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314 CHAPTER 9 Design Principles of Photovoltaic Irrigation Systems

where R s is the resistance coefficient of the discharge pipe, Q is the total discharge of
the pumping system, and m is the hydraulic exponent of the head loss equation.
Assuming that the flow regime is turbulent, it can be supposed that m is equal to
2(Eq. 9.13). In this equation, the resistance coefficient, R s , should include not
only the uniformly distributed losses in the pipe but also the local or minor head
losses.
Finally, the system curve is given in the following equation.
2
H ¼ Dz þ R s Q (9.14)
The shaft power supplied to the pump, P, can be calculated as a function of the
PV power, P PV, provided by the PV array.
P ¼ P PV $h $h C (9.15)
M
where h M and h are the efficiencies of the electric motor and the frequency con-
C
verter, respectively.
The shaft power can also be expressed as a function of the net power transferred
by the pump to the water, which depends on the head, the discharge supplied by the
pump, and the pump efficiency, h (Eq. 9.16).
P
P ¼ P PV $h $h ¼ g$Q$H=h P (9.16)
M
C
where g is the specific weight of the water.
Finally, solving the set of three equations made up by the generalized H-Q curve
of the pump (Eq. 9.11), the system curve (Eq. 9.14), and the power supplied by the
PV array (Eq. 9.16), the three unknown variables required to control the pumping
system can be calculated. These required variables are the output frequency of the
AC provided by the inverter (f), the discharge (Q) lifted by the pump when working
at this specific frequency, and the head (H) supplied by the pump in these working
conditions.

3.2.2.2 Direct Pumping
The simplest and most widely used procedure to control PV irrigation and match
renewable energy production to crop irrigation demand is to pump water to an
elevated storage tank and then distribute it to the plants by gravity. However, in
some cases, farms may be relatively flat, so perhaps no appropriate location is avail-
able to build a tank with sufficient elevation to take advantage of the gravitational
energy to distribute irrigation water. In any case, the construction costs must be
taken into account. To overcome these drawbacks, direct pumping PV irrigation sys-
tems can be used.
Regarding the type of the emitters in drip irrigation systems, two different groups
of emitters can be distinguished. Noncompensating emitters vary their discharge ac-
cording to the operating pressure. The inconvenience of noncompensating emitters
is that they may produce low irrigation uniformity when pressure variability in the
irrigation sector is high. To avoid this drawback, compensating emitters are used that
keep the discharge constant regardless of their working pressure. In these cases, the