1 Introduction11
Power system restructuring has transformed its monopolistic structure to
three different entities which are independent of each other. These
entities are generation companies (GENCOS), transmission companies
(TRANSCO) and distribution companies (DISCO) [1]-[3]. This has
led to paradigm shift in power sector and has introduced competition in
electricity market. Although the competition in electricity market is
limited to generation and distribution sides only, it has been proved
fruitful to operate the transmission system by a single entity due to
economy of scale and to keep the system secure and reliable. However due
to introduction of competition in electricity market and with the
increase in electricity demand globally, it enforces the transmission
system operator to operate the system near its operating limit. This
increases the prospect of system constraints violation and could hamper
the transmission system security. The violation of system operating
constraints causes the transmission line to become congested. Although
the paradigm shift in electricity market, due to introduction of
competition, promises greater benefits to all, the congestion of
transmission network due to violation of its any operating constraint
may wash away all the prospects of benefits of competitive electricity
market. Therefore, management of congestion is not only important for
secure operation of power system but is also vital in achieving the
power system economy.
A number of literatures are available for different congestion
management schemes developed to relieve the congestion efficiently
[4]-[8]. This includes generation rescheduling [9]-[10],
load curtailment [11], use of the FACTS devices [12],
distributed generations [13] etc.
In new scenario of electricity market, wherein several bilateral and
multilateral transactions co-exist, some of the lines may get overloaded
due to flow of power above their thermal limit while other lines may be
underutilized. It makes obligatory for the system operator to utilize
the available transfer capability properly and judiciously. This task
can be efficiently accomplished by installing FACTS devices which
regulate the power flow by altering the transmission line parameters
such as line reactance, voltage magnitude and voltage angle [14].
These devices can also be used for voltage stability improvement,
transient stability improvement, sub- synchronous resonance mitigation
etc. [15]. However, its power regulating feature as well as
advantages over other congestion management methods has made it popular
to utilize for managing congestion in deregulated environment of power
system [16]-[19].
FACTS devices involve heavy installation cost, therefore its optimal
location, size and setting plays a very vital role in maximizing the
social welfare which deregulation promises to the society. Therefore
this paper focusses on these aspects of implementation of FACTS devices
to manage congestion. A number of works have been reported in literature
on application of FACTS devices for congestion management
[20]-[23].
A novel method to place a TCSC and SVC is proposed by authors in
[24] for managing congestion in the system considering the static
security margin improvement. A method to maximize the social welfare by
optimal placement and size of TCSC for congestion management has been
presented in [25]. The similar aspect of TCSC is considered by
authors [26] to relieve congestion from the system. Authors have
presented a method based on real power performance index and total
system VAR power losses to optimally place the TCSC for congestion
management. A comparison has been presented by authors to place the
series FACTS device based on LMP difference across a line and total
congestion rent [27]. The authors have proposed the use of FACTS
devices along with demand response to relieve congestion from the
transmission lines [28] while in [29] a curtailment strategy
based on FACTS device has been proposed. In [30] and [31], line
outage sensitivity factor is utilized to optimally place the series
FACTS devices for congestion alleviation in deregulated environment.
However, the authors in [32] have considered dc load flow in their
problem.
It is evident from the literature survey that TCSC is one of the widely
used FACTS devices around the world. Its simple construction and
implementation as well as low cost compared to other FACTS devices make
it preferable over others for congestion management [32]-[34].
Therefore in this paper also, TCSC is considered for the purpose of
managing congestion. In this paper, line flow sensitivity factor based
on real power flow is proposed to find the optimal location and setting
of TCSC for congestion alleviation in deregulated electricity market.
Line limit violation factor and voltage limit violation factors are
introduced to penalize for congestion and hence the minimum generation
cost as well as cost of installation of TCSC is found by optimally
placing TCSC using line flow sensitivity factor.
2 Modelling and implementation of TCSC
All Power flow in a transmission network can be adjusted by varying the
net series reactance. Application of series capacitor to increase
transmission line capacity is a well-known method of series compensation
which helps to reduce net series reactance thereby allowing flow of
additional power through the lines. However, the conventional methods of
series compensation use capacitors with mechanical switches such as
circuit breakers over a limited range while compensation using thyristor
controllers rapidly controls the line compensation over a continuous
range with flexibility. Therefore TCSC is widely adopted to regulate the
power flow in a line. This section deals with the modelling and
implementation of TCSC.
Figure 1 shows a π-equivalent transmission line model connected between
bus-i and bus-j.
Fig. 1. Transmission line model
If \(V_{i}\angle\delta_{i}\) and \(V_{j}\angle\delta_{j}\) are the
voltages at bus-i and bus-j respectively, the equations for active and
reactive power flow from bus-i to bus-j can be given by equation (1) and
(2) respectively.
\begin{equation}
P_{\text{ij}}=\ V_{i}^{2}G_{\text{ij}}-V_{i}V_{j}[G_{\text{ij}}\cos\left(\delta_{\text{ij}}\right)+B_{\text{ij}}\sin\left(\delta_{\text{ij}}\right)]\ \ \text{\ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ (1)\nonumber \\
\end{equation}
\(Q_{\text{ij}}={-V}_{i}^{2}\left(B_{\text{ij}}+B_{\text{sh}}\right)\ \ \ -V_{i}V_{j}\left[G_{\text{ij}}\sin\left(\delta_{\text{ij}}\right)+B_{\text{ij}}\cos\left(\delta_{\text{ij}}\right)\right]\text{\ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ }\)(2)
where, δij = δi-δj.
Similarly, the power flows from bus-j to bus-i are given by equations
(3) and (4).
\begin{equation}
P_{\text{ji}}=\ V_{j}^{2}G_{\text{ij}}-V_{i}V_{j}[G_{\text{ij}}\cos\left(\delta_{\text{ij}}\right)+B_{\text{ij}}\sin\left(\delta_{\text{ij}}\right)]\ \ \ \ \ \text{\ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ (3)\nonumber \\
\end{equation}\begin{equation}
Q_{\text{ji}}={-V}_{j}^{2}\left(B_{\text{ij}}+B_{\text{sh}}\right)\ -V_{i}V_{j}\left[G_{\text{ij}}\sin\left(\delta_{\text{ij}}\right)+B_{\text{ij}}\cos\left(\delta_{\text{ij}}\right)\right]\text{\ \ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)\nonumber \\
\end{equation}
The application of TCSC in a transmission network can be visualized as a
control reactance connected in series to the specific transmission line.
Fig. 2 shows the transmission network model with a TCSC. During
steady-state condition, a TCSC can be taken as a static
capacitor/reactor with impedance -jXC [35].
Fig. 2 Transmission line model with TCSC
With the implementation of TCSC, the power flow from bus-i to bus-j is
modified as given in equation (5) and (6).
\begin{equation}
P_{\text{ij}}^{{}^{\prime}}=\ V_{i}^{2}G_{\text{ij}}^{{}^{\prime}}-V_{i}V_{j}[G_{\text{ij}}^{{}^{\prime}}\cos\left(\delta_{\text{ij}}\right)+B_{\text{ij}}^{{}^{\prime}}\sin\left(\delta_{\text{ij}}\right)]\ \ \ \ \ \ \ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }(5)\nonumber \\
\end{equation}\begin{equation}
Q_{\text{ij}}^{{}^{\prime}}={-V}_{i}^{2}\left(B_{\text{ij}}^{{}^{\prime}}+B_{\text{sh}}^{{}^{\prime}}\right)-V_{i}V_{j}\left[G_{\text{ij}}^{{}^{\prime}}\sin\left(\delta_{\text{ij}}\right)+B_{\text{ij}}^{{}^{\prime}}\cos\left(\delta_{\text{ij}}\right)\right]\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(6\right)\nonumber \\
\end{equation}
where,
\begin{equation}
G_{\text{ij}}^{{}^{\prime}}=\frac{r_{\text{ij}}}{r_{\text{ij}}^{2}+\left(x_{\text{ij}}-x_{c}\right)^{2}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\nonumber \\
\end{equation}
and
\begin{equation}
B_{\text{ij}}^{{}^{\prime}}=\frac{-\left(x_{\text{ij}}-x_{c}\right)}{r_{\text{ij}}^{2}+\left(x_{\text{ij}}-x_{c}\right)^{2}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\nonumber \\
\end{equation}
Generally, a congestion management problem employs static model of FACTS
device injecting power at sending and receiving end of line [36].
According to this model FACTS device can be represented as PQ element
injecting definite amount of power to the specific node. Fig. 3
shows the power injection model of TCSC.
The real power injected at bus-i and bus-j due to implementing TCSC is
given by equation (7) and (8).
\begin{equation}
P_{i}^{{}^{\prime}}=V_{i}^{2}G_{\text{ij}}-V_{i}V_{j}\left[G_{\text{ij}}\text{\ cos}\delta_{\text{ij}}+B_{\text{ij}}\text{\ sin}\delta_{\text{ij}}\right]\text{\ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ (7)\nonumber \\
\end{equation}\begin{equation}
P_{j}^{{}^{\prime}}=V_{j}^{2}G_{\text{ij}}-V_{i}V_{j}\left[G_{\text{ij}}\text{\ cos}\delta_{\text{ij}}+B_{\text{ij}}\text{\ sin}\delta_{\text{ij}}\right]\text{\ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ (8)\nonumber \\
\end{equation}
Fig. 3 Power injection model
Similarly the reactive power injected at bus-i and bus-j after
implementing TCSC is given by equation (9) and equation (10).
\begin{equation}
Q_{i}^{{}^{\prime}}={-V}_{i}^{2}B_{\text{ij}}-V_{i}V_{j}\left[G_{\text{ij}}\text{\ sin}\delta_{\text{ij}}+B_{\text{ij}}\text{\ cos}\delta_{\text{ij}}\right]\text{\ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ (9)\nonumber \\
\end{equation}\begin{equation}
Q_{j}^{{}^{\prime}}={-V}_{j}^{2}B_{\text{ij}}-V_{i}V_{j}\left[G_{\text{ij}}\text{\ sin}\delta_{\text{ij}}+B_{\text{ij}}\text{\ cos}\delta_{\text{ij}}\right]\text{\ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ (10)\nonumber \\
\end{equation}
where,
\begin{equation}
G_{\text{ij}}=\frac{x_{c}r_{\text{ij}}\left(x_{c}-2x_{\text{ij}}\right)}{\left(r_{\text{ij}}^{2}+x_{\text{ij}}^{2}\right)\left(r_{\text{ij}}^{2}+\left(x_{\text{ij}}-x_{c}\right)^{2}\right)}\nonumber \\
\end{equation}
and
\begin{equation}
B_{\text{ij}}=\frac{-x_{c}(r_{\text{ij}}^{2}-x_{\text{ij}}^{2}+x_{c}x_{\text{ij}})}{\left(r_{\text{ij}}^{2}+x_{\text{ij}}^{2}\right)\left(r_{\text{ij}}^{2}+\left(x_{\text{ij}}-x_{c}\right)^{2}\right)}\nonumber \\
\end{equation}
3 Problem formulation for congestion management using TCSC
The main objective of this work is to manage congestion by optimally
placing TCSC in the power system network which is achieved by minimizing
the cost of installation of FACTS device along with penalty for
violation of line flow limits and bus voltage limits due to congestion
as shown in equation (11).
\begin{equation}
\text{Min\ }\left[C_{i}\left(P_{i}\right)+C_{\text{TCSC}}+\lambda_{1}.VLV+\lambda_{2}\text{.FLV}\right]\text{\ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ (11)\nonumber \\
\end{equation}
where
\begin{equation}
C_{\text{TCSC}}=C_{t}\ x\text{\ S\ }x\ 1000\ \ \ (\$/hr)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (12)\nonumber \\
\end{equation}\begin{equation}
C_{t}=0.0015\ \text{S\ }^{2}+0.713S+153.75\ (\$/KVAR)\ \text{\ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ (13)\nonumber \\
\end{equation}\begin{equation}
S=\left|Q_{1}-Q_{2}\right|\text{\ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (14)\nonumber \\
\end{equation}
where, Ct is the unit cost of TCSC
S is the operating range of TCSC in MVAR
Q1 and Q2 are the reactive power flow in the line before and after
installation of TCSC
PL2 is the power flow in line k to
which TCSC is connected
λ1 and λ2 are the penalty coefficients
in the range of 105 to 108VLV is the voltage violation factor
FLV is the line flow limit violation factor
The objective function comprises of two parts. The first part is the
installation cost of TCSC whereas the second part is composed of penalty
cost due to violation of bus voltage limit and the line flow limit which
are given as:
\begin{equation}
VLV=\left(\frac{V_{b}-V_{\text{ref}}}{V_{\text{ref}}}\right)^{2},\ \ \ if\ V_{b}<V_{\text{ref}}^{\min}\text{\ or\ \ }V_{b}>V_{\text{ref}}^{\max}\text{\ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }(15)\nonumber \\
\end{equation}\begin{equation}
\ \ \ \ \ \ \ \ \ =0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if\ V_{\text{ref}}^{\min}<\ \ V_{b}<V_{\text{ref}}^{\max}\text{\ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ (16)\nonumber \\
\end{equation}\begin{equation}
FLV=\left(\frac{P_{\text{ij}}-P_{\text{ij}}^{\max}}{P_{\text{ij}}^{\max}}\right)^{2},\ \ \ if\ P_{\text{ij}}>P_{\text{ij}}^{\max}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ (17)\nonumber \\
\end{equation}\begin{equation}
\ \ \ \ \ \ \ =0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if\ P_{\text{ij}}<P_{\text{ij}}^{\max}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ (18)\nonumber \\
\end{equation}
The reactance of TCSC is chosen such that
Xckmin < Xck< Xckmax. For static model
of TCSC, the maximum compensation allowed is 70% of the reactance of
line [27]. The voltages Vrefmin is
taken as 0.94 pu while Vrefmax is
taken as 1.06 pu.
4 Optimal placement of TCSC
TCSC involves a heavy investment for its installation. Therefore, its
appropriate location and size play a very vital role in managing
congestion efficiently. Otherwise, it would not be proved beneficial as
compared to other methods of congestion management. Therefore, the size
and location of TCSC must be chosen with utmost care [37].
In this paper, the TCSC is optimally placed in the system considering
the line flow sensitivity factor which is defined as a change in real
power flow in a transmission line connected between bus-i and bus-j due
to change in control parameter of TCSC.
Since the real power flow of a transmission line changes with the change
in its reactance, the real power flow in the network paths changes due
to the change in series reactance of the line by placing TCSC. This
change in real power flow is a function of control parameter (i.e.
reactance setting) of TCSC. Thus change in real power flow of a line due
to change in control parameter of TCSC gives an indication for optimal
placement of TCSC in managing congestion.
Mathematically, the line flow sensitivity factor with respect to the
parameters of TCSC placed at line-k can be defined as:
\begin{equation}
\text{LSF}_{c}^{k}=\left.\ \frac{{\partial P}_{\text{LT}}}{{\partial X}_{\text{ck}}}\right|_{X_{\text{ck}}=0}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (19)\nonumber \\
\end{equation}
where, PLT is the real power flow in line connected
between bus i and bus j
Xck is the control parameter of TCSC
The lines with high negative values of line flow sensitivity factor are
the potential locations for placing TCSC in the network in order to
manage congestion efficiently. Equation (19) is calculated by
differentiating the power flow in a line with respect to TCSC control
parameters which is given as:
\begin{equation}
\text{LSF}_{c}^{k}=\left.\ \frac{{\partial P}_{\text{LT}}}{{\partial X}_{\text{ck}}}\right|_{X_{\text{ck}}=0}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(20\right)\nonumber \\
\end{equation}\begin{equation}
\ \ \ \ \ \ =\ C_{\text{ij}}\ \left[{-V}_{i}^{2}+V_{j}{V\cos\delta_{\text{ij}}}_{j}\right]-D_{\text{ij}}\left[V_{i}V_{j}\sin\delta_{\text{ij}}\right]\text{\ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }(21)\nonumber \\
\end{equation}
where
\begin{equation}
C_{\text{ij}}=\left(\frac{-2r_{\text{ij}}x_{\text{ij}}}{\left(r_{\text{ij}}^{2}+x_{\text{ij}}^{2}\right)^{2}}\ \right)\nonumber \\
\end{equation}\begin{equation}
D_{\text{ij}}=\left(\frac{r_{\text{ij}}^{2}-x_{\text{ij}}^{2}}{\left(r_{\text{ij}}^{2}+x_{\text{ij}}^{2}\right)^{2}}\ \right)\nonumber \\
\end{equation}
Once the optical location of TCSC is found, its optimal setting for
congestion alleviation is found using PSO algorithm described in
[38].
5 Results and discussions
The FACTS device should be placed on the most sensitive line. With the
sensitive factor computed for TCSC, the TCSC should be placed in a line
having the most negative line flow sensitivity factor. The proposed
method for congestion management using TCSC is implemented and tested
with IEEE 30-bus and IEEE 118-bus systems in order to analyse its
effectiveness and robustness. The proposed method has been also tested
with 33-bus Indian network and the results thus obtained are compared
with those presented in [39]. Optimizations are carried out with PSO
[38] developed in MATLAB language. The values of various parameters
taken for PSO are given in appendices.