Investigating the impacts of asynchronous torque affecting to the transient stability in multi-machine power system

A new algorithm simulating the impact of asynchronous torque to the transient stability of multi-machine power system is mentioned and some typical numerical examples are presented in this article. Based on the proposed algorithm this PC program uses the elements of the eigen-image matrix to bring the specific advantages for the calculations of the transient stability of multimachine power system.The symmetrical and unsymmetrical transient voltages of the VAR supporting devices such as the static var compensators, synchronous machines are simulated under condition of action of the automatic voltage regulation system and the transient frequencies relating to the action of the asynchronous torque are simulated for analyzing of the transient stability in multimachine power system.


INTRODUCTION
In recent years, transient stability problems have been reported in power systems which originated broad studies in many literatures concerning [4], [5].The Power system is affected by high electromechanical oscillations while a disturbance occurs that may lead to loss of synchronism of generators.The asynchronous regime is tested and allowable with small slip of speed of the synchronous machines operating in the power system.Excepting the test of behavior of the synchronous generators assuming the asynchronous operation, the most important is the problem of assessment of the transient stability in power system including the reclosing with asynchronism.The asynchronous speed of the synchronous machines may be engendered in a power system during swings caused by symmetrical or unsymmetrical faults, its effect may be taken into account to the analyzing of the transient stability.The main features of the synchronous operation are as follows: The current circulating through the synchoronous machines and power network has components of two frequencies ω and ωo.The synchronous machines assume the properties of an asynchronous machine, since the rotor current appears with a slip, the emf Eω varies with the slip, the reactances of power network assume new value Xω=X(ω/ωo).
There are three methods to assess the asynchronous parameters [3] such as the currents, powers, torques and emf.The first method is supperposition method applicating to a machine model consisting of two machine components.The second method is finding the asynchronous emf.The third method is using the Park-Gorev' equations to calculate all variations associating to the change in the emf and torque of synchronous machine.Using the first method, the instantaneous value of the asynchronous torques can be determined as follow and the average value of the asynchronous torque is where Tasf is the asynchronous torque component relating to the rotor winding TasD is the asynchronous torque components relating to the d-axis damper windings TasQ is the asynchronous torque components relating to the q-axis damper windings q q q asQ 2 q q q X ' X " sT " T ; (5 ) X ' X " 1 (sT " ) V is the voltage of observing busbar relating to the receiving system; is the slip of speed;

MATHEMATICAL MODELLING OF TRANSIENT STABILITY FOR MULTI-MACHINE POWER SYSTEM
The power system consists of the generators, exciters, governors, loads and other equipments such as the power transformers, the transmission lines, the synchronous condensers, the static var compensators...The equivalent models of the important components for the power system are illustrated below.
The figure 1 presents a multi-machine augmented network consisting of M generators with their pre-fault output power of turbine and of the j-th impedance loads ZLj The figure 2 presents a fault condition in the augmented multimachine network with a fault impedance ZF At the t-th time interval, some automatic protective technical action of system protection can remove the fault element from the network, the figure 3 presents the post-fault multi-machine augmented network

Trang 29
Referring to [1], [2], [3], [4], [5], [7], the mathematical model for analyzing of the transient stability in a multi-machine power system involving the asynchronous torque is developed from [7] as follow p is i-th variable damping factor depending on a set of different parameters such as the i-th elements of the eigen-image matrix, the phase angles t i  at the t-th time interval, the voltages t i V at the i-th observing bus in power network at the t-th tim interval, the subtransient time and the rated frequency of the power system.

Synchronous Machine Model
The synchronous machines will be taken into account with the transient and subtransient effects.It is assumed that the d-axis and q-axis all have damping coils.The parameters of the synchronous machine will be taken as the input data concerning a set of numbers such as the inertia constant MJ, the d-axis and q-axis synchronous, transient, subtransient, negative sequence reactances Xd, Xq, X'd, X'q, X"d, X"q, X2 and the time constant of machine windings under no-load condition Tfo.
The synchronous electrical powers at the ij is the transfer element (if i≠j) or the driving-point element (if j=i) of the equivalent augmented bus admittance matrix Ybus at t-th time interval; E' t i t i is the transient e.m.f. of i-th synchronous machine taking into account of AVR at t-th time interval.

Exciter Model
The transfer function of the automatic excitation system may be written as According to ( 8), the mathematical model simulating the action during the fault of automatic excitation system including the electromagnetic transient effects of i-th synchronous machine should be written as follow where E t i and E' t i are the synchronous and transient e.m.f taking into account of excitation control at t-th time interval; Tdoi is the d-axis time constant of machine windings under no-load condition;  is a function simulating the proportional action of automatic excitation system of i-th synchronous machine at t-th time interval; Ei T is the equivalent time constant of the automatic excitation system; t Ei K is the equivalent gain of excitation; where The E and The X are the thevenin e.m.f and thevenin reactance at the bus of SVC location in multi-machine power system.
Referring to [7], the main process of calculation of multi-machine transient stability by our PC program is developed and shown in the figure 5 as follows

NUMERICAL EXAMPLE
Let's survey the electro-mechanical transient process of a 38-bus power system consisting of 5 thermal stations, 1 synchronous condenser, 4 SVC stations and 28 composite loads.Total MW load power demand is 2682.5MW.Basic power is 100MVA.The linedata is given in table 1   Let's assume that the V/I characteristics of the SVCs in p.u. at the buses 29, 30, 31 and 32 are given for inputdata of this example and shown in the figures 6, 7 and 8 as follows The inputdata for the reactances and time constants of the synchronous machines are given in the tables 4 and 5 as follows  The typical surveying and obtained results A single high voltage transmission line (2-3) connecting two power plants 37 and 38 is chosen simulating some type of fault to test the transient stability of the power system.Let's suppose that a fault of three phase short circuit occurs at the line (2-3) near the bus 2, and the fault will be cleared at 0.1sec by a circuit breaker.The typical results of surveying of the transient states of the power system during 8sec are shown in the figures 9 -18 as follows In case of neglecting of asynchronous torques in the calculation of transient process, the network voltage profile and the frequency profile will oscillate with amplitudes larger and are graphically shown in the figures 19 and 20 as follows In case of three phase short circuit at line (2-3) near bus 2, the critical clearing time is 0.56sec.Let's suppose that the symmetrical fault at line (2-3) near the bus 2 will be late cleared at 0.57sec, the transient state of the power system will be very severe, the unstable transient process will lead to the voltage and frequency collapses.The collapses in power system are graphically shown in the figures 21 -28 as follows In case of occurring of unsymmetrical fault at a distance of one-level tranformer from the generator locations in power system, the e.m.f components calculating for the synchronous machines are only positive sequence, taking into account of the action of automatic excitation regulation, the three phase voltages at the terminals of the synchronous machines are almost equals.During short time of the fault, the frequency variation is very small, neglecting the high harmonics, only the fundamental frequency is taken into account, the asynchronous torque may be approximately determined as how we calculated for the symmetrical fault mentioned above.The phase voltages may be determined by application of the symmetrical sequence components.
Let's survey an unsymmetrical fault of phase to phase to ground short circuit occurring at the line (2-3) near the bus 2, the clearing time is 0.1sec, the phase voltage are shown in the figures 29 -31 as follows In this case, the critical clearing time is 0.7sec.If the clearing time is taken equal to 0.71sec, then the voltage and frequency will strongly oscillate leading to their collapses in a very short time.The collapses of voltage and frequency are shown in the figures 32 -35 as follows

CONCLUSION
The results of the surveys showed that the asynchronous torque effects are to damp the electromechanical transients process, to slightly increase the critical clearing time for the different type of fault occurring in the power system, to strengthen the transient stability of the multimachine power system.New algorithm calculating the transient stability of power system is formed to test a number of simulations which consist the participation of the eigen-image matrix elements into the calculation of the asynchronous torque and is proven having good effective by the results of the calculations surveying a number of the numerical examples for the different structures of power system.
the upper and under bounds of stator voltage allowing to decide a suitable action of automatic excitation regulation of i-th synchronous machine; t Gi V is the actual voltage at the terminal of synchronous machine under condition of excitation control at t-th time interval;Speed Governor ModelIn general, the impact of speed governor affects only slightly to the transient process.If the speed varies more than (1.5-2)% and the transient process is longer than (2-3)sec, then the turbine power variation caused by the speed governor should be taken into account as some turbine torque changement.The transfer function of the governor [3] can be written as  is the efficiency of the feedback; AEis the time constant of the amplifying element, ASG is the time constant of the servo-piston;According to (10), the torque of i-th turbine at the t-th time interval should be found as a t ) is the change of the i-th turbine power depending on a set of parameters drop and time constant of the i-th automatic frequency regulation (AFR) system;Static VAR System ModelA static VAR system is an aggregation of Static VAR Compensator and mechanical switched capacitors or reactors whose outputs are coordinated.The SVC can enhance the transient stability and the damping of system oscillations.The performance of the SVC [4] is instantaneously provide unlimited Q_power to hold the voltage at a specific bus in power network with its V/I characteristic showing in figure4as followThe composite characteristic of SVC -Power System, within the control range defined by the slope KS with reactance XSL may be expressed as

Table 2 .
Data of the bus loads

Table 5 .
Time and Inertia constants