Virtual Synchronous Machine
VSG
36
Virtual Synchronous Machine
❑ To make the interaction between grid and generator as in a remote power dispatch.
❑ To make the reaction to transients as well as the full electrical effects of a rotating mass
❑ To provide primary reserve
❑ From the grid point of view, to make wind and PV conversion systems to be regarded as conventional
power stations
❑ To make the inverter based DER to exhibit inertia and damping properties similar of conventional
synchronous machines for short time interval
37
Synchronous Machine
v  = rs i  +

1

2
Ks = 0
3

1
2

1
2
3
2
1
2
−
1 
2 

3
−
2 

1 
2 
−
dλ 
dt
λ  =  L s
i  
l sr   
 id 
 Ls
Ls = 
0
0
Ls 
v  = rs i  + L s
e
 Lsfd sin(r ) 
l sr = 

−
L
cos(

)
r 
 sfd
di 
+ e
dt
cos(r ) 
=  d r 

sin(

)
r 

Te = i 
T
l sr
id =  d i T
r
 d = Lsfd id
cos(r ) 
 sin( ) 
r 

38
Synchronous Machine
v  = rs i  + L s
e
di 
+ e
dt
cos(r ) 
=  d r 

sin(

)
r 

Te =  d i 
T
cos(r ) 
 sin( ) 
r 

39
Synchronous Generator
v  = −rs i  − L s
e
di 
dt
cos(r ) 
=  d r 

sin(

)
r 

Te =  d i 
T
+ e
cos(r ) 
 sin( ) 
r 

d r 1
= (TI − Te − D p r )
dt
J
Qgap =  d r i 
T
d r
= r
dt
 − sin(r ) 
 cos( ) 
r 

T
QPOC

 0 1 
=  v 
 i 

 −1 0 

40
Virtual Synchronous Generator
41
Dynamic model – Single phase equivalent VSG connected to a stiff grid
❑ The non-linear dynamic equation that describes the behavior of the
frequency, power angle, and flux is:

d 1 
V  sin() Po
=  − Dp  −
+
+ D p o 
dt J 
XL
o

d
=  − g
dt

d 1 
2 2 V
=  − DqV + Qo −
+
cos() + Dq Eo 
dt K 
XL
XL

42
Dynamic model – VSG connected to a Stiff Grid
❑ The non-linear dynamic equation that describe the behavior of the,
frequency, power angle, and flux of the VSG is:
d
= a11 + a12 sin() + b11
dt
d
=  + b12
dt
d
= a32 2  2 + a33 cos() + b31
dt
a11 =
− Dp
J
−V
a12 =
JX L
a32 =
−1
KX L
a33 =
V
KX L
b11 =
1 Po
( + D p o )
J o
b21 = −g
b31 =
1
(Qo + Dq ( Eo − V ))
K
43
Dynamic model – VSG connected to the Grid
0 = a11 + a12 sin() + b11
0 =  + b12
0 = a322 2 + a33 cos() + b31
This algebraic equation has an solutions:
(o , o ,  o )
The local qualitative behavior of the VSG can be analyzed from the eigenvalues of:
44
Example
Let us consider again a 33kVA single phase inverter connected to a 220V 60Hz grid where:
First, the droop coefficients can be found as:
Dp =
T
P
33000
=
=
= 4.64
 g  (260)(23)
Dq =
Q
33000
=
= 1500
E 0.1(220)
45
Example
Let us consider again a 33kVA single phase inverter connected to a 220V 60Hz grid where:
First, the droop coefficients can be found as:
Dp =
T
P
33000
=
=
= 4.64
  g  (2  60)(2 3)
Dq =
Q
33000
=
= 1500
 E 0.1(220)
46
Example
The time constants associated with the frequency and the voltage are:
J
f =
Dp
K
v =
g Dq
J =  f Dp
K = v g Dq
By selecting
 f = 0.1
v = 0.5
47
Qualitative Behavior
D p = 4.64
Dq = 1500
J = 0.464
K = 113110
X L = 0.257(pu)
48
Example
The relation between the states and the modes can be revealed from the participation
factors.
 1  -5.01 + j15.47 
  =  -5.01- j15.47 
 2 

  3  

-0.41
1
2
3
0.525 0.525
0

0.525 0.525
0

0
0
0.999 
49
Simulation
50
Damping Correction Action
d Te
D f ( )  ( −  g )
dt  d
S. Dong and C. Chen, "Adjusting Synchronverter Dynamic
Response Speed via Damping Correction Loop,“
publishied in 2018 IEEE Power & Energy Society General
Meeting (PESGM), Portland, OR, 2018, pp. 1-1”
51
Damping Correction Action
52
Simulation
53