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316 CHAPTER 9 Design Principles of Photovoltaic Irrigation Systems

required by the distribution system to supply a discharge Q can be decomposed into
the following components:

H ¼ Dz þ hf þ h (9.17)
where Dz is a constant value that represents the elevation change from the reservoir
to the most restricted point in the irrigation subunits (in meters), hf is the estimated
head losses in the network path between the head of the system and the most
restricted point (in meters), and h is the average pressure head of the emitters of
the system (in meters). If the system is composed of several heterogeneous irrigation
subunits, a different system curve should be calculated for each irrigation subunit. If
the irrigation subunits are relatively homogeneous, the system curve can be valid for
all of them. Head losses in this simplified system scheme will able to be calculated
by using a general head loss equation as a function of the pump discharge, as
explained in Eq. (9.13).
The discharge in a noncompensating emitter varies according to its working
pressure. The relationship between these two variables is given by the discharge
equation of the emitter [50]:
q ¼ kh x (9.18)
where k is the emitter discharge coefficient, which is a constant dependent on the
unit system used, h is the working pressure head at the emitter (in meters), x is
the emitter discharge exponent (nondimensional), and q is the discharge of the
emitter. Conventional labyrinth-type emitters usually work in a turbulent regime;
therefore the value of their exponent is close to 0.5. In the simplified irrigation
scheme proposed, an average h value is considered and the hydraulic variability
in the irrigation unit is neglected.
The total discharge in the irrigation system can be calculated by multiplying the
number of emitters in operation by their average discharge, which can be calculated
with Eq. (9.19).
x
Q ¼ q$n e ¼ n e $k$h (9.19)
Solving for the pressure head in Eq. (9.19), the average pressure at the emitters can
be expressed as a function of the total discharge of the system.
2
1 2 2
h ¼ Q ¼ K$Q (9.20)
n e k
And finally, substituting h in Eq. (9.17), the following equation for the system curve
can be derived:

H ¼ Dz þðR s þ KÞ$Q 2 (9.21)
Having identified the system curve (Eq. 9.21), the generalized H-Q performance
curve of the pump (Eq. 9.11), and the incoming power for the PV plant (Eq. 9.16),
the output frequency of the converter and the head and discharge of the pumping