Summary Electrical Power and Energy Systems | Universiteit Gent | 2025/26

Single phase AL
systems

FundamentaConcept

*
Voltage ,
Current and Power
~ F and V = Sinusoidal



Vejt
{ v(t) = VIV iin (wt + 2) =




(F(wt
òB
- vol Boz
campee ament
+ ie
ー
. =
comple phason
¿ lE ) 9 〆
= -

B
_
= VIas φ + VIG(2wt + d +
3)
〜 _
ct remm 2nd harmonic term
Complex Power S[VA] = active Power




yy
*
S =
VE -
+ S = 1S) = VI
v =
Ca) + jVisin 1=
apparent Power (
=




=

PtjQ
6
L Reactive
3
active Power PF(-)
P[W] Q CArJ
Power
-
Powver
factor
P =
VIcs9 Q =

VIsiny PF =
号 =
coo 9


Alternative
definitions ple =CaYIc
=




=
P(1 + G(2wt + 2x)) + qsin(2wt + 2x)
eliminable Part
VI
>
-Q =
of the
power
S ; P Soe oscillation with retention
! Q Ssor of
= -
-



4) PF E effectiveness of Power transfer) active Power P

Louseful effect of the Power
transfer

J2 knchofflor
Conservation holds
Tellegem's for
:疑
* theorem =>
of power ∵ _




S = P +
jQ i

4。
是…
武
…

* Active and reactive Current

Ʃ
Active Current
* Fp = I caf
* Reactive curent Fa = -
Isin D



P =
VIP
Q =
-

FQV




人

,*
Sign Conventions PJO
90
:


:
N Produces active Power
N Produces Reactive Power
PJO : Nansumes active Power
970 : Nansumer Reactive Power

∞ { }

Power
*
Transfer
* "d2"approc . not
really-Vbut ,
looks like
in DC
√
I .
3
@




Zl Ryt =




1V = vA Vzv(RG)
-
+
xsinf)I

.
m


*x
D jXI
* in inductive
case
of grids ⑬ ㎡
a RE
ㅌ
☆ ㎡


↳
↳
acce lines
high reactoner >
-
reduce loses
limited
IRCy Xeiny) + I




Voltage difference ( in
magnitude and phose angles) acros lines are




#P
mainly related to
phose-angle difference Es jxy/ purely reactive(
=




* maxcimum active Prver :
Pmax-
6 V
{
PB = sin (5a-5B)

VB/VA(a(5a-5r) VB)
related
-




mainly voltage difference QB
* a to =


Xe
leads to
=
transport of reactive Power
reactive
Voltage drope

* Per-Unit vollues

L Numerical values
of physical quantities -
comparison to a
reference
Absolute
reference
* (SI unit)

application dependent reference
-




Voltage (11 ,
Current (I) , Power /3) and impedance (2)


*
references Vref Fref Srf
:
, ,
and
Zref

같
ref
;
δ=
荊 :
= : 王
=
Choice of references : * 2
independent Commonly Veef and
Frfother determined such that

izofof
,



Pref -
Vof Fref


口

, Three-phase AC
systems
fundamental concepte
*
Voltage , Current and Power

In
A
√
-a
V VN
√ab
ー



α
\

ㄴ *
iRe
'




‰

fes

-



line-to-neutral
voltages


v 5 。 怎论了證 cveik atra
Nalt) = VEVw Co/wt + 2)
号 )=
_witha
ja3

, Protectioover
n tw
☆v
:
= e

=
a to = ejhty

linz-to-line
Voltager
v5/a es )
*

Ys)
V e5(
+


Jae = Va-Ver
_
= =




Verc = Ne- _
Vs= a2 fac ↳ k = V5KN

Via = =<- La = a ·Face




line currents




)
}
m 咖毙
(alt) = V& I CalWt + B)
-2
j( ) α
ieltl =VIIG/
B




Ut + B
-
2s = a 5
= a
4π()
VE 44/) [eJ/B
-




ic(t) =
[((wt +
B
-



Ec = =
@ In



Im
St nYa Three-phase balanced
systems
*I 뷰
all determined
I
currents
vollager
* ≡=±a and

µβ V and
D
by @
single Voltage
IRe Current E

Single-Phose repree. of
balanced
3f-syst .

, # Power


(t)
↑ =
Na(t) (a(t) + Ny(t) (g(t) + v,
(t)(2(t)
= 3VNI cos(2 -


B)
= 3UNICOY =
VIVL I cat


= Constant Power



p(t) = P
"usefull effect of Power transfer.



S =
32E *
=

PtjQ
P =
Scy 3UNIcy = =
VVICaf
& =
Seiny Inising =
= vVLFeinf

Some
↳
principles or with
Single-phase
Single-Phase system
#
equivalent

Consider : * 3-1 rym .
volt .
and current souce

*
3-f Sym .



grid components and loads
equal imped. in 3 phases
*
equal effects --b b a ,
- c
, - a

“ "

的 a x
-



,
c -> b
, b -
> a




and currents the balanced I three-phase
voltages Symmetrical
in
=> All
system are




The ?
=>
three-phase system can be reduced to an
equir .




Single-phase system

Va =
Zs EatEme tet Zma Ec

lEs
=
+ a Em + a Emc) Ea



lonologous for phases G and a


V
=> =

Eeg I


With at Em
Zeg = Es + + a EmC




sta _




Δ a =
VAa-IBa

=
Es Ea + Im Ie + Em Ec
=
(Es + at zm + aZm) Fa
=
1≡ - Zm ) Ea
0




Janalogous for phoss b and c


S上 Z

雪g
=
=



20 - m
=