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Location of Series-Connected Controllers to 
Reduce Proximity to Transient Instability based 
on a Trajectory Sensitivities Approach 


Enrique A. Zamora-Cardenas, Claudio R. Fuerte-Esquivel, Senior Member IEEE 


Abstract — Determining suitable locations of series-connected 
controllers is a practical problem when it is necessary to install 
them in modern power systems. The aim of this paper is to find 
the best location of series controllers in order to reduce the 
proximity to instability of a current operating point of a power 
system, from a transient stability viewpoint. In order to achieve 
this goal, a general approach has been developed based on an 
index of proximity to instability and trajectory sensitivity 
analysis. An efficient way to carry out multi-parameter 
sensitivities is formulated analytically and solved simultaneously 
with the set of differential-algebraic equations representing 
power systems dynamics within a single-frame of reference. 
Simulations are performed on 9-bus and 39-bus benchmark 
power systems for illustration purposes. Results show that the 
proposed approach provides the most effective location of series- 
connected controllers to improve the power systems transient 
behavior. 

Index Terms — Transient stability, trajectory sensitivities, 
sensitivity index, series compensation. 

I. Introduction 

A dvances in Flexible AC Transmission Systems (FACTS) 
controllers have led to their application in improving 
electric power systems’ controllability [1],[2], It has been 
recognized that the location of these controllers has a large 
impact on their performance with regard to the control 
objective to be fulfilled. The best allocation for one objective 
may be less suitable for another objective. This has motivated 
the development of several kinds of approaches for finding 
proper locations of FACTS controllers in order to improve the 
power system’s static or dynamic performance. 
Methodologies based on singular value decomposition [3], bus 
participation factor [4], augmented Lagrange multipliers [5], 
heuristic methods [6], [7], mixed integer linear programming 
[8] and sensitivity-based approach [9], [10] have been 
proposed to allocate controllers to satisfy suitable steady-state 
control objectives. On the other hand, proper locations to 

This work was supported by CONACyT, Mexico under the scholarship 
169415. 

A. Zamora-Cardenas and Claudio R. Fuerte-Esquivel are with the 
Electrical Engineering Faculty, Universidad Michoacana de San Nicolas de 
Hidalgo (UMSNH), Morelia, Michoacan, 58000, Mexico 
(email:reycoliman2002@hotmail.com, cfuerte@umich.mx). The first author is 
also with the Instituto Tecnologico Superior de Irapuato (ITESI), Gto., 
Mexico. 


improve damping of low frequency electromechanical 
oscillations have been determined based on modal analysis 
[1 1], [12], [13] and residue method [14], [15]. 

Transient stability analysis is also an essential study in the 
operation and planning of electric power system [16]. If this 
study determines that a rotor angle transient instability takes 
place due to large electromechanical oscillations among 
generation units and lack of synchronizing torque on the 
system, control actions have to be taken to prevent partial or 
complete service interruption. Among different preventive 
control measures, it is possible to apply series compensation 
in a proper place to regain an acceptable state of equilibrium 
after the disturbance by improving the system’s stability 
condition [16]. The stability condition can be computed by a 
time domain simulation which is applicable for arbitrarily 
complicated models, and it is feasible for large-scale power 
system analysis. However, this simulation only provides 
information about a single scenario, and repeated simulations 
have to be done to assess a degree of system’s stability or 
instability. 

Trajectory sensitivity (TS) analysis overcomes the need of 
repetitive simulation. In this approach, a linearization is 
carried out around a nominal trajectory rather than around an 
equilibrium point, such that it is possible to assess variations 
in the nominal transient trajectory resulting from small 
perturbations in the underlying parameters and/or initial 
conditions [17]. These results are only valid for the 
equilibrium point that defines the nominal trajectory. 
Applications of TS includes the study of parameter 
uncertainty in system behaviour [18], [19], determination of 
well-conditioned parameters for reliable estimation 
[20], [21], [22], and determination of critical machines which 
are likely to go unstable for a given loading conditions and a 
specified contingency [23], among others [24]. 

Owing to the fact that TS analysis provides a qualitative 
measure of how stable or instable a particular case may be, 
and valuable insights into the influence of parameters on the 
nominal transient trajectory of the system, this approach was 
applied to successfully analyze the Nordel power grid 
disturbance on January 1, 1997 [25]. Sensitivities of relative 
rotor angle with respect to line impedances were used to 
identify which line was most sensitive with respect to system 
stability and to indicate the effect of this line with respect to 
different generators in the system. Based on these results, 
authors suggested that the TS approach could be used to 


978- 1-4244-6551 -4/10/$26. 00 ©2010 IEEE 



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choose effective locations of FACTS devices. This idea is 
revisited and applied in this paper to determine the suitable 
locations of series compensators. 

Since the improvement of transient stability can be 
considered as a problem of controlling transient trajectories by 
a change in parameters, TS can be used to judge the 
effectiveness of FACTS controllers in improving stability. 
This study was performed in [26] considering static models of 
a thyristor controlled series compensator (TCSC) and a static 
synchronous compensator (STATCOM). The effects of these 
compensators were assessed by placing them individually at 
each transmission line, one at the time, and calculating a 
post-fault stability condition based on the numerical 
formulation of the trajectory sensitivity. In this case, 
sensitivities of state trajectories with respect to system 
parameters are found by perturbing the selected parameter 
from its nominal value, and making the division of the 
changes in the state variables over the parameter perturbation; 
i.e. Ax /A/?, at each time step of integration. This approach 
requires two time domain simulations for each sensitivity, and 
the selection of the size of the parameter perturbation. The 
later is selected heuristically such that the numerical 
sensitivity be very close to the analytically calculated 
trajectory sensitivity value. 

The most effective location of a series compensation can be 
obtained by the approach suggested in [26] for a given one 
fault scenario after the computation of F time domain 
simulations, where r=(l+number of parameters (Np)). As the 
number of fault scenarios increases by nf times, the required 
number of time domain simulations to assess the parameters’ 
effects is actualized by T=nj*F. 

A radical approach is applied in this paper to avoid the 
problem of performing a great number of time domain 
simulations to determine the most effective location of a series 
compensation. This consists of formulating the trajectory 
sensitivities analytically, such that only one time domain 
simulation is required to assess the effect of one parameter for 
a given fault scenario. The extension to multi-parameter 
sensitivities computations is straightforward. The number of 
simulations only increases directly proportional to the number 
of fault scenarios and is not a function of the Np parameters 
considered at each fault study. 

The remaining of the paper is organized as follows. Section 
II provides the mathematical models of the electric power 
components considered in the studies. Section III presents the 
detailed analytical formulation to carry out multi-parameter 
trajectory sensitivity studies, while Section IV describes the 
application of this formulation to select the most effective 
location of a series compensation. In Section V, two case 
studies based on the WSCC 9-buses 3 -generators system, and 
on the New England 39-buses 10-generators system, are 
analyzed and discussed in detail. Lastly, Section VI points out 
the conclusions. 

II. Power Systems Models 

An electric power system can be represented analytically by 
a set of parameter dependent differential equations 
constrained by a set of algebraic equations (DAEs), as given 


by (1), where x is a vector of the dynamic state variables, y is 
a vector of the algebraic variables (usually complex node 
voltages), and /? is a set of non-time varying system 
parameters 

x = f(x,y,0 ) f:9i" +m+p ^W 

0 = g(x,y,fi) g:9r m ^->9r (1). 

X6lc9i” jeTcOT /3<eJ3^W 

Due to the fact that transmission network dynamics are 
much faster than dynamics of the equipment or loads, it is 
considered that the variables y change instantaneously with 
variations of the x states. Hence, only the dynamics of the 
equipment, e.g. generators, controls, FACTS devices, and load 
at buses, are explicitly modeled by the set of differential 
equations (1). The set of algebraic equations express the 
mismatch power flow equations at each node. As the power 
system can be viewed as an interconnection of several electric 
power plant components, particulars of each model considered 
in this paper are given below. All variables are given in per 
unit, unless otherwise specified. 

A. Generator 

In this paper, the two-axis generator model is considered 
with a field and a damper winding on the d - axis as well as a 
damper winding on the g-axis. 

1) Rotor equations 

The rotor mechanical model is given by the swing 
equations. For the i ,h generator, these equations are 

4 =( 0 - 01 , 

(2) 

Pe i =Ki I « +E * I a + (Ki- X *V* I « 

where 2 Hj is the moment of inertia in seconds (sec), Z), is the 
damping constant, P ei is the generator’s electrical power 
associated with the internal voltage source, P Mi is the turbine 
mechanical power injection, Sj is the generator’s rotor angle 

in radians (rad), 0 \ t is the synchronous speed in rad/sec, and (>)■, 
is the actual rotor speed in rad/sec. 

The equations of the two electrical systems on the rotor are 

qi rp' 

m , (3) 

r . _~E di +{X qi -X qi )I qi 

h di ~ rp' 

No- 
where E di and E\ are the transient internal voltage 
magnitudes on d and q axes, respectively, I dj and / . are the 
stator currents on d and q axes, X ki and X kj are the 
synchronous steady state and transient state reactances on axes 
k=q, and k=d, respectively, T m and T qm are the constant time 
on d and q axes, respectively, lastly, E fdi is the DC controlled 
voltage field. 



3 


2) Stator equations 

The algebraic equations (4) can be obtained by neglecting 
both stator transients and stator resistance, where V l and 0 j 
are the magnitude and phase angle voltages at generator 
terminals on the network side, and the generated power is 
measured at terminals, 

E qi = V i cos(<5) -0j) + X di I di 

E'di = V, sinCJ, - 0 : ) ~X' qj I qi 

Pgei = J di V i sin (^/ - Oi ) + I q , V , COS (S { - 0, ) 

Q gei = I«V, cos {8, -0,)-!^ sin (8, -0,) 


B. Excitation system 

The excitation system is considered as shown in Fig. 1, 
which includes max/min ceiling limits, with its equation given 
by (5) 



Fig.l. Excitation system. 


F = - 

^ fdi 


KAKefi-VJ-E 


fdi 


(5) 


where E fdi is the DC controlled voltage field, V refi is the 
reference node voltage, V ti refers to the voltage at the 
generator terminal, K Ai is the control gain and T Ai is the 
excitation system time constant. 


C. Load 


The classical constant impedance load model is considered 
to capture the transient system trajectories. The impedance 
values are computed at the ith system node from the stable 
equilibrium point that define the pre-disturbance operative 
state as follows: 


P =P 

1 Li 1 Li 


f 1 


2 

( T 

^ \ 


; Q u = Qu 

r /0 

V 

y 


V 

y 


( 6 ) 


where P kI and Q kj are nominal demanded load and V° is the 
nominal voltage measured at the pre-disturbance state. 


D. Network 

This model consists of those equations expressing the active 
and reactive power balances at every system nodes. For the 
transmission element connected between nodes i and j, the 
active and reactive powers at the ith node are 

P„ = K 2 Ga + V,Vj (G, cos(0,. -0,) + B j sin(0 j - 0 j )) 

Qij = -V- Pa + V t Vj (G ij sin(0 l -0j)~ B 0 cos(0, - 0 j )) 


Assuming n g generator nodes and n pQ network nodes, 
V k and 0 k are the magnitude and phase angle voltages at 
network nodes, i = \,...,n g +n PQ , respectively, and 


Y = G + jB is the admittance of the transmission element 

y y J y 

connected between nodes i and j. 

The mismatch power equations at the network nodes are, 


A, = A + I P,J 

ye Q, 

Qgei = Qu + E Qij 

jsa, 

o = p u + E p, 

jeO, 

0 = Qu + S Qij 

y'e Q, 


V; = 1,... , n g 
V/ = 1, . . . , n g 
Vi = 1 + n g ,...,n PQ 
Vi = \ + n g ,...,n PQ 


(8) 


where Q ={y : j ± /} is the set of nodes adjacent to i, and P Li 
(Q u ) is the active (reactive) power demanded by the load 
embedded at the ith node. 


III. Trajectory Sensitivity Analysis 

The constrained nonlinear dynamics of the power system 
are calculated by solving (1). State trajectories associated to 
this dynamic behavior will vary with small changes in the 
system parameters or the variables’ initial conditions, and 
these variations can be quantified through a time domain 
sensitivity analysis. Nevertheless, there are two limitations in 
this approach, which relies on repeated simulations: (1) the 
computational burden required to calculate sensitivities, and 
(2) there is not a quantitative measure of the proximity to 
instability of the power system. 

It has been demonstrated that it is possible to find out in a 
rigorous approach how sensitive the trajectories of each state 
are to changes in system parameters or initial conditions. The 
theoretical treatment to obtain analytical equations of 
trajectory sensitivities has been reported for both ODEs [27], 
[28] and DAEs [27], [29]. The analytical system sensitivies of 
(1) w.r.t. system parameters can be found for the post- 
disturbance state by perturbing /3 from its nominal value j3 0 , 
and considering that f(x,y,j3 ) and g(x,y,j3) have 
continuous first partial derivatives w.r.t. x, y and [’> from the 
disturbance clearing time t cl to the end of the study time t end , 
for all (x, y, fi) E [t d , t end ] x9I" x "' v and initial conditions 
x(t d ) = x cl , y(t cl ) = y cl . 

A. Analytical Formulation 

Let /3 0 be the nominal values of /? , and assume that the 
nominal set of DAEs x = f(x,y,j3 0 ), 0 = g(x,y,/3 0 ) has a 
unique nominal trajectory solution x{t,x cl ,y d , J3 0 ) and 
y(t,x cl ,y d ,/3 n ) over ts[t cl ,t md ]. Then, for all /3 sufficiently 
close to j3 0 , the set of DAEs (1) has a unique perturbed 
trajectory solution x{t,x cI ,y cl ,/3 ) and y(t,x d ,y d ,fS) over 
t G \t d , t t-ml ] that is close to the nominal trajectory solution. 
This perturbed solution is given by 


4 


x(r) = x d + \'"‘f{x(-),y(-),P)ds 

(9). 

The sensitivities of dynamic and algebraic state vectors 
w.r.t. a chosen system parameter, Xp = dx(-)/d/3 and 

y p = dv(-)/dj6 , at a time t along the trajectory are obtained 
from the partial derivative of (9) w.r.t. P , i.e. 


8 - y 0 

d/3 


j* tend 
'll 


(%()& 8 /(-) 3 y jfflj 

v dx d/3 dy d/3 d/3 / 


ds 


dg(~) dx | 3g(-) dy | 3g(-) 
dx dp dy dp dp 


( 10 ). 


Lastly, the smooth evolution of the sensitivities along the 
trajectory is obtained by differentiating (10) w.r.t. t , i.e. 


Jill, + W V + W 

dx P dy P dP ; x fj {t c ,) = 0 (11) 

= f,x p + f y y p + f p 


and 


o-Ml,. + «!fcl *<■> 


dx 


dy 


dP ; yp(t d ) = 0 (12) 


= s x Xp + g y y p + gp 

where f„ f y , fp, g x , g y and gp, are time -varying matrices 
computed along the system trajectories. 


B. Numerical solution 

Trajectory sensitivities are computed by solving (11) and 
(12) simultaneously with (1) by using any numerical method. 
The staggered direct method (SDM) has been selected to solve 
sequentially the set of DAEs representing the power system 
and trajectory sensitivities [30]. The trapezoidal rule is applied 
to algebraize the differential equations, such that sets of DAEs 
(1), (11), and (12) are expressed by the following set of 
algebraic difference equations 


F l (3>=x M -x k -^{f M + f k ) = V 

M-) = g k+1=0 


F A)= x 7'~4~ 

M)=g k ;'M 


f fk+\ k+\ . rk+\ k+\ . fk+\\ 

Jx X p ^ J y yp J p 

+f k x k p+f k y k e+fe 


i __^+l , ,k -\- 1 | 

+ gy y P +1 


= 0 


(13) 

(14) 

0 (15) 
(16) 


where h is the integration time step, and the superscript k is an 
index for the time instant 4 at which variables and functions 
are evaluated, e.g. x k = x{t k ) and f k = f(x k ,y k ) . Nonlinear 

equations (13) and (14) are coupled to the DAEs (1) whereas 
(15) and (16) correspond to the set of linear time variant 
DAEs (11) and (12). 

The Newton-Raphson (NR) algorithm provides an 
approximate solution to (13) and (14) by solving for 


|^Ar* A y k ~] in the linear problem J' AX' = -F(X '), given in 

expanded form by (17), where J is known as the Jacobian 
matrix 


h rk + l 


•~2 f ‘ 

g X 


— — f k+l 
2 Jy 

g* +1 



Ax k 

‘ 

>.(■)] 


V. 


Ml 


sy f(-/ 


(17). 


For given values [x* , the method starts from an 

initial guess [x^ +1 = x k y k Q +l = y k J and updates the solution 

at each iteration i, i.e. [x t+1 = x k + Ax k y k+l = y k + Ay k J , 

until a convergence criterion is satisfied. The computational 
burden is directly associated to the size of the power system 
under study. 

Once the states have been computed for a new time step, 
the sensitivity trajectories are calculated from the set of linear 
equations (18), which is derived from (15) and (16) 


I--f k 

2 

g4 


—— 

2 - 

g k v' 


r/ +i i 

Ap 


k + 1 

\yfij 



4+j[f k x k p+f k y k p+fp+fp + 


„*+ 1 
-Sfi 


(18). 


J B 

The coefficient matrix on the left-side of (18) corresponds 
to the Jacobian matrix used in the final NR iteration to solve 
for x k+ ' and y k+l at (17). Based on this observation, the 


computational burden for the calculation of trajectory 
sensitivities is substantially reduced because the coefficient 
matrix is already factored, and only a forward/backward 
substitution is required for the solution of x k p l and y k p ' at 

each discrete time t k " of the integration period [29]. 


The solution approach described above can be extended to 
compute the trajectory sensitivities associated to Np 
parameters of the system at each discrete time. In this case 
(18) is expressed as (19), which is solved Np times for the 
solution of x k *' and y k p Vi = l,...,Np , one at a time at the 
same time step 



S B 


(19). 


IV. Location of series compensation 

Based on the observation that the more stressed the system 
is the larger its trajectory sensitivities are [17], it is possible to 
associate sensitivity information with the stability level of the 
system for a particular system parameter. The line 
susceptance’s effect on the system’s stability is measured by 
computing sensitivities of rotor angles trajectories w.r.t. 
transmission line susceptances, and measuring the proximity 
to instability. 

An index of proximity to instability is determined based on 
the sensitivity norm obtained for the period te f 01 ’ a 

« g -ma chines system given by [23], 



5 


SJt) 


n s 

z 


f dS k (t) 

{ w 


9^(0 Y 


fa^coY' 
l m ) , 


Mi = !,■••, Np 


where j denotes the reference machine. 


(20) 


The growth in the peak values of trajectory sensitivities 
indicates an underlying stability problem, and ideally S Ni (t) 
should be infinite at the point of the system’s instability. 
Bearing this in mind, the index to proximity to instability is 
defined as the inverse of S Ni (t) , T) i = I / max |5 vi (t)| , which 

will approach zero as the systems moves toward instability 
[23]. Based on this result, the most effective location of series 
compensation corresponds to a transmission line whose small 
perturbations in its susceptance produce the lowest value of 


Vi- 


V. Study Cases 

The WSCC 9 buses, 3 generators and New England 39 
buses, 10 generators systems shown in Figs 2 and 3, 
respectively, were analyzed to assess the effectiveness of the 
proposed approach. For the purpose of the studies presented in 
this paper, the two-axis generator model is considered which 
includes a simple faster exciter loop containing max/min 
ceiling limits, a field winding on d - axis and a damping 
winding on both af-axis and c/-axis. The constant impedance 
load model is considered in accordance to the model used by 
commercial grade programs to undertake studies of transient 
stability. Lastly, transmission network components are 
represented by their steady-state models, because the network 
variables change instantaneously with variations of the 
dynamic variables. 

The goal of the application is to determine the most 
appropriate line to be compensated to improve the transient 
stability for a set of possible faults taking place in a specified 
system’s geographic area. This improvement is quantified 
through the sensitivity index given in Section IV considering 
the sensitivities w.r.t. the series susceptance of transmission 
lines. The disturbance consisted of a self-cleared three phase 
to ground fault at one end of the transmission element. 

A. WSCC 9-buses, 3-generators system 

For this system, faults are applied at each system’s node, 
except at generator terminals, one at the time, and sensitivity 
indexes are computed for all transmission lines in order to 
determine which is the most suitable to be compensated. A 
stressed scenario was obtained for the kth fault by considering 
a clearing time (cl) value of 90% of the critical clearing time 
(CCT), t k cl = 0.9 t k cct . The results obtained by the proposed 
approach are reported in Table I. It must be pointed out that 6 
simulations were only required to obtain all results, whilst 42 
simulations would be required if the numerical TS method 
was used. 

From these results, it is observed that the stability index is 
most affected by changes in susceptances of lines connected 



g, 

Fig. 2. WSCC 9-buses, 3-generators. 



Fig. 3. New England 39-buses, 10-generators system. 


Table I. fj for each fault of the WSCC system. 


Line 


T] for a faulted node 



susceptance 

7 

8 

9 

6 

5 

4 

B l-S 

5.99 

8.51 

7.25 

10.3 

7.85 

9.82 

B S-9 

4.84 

6.77 

4.18 

6.46 

6.09 

6.93 

Ks 

0.74 

0.77 

1.07 

0.86 

0.82 

0.85 

B 9-6 

1.32 

1.26 

1.59 

1.35 

1.35 

1.39 

A-4 

3.33 

3.49 

4.84 

3.81 

3.57 

3.77 

B 6-4 

7.66 

7.31 

7.70 

7.68 

7.77 

7.82 


between nodes 7-5 and 9-6, B n _ s and B 9 _ 6 . Hence, the best 
improvement of the system’s transient behavior will be 
achieved by compensating the most sensitive line, which is 
connected between nodes 7-5. It must be pointed out that this 
line resulted in the best option for applying compensation for 
all cases. Therefore, a global stability control is accomplished 
by compensating this line. 

Trajectory sensitivity analysis only indicates which line has 
to be compensated, but it does not indicate the type and degree 
of its compensation. However, this problem has been solved 
by a simple comparison of the system transient behavior 
without and with a specified level of compensation. 


6 


Figure 4 shows three trajectories of the relative machine 
angle S 2 -<5j considering no compensation, as well as 30 % of 
inductive and capacitive compensation of line 7-5. A fault was 
located at bus 6 and cleared at t c f= 0.339 sec. From this figure 
it is clear that a capacitive compensation improves the 
transient stability by damping the first oscillation, whereas an 
inductive compensation has the opposite effect. 

Validation of the correct performance of the stability index 
is carried out by repeating the stability study stated above with 
a series capacitive compensation on the two lines with greater 
effect on the transient performance, i.e. lines 7-5 and 9-6, one 
at the time. The stability indexes for the base case indicated in 
Table I are t)(B 7 _ 5 ) = 0.86 and rj(B 9 _ 6 ) = 1.35 , respectively. 
Fig. 5 shows the transient trajectories for all cases, confirming 
that the compensation on the most critical line 7-5 improved 
the first oscillation, better than line 9-6. 

Lastly, a comparison of the CCTs for the base case and 
compensation on the line 7-5 case is shown in Table II 
regarding all faulted nodes, one at the time. It is observed in 
the last column of this table that in all cases the CCT 
improved when compensation is embedded in the transmission 
line, improving the transient stability margin. 

B. New England 39-buses, 10-generators system 

For the 39 buses 10 generators power system, the trajectory 
sensitivities have been used to increasing the transient stability 
margin in all nodes around a weak area in the power system. 
This area corresponds to the locations of faults with the lower 
critical clearing times computed by a contingency screening 
analysis on all non-generator nodes. For this network, the set 
of critical faulted nodes is given by 25 to 29. A fault in one of 
these nodes makes the generator embedded in the bus 38 lose 
synchronism with respect to the rest of generators, which in 
turn steers the system relatively faster to the instability than 
the remainder faulted nodes out of the mentioned area. As an 
example. Fig. 6 shows graphically how the generator at node 
38 moves the rest of the machines forward for a fault applied 
at node 29. 

The most effective location of series compensation is 
assessed by applying a fault at all nodes, except those with 
power injections, one at the time, and computing sensitivity 
indexes w.r.t. all line susceptances. Similarly to the study 
described in § N-A, the clearing time values were considered 
as 90% of the CCTs and the faults were self-cleared. In this 
case, the study required 29 simulations to identify the most 
influential line susceptance in the power system. It must be 
pointed out that if the same study was performed based on 
computing the trajectory sensitivities by using perturbed 
trajectories [26], 1015 time domain simulations would be 
required, significantly increasing the computational burden. 

Table III reports the stability indexes resulting from the 
trajectory sensitivity analysis. Only lines with the lower values 
of stability indexes are reported. It is interesting to observe 
that these indexes correspond to faults applied at buses 
making up the weak area, which indicates that once the weak 
area has been identified, or an area of interest has been 
determined, it is only necessary to carry out the study in the 
set of nodes that comprise this area. The study in a sub-region 


of power system reduces substantially the computational 
burden (number of operations as well as memory 
requirements) especially for cases of large-scale systems. 

Based on the results of Table III, the most effective location 
of series compensation is at the line with the most sensitive 
susceptance: the line connected between buses 26 and 29. This 
compensation will increase the “distance” of the system’s 
trajectories to the stability boundary. This statement is 
confirmed by results shown in Fig. 7 which shows the relative 
rotor angle trajectory of the generator connected at node 38 
w.r.t. the swing generator 39, J 3g _ 39 for cases of non 
compensation, and 30% of capacitive compensation at the 
lines B 26 _ 29 , B 26 _ 2S and B ls 26 , each one at the time. The fault 
was applied at bus 29 and cleared at its non compensation 
CCT given by t ccl = 0.197 sec. As was expected, all cases 

with series compensation improved the transient stability 
behavior, being the most effective when compensation is 
applied in line 26-29. 




Fig. 5 Effect of lines compensation in the first oscillation. 
Table II. CCTs with 30% compensation. 


Bus 

l CCT 

^CCT 

^ CCT 

7 

223 ms 

230 ms 

7 ms 

8 

297 ms 

31 1 ms 

14 ms 

9 

248 ms 

256 ms 

8 ms 

6 

377 ms 

404 ms 

27 ms 

5 

349 ms 

352 ms 

3 ms 

4 

301 ms 

310 ms 

9 ms 


7 



Finally, the comparison of the CCTs for the base case and 
compensation on line 26-29 is shown in Table IV regarding to 
faults at all nodes in the weak area, one at the time. It is 
observed in the last column of this table that in all cases the 
CCT improved when compensation was embedded in the most 
sensitive transmission line, improving the transient stability 
margin in the area. The best improvement was obtained when 
the fault is applied at bus 28, increasing the CCT in 20 ms, 
whereas the CCT is marginally improved by 9 ms when the 
fault is applied at bus 25. 

VI. Conclusions 

In this paper, a systematic trajectory sensitivity-based 
approach has been proposed to determine the proper allocation 
of series-connected controllers in order to improve the 
transient stability margin of power systems. The approach is 
completely general and its application does not depend on the 
kind of series controller to be installed in the system. 
Guidelines to identify the most suitable location are given 
according to the relation between sensitivities of relative rotor 
angle with respect to line susceptances and an index to 
proximity to instability. Numerical examples on two 
benchmark power systems are provided and confirm the 
feasibility as well as the validity of the formulation. The 
proposed approach is an effective and practical method which 
could be used for large-scale power systems planning and 
operation. 

VII. References 

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Table III. rj for each faulted node of the new-england system. 


Line 

Reactance 

Bus 25 

ri fo 
Bus 
26 

a faulted 

Bus 

27 

node 

Bus 

28 

Bus 

29 

D 

D 26-29 

6.56 

4.53 

4.37 

4.54 

4.57 

D 

D 26-28 

13.34 

9.13 

8.87 

9.08 

9.13 

B 25-26 

22.27 

13.30 

12.23 

12.79 

13.05 

D 

° 2&-29 

69.54 

49.83 

47.23 

50.62 

50.93 

D 

D 26-27 

58.43 

56.51 

58.67 

54.17 

53.49 



Table IV. CCTs with compensation. 


Bus 

t° 

1 CCT 

^CCT 

St CCT 

25 

300 ms 

309 ms 

9 ms 

26 

227 ms 

238 ms 

1 1 ms 

27 

307 ms 

323 ms 

16 ms 

28 

230 ms 

250 ms 

20 ms 

29 

197 ms 

209 ms 

12 ms 


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VIII. Biographies 

Enrique A. Zamora-Cardenas received the B.Eng. (Hons.) degree in 1999 
from the University of Colima, Colima, Mexico, and the MS degree in 2004 
from the Universidad Michoacana de San Nicolas de Hidalgo (UMSNH), 
Morelia, Mexico, in 2004. He is currently pursuing the PhD degree in 
UMSNH in the area of dynamic and steady-state analysis of FACTS. He is 
also an Associate Professor at the Institute Tecnologico Superior of Irapuato, 
at Irapuato, Guanajuato, Mexico. 

Claudio R. Fuerte-Esquivel (M’91) received the B.Eng. (Hons.) degree from 
the Institute Tecnologico de Morelia, Morelia, Mexico, in 1990, the MS 
degree ( summa cum laude) from the Institute Politecnico Nacional, Mexico, in 
1993, and the PhD degree from the University of Glasgow, Glasgow, 
Scotland, U.K., in 1997. Currently, he is an Associate Professor at the 
Universidad Michoacana de San Nicolas de Hidalgo (UMSNH), Morelia, 
where his research interests lie in the dynamic and steady-state analyzes of 
FACTS.