Let us consider a PV array affected by shading. At any instant, we
can state
Ps = Pns
(1-Fes) Eq.
(2)
where Ps and Pns represent the power delivered by the PV array with
and without shading, respectively, and Fes so-called here as
effective shading factor, whose value determines the power decrease.
A first possible Fes estimation consists of assuming that the power
reduction is just equal to shaded array fraction. This is the
geometrical shading factor Fgs:
Fes =
Fgs Eq. (3)
Obviously, this approximation represents a minimum limit for power
reduction. Hence, it is always optimistic.
A second approximation, this time pessimistic, is to assume that any
shadow fully cancels power:
Fgs > 0 then Fes =
1 Eq.
(4)
A better approximation is obtained by taking into account the
shaded blocks. A "block" is here defined as a group of
cells protected by one bypass diode. A block is shaded when at least
one of its cells is shaded. A first possibility is to consider that
the power of a block is fully cancelled when the block is shaded.
Hence
(1-Fes) = (1-
Nsb/Ntb) Eq.
(5)
where Ntb is the total number of blocks inside the concerned array
and Nsb is the number of shaded blocks. A priori, Eq. (5) tends to be
optimistic because the power losses are usually greater than the
power of the shaded blocks. For example, when a block is shaded and
its bypass diode is ON the output power of the block is cancelled.
Besides, if there are other unshaded strings connected in parallel
their operating voltage will be reduced, causing additional power losses.
Another example: if a block is shaded and its bypass diode is OFF the
string current is limited by this block, which reduces the power of
the remaining unshaded blocks connected in series. Later in this
paper we will show that Eq. (5) actually leads to an optimistic estimate.
(1-Fes) =
(1-Fgs)(1-Nsb/(Ntb+1)) Eq.
(6)
The number " 1 " added in the denominator has not direct
physical sense: it is a mathematical trick to avoid fully cancel
power when a shadow affects all the array blocks (Nsb=Ntb\ but still
keeps a significant illuminated area (low Fgs). It is worth stressing
that Eq. (6) is purely experimental and its physical interpretation
may lack sense.For example, for a large value of Ntb the ratio
Nsb/(Ntb+1) tends toward Fgs. Hence (1-Fes) ~ (1-Fgs)².
Another example: when all blocks are shaded (Nsb=Ntb) the ratio
Nsb/(Ntb+1) varies between 0.5 (Ntb=1) and 1 ( Ntb >> 1), which
is unreal because it implies that the power losses caused by the same
shadow repeated on several PV modules increase as the number of PV
modules increases (actually, the power losses could be equal).
The simplicity of this model does not allow taking into consideration
the electrical characteristics of the PV array, which would require
the simulation of the I-V curve. However, and despite its
limitations, the model performs well and better than the others.
Reference:
"Experimental model to
estimate shading losses on PV arrays"
F. Martinez-Moreno, J.
Muñoz, E. Lorenzo
Instituto de Energía
Solar—Universidad Politécnica de Madrid (IES-UPM),
E.T.S.I. Telecomunicación, Ciudad Universitaria, s/n 28040,
Madrid, Spain