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 = δij. 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.