CHAPTER 4: OPTIMAL ENERGY CONTROL OF A GRID-CONNECTED SOLAR-WIND
4.3 DESIGN MODEL
The PV and The PW are described in detail in the previous chapter.
4.3.1 Pump hydro storage
In this chapter, the PHS system is selected as the energy storage system and consists of a pump unit and a turbine generator unit.
4.3.1.1 Pump unit
The pump is directly supplied by the main bus where the renewable energy source is connected, and the flow rate drawn from the lower reservoir by the pump is expressed by equation (4.1),
) ( ) .
( .
3
3 c P k
gh k qin p P p
(4.1)
where P3(k) is the power supplied by the main bus to the pump, h is the elevating head (m), g is the acceleration due to gravity
9 . 8
m/
s2 ,
is the density of water 1000
kg/
m3
,
p is the overall pumping efficiency and cp is the water pumping coefficient of the pump unit(m3/kWh)4.3.1.2 Turbine generator
When the renewable energy source cannot meet the demand, the upper reservoir is responsible for drawing the water to operate the generator and supply the main bus in order to turn the turbine.
This can be expressed by equation (4.2):
out t q c k
P4
( )
.
(4.2)36
Where:
P
4( k )
is the power supplied by the generator to the main bus, qout is the water volumetric flow rate input into the turbine(
m3/
s)
and ct is the turbine generating coefficient
kWh/
m3
.4.3.1.3 Water storage reservoir
The water storage reservoir is modelled to meet the load demand during the period when the renewable energy source is unavailable. In addition, the power output from the renewable energy and the load demand at an hour t, determines the charge (pumping) or discharge (generating) power into and out of the reservoir respectively, k is an integer representing the kth hour interval.
Thus, the flow of the water in the reservoir at any hour k, v(k), depends on the flow of the water in the previous hour. At any given hour, the water flow in the reservoir will be rendered by the expression:
1 ( )
3( ) 1
P4(
k)
k cP c t k V k
V
t
p . (4.3) The following general expression derived from equation (4.3) applies to the reservoir dynamics:
k
t i k
p i P k
tc k P c t V
k V
1 4
1 3( ) 1 ( )
. 0
1 (4.4)
where V(0) is considered the initial level of the water in the reservoir,
k
p i P k
c t
1 3( ) .
represents the pumping power at any time, and
k
t i
k c P
t
1 4( )
1 is the generating power at any time. The water in the upper reservoir must be maintained at a certain limit; therefore it must not be less than the minimum allowable level Vmin and not higher than the maximum allowable level of the water in the reservoirVmax.
max min V(k) V
V (4.5)
4.3.2 Optimisation
The TOU tariff scheme has been employed in the mining sector to facilitate better system performance and to save on the cost of electricity. In [223] the efficiency of applying a conveyor belt has been emphasised in the application of TOU to show that a great saving on energy costs is achievable. To maximise the energy supply to the grid by using renewable energy sources, optimal control must be introduced and the energy cost must be taken as a typical indicator to measure the sales performance of the grid. The total electricity cost in a period is the related energy cost in the period, which can be expressed as:
tf
t
C p k p k dk
E
0
) ( )
6
(
(4.6)37
where
[
t0,
tf]
is the period of a total cost calculation, p6(
k)
is the power flow to the grid from the sampling timeN t t tf o
, and p(k)is the TOU tariff function. The equation (4.6) is discretised by equation (4.7):
) ( ) (
1 p6 k p k t
J k
i
(4.7) where t is the sampling time and p(k) is the electricity price at the kthsampling time.
4.3.2.1 Objection function
The first objective is to maximise the sale of energy to the grid by taking advantage of the TOU electricity tariff, which is also used as input to the proposed model control strategies. The TOU tariff is an important parameter in this model. As stated earlier, the maximisation of energy sales is achieved by selling more energy to the national grid during peak periods when the cost of energy is higher than during off-peak and standard periods.
The classification of the TOU input parameter is off-peak, standard and peak, which also differs during the low and high demand seasons. Consequently, the objective function is presented by the following expression:
k
i
k P k P
1 6( ) ( )
max . (4.8) The second objective is to minimise the energy supplied by the main bus to the pump
(
P3)
and is expressed by:
ki
k P
1 3( )
min . (4.9) The combination of the two objective functions renders the multi-objective function for the purpose of optimisation and is expressed by:
k
i k
i
k p k
p k p
1 3
1 6( ) min ( )
max
. (4.10)
4.3.2.2 Constraints
The constraints of this optimal control problem are listed as follows:
1. Equation (4.11) implies that the power supplied by the PV, PW and the hydro pump is equal to the total power supplied and the Kirchhoff law, which is applied to determine the total power in the main bus.
P1(k)P2(k)P4(k)P3(k)P5(k)P6(k) (4.11)
38
2. Equation (4.12) ensures that the power supplied by the PV, PW and the hydro pump at any hour of the day equals that of the local load and the load to be supplied to the grid during the peak period.
P1
(
k)
P2(
k)
P4(
k)
P5(
k)
P6(
k)
(4.12) 3. Equation (4.13) ensures that the power supplied by the PV, PW and the hydro pump directlyto the local load or to the grid is greater than or equal to zero.
P1
(
k)
0 ;
P2(
k)
0 ;
P4(
k)
0
(4.13) 4. Each energy source i is constrained by minimum and maximum values as specified inequation (4.14):
Pimin Pi
(
k)
Pimax (4.14) 5. Equation (4.15) ensures that the water in the upper reservoir will not be less than theminimum value or higher than the maximum value
4 max
1 1 3
min (0) . ( ) .1 P (k) V
t c k P c t V
V k
t i k
p i
. (4.15)
For all t=1, ….N is the sampling time.