INTRODUCTION
The electricity supply industry is undergoing a profound transformation worldwide.
Market forces, scarcer natural resources and an everincreasing demand for electricity
are some of the drivers responsible for such an unprecedented change. Against
this background of rapid evolution, the expansion programmes of many utilities
are being thwarted by a variety of wellfounded, environmental, land use and
regulatory pressures that prevent the licensing and building of new transmission
lines and electricity generating units. An indepth analysis of the options
available for maximizing existing transmission assets, with high levels of reliability
and stability, has pointed in the direction of power electronics. There is general
agreement that novel power electronics equipment and techniques are potential
substitutes for conventional solutions, which are normally based on electromechanical
technologies that have slow response times and high maintenance costs (Hingorani
and Gyugyi, 2000).
Until recently, active and reactive power control in AC transmission networks
was exercised by carefully adjusting transmission line impedances, as well as
regulating terminal voltages by generator excitation control and by transformer
tap changes. At times, series and shunt impedances were employed to effectively
change line impedances. The FACTS technology is most interesting for transmission
planners because it opens up new opportunities for controlling power and enhancing
the usable capacity of present, as well as new and upgraded, lines. The possibility
that current through a line can be controlled at a reasonable cost enables a
large potential of increasing the capacity of existing lines with large conductors
and use of one of the FACTS controllers to enable corresponding power to flow
through such lines under normal and contingency conditions (Hingorani,
1993; Mathur and Varma, 2002).
POWER FLOW ANALYSIS
Planning the operation of power systems under existing conditions, its improvement and also its future expansion require the load flow studies, short circuit studies and stability studies.
Through the load flow studies we can obtain the voltage magnitudes and angles
at each bus in the steady state. This is rather important, as the magnitudes
of the bus voltages are required to be held within a specified limit. Once the
bus voltage magnitudes and their angles are computed using the load flow, the
real and reactive power flow through each line can be computed. Also based on
the difference between power flow in the sending and receiving ends, the losses
in a particular line can also be computed. One of the main strengths of the
Newton Raphson method is its reliability towards convergence. Contrary to non
Newton Raphson solutions, convergence is independent of the size of the network
being solved and the number and kinds of control equipment present in the system.
So, this is the most favored power flow method (Acha et
al., 2004; FuerteEsquivel and Acha, 1997).
The Newton Raphson algorithm: In largescale power flow studies, the
Newton raphson has proved most successful owing to its strong convergence characteristics.
The power flow Newton Raphson algorithm is expressed by the following relationship
(Milano, 2009; Bonert, 1998).
It may be pointed out that the correction terms ΔV_{m} are divided by V_{m }to compensate for the fact that jacobian terms (∂P_{m}/∂V_{m})V_{m} and (∂Q_{m}/∂V_{m})V_{m} are multiplied by V_{m}. It is shown in the directive terms that this artifice yields useful simplifying calculations. Consider the lst element connected between buses k and m in Fig. 1, for which self and mutual Jacobian terms are given below:
For k ≠ m.

Fig. 1: 
Equivalent impedance 
For k = m:
The mutual elements remain the same whether we have one transmission line or n transmission lines terminating at the bus k.
The sample 5 bus system: In case study we have considered the five bus
system as shown in Fig. 2. The input data for the considered
system are shown in Table 1 for the bus and Table
2 for transmission line. Assuming base quantities of 100 MVA and 100 KV.
The result for the above system is shown in Table 3. All the
nodal voltages are achieved to be within acceptable voltage magnitude limits.

Fig. 2: 
The fivebus network 
Table 1: 
Input bus data for the study system (p.u.) 

Table 2: 
Input transmission line data for the study system (p.u.) 

Table 3: 
Power flow result of study system without any FACTS devices 

POWER FLOW MODEL OF FACTS DEVICES
Power flow model of SVC Shunt variable susceptance model: In practice
the SVC can be seen as an adjustable reactance with either firing angle limits
or reactance limits. The equivalent circuit shown in Fig. 3
is used to derive the SVC nonlinear power equations and the liberalized equations
required by the Newton’s method (Bijwe and Kelapure,
2003). With reference to Fig. 3, the current drawn by
the SVC is I_{SVC}=jB_{SVC}V_{k }and the reactive power
drawn by the SVC, which is also the reactive power injected at bus k, is Q_{SVC}
= Q_{k} = V_{k}^{2}B_{SVC . }At the end of
each iteration, the variable Shunt Susceptance B is updated according to:
The changing Susceptance represents the total SVC Susceptance necessary to maintain the nodal voltages at specified value.
Firing angle model: An alternative SVC model, which circumvents the
additional iterative process, consists in handling the thyristor controlled
reactor (TCR) firing angle α as a state variable in the power flow formulation.
The positive sequence Susceptance of the SVC, is given by:

Fig. 3: 
Variable shunt susceptance 
From the above equation, the linearised SVC equation is given as:
At the end of iteration (i), the variable firing angle α_{SVC} is updated according to α_{SVC}^{ (i)}_{ = }α_{SVC}^{(i1)} + Δα_{SVC}^{(i)}
Power flow model of TCSC: Two alternative power flow models to assess
the impact of TCSC equipment in network wide applications are presented here.
The simpler TCSC model exploits the concept of a variable series reactance.
The series reactance is adjusted automatically, within limits, to satisfy a
specified amount of active power flows through it. The more advanced model uses
directly the TCSC reactancefiring angle characteristics, given in the form
of a non linear relation. The TCSC firing angle is chosen to be the state variable
in the NewtonRaphson power flow solution.
Variable series impedance power flow model: The TCSC power flow model
presented in this section is based on the simple concept of a variable series
reactance, the value of which is adjusted automatically to constrain the power
flow across the branch to a specified value. Nelson et
al. (1995). The amount of reactance is determined efficiently using
Newton’s method. The changing reactance X_{TCSC, }shown in Fig.
4, represents the equivalent reactance of all the seriesconnected modules
making up the TCSC, when operating either in inductive or in the capacitive
regions. The transfer admittance matrix of the variable series compensator shown
in Fig. 4 is given by:
For inductive operation, we have:

Fig. 4: 
Thyristor controlled series compensator equivalent circuit,
(a) Inductive and (b) capacitive operative regions 
For capacitive operation the signs are reversed. The active and reactive power equations at bus k are:
For the power equations at bus m, the subscripts k and m are exchanged in the above equations. The state variable X_{TCSC} of the series controller is updated at the end of each iterative step according to:
Firing angle power flow model:The model presented in this section uses
the concept of an equivalent series reactance to represent the TCSC. Once the
value of reactance is determined using Newton’s method then the associated
firing angle α_{TCSC} can be calculated. However, such calculations
involve an iterative solution since the TCSC reactance and the firing angle
are nonlinearly related. One way to avoid the additional iterative process is
to use the alternative TCSC power flow model presented in this section. The
fundamental frequency equivalent reactance X_{TCSC} of the TCSC module
shown in Fig. 5 is:

Fig. 5: 
Thyristor controlled series compensator firing angle module 
where:
The TCSC active and reactive power equations at bus k are:
where, .
Power flow model of static synchronous compensator: The Static Synchronous
Compensator (STATCOM) is represented by a synchronous voltage source with minimum
and maximum voltage magnitude limits. The bus at which STATCOM is connected
is represented as a PV bus, which may change to a PQ bus in the events of limits
being violated. In such case, the generated or absorbed reactive power would
correspond to the violated limit. The power flow equations for the STATCOM are
derived below from the first principles and assuming the following voltage source
representation (Gyugi, 1994).
Based on the shunt connection shown in Fig. 6, the following equation may be written:
where, * represents the complex conjugate.

Fig. 6: 
Static compensator (STATCOM) equivalent circuits 

Fig. 7: 
Unified power flow controller equivalent circuit 
The following are the active and reactive power equations for the converter
at bus k,
Power flow model of unified power flow controller: The UPFC equivalent
circuit consists of two coordinated synchronous voltage sources for the purpose
of fundamental frequency steady state analysis. Such an equivalent circuit is
shown in Fig. 7.
The UPFC voltage sources are:
where, V_{vR} and δ_{vR} are the controllable magnitude
(V_{vRmin }≤ V_{vR }≤ V_{vRmax}) and phase angle
(0 ≤ δ_{vR}≤ δ2π) of the voltage source representing
the shunt converter. The magnitude V_{cR} and phase angle δ_{cR}
of the voltage source representing the series converter are controlled between
limits (V_{cRmin }≤ V_{cR }≤ V_{cRmax}) and (0≤δ_{cR}
≤2π), respectively. The phase angle of the series injected voltage determines
the mode of power flow control (Kannan et al., 2004;
Gyugyi et al., 1995). If δ_{cR }is
in phase with the nodal voltage angle ө_{k}, the UPFC regulates
the terminal voltage. If δ_{cR} is in quadrature with ө_{k},
it controls active power flow, acting as a phase shifter. If δ_{cR}
is in quadrature with line current angle then it controls active power flow,
acting as a variable series compensator. At any other value of δ_{cR},
the UPFC operates as a combination of voltage regulator, variable series compensator
and phase shifter. The magnitude of the series injected voltage determines the
amount of power flow to be controlled. Based on the equivalent circuit shown
in Fig. 7 the active and reactive power equations are:
At bus k:
At bus m:
Series converter:
Shunt converter:
Assuming lossless converter values, the active power supplied to the shunt converter, P_{vR}, equals the active power demanded by the series converter, P_{cR }, i.e., P_{vR}+P_{cR} = 0. Further more, if the coupling transformers are assumed to contain no resistance then the active power at bus k matches the active power at bus m. Accordingly, P_{vR}+P_{cR} = P_{k}+P_{m} = 0. The UPFC power equations are combined with those of the AC network.
CASE STUDIES WITH FACTS CONTROLLERS
Power flow study with SVC: The SVC is included in the bus 3 (Fig.
8) of the sample system to maintain the nodal voltage at 1 p.u.
The SVC data:
• 
The initial susceptance: 0.02 p.u 
For the firing angle model:
• 
The initial firing angle:140 degrees 
• 
Inductive reactance: 0.288 p.u 
• 
Capacitive reactance: 1.07 p.u 
The result for the voltage magnitude and phase angle obtained for the above
system for both variable Susceptance and Firing angle model is shown in Table
4. Due to the inclusion of SVC in the third bus, its voltage is maintained
at 1 p.u.
Power flow study with TCSC: The original fivebus network is modified
to include one TCSC (Fig. 9) to compensate the transmission
line connected between bus 3 and 4. The TCSC should maintain the real power
flow in transmission line 6 as 21 MW.

Fig. 8: 
Study System with SVC included 
TCSC data:
• 
Initial capacitive reactance: 0.015 p.u. 
In the firing angle model:
• 
Firing angle:145 degrees 
The result for the voltage magnitude and phase angle obtained for the above
system for both variable reactance and firing angle model is shown in Table
5. Due to the inclusion of TCSC, the target power flow of 21 MW in line
6 is maintained.
Power flow study with statcom: The STATCOM is included in the bus 3
(Fig. 10) of the sample system to maintain the nodal voltage
at 1 p.u.
Statcom data:
• 
The initial source voltage magnitude: 1 p.u. 
• 
Phase angle: 0 degrees. 
• 
The converter reactance: 10 p.u. 
The power flow result indicates that the STATCOM generates 20.5 MVar in order
to keep the voltage magnitude at 1 p.u. at bus3. Use of STATCOM results in an
improved network voltage profile as shown in Table 6.
Table 4: 
Result with SVC included in Bus 3 

Table 5: 
Result with TCSC included in line 6 


Fig. 9: 
Study System with TCSC included 

Fig. 10: 
Study System with STATCOM included 

Fig. 11: 
Study System with UPFC included 
Table 6: 
Result with STATCOM included in Bus 3 

Table 7: 
Result with UPFC included in line 6 

Power flow study with UPFC: The original fivebus network is modified
to include one UPFC to compensate the transmission line linking bus 3 and bus
4 (Fig. 11). UPFC should maintain real and reactive power
flowing towards bus 4 at 40 MW and 2 MVar, respectively. The UPFC shunt converter
is set to regulate the nodal voltage magnitude at bus 3 at 1 p.u. as shown in
Table 7.
UPFC data: The starting values of the UPFC shunt converter are:
• 
Voltage magnitude: 1 p.u. 
• 
Voltage magnitude: 0.04 p.u. 
• 
Phase angle: 87.13 degrees 
• 
Reactance for both the converters: 0.1 p.u. 
RESULTS
The power flow for the five bus system was analyzed without and with FACTS devices performing the NewtonRaphson method. The largest power flow takes place in the transmission line connecting the two generator buses: 89.3 MW and 74.02 MVar leave bus1and 86.8 MW and 72.9 MVar arrive at bus2. The operating conditions demand a large amount of reactive power generation by the generator connected at bus1 (i.e., 90.82 MVar). This amount is well in excess of the reactive power drawn by the system loads (i.e., 40 MVar). The generator at bus2 draws the excess of reactive power in the network (i.e., 61.59 MVar). This amount includes the net reactive power produced by several transmission lines, which is addressed by different FACTS devices. Thus SVC upholds its target value and as expected identical power flows and bus voltages are obtained for both Shunt Variable Susceptance Model and Firing angle power flow models.
The TCSC variable series compensator model is used to maintain active power flowing from the extra fictious bus 6 towards bus 3 at 21 MW. The starting value of the TCSC is set at 50% of the value of transmission line inductive reactance. The TCSC upholds the target value of 21 MW, which is achieved with 70% series compensation of the transmission line 6. In the case of the firing angle model the initial value of firing angle is set at 145° and the TCSC upholds the target value of 21 MW.
The power flow result indicates that the STATCOM generates 20.5 MVar in order to keep the voltage magnitude at 1 p.u., at bus 3. Use of STATCOM results in an improved network voltage profile, except at bus 5, which is too far away from bus 3 to benefit from the influence of STATCOM.
The original fivebus network is modified to include one UPFC to compensate the transmission line linking bus 3 and bus 4 The UPFC is used to maintain active and reactive powers leaving UPFC, towards bus 4, at 40 MW and 2 MVar, respectively. Moreover, the UPFC shunt converter is set to regulate the nodal voltage magnitude at bus 3 at 1 p.u. There is a 32% increase of active power flowing towards bus3. The increase is in response to the large amount of active power demanded by the UPFC series converter. Thus from the above analysis we find that within the framework of traditional power transmission concepts, the UPFC is able to control, simultaneously or selectively, all the parameters affecting power flow in the transmission line (voltage, impedance and phase angle) and this unique capability is signified by the adjective unified in its name.

Fig. 12: 
Single line circuit diagram of the IEEE 30bus system 

Fig. 13: 
Single line circuit diagram of the IEEE 118bus system 
To verify the proposed different FACTS steady state model, numerical studies
have been carried out on the IEEE 30bus system (Fig. 12)
and IEEE 118bus system (Fig. 13). In the tests, a convergence
of 1.0e12 p.u., is used for maximal absolute bus power mismatch. The convergence
of result is achieved within 4 to 5 iterations.
CONCLUSION
The model developed for different FACTS devices are used in the 5 bus, IEEE 30 bus and IEEE 118 bus system to perform power flow solutions. The effectiveness of the models developed is reflected in quick convergence during the power flow study.
ACKNOWLEDGMENT
The authors are thankful to the management of SSN College of Engineering, Kalavakkam, Chennai for providing the Computational facilities to carryout this study.