Problem 1.1) Assume the AC frequency is 50 Hz. Convert these phasors to signals (remember that
phasor magnitudes are RMS values of the signals)
(a) V = 90∠(π/2) V
It’s voltage is RMS value, so we need to convert maximum value. 90=x/(√2) => x=127,3Vmax
π/2 is radian value, so we can convert it to degree. π/2=90 °
w=2πf = 2*π*50=100 π=314
V=127,3.cos(314t+90°)V
(b) I = 8 − 2j A
√(8^2+(-2)^2)=8,25
Ɵ=arctan(-2/8)= -14 °
I=8,25∠(-14°) A
It’s current is RMS value, so we need to convert maximum value. 8,25=x/(√2) => x=11,66 Amax
I=11,66.cos(314t-14°) A
Problem 1.2) Convert these signals to phasors (remember to use RMS values for the phasor
magnitude)
a) v(t) = 230 cos(314t + π/4) V
230 is a maximum value of voltage, we must convert it to RMS value.
230/√2=162,63V
w=2πf = 314=> f=49,97Hertz
V=162,63∠(π/4) V, at f=49,97Hertz
b)
i(t) = 3 cos(314t + π/8) A
3 is a maximum value of current, we must convert it to RMS value.
3/√2=2,12A
w=2πf = 314=> f=49,97Hertz
A=2,12∠(π/8) A, at f=49,97Hertz
Problem 1.3) Suppose the voltage across a load is v(t) and the current drawn is i(t) where
v(t) = 230 cos(314t + π/4)V
i(t) = 3 cos(314t + π/8)A
(a) calculate the complex power, real power, reactive power and apparent power
Vm=230V, Im=3A, Vrms=Vm∠(Ɵv) /√2=162,63∠(π/4)V
Irms=Im∠(Ɵi) /√2=
2,12∠(π/8)A
Complex power s=Vrms*(Irms’)=344,77∠(π/8) VA
Real power p=344,77*cos(π/8)=3118,52V
Imaginer power Q=344,77*sin (π/8)=131,9VAR
Apperant power=344,77VA
(b) calculate the power factor and indicate if it is leading or lagging
power factor= real power/apperant power => 318,52V/344,77VA=0,923 (uniteless), it is
lagging because angle of phasor positive, π/8.
Problem 4) Subsidy Reform
There are two groups of DEWA residential customers: Nationals and Expats. The total Dubai
population in 2018 was 3.14 million. Nationals constitute about 15% of Dubai population. The total
electricity consumption for DEWA residential customers is 12795 GWh. On average, Dubai National
consumers use 0.7 kW or electricity every hour of the day per person. Nationals and Expats are
charged different electricity tariffs. In reality, there is a 4 tiered electricity tariff structure where
prices depend on usage levels. But for simplicity, assume fixed electricity rates that do not depend on
usage. After Dubai Electricity Subsidy Reform (2011), Nationals pay $0.03/kWh and Expats pay
$0.11/kWh.
Problem 4.1) Demographics and Electricity Consumption Use the data provided to complete the
table below.
Population calculations
Total Dubai population: 3,14Million
Nationals: %15*3,14Million=0,471Million
Expats:
%75*3,14Miliion=2,355Million
Electric consumption calculations
Nationals average consumption per person: 0,7kWh
Nationals average consumption for all people: 0,7kWh*0,471M=329,7MWh
Nationals average consumption for all people for year: 329,7MWh*24H*365D= 2888,17GWh
Expacts average consumption for all for year: 12795GWh-2888,17GWh=9906,83GWh
Expacts average consumption for all year per person: 9906,83GWh/2,355M=4,206MWh
Expacts average consumption for per person: 4,206MWh/(24H*365D)=0,48kWh
Population
Millions
Expacts
Nationals
Dubai Total
2,355M
0,471M
3,14M
Annual
Consumption
GWh
9906,83GWh
2888,17GWh
12795GWh
Annual per capita
consumption
kWh
0,4206GWh
0,3297GWh
0,750,GWh
Hourly per capita
consumption
kWh
0,48kWh
0,70kWh
1,18kWh
Problem 4.2) Existing Tariff: Revenue, Costs, and Profit The average cost of generating electricity for
DEWA is $0.08/kWhr. Use your results from the previous part to complete the table below
Retail Price
Expacts
Nationals
Dubai Total
$0.11/kWh
$0.03/kWh
-
DEWA annual
revenue M$
1089,75 M$
86,64 M$
1176,39 M$
DEWA annual
gen cost M$
792,54 M$
231,05 M$
1023,59 M$
DEWA annual
profit M$
+297,21 M$
-144,41 M$
+152,8 M$
Problem 4.3) Analysis
(a) How much does the current policy of subsidizing Dubai Nationals cost DEWA (gen cost revenue from Nationals)?
The current policy of subsidizing Dubai Nationals are cost 144,41M$, nevertheless, in the final
situation, DEWA's total profit 152,8M$.
(b) What is the average annual value of the subsidy for a family of 4 Nationals?
-144,41M$/(0,471Mpeople)=-306,60$ per 1 person for 1 year.
For 4 Nationals, it will be cost -306,60$*4=-1226,4$ found.