FACULTY OF ENGINEERING
LAB SHEET
TRANSIENT STABILITY OF POWER SYSTEM
EET2056
TRIMESTER II (2011/2012)
TS1 - TRANSIENT STABILITY ANALYSIS OF ONE-MACHINE-
INFINITE-BUS-SYSTEMS
TS2 - TRANSIENT STABILITY ANALYSIS OF
MULTIMACHINE SYSTEMS
*Note: On-the-spot evaluation may be carried out during or at the end of the experiment. Students are advised to read through this lab sheet before doing experiment. Your performance, teamwork effort, and learning attitude will count towards the marks.
Experiment # 1
TRANSIENT STABILITY ANALYSIS OF ONE-MACHINE-
INFINITE-BUS SYSTEMS
Objectives
The objectives of this experiment are:
1. To study the stability behaviour of one machine connected to a large power system subjected to a severe disturbance (3-phase short circuit)
2. To understand the principle of equal-area criterion and apply the criterion to study the stability of one machine connected to an infinite bus
3. To determine the critical clearing angle and critical clearing time with the help of equal-area criterion
4. To do the stability analysis using numerical solution of the swing equation
Introduction
The tendency of a power system to develop restoring forces to compensate for the disturbing forces to maintain the state of equilibrium is known as stability. If the forces tending to hold the machines in synchronism with one another are sufficient to overcome the disturbing forces, the system is said to remain stable.
The stability studies which evaluate the impact of disturbances on the behaviour of synchronous machines of the power system are of two types – transient stability and steady state stability. The transient stability studies involve the determination of whether or not synchronism is maintained after the machine has been subjected to a severe disturbance. This may be a sudden application of large load, a loss of generation, a loss of large load, or a fault (short circuit) on the system. In most disturbances, oscillations are such magnitude that linearization is not permissible and nonlinear equations must be solved to determine the stability of the system. On the other hand, the steady-state stability is concerned with the system subjected to small disturbances wherein the stability analysis could be done using the linearized version of nonlinear equations. In this experiment we are concerned with the transient stability of power systems.
A method known as the equal-area criterion can be used for a quick prediction of stability of a one-machine system connected to an infinite bus. This method is based on the graphical interpretation of energy stored in the rotating mass as an aid to determine if the machine maintains its stability after a disturbance. The method is applicable to a one-machine system connected to an infinite bus or a two-machine system. Because it provides physical insight to the dynamic behaviour of the machine, the application of the method to analyze a single-machine system is considered here.
Test System and Data
A typical one-machine system connected to an infinite bus is shown in Fig. 1. The system consists of one synchronous machine, one transformer, two parallel transmission lines and an infinite bus. The various data are:
· Inertia constant of the synchronous machine, H = 5 MJ/MVA
· Direct axis transient reactance of the generator, Xd’ = 0.3 p.u.
· Transformer leakage reactance, Xt = 0.2 p.u.
· Line reactance, XL1 = XL2 = 0.3 p.u.
· Supply frequency, f = 50 Hz
· Real and reactive power delivered to the infinite bus, Pe = 0.8 p.u. and Qe = 0.074 p.u.
· The infinite bus voltage, V = 1.0 p.u.
Fig. 1 One-machine system connected to an infinite bus
Different Cases of Study
Case 1
A temporary three-phase symmetrical fault occurs at point P for a short time and then the fault is cleared. Both lines are intact.
Case 2
A three-phase symmetrical fault occurs at Q, the middle point of one of the lines shown in Fig.1. The fault is cleared by isolating the faulted line by opening the circuit breakers at both ends.
Case 3
A three-phase symmetrical fault occurs at point R shown in Fig. 1. The fault is cleared by isolating the faulted line by opening the circuit breakers at both ends.
Case 4
One of the transmission lines (say, line SR) is snapped at point S and falls on the ground creating a three-phase short circuit. The fault is cleared by isolating the faulted line.
Formulas
Electrical power, Pe = where , X is the reactance between E and V.
The critical clearing angle for case 1
The critical clearing angle for cases 2, 3 and 4
The critical clearing time for case 1
where are initial rotor angle, maximum rotor angle, synchronous speed and mechanical power input to the machine, respectively. P2max and P3max are the maximum electrical power during the fault and after clearing the fault, respectively.
There is no analytical expression to calculate the critical clearing time for other cases.
Problem Statement
Given the one-machine-infinite-bus system shown in Fig. 1 determine the system stability for all the four cases. Also find the critical clearing time and critical clearing angle for all the cases.
Software Used
MATLAB functions used are eacfault, and swingmeu.
Function eacfault(Pm, E, V, X1, X2, X3)
This function obtains the power angle curves for the one-machine system before the fault, during the fault and after the fault is cleared. The equal area criterion is applied to find the critical clearing angle for the machine. Also it computes the critical clearing time for case 1.
The function arguments are:
Pm = Generator output power in p.u. at steady state which is equal to the generator mechanical power input.
E = Generator e.m.f. in p.u. It is the voltage behind the transient reactance of the machine.
V = Infinite bus-bar voltage in p.u.
X1 = Reactance in p.u. between E and V before fault
X2 = Reactance in p.u. between E and V during fault. (If the power transfer to the infinite bus during fault is zero then X2 = inf)
X3 = Reactance in p.u. between E and V after fault is cleared.
Function swingmeu(Pm, E, V, X1, X2, X3, H, f, tc, tf, Dt)
This program solves the swing equation of a one-machine system when subjected to a three-phase fault with subsequent clearance of the fault using the modified Euler method. The swing cure is displayed. This is used to determine the system stability for a particular fault clearing time.
The function arguments are:
Pm, E, V, X1, X2, X3 are as defined earlier.
H = Generator inertia constant in second (MJ/MVA)
f = System frequency in Hz.
Dt = Integration time interval
tc = Fault clearing time in seconds
tf = Final time of integration
Experimental Procedure
1. Launch the MATLAB and study the above functions to understand their working.
2. To study the stability for case 1, determine the following function arguments.
Pm = E =
V = X1 =
X2 = X3 = H =
To compute E, use the following formula:
S* = 0.8 – j0.074 (Given)
I = S*/V* =
E = V + jX1*I = 1.1726.3877o
3. Run the program using the function eacfault. Supply all the data. You will observe the power-angle curve with area A1 (during acceleration period) and area A2 (during the deceleration period) marked in different colours. Observe and record the following output values:
Initial power angle =
Maximum angle swing =
Critical clearing angle =
Critical clearing time =
Draw the power-angle curve with areas A1 and A2 marked. Also show in the plot the initial angle, maximum angle swing and critical clearing angle.
4. Repeat steps 2 and 3 for case 2.
The function arguments are:
Pm = E =
V = X1 =
X2 = X3 =
The output values:
Initial power angle =
Maximum angle swing =
Critical clearing angle =
5. Repeat step 4 for case 3
The function arguments are:
Pm = E =
V = X1 =
X2 = X3 =
The output values:
Initial power angle =
Maximum angle swing =
Critical clearing angle =
Critical clearing time =
6. Repeat step 5 for case 4
The function arguments are:
Pm = E =
V = X1 =
X2 = Inf X3 =
The output values:
Initial power angle =
Maximum angle swing =
Critical clearing angle =
7. Solve the swing equation for all the four cases using the function swingmeu. Create a Matlab file TS1.m with the following statement and store it in the MATLAB work.
global Pm f H E V X1 X2 X3
Pm =
E =
V =
X1 =
X2 =
X3 =
H =
f =
tf = 1.0;
Dt = 0.01;
tc =
swingmeu(Pm, E, V, X1, X2, X3, H, f, tc, tf, Dt)
Run the program by using the command TS1. Choose at least two fault clearing time, tc for each case, one less than the critical clearing time and the other more than the critical clearing time. The additional function arguments are:
H = 5 tc =
tf = 1.0 Dt = 0.01
The time interval and the corresponding power angle δ in degrees and the speed deviation Δω in rad/s are displayed in a tabular form. The swing curve is also plotted. The swing curve for the stable case shows that the power angle returns after a maximum swing indicating that with inclusion of system damping, the oscillations will subside and a new steady state condition will be reached. For unstable case, the swing curve shows that the power angle is increasing without limit. From the simulation, determine the critical clearing time for all the four cases and record them. For case 1, compare the critical clearing time with that obtained in step 3. Also plot the swing curve for one stable case and one unstable case.
Observations
(a) Plot power-angle curves for Steps 3, 4, 5 and 6
(b) The critical clearing time for Case 1:
The critical clearing time for Case 2:
The critical clearing time for Case 3:
The critical clearing time for Case 4:
(c) Sketch the swing curves for one stable system and one unstable system.
Exercise
1. Explain the usefulness of equal-area criterion for stability analysis of one-machine-infinite-bus system.
2. Is it possible for applying equal-area criterion for two machine system?
3. How do you check the stability of the system from simulation results?
4. Briefly explain the modified Euler method of solving a differential equation.
5. What do you learn from this experiment?
References
1. Hadi Saadat, “Power System Analysis”, McGraw-Hill, 2004
2. John J. Grainger and William D. Stevenson, Jr. “Power System Analysis”, McGraw-Hill, 1994
Experiment 2
TRANSIENT STABILITY ANALYSIS OF MULTIMACHINE SYSTEMS
Objectives
The objectives of this experiment are
1. To analyze the transient stability of multimachine systems by solving swing equations by numerical integration.
2. To determine the stability of the given multimachine system for various fault clearing times.
3. To determine the critical clearing time using swing curves.
Introduction
The equal-area criterion used in Experiment 1 cannot be directly used to determine the stability of multimachine systems. Although the physical phenomena observed in one-machine-infinite-bus system are basically the same as in the maultimachine case, the complexity of the numerical computation increases with the number of machines increases. In order to reduce the complexity some simplifying assumptions (similar to the ones assumed in Single-Machine-Infinite-Bus systems) are made as follows.
(i) Each machine is represented by a constant voltage source behind the direct axis transient reactance.
(ii) The input mechanical power is assumed constant during the entire period of simulation.
(iii) All loads are converted to equivalent admittances to ground using the prefault bus voltages and assumed to remain constant.
(iv) Damping or synchronous powers are neglected.
(v) The mechanical rotor angle of each machine coincides with the angle of the voltage behind the machine transient reactance.
(vi) Machine belonging to the same station swing together and are said to be coherent.
The first step in the transient stability analysis is to solve the initial load flow and determine the initial bus voltages. The machine currents prior to disturbance are calculated from
i = 1, 2, … m (1)
where m is the number of generators., Vi is the terminal voltage of the ith generator, Si = Pi + jQi is the complex power of the generator i. The generator resistance is neglected and the generator voltage behind the transient reactance is obtained as
(2)
Equivalent admittances of loads
(3)
The m generator internal buses with voltages Ei’ are added to the n-bus power system network and the resultant Ybus matrix is formed. Then all buses other than the generator internal buses are eliminated using the Kron reduction formula. The reduced bus admittance matrix of dimension (m x m) be denoted as
Then the electrical power output
(4)
where Ei’ = |Ei’|δi and Yij = |Yij|θij and Yij is the ij th element of Ybus.
A three-phase fault at bus k results in the bus voltage Vk = 0. This is simulated by removing the the kth row and column from the prefault bus admittance matrix. The new bus admittance matrix is reduced by eliminating all buses except the internal generator buses. The generator excitation voltages during the fault and post fault condition are assumed to remain constant. The electrical power of the ith generator in terms of the reduced bus admittance matrix is obtained from (4). The swing equation for machine i becomes