 cells in series ^ Per cell: Z|C — 3A, VOC — 0.6 V and PM — 1.35 W and assuming m — 1; Vt(V) — 0.025 x (273 + 25)/300 — 0.0248V ^ vOC — 0.6/0.0248 — 24.19 > 15

then, FF0 — (24.19 – ln(24.91))/25.19 — 0.833; FF — 1.35/(0.6 x 3) — 0.75 and rs — 1 – 0.75/0.833 — 0.0996 < 0.4 ^ RS — 0.0996 x 0.6/3 — 19.93mfl

a — 20.371; b — 0.953 ^ VM/VOC — 0.787 and ZM//SC — 0.943

It is worth noting that these values lead to a value of FF — 0. 742, slightly different from the starting value. This error shows the precision available by the method, better than 1% in this instance. Sometimes values of m — 1.2 or 1.3 give a better approximation.

2. Determination of the temperature of the cells under the operating conditions considered (Equations 22.68 and 22.69):

Ct — 23/800 — 0.0287 °Cm2/W ^ Tc — 34 + 0.0287 x 700 — 54.12 °C

3. Determination of the characteristic parameters of the cells under the operating conditions considered (Equations 22.66 and 22.67):

ZSC(700 W/m2) — 3 x (700/1000) — 2.1A VOC(54.12 °C) — 0.6 – 0.0023 x (54.12 – 25) — 0.533 V

With RS considered constant, these values lead to:

Vt = 27.26mV; vOC = 19.55; rs = 0.0785; FF0 = 0.805; FF = 0.742; PM = 0.83 W

4. Determination of the characteristic curve of the generator, (IG, Vg): Number of cells in series 330; Number of cells in parallel: 4. Then:

ISCG = 4 x 2.1 A = 8.4 A; Vocg = 330 x 0.533 V = 175.89 V; RSG = 1.644Я;

To calculate the value of the current corresponding to a given voltage, we may solve this equation iteratively, substituting IG for 0.9ISCG on the first step. Only one iteration is required for Vg < 0.8Vocg. By way of example, the reader is encouraged to do it for Vg = 140 V and Vg = 150 V. The solution is Ig(140V) = 7.77 A and Ig(150V) = 6.77 A.

5. Determination of the maximum power point:

a = 17.48; b = 0.9458; Vm/Voc = 0.7883; IM/ISC = 0.9332 Vm = 138.65 V; IM = 7.84 A, PM = 1087 W.

Note that the ratio PM /PM = 0.661, while the ratio Geff/G* = 0.7. This indicates a decrease in efficiency at the new conditions compared to STC, primarily due to the greater solar cell temperature, Tc < Tf. An efficiency temperature coefficient can now be obtained by

J_ dП_ = (Рм G/_ _ (_____________ 1_

h* dPc Um ‘ Geff )TC-T*

This means the efficiency, hence the power decrease is about 0.4% per degree of temperature increase, which can be considered as representative for c-Si. This value, calculated for typical parameters with some assumptions, agrees very well with the range of values commonly reported on module data sheets from —0.45 to —0.5%/°C.

In fact, this relation between power and solar cell temperature can always be directly employed. In this way:

G

Pm(Geff, Tc) = P*-AF[l-у(Tc-Tc*)] (22.70)

or

n(Tc) = n*[1 — Y(Tc — о (22.71)

where y is the so-called temperature power coefficient. Note that these expressions allow for calculation of power in any operation conditions without the need of previous calculation of the full I – V curve. This is very useful for simulation purposes, because the computing time is greatly reduced.

It should be noted that, depending on the input data availability, other PV generator modelling possibilities exist. For example, commonly and are given in the specifications in addition to their product Then, the series resistance can directly be estimated from Equation (22.60). This leads to

^oc-^M + ^tln(l + ^)

Rg =—————————– ;— ————- ^CZ_ (22.72)

Iiv

It should be noted that a similar but different set of I – V translations are found in Section

17.2 of Chapter 17 of this book.