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TRANSIENT STABILITY -OBJECTIVE TYPE QUESTIONS 1. The critical clearing angle of a given power system for a certain fault is a. b. c. proportional to the Inertia Constant M proportional to the Inertia Constant M independent of M. Ans.: © 1. Equal area criterion of stability is applicable to a. b. c. a machine infinite bus system only both to a machine-infinite bus and a two-machine system a multi-machine system. Ans.: (b) 1. In a two-machine power system, machine A delivers power to machine B. A 3-phase fault occurs at the terminals of machine A. Initial acceleration of machine A is (a) positive (b) zero a. negative Ans.: (a) 1. Which one of the following enhances the transient stability of a system the most a. b. c. proper choice of make and break capabilities of the circuit breakers installation of 3-pole auto-reclose circuit breakers installation of single pole auto-reclose circuit breakers Ans.: (c ) TOP Inertia Constant H: Inertia constant H is different from inertia constant M. For a synchronous machine inertia constant H is frequently specified. It is defined as the ratio of the stored K.E at rated speed to the rated apparent power of the machine, i.e. H= Stored K.E in MJ at synchronous speed / machine rating in MVA (5) Swing equation (4) reduces to the form (2H/ws) d2 /dt2= Pm -Pe where Pm and Pe are pu powers, and ws should have consistent units Linearization of swing equation For small perturbations, the dynamic behaviour of the system can be studied by linearising the swing equations around the nominal operating point. The linearised swing equation is d2 /dt2 + s . S. = 0 2H where = small change in nominal operating angle S = Synchronizing power coefficient = d Pe (at = o) d TOP a synchronous generator? Equal area criterion (EAC) of stability To determine whether a power system is stable after a disturbance, it is necessary, in general, to plot and inspect the swing curves. If these curves show that the angle between any two machines tends to increase without limit, the system of course is unstable. If, on the other hand, after all the disturbances including switching have occurred, the angles between the two machines of every possible pair reach maximum values and thereafter decrease, it is probable, although not certain, that the system is stable. Occasionally in a multi-machine system one of the machines may stay in step on the first swing and yet go out of step on the second swing because the other machines are in different positions and react differently on the first machines. In a two-machine system, under the usual assumptions of constant input, and constant voltage behind transient reactance, the angle between the machines either increases indefinitely or else, after all disturbances have occurred, oscillates with constant amplitude. In other words the two machines either fall out of step on first swing or never. Under these conditions the observation that the machines come to rest with respect top each other may be taken as the proof that the system is stable. There is a simple graphical method of determining whether the machines come to rest with respect to each other. This method is known as the equal area criterion of stability. When this criterion is applicable its use wholly or partially eliminates the need of computing swing curves and thus saves considerable amount of computation time. It is applicable to any two-machine system for which the assumptions stated above may be made. Application of EAC Consider a machine infinite bus system. The swing equation of the system is M d2 /dt2= Pm -Pe = Pa Multiplying both sides of the equation by (2/M) d /dt, we get 2 d /dt .d2 /dt2= (2Pa/M) d /dt (6) Since d /dt. (d /dt)2 = 2 d /dt .d2 /dt2, Equation (6) reduces to d /dt. (d /dt) 2 = (2Pa/M) d /dt Next, multiply each side by dt, obtaining d[. (d /dt) 2]= (2Pa/M) d Integrating this equation, we get (d /dt) 2 = (2/M) ƒ Pa d o (d /dt) 2 = [(2/M) ƒ Pa d ] o When the machine comes to rest with respect to the infinite bus- a condition, which may be taken to indicate stability-requiring that m ƒ Pa d =0 o This integral may be integrated graphically (Fig.9) as the area under a curve of Pa plotted against between limits o, the initial angle, and m, the final angle. Area A1 and A2 may be interpreted in terms of kinetic energy gained /lost by the synchronous generator. Illustrate the application of equal area criterion by applying it (to the following two simple cases) for a synchronous generator connected to infinite bus through a double-circuit line. Sustained fault Line fault cleared after the lapse of a certain time by the simultaneous opening of the circuit breakers at both ends of the line. Transient stability limit Of a two-machine system is defined as the maximum power that can be transmitted from one machine to the other without loss of synchronism for a specified, sudden, severe, unrepeated shock. Illustrate the concept of transient stability limit using equal area criterion of stability. Obtain an equivalent one-machine infinite bus system of a two finite machines system Obtain an expression for the critical clearing angle What is the effect of location and duration of fault on the transient stability limit Numerical technique for solution of swing equation The transient stability analysis requires the solution of a system of coupled non-linear differential equations. In general, no analytical solution of these equations exists. However, techniques are available to obtain approximate solution of such differential equations by numerical methods and one must therefore resort to numerical computation techniques 9commonly known as digital simulation0. Some of the commonly used numerical techniques for the solution of the swing equation are: Point by point method Euler's method Euler's modified method Runge-Kutta method, etc. Point -by -point method Point by point solution, also known as step-by-step solution is the most widely used way of solving the swing equation. The following two steps are carried out alternately. 1. 2. First, compute the angular position , and angular speed d /dt at the end of the time interval using the formal solution of the swing equation from the knowledge of the assumed value of he accelerating power and the values of and d /dt a the beginning of the interval Then compute the accelerating power of each machine from the knowledge of the angular position at the end of the interval as computed in step 1. There are two different point-by-point methods. Method 2 is more accurate compared to method 1. Method 2 In this method the accelerating power during the interval is assumed constant at its value calculated for the middle of the interval. The desired formula for computing the change in during the nth time interval is n = n-1 + [( t) 2 /M] Pa(n-1) where, n = change in angle during the nth time interval n-1 = change in angle during the (n-1)th time interval t= length of time interval Pa(n-1)= accelerating power at the beginning of the nth time interval Due attention is given to the effects of discontinuities in the accelerating power Pa which occur, for example, when a fault is applied or removed or when any switching operation takes place. If such a discontinuity occurs at the beginning of an interval, then the average of the values of Pa before and after the discontinuity must be considered. Thus, in computing the increment of angle occurring during first interval after a fault is applied at t=0, the above equation becomes: 1 =[( t) 2 /M] Pa0+/2 where Pa0+ is the accelerating power immediately after the occurrence of the fault. If the fault is cleared at the beginning of the mth interval, then for this interval, Pa(m-1) = 0.5 [Pa(m-1)- + Pa(m-1) +] Where Pa(m-1)- is the accelerating power before clearing and Pa(m-1) + is that immediately after clearing the fault.. If the discontinuity occurs at the middle of the interval, no special treatment is needed. Improvement of power system stability System parameters affecting stability: Synchronous machine parameters Transmission line parameters Circuit breaker & relay characteristics System layout Excitation system and governor characteristics Neutral grounding Discrete Controls: Auto -reclosing of circuit breakers Single-pole switching Resistor breaking Generator dropping Load shedding Series capacitor switching Shunt reactor/capacitor switching Boosting power on D.C links Fast Valving Discuss the effect of variation of M and auto -reclosure on the transient stability limit STEADY-STATE STABILITY- OBJECTIVE TYPE QUESTIONS 1. The power angle characteristic of a machine-infinite bus system is P= 2 sin (pu). The initial operating angle is 60 deg., inertia constant H= 5 sec. System frequency is 50 Hz. The angular frequency of oscillation following small perturbation will be a. sqrt(31.4) rad/sec b. sqrt(15.7) rad/sec c. sqrt(62.8) rad/sec Ans.: (a) 2. 2. In an interconnected power system ,the frequency of electro-mechanical modes of oscillation lies in the range (a) 0.5-2.5 Hz (b) 1-10 Hz (c) 30-60 Hz Ans.: (a) Small and large perturbations 3. 3. If the sending end and receiving end voltages for a 3-phase transmission line are each 33kV(line), and if the reactance of the line is 13 ohms per phase, the maximum power transmitted per phase will be a. b. c. d. a. b. c. d. 60 MW 30 MW 29 MW 28 MW Ans. d 4. 4. The torque angle corresponding to the steady-state stability limit of a salient-pole alternator is a. b. c. a. b. c. Less than 90 deg. Greater than 90 deg. Equal to 90 deg. Ans. a 5. 5. When the alternator stalls (near the stability limit) the armature current is a. b. c. a. b. c. Less than the rated value Greater than the rated value Equal to the rated value. Ans. a Introduction Power system stability is a term applied to alternating current electric power systems, denoting a condition in which the various synchronous machines of the system remain in synchronism, or "in step" with each other. Conversely, instability denotes a condition involving loss of synchronism, or falling "out of step". The AIEE standard definition of stability is as follows" Stability when used with reference to a power system, is that attribute of the system, or apart of the system which enables it to develop restoring forces between the elements thereof, equal to or greater than the disturbing forces so as to restore a state of equilibrium between the elements". Small and large perturbations A power system is subjected to a variety of disturbances. These are classified into two categories: Small perturbations Large perturbations Small Perturbations Perturbations are characterized as small if the changes in system states are small due to these perturbations. The magnitude of perturbation is small enough o allow the use of linearized state equations obtained by linearizing the nonlinear differential equations around the operating point for studying he dynamics of the system. Random changes in load which occur in the system continuously is a n example of small perturbation. Stability problem associated with small perturbation is known as dynamic stability. Large perturbations A power system may be subjected to large perturbations such as : Occurrence of faults on the line Loss of large generating units Loss of major transmission facilities Loss of large loads The stability problem associated with large perturbations is known as transient stability. Derive and explain the power angle characteristic of Machine infinite-bus system Two machine system Discuss the concepts of stable & unstable equilibrium points. What is meant by steady state & transient stability of a system? Voltage Stability-ModuleC(.xls file) This module introduces the static voltage instability problem. The module uses a simple generator connected to a load through two parallel transmission lines. The student can plot PV, QV, or PQ curves for different values of line reactance, generator terminal voltage, and power factor. Parameters can be changed under the parameter column. Different plots can be obtained from the Plot Command. Clicking the Introduction button provides student with basic material to understand the concept of maximum power transfer. The information button includes the instructions needed to plot various curves. The student is asked to calculate the reactive power required maintaining a certain voltage Vr at the load bus for a given Pr, Vs, and Xl. The student is asked to calculate the reduction in maximum power at the load bus with one line out.