TOU Homeflex structure8 for residential consumers given below is
used [46].
peðtÞ ¼
poff ¼ 0:6281 R=Kwh ift 2 ½0; 6 [ ½10; 18 [ ½20; 24
ppeak ¼ 1:9935 R=Kwh ift 2 ½7; 10 [ ½18; 20
(
ð49Þ
where poff is the off peak price, ppeak is the peak time price, R is the
South African currency, Rand, and t is the time of day in hours. The
tariff has five charge components as service charge, net-work
charge, environmental levy, peak charge and off-peak charges [47]
3.3. Water tariffs
The city of Tshwane9 has various water tariffs for different
classes of consumers and the domestic consumers are charged using
the rates in Table 1. The monthly amount of water used for irrigation
is assumed to be less than 6 m3 meaning the unit price of water used
pw ¼ 6:81 ðR=m3Þ.
3.4. Water tank and pump
There are various companies supplying water tanks in South
Africa. Jojo’s10 1000 l cylindrical water tank with a diameter and
height of 1.1 and 1 m respectively is used chosen. The lower and
upper levels of the water in the tank are set as 0.12 and 1 m, respectively
to avoid spilling the water from the tank as well as running it
completely empty.
The pump chosen for this model is the Al-Ko HW 3000 Classic,11
with a power rating of 650W and a flow rate of 3.1 m3=h. It is ideal
for small sprinklers and domestic applications.
4. Simulation results and discussion
4.1. Optimal scheduling without rooftop water harvesting
Fig. 3a shows the optimal irrigation schedule. The schedule has
three switching regimes with two taking place early in the morning.
Another schedule takes place in the evening taking advantage
of the low evapotranspiration rates as well as obey the by-laws.
The water content in the soil, shown in Fig. 3b, is maintained
within the required range, ensuring that there is no drainage from
the soil as well as the soil doesn’t become too dry.
4.2. Comparison of the switching strategies
Fig. 4 shows the comparison of the two switching minimization
strategies used to reduce the pump switching frequency. The
legend used is the same throughout this paper. The Pretoria
method (Fig. 4a) optimally reduces the switching to the minimum
feasible times. In this case, the strategy reduces the switching to
two times during the entire control horizon. On the contrary, the
constraint method (Fig. 4b), which explicitly sets the maximum
number of possible switching (smax ¼ 3), actually switches the
pump 3 times during the control horizon. The constraint method
however runs at a risk of infeasibility if the irrigation demand
increases to a level where the pump must switch on more than
the set smax. This shows that the Pretoria method is more effective
in reducing the maintenance cost than the constraint method and
it is therefore used in the later sections. It is important to note that
both strategies incur the same cost of energy during the period, as
the same amount of water is pumped for irrigation. Further,
although in one day, the extra switching regime in the constraint
method may not have very high difference in maintenance cost,
over a long period of time, the extra switching will affect the maintenance
cost by lowering the life cycle of the pump.
4.3. Optimal scheduling with rooftop water harvesting
Two scenarios are analyzed when incorporating rooftop water
harvesting. The optimal schedules of the valve and the pump are
shown in Fig. 5 in cases where there is no precipitation and when
1-mm of precipitation event occurs. The valve controlling the
municipal water into the tank (Fig. 5a and c) uses negligible
amount of power, hence it is allowed to operate throughout
the control horizon irrespective of whether it is peak or off-peak
in the TOU tariff.
This model has two state variables; height of water in the tank
and depth of water in the soil. The height of water in the tank,
hðjÞ ¼ vðjÞ Atank
, is obtained using Eq. (25) while the soil water variation
is obtained using Eq. (10). The variation of these variables during
the control horizon is shown in Figs. 6 and 7 respectively. In both
figures, left vertical axis represents the water level either in the
tank or soil while the rainfall event is shown by the right axis.
4.3.1. Scenario 1: Optimal schedule with no precipitation
With no precipitation, or zero rainfall recorded, the tank is
purely filled with utility water with the valve optimally operating
as shown in Fig. 5a. This results from the demand of water in the
lawn which is met by optimally operating the pump as shown in
Fig. 5b. The optimal schedule (Fig. 5a and b) have three tank filling
and two irrigation regimes to ensure that the water level in the
tank and the soil remain within the required height and depth
respectively. The solenoid valve switches on at 00:30 causing the
water level in the tank to rise to 0.65 m. The pump then switches
on between 01:00 and 01:30 leading to the water height in the
tank dropping to 0.18 m while the water level in the soil rises to
29.35 cm. The water height in the tank then rises again to 0