2005 IEEE/PES Transmission and Distribution
Conference & Exhibition: Asia and Pacific
Dalian, China
1
Modeling of static synchronous series
compensator in Newton power flow calculation
in PSASP
ZHAO Jianjun, GUO Jianbo and ZHOU Xiaoxin, Fellow, IEEE
Abstract-- This paper investigated a model of SSSC in power
flow calculation which considering characteristic of injected
voltage of SSSC and the control constraints of inverter in Newton
power flow analysis in PSASP (Power system analysis software
package). The key problem for developing a suitable model of
SSSC for power system and power flow calculation in bulk
power grid is resolved. The UD (User-defined model) and UPI
(User program interface) function of PSASP have been used for
generating the controllable injection voltage. The modeling could
provide both capacitive and inductive compensation through
regulating phase angle of injection voltage. The analysis results
based on EPRI-7 system are presented to demonstrate the
performance of SSSC power flow model and the satisfactory
steady state performance also shown the effective regulation of
SSSC.
Index Terms-- static synchronous series compensator, model,
steady state performance, power flow control
I. INTRODUCTION
The controllable series compensation technology is more
efficient in transmission line power flow control, enhancing
system stability and damping SSR, therefore we can fully
utilize the existing transmission facilities without reducing
system stability and security margins by applying this
technology.
SSSC is a new controllable series compensation technology
which inserts a controllable series voltage into the
transmission line [1]. The magnitude of almost sinusoidal
injected voltage can be adjusted to influencing the
transmission line power flow and the phase angle of voltage is
almost in quadrature with the line current. SSSC can adjust
power flow of transmission line even to change the direction.
Power flow calculation is a basic of operating, planning,
and controlling of power grid. The size and complexity of
bulk power system in China are increased over past several
years and will be the largest interconnected power grid in
future. SSSC is a new controllable series compensation device
This research is supported by National Natural Science Foundation of
China (No.50477055).
ZHAO JianJun is with the China Electric Power Research Institute,
QingHe,Beijing ,100085,P.R.CHINA (e-mail:zhaojj@epri.ac.cn)
GUO Jianbo is with the China Electric Power Research Institute,
QingHe,Beijing ,100085,P.R.CHINA (e-mail:jbguo@epri.ac.cn)
ZHOU Xiaoxin is with the China Electric Power Research Institute,
QingHe,Beijing ,100085,P.R.CHINA (e-mail:xxzhou@epri.ac.cn)
0-7803-9114-4/05/$20.00 ©2005 IEEE.
1
with technique benefits and the analysis of characteristics of
SSSC is an essential issue.
A power flow model and optimized method of IPFC and
GUPFC were studied by X.P, Zhang. The performance of
IPFC for power flow control is investigated [9][11]. An
injection model of series devices were presented,
mathematical models and the performance of series devices is
derived [8]. An injection model of power flow calculation of
UPFC is applied by M.Noroozian and a power flow optimized
method through UPFC is presented. The proposed model is
used to demonstrate some of the features of UPFC for optimal
power flow control application [10]. Literatures [1-7] are
focused on mathematical model and steady state
characteristics of SSSC, the model used in power flow
analysis is not considered.
This paper introduces the essential principles and
characteristics of SSSC which are also the basic of
performance of steady state study and detailed mathematic
model. A method of injecting current from additional nodes is
presented by analyzing the influence of nodes power equation
by injection voltage and the modification measure of
admittance matrix and Jacobin matrix, the benefits of the new
method are also analyzed which make the calculation of
matrix modification simplified.
This paper, based on the theory analysis, investigated a
power flow calculation model of SSSC in PSASP. PSASP is a
wildly used power system analysis program in china and the
UD/UPI function, the user interface provided by PSASP, are
the basic of modeling. The correctness of theory analysis and
the power control function of SSSC model are certificated
through simulating in EPRI-7 nodes system.
II. BASIC PRINCIPLES AND TRANSMISSION CHARACTERISTICS
OF SSSC
A. Basic principles
SSSC is a new controllable series compensation technology
which can adjust line power flow through injected voltage
which is almost in quadrature with the line current. Fig.1
shows a single line diagram of a simple transmission line with
SSSC.
2
X1
I
III. POWER FLOW ANALYSIS IN POWER GRID WITH SSSC
X2
Vss
SSSC is a compensation measure aimed to longitudinal
elements of transmission lines, it’s control object is not
modify the branches parameters, as TCSC done, but is
materialized the control effect of injection voltage. A method
of modifying admittance matrix through additional nodes and
modifying Jacobin matrix by additional injection quantity is
presented which resolve the problem satisfactorily.
V2
V1
Converter
Controller
A. Modifying admittance matrix
Fig.1 Single line diagram of SSSC
The allocation of SSSC in transmission line is arbitrary
which could be achieved by added two nodes in compensated
transmission line and the admittance matrix is also modified.
It could be supposed that SSSC is installed at send end of
'
'
transmission line, show in Fig.4, nodes i and j are added in
branch ij, y ss is additional admittance which consider the
loss of series transformer and inverter circuit of SSSC.
B. Transmission characteristics
Line current I has not influence on the amplitude of
injected voltage U in . The phase angle of injected voltage,
lagging phase angle of line current 90˚ in capacitive
compensation mode, makes line current increasing and
thereby enhancing power flow without changing power angle.
In inductive compensation mode the condition is reversed, it is
showed in Fig.2. Therefore, SSSC can control line power flow
flexible and improving power system stability obviously.
V1
V•sj
jIX 2
•
V2
•
Vsi
jIX 2
•
Vsj
Vsi
δ
θ
•
•
Vss
δ
θ
•
V2
•
I
I
No compensation
•
•
I
Capacitive compensation
jIX 1
Power system
•
Vss
Yij1
jIX 2
V2
i
Inductive compensation
Active power and reactive power of transmission line with
SSSC are described by (1) and (2) respectively.
(1)
Vss
V1V2 sin δ
Q=
x1 + x2
(1 −
V12 + V22 − 2V1V2 cos δ
(2)
Fig. 3 shows the transmission characteristics of SSSC, the
changing of Vss can improve transmission characteristics,
increase power system transfer capability and enhance system
stability.
1.4
0.9
0.4
0
10
20
30
40
50
60
70
80
Yij
‘
i
‘
j
j
Despite two additional nodes, a little amount of calculation
of modification admittance matrix is needed. The modified
admittance matrix is presented in (3).
0
0 
Y11 L Y1i L Y1 j L Y1n
 M
M
M
M
M
M 


0 
Yi1 L Yii' L 0 L Yin Yii '


M
M
M
M
M 
(3)
 M
'
'
0 Y jj ' 
Y B = Y j1 L 0 L Y jj L Y jn


M
M
M
M
M
M 

Yn1 L Yni L Y nj L Ynn
0
0 


 0 L Y i ' i L 0 L 0 Yi ' i ' Y i ' j ' 
 0 L 0 L Y ' L 0 Y '' Y 
j j
ji
j' j' 

Where the modify value is:
'
Yii' = Yii − yij + yij1 ； Y jj = Y jj − yij + yij 2
)
Vss
V1 (V1 −V2 cosδ )
)(1 −
)
x1 + x2
V12 + V22 − 2V1V2 cosδ
-0.1
Yij 2
Fig.4 The admittance matrix is modified by adding two additional
nodes in transmission line
Fig.2 Phasor diagram of SSSC
P=
j
Yij
•
V1
jIX 1
•
jIX 1
θ
i
•
•
V1
δ
Power system
Yii ' = Yi ' i = − yij1 ； Y jj = Y j j = − yij 2
'
90 100 110 120 130 140 150 160 170 180
'
Yi 'i ' = y ij1 + y ss ； Y j j = y ij 2 + y ss
' '
-0.6
L-0.4
without
C-0.4
Yi ' j ' = Y j 'i ' = y ss
Fig.3 SSSC Transmission characteristics
The other elements were not changed due to have no
relationship with additional nodes.
2
3
B. Power equation
According to Fig.4, the injection voltage Vss∠θ ss , which is
∆Pi ' j ' = Pi ' j ' − Pijref , ∆Qi ' j ' = Qi ' j ' − Qijref
Two state variables, θss and V ss , are added, θss is changed
derived by SSSC, will affect the power equation of nodes i '
and j ' while the other nodes power equation have not
along with current of corresponding transmission line, and
θin = θij ± 90° where θ ij is the phase angle of current of
changed. The power equation of nodes i ' and j ' are given by
corresponding transmission line. The constraints of power
flow is
Vss min ≤ Vss ≤ Vss max
Pi ' = Vi ' Gi 'i ' +Vi ' [Vi (Gi 'i cosδi 'i + Bi 'i sinδi 'i )
2
+V j ' (Gi ' j ' cosδi ' j ' + Bi ' j ' sinδi ' j ' )]
θ ss = θij ± 90° ， − π ≤ θ ij ≤ π
+Vi 'Vss [Gi ' j ' cos(δi ' −θss ) + Bi ' j ' sin(δi ' −θss )]
Pij = Pijref , Qij = Qijref
Qi ' = −Vi '2 Bi 'i ' + Vi ' [Vi (Gi 'i sin δ i 'i − Bi 'i cos δ i 'i )
(4)
+ V j ' (Gi ' j ' sin δ i ' j ' − Bi ' j ' cos δ i ' j ' )]
IV. MODELING OF POWER FLOW CONTROLLER OF SSSC IN
PSASP
+ Vi 'Vss (Gi ' j ' sin(δ i ' − θ in ) − Bi ' j ' cos(δ i ' − θ in ))
Pi ' j ' = Vi '2Gi 'i ' +Vi 'V j ' (Gi ' j ' cosδi ' j ' + Bi ' j ' sinδi ' j ' )]
A. The method of injecting current from additional nodes
SSSC is a compensation measure aimed to longitudinal
elements of transmission lines, it’s control object is not
modify the branches parameters but is materialized the control
effect of injection voltage. The theory of voltage source and
current source exchanging is adopted so that the injection
voltage could be equal to injection current which is injected in
additional nodes. It is showed in Fig.5 and Fig.6.
+Vi 'Vss [Gi ' j ' cos(δi ' −θss ) + Bi ' j ' sin(δi ' −θss )]
Qi ' j ' = −Vi '2 Bi 'i ' + Vi 'V j ' (Gi ' j ' sin δ i ' j ' − Bi ' j ' cos δ i ' j ' )]
(5)
+ Vi 'Vss (Gi ' j ' sin(δ i ' − θ ss ) − Bi ' j ' cos(δ i ' − θ ss ))
It could derived from equation (5)
P =P
+ ∆P ； Q = Q
+ ∆Q
i' j '
i' j' ( 0)
i' j'
i' j'
i' j' ( 0)
i' j'
Where P ' ' is transmission line power flow without SSSC,
i j (0)
∆Pi ' j ' is increment of power flow after compensation and is
Il
also the injection power of SSSC.
C. Modifying Jacobin matrix
The injection voltage effeteness of Jacobin matrix can be
presented as
 ∂P1
 ∂δ
 1
 ∂Q1
 ∂δ
 1
 M
∂P
 ∆P1   i
 ∆Q   ∂δ1
 1   ∂Q
 M   i
  ∂δ1

 ∆Pi   M
 ∆Qi   ∂Pj
 

 M   ∂δ1
 ∆Pj  ∂Q
= j

 ∆Qj   ∂δ1
 M   M
  ∂P

 ∆Pn   n
 ∆P'   ∂δ1
 i   ∂P'
 ∆Pj'   i
 ∆P   ∂δ1
 i' j'   ∂P '
∆Qi' j'   j
 ∂δ
 1

 0


 0

∂P1
V1
∂V1
∂Q1
V1
∂V1
M
∂Pi
V1
∂V1
∂Qi
V1
∂V1
M
∂Pj
V1
∂V1
∂Qj
V1
∂V1
M
∂Pn
V1
∂V1
∂Pi'
V
∂V1
∂Pj'
V1
∂V1
0
0
∂P1
∂P1
Vi
∂δi
∂Vi
∂Q1 ∂Q1
L
Vi
∂δi
∂Vi
M
M
M
∂Pi
∂Pi
L
Vi
∂δi
∂Vi
∂Qi ∂Qi
L
Vi
∂δi
∂Vi
M
M
M
∂Pj
∂Pj
L
Vi
∂δi
∂Vi
∂Qj ∂Qj
L
Vi
∂δi
∂Vi
M
M
M
∂Pn
∂Pn
L
Vi
∂δi
∂Vi
∂Pi'
∂Pi'
M
M
Vi
∂δi
∂Vi
∂Pj' ∂Pj'
L
Vi
∂δi
∂Vi
∂Pi' j' ∂Pi' j'
L
Vi
∂δi
∂Vi
∂Qi' j' ∂Qi' j'
L
Vi
∂δi
∂Vi
L
∂P1
∂δ j
∂Q1
L
∂δ j
M
M
∂Pi
L
∂δ j
∂Qi
L
∂δ j
M
M
∂Pj
L
∂δ j
∂Qj
L
∂δ j
M
M
∂Pn
L
∂δ j
∂Pi'
M
∂δ j
∂Pj'
L
∂δ j
∂Pi' j'
0
∂δ j
∂Qi' j'
0
∂δ j
L
∂P1
Vj
∂Vj
∂Q1
Vj
∂Vj
M
∂Pi
Vj
∂Vj
∂Qi
Vj
∂Vj
M
∂Pj
Vj
∂Vj
∂Qj
Vj
∂Vj
M
∂Pn
Vj
∂Vj
∂Pi'
Vj
∂Vj
∂Pj'
Vj
∂Vj
∂Pi' j'
Vj
∂Vj
∂Qi' j'
Vj
∂Vj
∂P1
∂δn
∂Q1
L
∂δn
M M
∂P
L i
∂δn
∂Qi
L
∂δn
M M
∂Pj
L
∂δn
∂Qj
L
∂δn
M M
∂P
L n
∂δn
∂Pi'
M
∂δn
∂Pj'
L
∂δn
L
0
0
0
0
∂P1
∂δi'
∂Q1
∂δi'
M
∂Pi
∂δi'
∂Qi
∂δi'
M
∂Pj
∂P1
∂δ j'
∂Q1
∂δ j'
M
∂Pi
∂δ j'
∂Qi
∂δ j'
M
∂Pj
M
∂Pi
∂θss
∂Qi
∂θss
0
∂Pj
∂δi'
∂Qj
∂δ j'
∂Qj
∂θss
∂Qj
∂δi'
M
∂Pn
∂δi'
∂Pi'
∂δ j'
M
∂Pn
∂δ j'
∂Pi'
∂θss
0
∂δi'
∂Pj'
∂δi'
∂Pi' j'
∂δi'
∂Qi' j'
∂δ j'
∂δ j'
∂Pj'
0
0
0
∂Pi'
∂θss
∂Pj'
∂δ j'
∂θss
∂Pi' j'
∂Pi' j'
Vi'
∂Vi'
∂θss
∂Qi' j'
∂Qi' j'
Vj'
∂Vj'
∂θss
j
i
Vss
Xs
Fig.5 The injection voltage scheme of SSSC
i



0 

M  ∆δ1 
 ∆V 
∂Pi
Vss  1 
∂Vss  V1 

∂Qi  M 
Vss 

∂Vss  ∆δi 
 ∆V
0  i 
∂Pj  Vi 
Vss 

∂Vss  M 

∂Qj  ∆δ j 
Vss  ∆V 
∂Vss  j 

0  Vj 
 M 
0 
 ∆δn 


 ∆δ ' 
∂Pi'
Vss  i 
∂Vss ∆δ j' 

∂Pj' ∆θss 
Vss ∆V 
ss
∂Vss


∂Pi' j'  Vss 
Vss 
∂Vss 
∂Qi' j' 
Vss 
∂Vss 

i'
0
I ss
j
i'
bss
I ss
Fig.6 The injection current scheme of SSSC by adding two additional
nodes
The injection voltage of SSSC is presented by injection
current of bus i ' and bus j ' which could derived by
•
Il
I ss = ±bss k
Il
•
(7)
Where bss is admittance of series transformer and DC
circuit of SSSC.
•
I l is current of transmission line which is compensated
by SSSC.
k is proportion factor of injection voltage and equivalent
injection current.
The method of injecting current from additional nodes
increase dimension of admittance matrix so that the amount of
modify calculation is more than those in directly modify
admittance of branches, but the amount of modify calculation
of Jacobin matrix is simplified because the influence of
(6)
Where ∆Pi and ∆Qi are mismatch value of active power
and reactive power of bus i, Pi and Qi are active power and
reactive power injected into bus i (i=1-n). The control effect
could be presented by the last four rows and the last four
columns in (6), ∆P ' ' and ∆Q ' ' are active and reactive power
i j
i j
control portion respectively.
'
injection current is limited in nodes i , i , j and j ' which could
3
4
be derived from (4). If injection voltage is directly added in
branch ij that Power equation of all branches corresponding to
node i and node j will have an append term Vss , the amount of
14
12
10
8
calculation is increased.
6
B. The power flow program of SSSC in PSASP
PSASP is broadly used as power system analysis soft ware
package in China which including models of SVC, TCSC,
HVDC in power flow program and transient analysis
program[12][13]. The control function are achieved by special
subprogram which exchange datum with main power flow
program through interface program so that it could develop
new function by commerce software package without
developing full power flow software.
PSASP provide convenient user interface function which
could make user achieved additional calculate and control
function through UPI and UD. The model of SSSC of power
flow analysis is achieved through control element and I/O
interface variables provided by UD. The UD model has the
characteristic of independent and ease to using, so the SSSC
model could be used conveniently in any power system. Fig.7
is the frame of power flow control module.
4
2
0
0
2
4
6
8
10
Pl1
12
14
16
18
Pl2
20
22
24
26
Pl3
28
30t(s)
28
30
Pl5
Fig.9 The power flow control by SSSC
0
2
4
6
8
10
12
14
16
18
20
22
24
26
0.2
0.15
0.1
0.05
0
-0.05
-0.1
t(s)
Fig.10 The injection voltage of SSSC
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
12
10
8
6
4
Input datum of UD mod el
2
Main program
Controller of
UD model
of PSASP
power flow
0
t(s)
-2
Qij
softwarer
Output datum of UD mod el
Pij
Il1
Fig.11 The injection active and reactive power of SSSC
Fig.7 The power flow control frame of SSSC
The active power control is speedily and exactly by SSSC.
SSSC could redistribute active power as needed between two
shunt lines. The value of Vss is 0.20 which increase the active
EPRI-7 system has two machines and an infinite bus which
connected through double circuit and a HVDC line, showed in
Fig.8. SSSC is installed at L1 near bus B1-500 which controls
the power flow of L1.
G1
B1 - 500
1 : 1.05
L2
0.0038+ j0.0406/1.088
Bj
Bi
0.0 + j0.01
B4 - 500
L4
B3 - 500
0.0038+ j0.0406/1.088
1 : 1.05
0.0038+ j0.0406/1.088
0.0038+ j0.0406/1.088
L3
S1
0.0 + j0.0067
0.0 + j0.02
1 : 1.0
j0.6666
L1
power of L1 from 680MW to 1000MW and transfer capability
increased about 47% while the injection power of SSSC is
195Mvar.
The phasor diagram of corresponding node voltages is
showed at Fig.12 (a). Not only the transfer capability is
increased but also the difference of angle between sending end
and receiving end is decreased from 19 .97 ° to 11 .47 ° .
Due to influence of injection voltage, the voltage between
node i and node j is reversed which has showed in Fig.12(b)
and the simulation result also testify the correctness of model.
B2 - 500
G2
1 : 1.05
Converter
Controller
SSSC
L5
0.8
0.0 + j0.0067
VB1
0.7
VBj
VBj
0.6
Fig.8 The EPRI-7 power system with SSSC
VB1
0.5
VB4
0.4
C. Case analysis
0.3
Case1：Capacitive compensation
The control target of active power of L1 is 1000MW and
the simulation result is showed in Fig.9 to Fig. 11.
VB4
0.2
0.1
0
0
0.2
Initial condition
0.4
0.6
0.8
1
With SSSC
（a）
4
5
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
are compared. SSSC could achieve the same control purpose
with lower output power and the benefits of SSSC are more
evident than TCSC especial in lower transmission power
condition. The main works of this paper as follows.
The characteristics and mathematic model are analyzed.
A method of injecting current from additional nodes is
presented which materialized the modification of admittance
matrix and Jacobin matrix.
The model of SSSC used in power flow analysis is
presented and the correctness is validated.
The characteristics of SSSC are compared with those of
TCSC and the benefits of SSSC are illuminated by simulation
results.
0.15
0.5
0.4
3
1
0.3
3
1
0.2
0.1
2
0
-0.1
2
-0.2
Initial condition
•
With SSSC
•
•
VI. REFERENCES
（b）
•
•
[1]
•
1＝ V B1 − V B 4 、2＝ V Bi − V Bj 、3＝ V Bj − V B 4
Fig.12 The influence of nodes voltage by SSSC
[2]
Case2：Inductive compensation
The simulation result of inductive compensation is same as
capacitive compensation which is showed in Fig.13 and
Fig.14.
[3]
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
[4]
30
8
7
6
5
4
3
2
1
0
-1
-2
[5]
[6]
t(s)
Qij
[7]
Pi
Fig.13 The power flow control by SSSC
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
[8]
0.2
0.15
[9]
0.1
0.05
0
[10]
t(s)
-0.05
Fig.14 The injection voltage of SSSC
[11]
V. CONCLUSIONS
[12]
This paper investigated a practical model of SSSC in power
flow analysis. The method of injecting current from additional
nodes is presented by a modification measure of admittance
matrix and Jacobin matrix, and the benefits of the new method
are analyzed. A model of SSSC in power flow analysis is
realized through the user interface programmer of PSASP and
the problem of power flow analysis in bulk power system with
SSSC is resolved. The correctness of theory analysis and
power control function of SSSC model are certificated
through simulation in EPRI-7 nodes system and control
characteristics of TCSC and SSSC in same control condition
[13]
5
Kalyan K. Sen, SSSC-Static Synchronous Series Compensator : Theory,
Modeling, and Applications[J], IEEE Trans. on Power Delivery, Vol. 13,
No.1, January 1998,241-246
Laszlo Gyugyi, Colin D. Schauder, Kalyan K. Sen, Static Synchronous
Series Compensator: A Solid-State Approach to The Series
Compensation of Transmission Lines [J] ， IEEE Trans. on Power
Delivery，Vol.12,No.1,January 1997,406-417
R. Mihalič,I. Papič, Mathematical Models and Simulation of a Static
Synchronous Series Compensator[C] ， University of Ljubljana
Ljubljana, Slovenia
Kalyan K.Sen, Eric J Stacey, UPFC-Unified power flow controller
Theory, Modeling and Applications[J]，IEEE Trans. on Power Delivery,
Vol. 13, No. 4, October 1998.1453-1460
Edvina Uzunovic, Claudio A. Canizares, John Reeve, Fundamental
Frequency Model of Unified Power Flow Controller[C] ， North
American Power Symposium (NAPS), San Luis Obispo, California,
October 1999
L. Sunil Kumar, Arindam Ghosh, Modeling and control design of a
static synchronous series compensator, IEEE Trans. on Power
Delivery，Vol.14, No.4 , pp.1448-1453, 1999
A. Nabavi-Niaki and M.R. Iravani, Steady-State and Dynamic Models of
Unified Power Flow Controller (UPFC) for Power System Studies [J]，
IEEE Trans. on Power Systems, Vol. 11, No.4, November 1996,
pp.1937-1943.
Mehrdad Ghandhari,Göran Andersson,Ian A. Hiskens, Control
Lyapunov Functions for Controllable Series Devices，IEEE Trans. on
Power Systems, VOL. 16, NO. 4, Nov.2001
X.P.Zhang, Modeling of the interline power flow controller and the
generalized unified power flow controller in Newton power flow，IEE
Proc.-Gener. Transm. Distrib., Vol. 150,No.3,May 2003
M.Noroozian,L.Angquist,M.Ghandhari,G.Andersson，Use of UPFC for
optimal power flow control，IEEE Trans. on Power Delivery，Vol.12,
No.4, 1997, pp.1629-1634
Xiao-Ping Zhang,Edmund Handschin, Modeling of the Generalized
Unified Power Flow Controller (GUPFC) in a Nonlinear Interior Point
OPF，IEEE Trans. on Power System，VOL. 16, NO. 3, AUG 2001.
Power system analysis software package, load flow calculation user’s
manual，Electric Power Research Institute of China.
Power system analysis software package, User-defined model& User
program interface, Electric Power Research Institute of China.