Advanced SVC models for newton-raphson load flow and ... - ITCJ

IEEE TRANSACTIONS ON POWER SYSTEMS. VOL. 15. NO. 1, FEBRUARY 2000 129
Advanced SVC Models for Newton-Raphson Load
Flow and Newton Optimal Power Flow Studies
H. Amhriz-PBrez, E. Acha, and C. R. Fuerte-Esquivel
Abstract-Advanced load flow models for the static VAR cumpensator
(SVC) are presented in this paper. The models are incorporated
into existing load flow (LF) and optimal power flow (OPF)
Newton algorithms. Unlike SVC models available in open literature,
the new models depart from the generator representation of
the SVC and are based instead on the variable shunt susceptance
concept. In particular, a SVC model which uses the firing angle
as the state variable provides key information for cases when the
load flow solution is used to initialize other power system appliedtions,
e.g., harmonic analysis. The SVC state variables are combined
with the nodal voltage magnitudes and angles of the network
in a single frame-of-reference fur U unified, iterative solution
through Newton methods. Both algorithms, the LF and the OPF
exhibit very strong convergence characteristics, regardless of network
size and the number of controllable devices. Results are presented
which demonstrate the prowess of the new SVC models.
Index Terms-FACTS, Newton method, OPF, SVC, voltage control.
I. INTRODUCTION
N ELECTRIC power systems, nodal voltages are signifi-
I cantly affected by load variations and by network topology
changes. Voltages can drop considerably and even collapse
when the network is operating under heavy loading. This
may trigger the operation of under-voltage relays and other
voltage sensitive controls, leading to extensive disconnection
of loads and thus adversely affecting consumers and company
revenue. On the other hand, when the load level in the system
is low, over-voltages can arise due to Ferranti effect. Capacitive
over-compcnsation and over-excitation of synchronous
machines can also occur [I]. Over-voltages cause equipment
failures due to insulation breakdown and produce magnetic
saturation in transformers, rcsulting in harmonic generation.
Hence, voltage magnitudc throughout the network cannot
deviate significantly from its nominal value if an efficient and
reliable operation of the power system is to bc achieved.
Voltage regulation is achieved by controlling the production,
absorption and flow of reactive power throughout the network.
Reactive power flows are minimized so as to reduce system
losses. Sources and sinks of reactive power, such as shunt capacitors,
shunt reactors, rotating synchronous condensers and
SVC’s are used for this purpose. Shunt capacitors and shunt
Manuscript received December 8, 1997; revised September 30,1998. H. Ambriz-P6rez
and C. R. Fuerte-Esqoivel were financially supported hy the Consejo
Nacional de Ciencia y Tecnologiil, MCxica.
H. Ambriz-PCrw and E. Acha are with the Department of Electronics and
Electrical Engineering, University of Glasgow, Scotland, UK.
C. R. Fuerte-Esquivel is with the Departamentu de Ingenieria EICctrica y
Electrhica, Institute Tccnol6gico de March, Mexico.
Publishcr Item Identifier S 0885-8950(110)01862-9.
reactors are either permanently connected to the network, or
switched on and off according to operative conditions. They
only provide passive compensation since their productionlabsorption
of reactive power depends on their ratings and local bus
voltage level. Conversely, the reactive power suppliedabsorbed
by rotating synchronous condensers and SVC’s is automatically
adjusted, attempting to maintain fixed voltage magnitude at the
connection points.
This paper focuses on the development of new SVC models
and their implementation in Newton-Raphson load flow and optimal
power flow algorithms. The SVC is taken to be a continuous,
variable-shunt susceptance, which is adjusted in order
to achieve a specified voltage magnitude while satisfying constraint
conditions.
Two models are presented in this paper:
I) SVC total susceptance model. A changing susceptance
R,,, represents the fundamental frequency equivalent
susceptance of all shunt modules making up the SVC.
This model is an improved version of SVC models
currently available in open literature.
2) SVC firing angle model. The equivalent susceptance Be,
which is function of a changing firing angle (Y, is made
up of the parallel combination of a thyristor controlled reactor
(TCR) equivalent admittance and a fixcd capacitive
susceptance. This is a new and more advanced SVC representation
than those that are currently available in open
literature. This model provides information on the SVC
firing angle required to achieve a given level of compensation.
The SVC models have been tested in a wide range of power
networks of varying sizes. Io this paper, an actual power network
consisting of 26 generators, 166 buses, 108 transmission lines
and 128 transformers is used as the test network.
11. STATIC VAR COMPENSATOR‘S EQUIVALENT SUSCEPTANCB
Advances in power electronics technology together with sophisticated
control methods made possible the development of
fast SVC‘s in the early 1970’s. The SVC consists of a group of
shunt-connected capacitors and reactors banks with fast control
action by means of thyristor switching. From the operational
point of view, the SVC can be secn as a variable shunt reactance
that adjusts automatically in response to changing system
operative conditions. Depending on the nature of the equivalent
SVC’s reactance, i.e., capacitive or inductive, the SVC draws
either capacitive or inductive current from the network. Snitable
control of this equivalent reactance allows voltage magnitude
regulation at the SVC point of connection. SVC’s achieve
0885-8950/00$10.00 0 2000 IEEE

XC fi'
Fig. I. SVC stmctiirc.
60 -
- E
J
40 -
Reactive region
-
6
20 -
U
"
c -
4 o-
n
r
+ -20 -
-
0
c3
.f -40 -
w LT
Fig. 2.
-60 1 , , , , , , , ,
- -
Capacitive region
90 100 110 120 130 40 150 160 170 180
Firing angle (degrees)
SVC equivalent reactancc as function of firing angle.
their main operating Characteristic at the expense of generating
harmonic currents and filters are employed with this kind of devices.
SVC's normally include a combination of mechanically controlled
and thyristor controlled shunt capacitors andreactors [U,
[Z]. The most popular configuration for continuously controlled
SVC's is the combination of either fix capacitor and thyristor
controlled reactor or thyristor switched capacitor and thyristor
controlled reactor [3], [41. As far as steady-stale analysis is concerned,
both configurations can be modeled along similar lines.
The SVC structure shown in Fig. I is used to derive a SVC
model that considers the TCR firing angle (Y as state variablc.
This is a new and more advanced SVC representation than those
currently available in open literature.
The variable TCR equivalent reactance, X.ce,, at fundamental
frequency, is given by [31,
wherc IY is the thyristor's firing angle.
The SVC effective reactance X,, is determined by the parallel
combination of X

AMBRIZPkREZ ctnl.: ADVANCE0 SVC MOUE1.S FOR NEWTON-RAPHSON LOAD FLOW AND NEWTON OPTIMAL POWER FLOW STUDIES 131
v4
Capacitive
ratine - Ir
Inductive
rating
I
I
IMlN 0 IM4X
System reactive
characteristics
b Is
-
112r
110-
108-
&? 106-
' 104 -
I
102-
-
e
g 100
- Generator model /'
--- Susceptonce model //
/'
,
Fig. 4. Comparison between actual and idcalized SVC voltage c~irrcnt
characteristic.
changing the SVC representation to a fixed reactive susceptance.
This combined model yields accurate results. However,
both representations require a different number of nodes.
The generator uses two or three nodes [4] whereas the fixed
susceptance uses only one node [I]. In Newton-Raphson load
flow implementations this may require Jacobian reordering and
redimensioning during the iterative solution. Also, extensive
checking becomes necessary in order to verify whether or not
the SVC can return to operation inside limits.
It must be remarked that for operation outside limits, it is important
to model the SVC as a susceptance and not as a generator
set at its violated limit, QvioliLtcc~. Ignoring this point will
lead to inaccurate results. The reason is that the amount of reactive
power drawn by the SVC is given by the product of the fixed
susceptance, nfiXed, and the nodal voltage magnitude, V, , Since
V, is a function of network operating conditions, the amount of
reactive power drawn by the fixed susceptance model may differ
from the reactive power drawn by the generator model, i.e.
Qviolalcd f - Rfixcd vb2. (4)
This point is exemplified in Fig. 5 where the reactive power
output of the generator was set at 100 MVAR. This value is constant
as it is voltage independent. The result given by the constant
susceptance model varies with nodal voltage magnitude. A
voltage range of 0.95-1 .OS was considered. A susceptance value
of 1 pa. on a 100 MVA base was used.
The SVC model presented above, based on the generator and
fixed susceptance representations, is better handled as a susceptance
model only. It takes the form of a variable susceptance
when the SVC is operating within reactive limits and it takes
the form of a fixed susceptance otherwise.
Moreover, a new and more advanced SVC representation then
becomes possible, wherc the thyristor firing angle mechanism is
represented explicitly as a function of network operating conditions.
In contrast to all SVC models proposed heretofore, this
model assesses ranges of firing angle operation leading to intrinsic,
internal SVC resonances. Both representations, the total
susceptance and the firing angle models, are presented below.
Fig. 5.1.
-
Variable shunt susceptance.
IV. NEW SVC LOAD FLOW MODELS
In practice the SVC can be seen as an adjustable reactance
with either firing angle limits or reactance limits. The circuit
shown in Fig. 5.1 is used to derive the SVC's nonlinear power
equations and the linearized equations required by Newton's
method.
In general, the transfer admittance equation for the variableshunt
compensator is,
and the reactive power equation is,
/ = jov, (4)
Q, = -v;u. (3
Linearized, positive sequence SVC models for
Newton-Raphson load flows are presentcd in this section
whereas SVC models optimal power flows are presented in
Scction V.
A. SVC Total Suscnptance Model (H = BSVC;)
The linearized equation ofthe SVCis given by (6), wherc the
total susceptance BSVC is taken to be the state variable,

~ iti
I12 IEEE TRANSACTIONS ON POWhK SYSTEMS, VOL IS. NO I, rEBRUARY 2000
At the end of iteration i, the variable shunt susceptance Hsvt:
iq updated according to (7),
This changing susceptance represents the tolal SVC susceptance
necessary to iiiaintsin the nodal voltage magnitude at the specified
value.
Once the level of compensation has been determined, the
firing angle requircd to achievc such compensation level can
be calculated. This assumes that the SVC is represented by the
slriicture shown in Fig. 1. Since the SVC susceptance given hy
(3) is a transcendental equation, tlie coinpulation of the tiring
angle valirc is dctcrmincd by iteration.
R. SVC Firing Angle Model (U = De,)
Provided the SVC can be represented by the structure shown
in Fig. 1, it is possihle to consider the firing angle to be the state
variable. In this casc, the lincarized SVC equation is given as,
where
At the end oC iteration i, lhc variahle firing angle 0 is updated
according to ( I 0),
nitl ~
4- ACV‘
(10)
and the new SVC susceptancc 1Lq is calculated from (3).
It should be remarked that both models, the total susceptance
model and the tiring anglc tnodcl obscrve good numerical properties.
However, the Cortncr model requires of an additional itcrativc
procedure, after the load flow solution has converged, to
determine the tiring angle. These models were developed to satisly
similar requirements, hut different parametcrs are adjusted
during the iterative process. Hence, heir inathcniatical formulalions
arc quite different. Accurate information about the SVC
tiring angle, as given by the load tlow solution, is of paramount
important in harmonics and clcctromagnetic transients studies
~51.
C. Nodal Voltage Magnitude Controlled by SVC
The implemeiitation of the variahlc shunt susceptance models
in a Newlon-Raphson load flow program has required the incorporation
of a nonstandard type of bus, namcly PVB. This
is a controlled bus where the nodal voltage inagnitiide and active
and rcactive powers are specified whilst either the SVC’s
firing anglc (I or the SVC’s total susceptance Usvc are handled
as state variables. If a or l l s arc ~ ~ within ~ ~ limits, the specified
voltagc magnitude is atlained and the coiitrolled bus remains
I’VD-type. However, ii CY or Dsv,: go out of limits, they are
fixed at the violatcd limit and tlie bus becomes’ PCJ-type. This
is, of course, in the absence of other FACTS devices capable ol
providing reactive power support.
D. Revision @SVC Limits
The revision criterion of SVC limits is based mainly on the
rcactive power mismatch values at controlled buses. SVC limits
revision startsjust after the reactive power at the controlled node
is lcss than a specified tolerance. Hcrc this value was set at
p.u.
If the SVC violates limits, the SVC’s state variable is fixed at
the offending limit. In this case the node is changed from PVB
to PQ type. In this situation the SVC will act as an unregulated
voltage compensator whose productionhibsorption reactive
power capabilities will be a function of the nodal voltage at
the SVC point of connection.
At the end of each iteration, after the network and SVC’s state
variables have been updated, the voltage magnitudes of all nodes
transformed from I’VU to I’& type are revised. The purpose
of this exercise is to check whether or not it is still possible
to maintain the SVC’s firing angle or SVC’s total susceptance
fixed and to check whether 01’not the node has changed back to
thc original PVR type without exceeding a or Bsvc: limits. The
nodal voltagc magnitudes of the converted nodes are compared
against the specified nodal voltage magnitudes.
The node is recoiivcrted to PVR type if any of the following
conditions are satisfied:
I) The SVC violates its upper n or Dsvc limit and the actual
nodal voltage magnitude is larger than the specified
voltage.
2) The SVC violates its lower (Y or Usvc limit and the actual
nodal voltage magnitude is lower than ‘the specified
voltage.
After the node is returned to PVB type, the voltage magnitude
is fixed at the original target value.
E. Control Coordination Between Reactive Sources
In cases when different kinds of reactive power sources are
set to control voltage magnitude at the same node, they have to
be prioritized in order to have a single control criterion. Synchronous
reactive sources have been chosen to be the regulating
components with the highest priority order, holding all other reactive
power sources fixed at their initial condition so long as the
synchronous Renerators and condensers operate within limits.
After these rotating sources start to violate reactive limits, they
arc fixed at the violated limit and auother kind of reactive power
source, e.g., SVC’s, are activated. In this case, the node is transformed
from PV type to PVH type.
v. LOAD FLOW TEST CASE
An ohject oriented, Newton-Raphson load flow program
[61 has been extended in order to incorporate the models and
methods presented above. A real life power network, consisting
of 166 buses, 108 transmission lines and 128 transformers [7],
has been used to show the capab es of the proposcd SVC
models.
Three SVC’s have been embedded in key locations ofthe net-
‘work, resulting in significant improvement of the voltage magnitude
profile. The SVC’s were set to control nodal voltage mag-
nitude at I p.u. The relevant SVC’s parameters are given in
Section 11.

AMRKIZPEREZ eta!.: ADVANCE0 SVC MODELS FOR NF.WTON~RAP1iSON LOAD Fl.OW AND NEWTON OPTIMAL POWER PLOW STUDIFS
133
static VAR
Compensator
SVCl
svc2
svc3
susceptance Firing angle
Model
model
Bsvc @.U,) a (degrees) Bdp.u.)
0.2378 136.038 0.2378
0.0848 136.016 0.0848
0.1709 136.028 0.1709
state variables are combined with the AC nctwork nodal state.
variables for a unified, optimal solution via Newton's method.
The SVC state variables are adjusted antoinatically so as to satisfy
specified power flows, voltage magnitudes and optimality
conditions as given by Kuhn and Tuckcr [Ill.
The first stcp in finding thc optimal power flow solution is
to build a Lagrangian function corresponding to the active and
reactive power flow mismatch equations at every nodc, SVC
nodes included. The contribution of the SVC to the Lagrangian
function is cxplicitly modeled in OPF Newton's method as an
equality constraint given by the following equation:
h(h, A) = XqrBr (1 1)
where &e is the reactive power injected by the SVC at nodc le
as defined in (5). Xqe is the Ltigrange multiplier at node le.
The SVC linearized systcm of equations for minimizing the
Lagrangian function via Newton's method is given by [SI and
PI,
[1/1 2 [WI[Azl (12)
L-
1 2 3 4
lteroiions
Fig. 6. Power mismatches as function of number of iterationb.
In order to compare the virtues and limitations of the new
SVC total susceptance model and the SVC firing anglc model
load flow solutions were carried out using both models. The
number of iterations required by the load flow to converge war
5 iterations in all cases. The same final values of SVC's total
reactance required to achieve the specified nodal voltage mag
nitude control were arrived at. These values are shown in Tablc I
The behavior of the lnaximum absolute power mismatches, as
function of the number of iterations, is plotted in Fig. 6 for both
cases.
VI. NEW SVC MODEL FOR OPTIMAL POWER FLOW
Linearized, positive sequence SVC models suitable for OPF
solutions are described in this section, Similarly to the SVC load
flow model implcmentations described in Section IV, the SVC
where matrix [W] contains second partial derivatives of
the Lagrangian function /,h(Vh, 13, Xqk) with respect to
statc variables V,, H, and Xqr. The gradicnt vector [g] is
[VVk VH VAJt. It consists of first partial derivative
tcrms. [Az] is the vector of corrcction terms, given by
[Allh AR Also, [Az] = [%Ii - [z]"', where i is the
itcration counter.
Once (12) has been assembled and combined with matrix [W]
and gradient vector [g] of the entirc network then a sparsityoriented
soltition i s carried out. This process is repeated until a
small, prespecified tolerance is reached.
The Dartial derivativcs, with resoect to the variable shunt susqt:
rice /?, are function of the svc State variable to be adjusted,
i.c., Ilsvc: or o, in order to satisfy the network constraints.
A. SVC ?btu1 Susceptance Model (H = I&; J
In this casc, (12) takes the following form, as shown in (13)
at the bottom of the page. The partial derivativcs corresponding
to the SVC are derivcd from (5) and (I I). Terms written in bold

134
IEEE TRANSACTIONS ON POWBK SYSTEMS, VOL. 15, NO. I. PCDRUARY znnn
TAULE If
OWIMAI. GENERATION COST AND SYSrF.M LOSSES
- aL -I
-~
asi,
a I,
a C%
BL
--
-- -
ilX,i
dL
--
a&
aL -_
- act -
-
Base
CaSe
cost (Sh) 264.137
Losses (MW) 13.48
Compensator
svc 1
svc2
svc3
Susceptance Firing angle
Model model
264.135 264.136
13.47 13.475
TABLE Ill
COMPARSION OF OPTIMAI. SUSCEPTANCBS FOR BOT?[ SVC MOIX1.S
I Staticvm I ~usceptanc I Firing angle I
e Model
model
Bsvc (P.U.) a (degrees) Blb (P.U.)
0.1905 136.28 0.1908
0.0754 136.11 0.0751
0.1194 136.18 0.1242
(1 8)
wheregi(x) isthestatevariablevalue, i.e., Hsvc orru,attheend
of each iteration. 4 and g are the maximum and minimum limits
of the SVC state variable. p is a multiplier term and c is a pcnalty
term that adapts itself in order to force inequality constraints
within limits while minimising the objective function.
D. Lugrange Multiplier
The Lagrange multiplier for active and reactive powcr flow
mismatch equations are initialized at 1 and 0, respectively. For
the SVC Lagrangc multiplier the initial valiic of A,,, is set equal
to 0.
(14)
The relevant partial derivative terms are derived from (5)
and (I I). The derivative terms corrcsponding to ineqtiality
constraints are not required at the beginning of the iterative
solution, they are introduced into matrix (I 3) or (14) only after
limits arc enforced. Further explanation is given below.
C. Handling Limits of SVC Controllable Variable
In the OPF forinnlation, voltage magnitude and active power
limits are included in the inequality constraints set. The inultiplier
method 1101, [I I] is used to handle this set. Here, a penalty
term is added to the Lagrangian function, which then becomes
the augmented Lagrangian function. Variables inside bounds are
ignored whilst binding ineqtiality constraints become part of the
augmented Lagrangian function and, hence, hecome enforced.
The handling of SVC inequality constraints can be carried out
by using the following generic function 1101,
di(Si(Xj, lJij
3;)' ifpi + c(gi(z) - B ~) >_ o
c
+,(Si(xj -G)~ ifpi + c(s;(z) - -%9.) 2 U
E. SVC Control of Voltage Magnitude ut a Fixed Value
In OPF studies it is normal to asstime that voltage magnitudes
arc controlled within certain limits, e.g., 0.95-1.10 p.u. However,
if the voltage magnitude is to he controlled at a Fixed value,
then matrix [W] is suitably modified to reflect this opcrational
constraint. This is done by adding to the second derivative term
of the Lagrangian function with respect to the voltage inagnitude
Vi, the second derivative term of a large, quadratic penalty
factor. Also, the first derivative term of the quadratic pcnally
function is evaluated and added to the corresponding gradient
element. Hence, the diagonal element corresponding to voltage
magnitude Vi, will have a very large value, resulting in a null
voltage increment, A l/k. This is equivalent to deactivating the
equation of partial derivatives of the Lagrangian function with
respect to K:, from matrix [I&'].
VI1. OPF TRST CASES
The SVC models described above have been implemented
in an OPE To test the robustness of the algorithm, the electric
system described in Section IV-F was used [7]. The objective
function to be minimized is mainly active power generation cost.
A. Variable Susceptance and Firing Angle Models
The optimal generation cost for the base and modified cases
(SVC upgraded) are given in Table 11, together with system
losses. The optimal susceptance values of SVC's are given in
Table 111. The SVC's were embedded in nodes with low voltage

AMBRIZ-PBRBZ ADVANCED svc MODEIS FOR NEWTON-RAPHSON MAD FLOW AND NEWTON OPTIMAL POWER FLOW STUDIES 135
.-
2
"
1 300 -
g 290 -
.s 280 -
+
D
t 270 -
!?
2
260 -
a
SVC toto1 susceptanre model
$ 250. SVCfiring ongle model
?.
4' 2 4 C L ' ' ' ' '
0 1 2 3 4 5 6
34 1
32 -
30-
YI 28-
26-
3 24-
g 22-
a 20.
%!
z 18-
' 16-
14 -
lterotions
Fig. 7. Active power generiltion cos1 pr~fil~s
Fig. X.
\ ---
-
Base case
N C total susceptonce model
. SVC iiring ongle model
1 2 5 r - 7 ' 1 " 3 " 4 " 5 ' 6
Iterations
Active puwer lasses profiles.
magnitudes and, as expected, the voltage profiles improved in
such nodes (not shown). SVC voltage magnitudes were subjected
to inequality constraints within 0.95-1.10 pa.
In all cases, solutions were achieved in 6 iterations. Fig. 7
shows active power generation cost as a function of the iteration
number whilst Fig. 8 shows the active power losses as a
function of the iteration number. Oscillations can be observed in
the cost and losses profiles in the early iterations. This is due to
large variations in both the active set constraint and the penalty
weighting factors during the early iterations.
The SVC control specifications used above werc modified
in order to assess the SVC inodcl capability to control voltage
magnitude at fixed values. In this case the SVC's were set to
control voltage magnitudes at I p.u.
The optimal SVC state variable values for both models, i.c.
susceptance and firing angle models, are shown in Table IV. As
expected, these values difler from those given in Table 111.
It is interesting to note that both SVC operating control
modes, i.e. fixed and free, produce similar active power generation
cost and active power losses.
B. Transmission Lusses Minimization
The algorithm is now applied to investigate the problem of
transmission loss tniniinimtion. The active power cost optimization
and the transmission losses minimization procedures
TABLE 1V
COMPARISON OmlMAL SUSCEPPANCE FOR BOTH SVC MODELS
Firing angle
0.2150 136.32 0.2151
0.0741 136.10 0.0741
0.1376 136.20 0.1377
TABLE V
OI'TIMAL GENERATION COST ANU SYSTEM LOSSES
Base Susceptance Firingangle
case Model model
cost (Sib) 264.136 264.134 264.134
I7 475 11AKl
1.018L-12.23 1.021L-12.23
I I
&40'00 6141.736
+
Fig. 9. Comparison between the gewmov and susceptiince SVC ,nod& for
the case when upp~
limits have bcen reached.
are carried out sequentially. Transmission loss minimization is
amenable to a redistribution of the reactive power throughout
the network, which in turn induces changes in the active power
generated by the slack generator. This may then result in an
additional reduction of the active power generation cost. Table
V presents results obtained with the combined optimization
algorithm for the case when the SVC's are not set to maintain
nodal voltage magnitude at a specified value. It can be
observed that the second optimization exercise, i.e., network
loss minimization, has only a minor effect on the active power
generation cost. The network losses were reduced in only
0.15%, but a more uniform voltage profile was observed in all
nodes (not shown). The reason for such a small variation in
the network losses is due to the fact that the first optimization
exercise, active power cost optimization, indirectly reduces the
transmission network losses, i.e., large network losses means
more active generation. The network loss minimization cases
converged in 2 iterations
C. Limitations of the Generator Model
In order to show the limitations of the generator representation
of an SVC, compared to the two SVC models introduced
in this paper, a case is presented below where the SVC hits its
upper reactive limit. SVC 1 was assumed to have a 40 MVAR
upperlimit. Fig. 9 shows thereactive powers contributed by both
models, where the network losses objective function was minimized.
It can be observed that the reactive powers drawn by bath
SVC models differ. The reason is that the reactive power is not
really constant, as taken to be by the generator representation,
but afunction of nodal voltage magnitude, as correctly indicated

116 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL 15. NO. I. PEBRUARY 2000
by the variable susceptance representation. As expected, both
representations have a minimum impact on network losses, but
at the point of SVC connection differences can be observed in
the voltage profile and in the reactive power contributed by both
models.
As discussed in Section 111, the compound generator-constunt
susceptance representation provides an accurate load flow
represetitation of SVC's for the complete range of operation.
However, inefficiencies may be observed owing to the need to
reorder and to redimension the linearized systems of equations.
In the OPF formulation, further complications may arise with
this compound representation. The multiplier method, which
has shown to be very effective in enforcing variables outside
limits, may have difficulties in checking whether or not a SVC
is operating within limits when represented as a constant susceptancc.
VIII. CONCLUSIONS
Comprehensive SVC models suitable for conventional
and optimal power flow analysis have been presented in
this paper, namely SVC total suscepiance model and SVC
firing angle model. In contrast to SVC models reported in
the open literature, the proposed models do not make use
of the generafor concept normally employed for the SVC
representation. Instead, they use the variable shunt susceptance
concept. Arguably, this has the advantage of representing actual
SVC operation more realistically. Moreover, since SVC shunt
susceptance models only make use of one node to represent
SVC's operating inside and outside ranges then Newton based
power flow solutions become more efficient, compared to cases
when generafor based models of SVC's are used in Newton
algorithms. An SVC model which uses the thyristor firing angle
as the state variable has shown to provide fuller information
that existing SVC models. The firing angle required to achieve
a specified level of compensation becomes readily available
from the power flow solution, as well as the fundamental
frequency, internal SVC resonant points. A Newton-Raphson
load flow and a Newton's OPF algorithms have been upgraded
to incorporate the new SVC models. A real-life, bulk transmission
system has been used as the test case. Conventional and
optimal solutions were obtained in less than 6 iterations.
ACKNOWLEDGMENT
H. Ambriz-PBrez would like to thank Comisi6n Federal de
Electricidad, MBxico for granting him study leave to carry out
Ph.D. studies at the University of Glasgow, Scotland, UK.
- .
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It. Aeha was horn in Mexico. He graduated from University of Michoilcdn
in 1979 and rcceiverl the Ph.D. degrcc from the University of Canterbury,
Christchurch, New Zealand, in 1988, He was a postdoctoral Fellow at the
University of Toronto, Cantidti and the Univcrsity of Durham, England. He
is currently n Senior Lecturer ut the University 01 Glasgow, Scotland, wherc
hc lectures and conducts mseilrcli on power systems analysis i~nd power
electronics applications. His research interests arc in the aceus of FACTS,
custom power, and real-time modeling and iinalysis.
C. R. Focrte-Esqaivel was barii in Mexico in 1964. He received the B.Eng. degree
(Hans) from Institute Tecnol6gicn de March, Mexico in 1990, the MSc.
degree 1wm lnstiluto Politecnico Naciunal, Mexico in 1993, and thc Ph.D degree
from the University dGlasgaw, Scotland, UK, in lY97. He is currently :NI
Assistant Professor at thc Institute TecnolOgico de March His main research
interests lie oil the dynamic and steady-state analysis of FACTS, custom power,
and real-time modeling and analysis.