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312 CHAPTER 9 Design Principles of Photovoltaic Irrigation Systems

3.2.1 Pumping System
Pumping systems are usually composed of one or more than one centrifugal pumps
arranged in parallel. Early PV pumping systems used fixed speed pumps driven by
electric motors, and their design and management was considerably simpler.
Today, to pump variable flowrates to different total heads efficiently, variable
speed pumps are commonly used in these systems. Variable speed pumps are pow-
ered by an asynchronous motor and an electronic frequency converter that allows the
pump to work under different angular velocities. The frequency converter can be
considered as an inverter that is able to modulate the output signal to the desired fre-
quency and amplitude.
Because the power supplied by the PV array is variable throughout the
day depending on the incoming radiation, the power required by the pump can be
modified by modifying the angular velocity of the pump to match the incoming
power.
To select the most appropriate pump for a specific application, the performance
curves (Discharge-Head-Efficiency) of the pump provided by the manufacturer and
the affinity laws that relate the shaft power, P, the head (H), and the discharge, Q,
provided by the pump to its rotation speed (n) are required.
By simultaneously applying the affinity laws (Eq. 9.9) to an operating point, H 1 -
Q 1 , belonging to the Discharge-Head curve of a pump working at a specific pump
speed (N1), the new operating point, H 2 -Q 2 , of that pump at another specific velocity
(N2) can be calculated. The efficiency of the pump is assumed to be constant in these
two homologous operating points.
2 3
Q 2 N 2 H 2 N P 2 N h 2
2
2
¼ ¼ ¼ ¼ 1 (9.9)
2
3
Q 1 N 1 H 1 N P 1 N h 1
1 1
where subscripts 1 and 2 refer to two different rotation speeds.
Considering that the rotational speed of the pump is proportional to the fre-
quency of electric current, affinity laws can be expressed as a function of the fre-
quency rather than the speed of the pump.
2 3
Q 2 f 2 H 2 f P 2 f h 2
2
2
¼ ¼ 2 ¼ 3 ¼ 1 (9.10)
Q 1 f 1 H 1 f P 1 f h 1
1 1
Assuming that the H 0 -Q 0 performance curve of a commercial pump is given by
the manufacturer at a nominal AC frequency (f 0 ) (which in EU countries is 50 Hz),
the generalized H-Q performance curve of the pump for any other frequency, f, can
be obtained from the nominal H 0 -Q 0 curve by applying the affinity laws. The gener-
alized H-Q equation is given in Eq. (9.11) and the resulting H-Q curves are depicted
in Fig. 9.18.

f 2
H ¼ A BQ (9.11)
f o
3.2.2 On-Farm Irrigation Network
To calculate the operating point (discharge and head) of the PV irrigation system as a
function of the power supplied by the PV plant, not only the Head-Discharge of the