Transient Stability of the Power System with the Exact Long Transmission Line Model

Problem statement: The exact long transmission line model consists of the lump of the series resistance, reactance and shunt capacitance. With the consideration the actual long transmissi on line model, it causes in the difficulty of deriving the mathematical model. Approach: This study investigates the transient stability of power syste m with consideration the exact long transmission li ne model. The concept of two-port network is applied i n this study. The generator, transformer and short transmission line are represented by two-port netwo rks. With the combination principles of the series and shunt connection, the mathematical model is ach ieved in a much simpler way. The proposed method is tested on the sample system and compared on various cases. Results: The first swing of rotor angle curve of the faulted system without the resistance is obviously higher than that of with t he resistance whereas the second swing of the faulted system without the resistance is slightly less than that of with the resistance. The critical clearing time of the system with the resis tance is better than that of with resistance. Conclusion: It was found from the simulation results that the resistance of the line provides the improvement of the first swing but not for the second swing. It was found from this study that for practical long line, the resistance is ver y import parameters to determine the critical clear ing time of the single machine infinite system whereas shunt capacitance insignificantly affects on the critical clearing time of the single machine infini te bus system.


INTRODUCTION
Nowadays, the demand of electricity has dramatically increased and a modern power system becomes a complex network of transmission lines interconnecting the generating stations to the major loads points in the overall power system in order to support the high demand of consumers. It is becoming increasingly important to fully utilize the existing transmission system assets due to environmental legislation, rights-of-way issues and cost of construction and deregulation policies that introduced in recent years. A number of Flexible AC Transmission System (FACTS) controllers, based on the rapid development of power electronics technology, have been proposed for better utilization of the existing transmission systems (Li et al., 2010;Magaji and Mustafa, 2009;Kumkratug, 2010;Omar et al., 2010;Padma and Rajaram, 2011;Valarmathi and Chilambuchelvan, 2011;Zarate-Minano et al., 2010).
The evaluation of Critical Clearing Time (CCT) of power system is one of the most important research areas for power engineers because it indicates the robustness of the faulted power system. The dynamic behavior of synchronous generator plays very important role to determine the CCT of power system.
The transmission line is one of the most important parts in power system components. The transmission line is generally divided into three major categories; short, medium and long model whose distance are about 80 km, above 80-250 km and above 250 km, respectively. Many previous researches used simple transmission line model by neglecting its resistance or capacitance. To fully utilization the existing the transmission, the exact transmission line is needed to studied.
This study will investigate the critical clearing time of the system with the exact long transmission line model. The concept of two-port network is applied to simplify the mathematical model of the power system. The proposed method is tested on sample system and compared on various cases.

Mathematical model:
The transmission line is considered as long line when its distance is beyond 150 km. The basic model of long transmission line consists of the lump of medium transmission line model with serial π models as shown in Fig. 1a.
Its simpler model is represented by a π model as shown in Fig. 1b Eq. 1 and 2.
The Z'and Y'are: Here: This study applies the concept of the two-port network to simplify the equivalent in Fig. 2b. Each component of power system can be represented the two-port networks (A, B, C and D) as shown in Fig. 2d and given by Eq. 3-8: It can be seen from the Fig. 2c that some ports are in series and in shunt connection. For example, a port 1 and port 2 are in series connection whereas port 3 and port 4 are in shunt connection. Thus with the series combination of port 1 and port 2, a new port is given by Eq. 9-12: Similarly, with the shunt combination of port 3 and port 4, a new port is given by Eq. 13-16: With the above concepts, the net two-port network diagram is shown in Fig. 2d. Here A eq , B eq , C eq and D eq are the element in net matrix of net two-port networks. The output electrical power of synchronous machine (P e ) is given by Eq. 17: Here eq eq Aeq eq eq Beq The dynamic equation for evaluating critical clearing tine of the system in Fig. 2a Here δ, ω and P m are the rotor angle, speed, mechanical input power and moment of inertia, respectively of synchronous machine.  Thus the impedance (Z) and the admittance (Y) of the long line at fundamental frequency are Z = 19.588 + j187.09 ohm and Y = 0 + j0.0021 Siemens, respectively. The percentage of R/X and B c /X of the line are = 10.47 % and B c /X = 0.00112 %, respectively.

Consider the diagram of sample system is shown in
With the given reactance of the line (X) = 0.5 pu, the per unit of R and B c are 0.0524 and 0.002 pu, respective.
It is considered that three phase fault appears at line 2 near bus m and the fault is cleared by opening circuit breakers at the end of the line. Figure 3 shows the rotor angle of the system with clearing time (t cl ) for 170 msec and the detail of the simulation and data is summarized in Table 1. Figure 4 shows the rotor angle of the system with 5pecent for t cl = 191msec. Table 1 summarizes the critical clearing time (t cr ) of the system with various R/X.

DISCUSSION
It can be seen from the Fig. 3 and Table 1 that resistance of the long line provides the improvement of the first swing stability but not for the second swing. Without the resistance of the long line, the maximum and the minimum rotor angle are 107.80 and-0.63°, respectively whereas with the resistance, the maximum and the minimum rotor angle are 104.20 and-2.40°, respectively. In practical long line, the resistance is very import parameters to determine the critical clearing time of the system whereas shunt capacitance would not affect on critical clearing time of single machine infinite bus as can be seen in Fig. 4 and Table 2. Without resistance, the critical clearing time of the system is 190-191 m sec whereas with resistance the critical clearing is increased to around 204-205 m sec. With t cl = 191 m sec, the system without consideration of line resistance can be considered as unstable but for with line resistance is stable as shown in Fig. 4.

CONCLUSION
This study investigated the critical clearing time of the system with consideration various parameters of long transmission line. The concept of two-port network is applied to simplify the mathematical model of the power system. The reactance of generator, transformer and exact long line model can be represented by a net two-port network.
It was found from the simulation results that the resistance of the line provides the improvement of the first swing but not for the second swing. It was found from this paper that for practical long line, the resistance is very import parameters to determine the critical clearing time of the single machine infinite system whereas shunt capacitance would not affect on critical clearing time of single machine infinite bus system.