Transient Analysis of SMIB System

Power System Simulation Lab - 2
M.E (Power Systems Engineering) MATHANKUMAR.S, AP/EEE
EXPERIMENT: Date:
TRANSIENT ANALYSIS: SINGLE - MACHINE
INFINITE BUS SYSTEM
AIM
To become familiar with various aspects of the transient analysis of Single-
Machine Infinite Bus (SMIB) system.
OBJECTIVES
i. To understand modelling and analysis of transient stability of a
SMIB power system.
ii. To examine the transient stability of a SMIB and determine the
critical clearing time of the system through simulation by trial and
error method and by direct method.
iii. To determine transient stability margin (MW) for different fault
conditions.
iv. To obtain linearised swing equation and to determine the roots of
characteristics equation, damped frequency of oscillation and
undamped natural frequency.
SOFTWARE REQUIRED
TRANSIENT -SMIB module of AU Powerlab or equivalent
THEORETICAL BACK GROUND
Stability:
Stability problem is concerned with the behaviour of power system
when it is subjected to disturbances and is classified into small signal
stability problem if the disturbances are small and transient stability
problem when the disturbances are large. The description of the problems
are as follows.

Transient Stability
When a power system is under steady state, the load plus transmission
loss equals to the generation in the system.
The generating units run at synchronous speed and system frequency,
voltage, current and power flows are steady. When a large disturbance
such as three phase fault, loss of load, loss of generation etc., occurs the
power balance is upset and the generating units rotors experience either
acceleration or deceleration. The system may come back to a steady state
condition maintaining synchronism or it may break into subsystems or one
or more machines may pull out of synchronism. In the former case the
system is said to be stable and in the later case it is said to be unstable.
Mathematical Modelling For Transient Stability
Consider a single machine connected to an infinite bus shown in
fig. 1. An infinite bus is a source of invariable frequency and voltage.
Fig. 1. Single machine connected to infinite bus system
The equivalent circuit with the generator represented by classical model
and all resistances neglected is shown in fig. 2.
Fig. 2. Equivalent circuit

E‟= E t + jX‟dIt
X = X‟d +XE where XE = Xtr + X1 || X2
Pe = |E‟||E B max (1)
X
Where, E‟= e.m.f behind machine transient reactance
δ = rotor angle with respect to synchronously rotating reference
phaser EB L00. E‟ leads EB by δ
ωo = synchronous speed of rotor
Pe = electrical power output of generator in p.u
(2)
(2)
(3)

(4)
(5)
Swing Equation
During any disturbance in the system, the rotor will accelerate or decelerate
with respect to synchronously rotating axis and the relative motion begins. The
equation describing the relative motion is called as swing equation.
The following assumptions are made in the derivation of swing equation
1. Machine represented by classical model
2. Controllers are not considered
3. Loads are constants
4. Voltage and currents are sinusoids
The fundamental equation of motion of the rotor of the synchronous machine is
given by
(6)

Where, Pm = Mechanical power input in p.u
Pmax = Max. Electrical power out in p.u
H = Inertia constant in seconds.
δ = rotor angle in electrical radian measured from synchronous
by rotating reference frame.
ωo = Synchronous angular velocity rad / sec.
Rewriting equation (6) in state variable form
(7)
Changing rotor speed in to per unit and introducing damping torque, equation (7)
become
(8)
Where, Δωr = rotor speed deviation in p.u
Pm = mechanical input in p.u
KD = damping co-efficient in p.u
Numerical Integration Techniques
The differential equations (8) are to be solved using numerical techniques. There are
several techniques available and two of them are given below.
I. Modified Euler Method
Consists of the following steps
(i) Compute the derivative at t: PX(t) = f[X(t), Δt]
(ii) Compute first estimate : X1(t+Δt) = X(t) + PX(t) Δt
(iii) Compute the derivative : PX (t+Δt)= f[X1(t + Δt), t+Δt]
(iv) Compute the average derivative : PXav (t) = ½ [PX(t) + PX(t+ Δt)
(v) Compute the final estimate : X(t+Δt) = X(t) + PXav(t) Δt
II. Fourth order Runge-Kutta Method
This is an explicit algorithm. The general formula giving the value of X
for the (n+1)th step is
Xn+1 = Xn + 1/6 (K1+ 2K2 +2K3 + K4) (9)
K1 = f(Xn, tn) Δt
K2 = f(Xn + K1/2, tn+ Δt/2) Δt
K3 = f(Xn + K2/2, tn + Δt
K4 = f (Xn + K3, tn + Δt) Δt

Determination of Critical Clearing Time
Critical clearing time is the maximum allowable time between the occurrence of a
fault and clearing of the fault in a system for the system to remain stable. For a given
load condition and specified fault, the critical clearing time for a system is found out
as follows. Choose a large clearing time say 30 cycles and decrease the clearing time
in steps of say two cycles and check for stability at each time step until the system just
becomes unstable. Vary the clearing time around this point in small step till you find
the clearing time which is just critical. The clearing time margin for a fault may be
defined as
Clearing time margin = critical clearing time – clearing time specified
= tc (critical) - tc (clearing)
Stability Margin in MW
Consider that the machine connected to infinite bus delivers Po MW (Fig. 1) and a
fault is specified at the end of line no. 1 with a clearing time tc = 0.3 seconds.
Suppose the MW output of the machine is increased in steps and stability is checked
for each step of load with the same clearing time and fault, then the system becomes
just stable at a loading say Pm and a small increase in load beyond Pm
causes instability; then the MW stability margin is defined as Ps = Pm - Po
Critical Clearing Time and Clearing Angle from Equal Area Criteria
This method can be used for quick prediction of stability but is applicable only to
single machine connected to infinite bus. The fundamental concepts and principles of
stability can be explained very well. Consider the system shown in fig. 1. and its
model in .5. The terminal power is given by equation (1) and the power angle curves
for various operating condition is given in fig. 3.
Fig.3. Power angle curve
The steady state operating condition is given by point a and the corresponding
rotor angle is δo. Consider a three phase fault at location F on line 2 as shown in
fig. 1. The fault is cleared by opening the circuit breakers at both ends of the line.
The p-δ plot for three network conditions are shown in fig. 3.
When the fault occurs, the operating point changes from a to b. Since Pm >
Pe, the rotor accelerates until the operating point reaches c where the fault is
cleared at δ1. The operation shifts to e. Now Pe>PM the rotor decelerates, but δ
continues to increase until the kinetic energy gained during the period of
acceleration (Area A1) is transferred to the system.

The operating point moves from e to f such that area A2 is equal to A1.The rotor
angle will oscillate back and forth around at its natural frequency such that
|area A1| = |area A2|. This is known equal area criterion.
Fig. 4. Equal area criterion for critical clearing angle
Critical Creating Angle an Time
With delayed fault clearing as shown in fig. 4 , the area A2 just equals to A1
at clearing angle equal to δc. Any further delay in clearing causes area A2 above
Pm less than A2 resulting in loss of synchronism, this angle δcfor which A1 = A2
is called critical clearing angle. The critical clearing angle can be computed as
Applying equal area criterion to fig. 4.
Integrating both sides and solving for δc
(10)
The corresponding critical clearing time is given by
Similarly you can find out δc and tc for the different types of faults.

Exercise
A power system comprising a thermal generating plant with four 555 MVA, 24kV,
and 60HZ Units supplies power to an infinite bus through a transformer and two
transmission lines
FIG: Single Machine Infinite Bus System
The data for the system in per unit on a base of 2220 MVA, 24 kV is given below:
An equivalent generator representing the four units, characterized by classical model:
Xd‟ = 0.3 p.u ;
H= 3.5 MW-s/MVA ;
Transformer: X = 0.15 p.u
Line 1 : X = 0.5 p.u ;
Line 2 : X = 0.93 p.u
Plant operating condition:
P = 0.9 p.u ;
Power factor: 0.9 lagging;
Et = 1.0 p.u
It is proposed to examine the transient stability of the system for a three-phase-to ground
fault at the end of line 2 near H.T bus occurring at time t= 0 sec. The fault is cleared at 0.07
sec. by simultaneous opening of the two circuit breakers at both the ends of line 2.

Viva Questions
1. Define critical clearing time.
2. What is transient stability limit?
3. Define critical clearing angle.
4. Write the expression for the critical clearing angle
5. What is power angle equation?
6. Define transient stability.
7. Write any three assumptions made upon transient stability.
8. Define Modified Eulers method.
9. List the methods of improving transient stability limit of a power system?
10. Define power angle diagram.
11. What is Voltage Collapse?
12. Define Runge – Kutta method.
13. What is meant by voltage instability?
14. What are the essential factors affecting the stability?
15. How will the transient stability limit of power system can be improved?
16. When is a power system said to be transiently stable?
17. What is transient state of power system?
18. What are the factors that affect the transient stability?
19. List the types of disturbance that may occur in a single machine infinite bus bar
system of the equal area criterion stability.
20. What are various faults that increasing severity of equal area criterion?

Result:
Marks split-up
Marks
Secured
Marks
Awarded
Basic understanding 15
Theoretical Calculation 20
Conducting 15
Software output with graph 20
Comparison Results 10
Record 10
Viva - voce 10
Total Marks 100

EXPERIMENT: Date:
SMALL SIGNAL STABILITY ANALYSIS: SINGLE
- MACHINE INFINITE BUS SYSTEM
AIM
To become familiar with various aspects of the small signal stability analysis of
Single-Machine Infinite Bus (SMIB) system.
OBJECTIVES
i. To understand modelling and analysis of small signal stability of a SMIB
power system.
ii. To examine the small signal stability of a SMIB and determine the critical
clearing time of the system through simulation by trial and error method
and by direct method.
iii. To obtain linearised swing equation and to determine the roots of
characteristics equation, damped frequency of oscillation and undamped
natural frequency.
SOFTWARE REQUIRED
SMALL SIGNAL STABILTIY - SMIB module of AU Power lab or equivalent
THEORETICAL BACK GROUND
Stability:
Stability problem is concerned with the behavior of power system when it is
subjected to disturbances and is classified into small signal stability problem if
the disturbances are small and transient stability problem when the disturbances
are large. The descriptions of the problems are as follows.
Small Signal Stability
When a power system is under steady state, normal operating condition, the
system may be subjected to small disturbances such as variation in load and
generation, change in field voltage, change in mechanical torque etc. The
nature of system response to small disturbances depends on the operating
condition, the transmission system strength, types of controllers etc.
Instability that may result from small disturbances may be of two forms
(i) Steady increase in rotor angle due to lack of synchronising torque.
(ii) Rotor oscillations of increasing magnitude due to lack of sufficient damping
torque.

Lack of sufficient synchronising torque results in instability through non-
oscillatory mode shown in fig.1., Fig.2. Shows the instability of a synchronous
machine through oscillations of increasing amplitude.
Fig.1 Non-Oscillatory instability Fig.2. Oscillatory instability
Swing Equation
During any disturbance in the system, the rotor will accelerate or decelerate
with respect to synchronously rotating axis and the relative motion begins. The
equation describing the relative motion is called as swing equation.
The following assumptions are made in the derivation of swing equation
1. Machine represented by classical model
2. Controllers are not considered
3. Loads are constants
4. Voltage and currents are sinusoids
The fundamental equation of motion of the rotor of the synchronous machine is
given by
(1)
Where, Pm = Mechanical power input in p.u
Pmax = Max. Electrical power out in p.u
H = Inertia constant in seconds.
δ = rotor angle in electrical radian measured from synchronous
by rotating reference frame.
ωo = Synchronous angular velocity rad / sec.
Rewriting equation (1) in state variable form
(2)

Changing rotor speed in to per unit and introducing damping torque, equation (2)
become
(3)
Where, Δωr = rotor speed deviation in p.u
Pm = mechanical input in p.u
KD = damping co-efficient in p.u
Modelling For Small Signal Stability
The electrical power output of the generator in p.u. is
Pe = |E‟ EB| sinδ (4)
X
In p.u. the air-gap torque is equal to air-gap power, Hence
(5)
Linearising equation (5) about in initial operating condition at δ = δ0
(6)
ΔTe = KS ΔS
Where, , called synchronising coefficient.
The state equations (3) are rewritten as
(7)
(8)
Where Tm , Te are in p.u. Δωr is per unit speed deviation, δ is the rotor angle in
electrical radians, ω0 is the base (rated) rotor speed in electrical radians per second,
KD is the damping coefficient in p.u., H is in p.u (seconds).

Linearising equation (7) and using (8) we get,
(9)
(10)
Fig.3 Block Diagram for Equation (10)
Taking Laplace transform for the above equation,
(12)
(13)
(14)
(15)

Damping ratio = ξ
The roots of characteristic equation are
S1 = ξ ωn + jωd
S2 = ξ ωn - jωd
Where, ωd is the damped frequency of oscillation given by
ωd = ωn - √1-ξ2
Taking inverse Laplace transform of equation (14) and (15) and takingΔδo = 10o
= 0.1745 radians; we get the equation for motion of rotor relative to
synchronously revolving field and the rotor angular frequency
(16)
(17)
Where θ = cos-1ξ
The response time constant τ = (1/ ξωn) = 2H / πf0 D)

EXERCISE
A power system comprising a thermal generating plant with four 555 MVA, 24kV,
60HZ units supplies power to an infinite bus through a transformer and two
transmission lines
FIG : Single Machine Infinite Bus System
The data for the system in per unit on a base of 2220 MVA, 24 kV is given below:
An equivalent generator representing the four units, characterized by
classical model:
Xd‟ = 0.3 p.u
H= 3.5 MW-s/MVA
Transformer: X = 0.15 p.u
Line 1 : X = 0.5 p.u
Line 2 : X = 0.93 p.u
Plant operating condition:
P = 0.9 p.u ; Power factor: 0.9 lagging;
Et = 1.0 p.u
It is proposed to examine the small-signal stability characteristics of the system
given in this problem about the steady-state operating condition following the loss
of line 2; Assume the damping coefficient KD = 1.5 p.u torque / p.u speed
deviation.
(a) Write the linearized swing equation of the system. Obtain the characteristic
equation, its roots, damped frequency of oscillation in Hz, damping ratio and
undamped natural frequency. Obtain also the force-free time response ∆δ(t) for an
initial condition perturbation ∆δ(0) = 5˚ and ∆ω(0)=0,using available software.

Viva Questions
1. Define power system stability.
2. What are the methods of maintaining stability?
3. How stability studies are classified, what are they?
4. What is steady state stability limit?
5. Define steady state stability.
6. What is stability study?
7. Given an expression for swing equation. Explain each term along with their units.
8. What are the assumptions that are made in order to simplify the computational
task in stability studies?
9. How is the machine connected to infinite bus?
10. What is meant by an infinite bus?
11. Give the control schemes included in the stability control techniques.
12. What are various faults that increase severity of equal area criterion?
13. State the applications of the equal area criterion.
14. What are the essential factors affecting the stability?
15. Define Inertia constant (H) & Movement of inertia (M).
16. What are the machine problems seen in the stability study?
17. What is small disturbance? Give some example.
18. How to you classify steady state stability limit. Define them.
19. When is a power system said to be steady state stable?
20. What are the system design strategies aimed at lowering system reactance?

EXPERIMENT: Date:
LOAD-FREQUENCY DYNAMICS OF SINGLE-AREA
POWER SYSTEMS
AIM
To become familiar with the modelling and analysis of load-frequency and
tie-line flow dynamics of a power system with load-frequency controller (LFC)
under different control modes and to design improved controllers to obtain the
best system response.
OBJECTIVES
i. To study the time response (both steady state and transient) of area
frequency deviation and transient power output change of regulating
generator following a small load change in a single-area power system with
the regulating generator under “free governor action”, for different
operating conditions and different system parameters.
ii. To study the time response (both steady state and transient) of area
frequency deviation and turbine power output change of regulating
generator following a small load change in a single- area power system
provided with an integral frequency controller, to study the effect of
changing the gain of the controller and to select the best gain for the
controller to obtain the best response.
iii. To analyse the time response of area frequency deviations and net
interchange deviation following a small load change in one of the areas
in an inter connected two-area power system under different control
modes, to study the effect of changes in controller parameters on the
response and to select the optimal set of parameters for the controller to
obtain the best response under different operating conditions.
SOFTWARE REQUIRED
„LOAD FREQUENCY CONTROL‟ module of AU Powerlab or equivalent.
THEORETICAL BACKGROUND
Introduction
Active power control is one of the important control actions to be performed
during normal operation of the system to match the system generation with the
continuously changing system load in order to maintain the constancy of system
frequency to a fine tolerance level. This is one of the foremost requirements in
providing quality power supply. A change in system load causes a change in the
speed of all rotating masses (Turbine – generator rotor systems) of the system
leading to change in system frequency. The speed change from synchronous
speed initiates the governor control (Primary control) action resulting in all the
participating generator – turbine units taking up the change in load, stabilizing
the system frequency. Restoration of frequency to nominal value requires

secondary control action which adjusts the load-reference set points of selected
(regulating) generator – turbine units. The primary objectives of automatic
generation control (AGC) are to regulate system frequency to the set nominal
value and also to regulate the net interchange of each area to the scheduled
value by adjusting the outputs of the regulating units. This function is referred
to as load – frequency control (LFC). The details of modelling and analysis of
LFC are briefly presented in the following sections.
Load-Frequency Control in an Interconnected Power System
An interconnected power system is divided into a number of “control areas” for
the purpose of load- frequency control. When subjected to disturbances, say, a
small load change, all generator – turbine units in a control area swing together
with the other groups of generator – turbine units in other areas. Hence all the
units in a control area are represented by a single unit of equivalent inertia and
characterized by a single (area) frequency. Since the area network is “strong”(all
the buses connected by adequate capacity lines), all the bus loads in a control
area are assumed to act at a single load point and characterized by a single
equivalent load parameter. The different control areas are connected by relatively
“weak” tie-lines. A typical n-area power system is shown in Fig.1.
Area 1
Other areas
Area i PNIi
Area n
Fig .1 Multi- Area Power System
For successful operation of an interconnected power system the following
operating principles are to be strictly followed by the participating areas:
i. Under normal operating conditions each control area should strive to meet
its own load from its own spinning generators plus the contracted
(scheduled) “interchange” (import / export) between the neighboring
areas.
ii. During emergency conditions such as sudden loss of generating unit, area
under emergency can draw energy as emergency support from the
spinning reserves of all the neighboring areas immediately after it is
subjected to the disturbances but should bring into the grid the required
generation capacity from its “hot” and “cold” reserves to match the lost
capacity and to enforce operating principle

(i) Satisfaction of principle (ii) during normal operation requires a load-
frequency controller for each area which not only drives the area
frequency deviation to zero but also the “net interchange” of that
area to zero under steady- state condition. “Net interchange” of area i, NIi
is defined as the algebraic sum of the tie- line flows between area i and other
connected areas (Fig .1) with tie-line flow out of area i taken as positive and
is given by
NIi = ΣPij (1)
j € αi
where αi is the set of all areas connected to area i
Modelling of Governor and Turbine
Governor with speed – droop characteristics
Governor is provided with a speed- droop characteristics so as to obtain stable
load deviation between units operating in parallel. The ideal steady- state speed
versus load characteristics of the generating unit is shown in Fig.2.
fNL
Frequency or
speed (p.u) Slope = -R
fFL
0 1.0
Power output (or) valve / gate position (p.u)
Fig .3 Steady-State Speed-Load Characteristics of
a Governor with Speed Droop
The negative slope of the curve, R, is referred to as “Percent speed regulation or
droop”and is expressed as
Percent R = Percent speed or frequency change x 100
Percent power output change
=((fNL – fFL) / f0 ) x 100 where
fNL = steady-state frequency at no load
fFL = steady- state frequency at full load
f0
= nominal or rated frequency
For example a 5% droop means that a 5% frequency deviation causes 100%
change in valve position

Control of generating unit power output
The output of a generating unit at a given system frequency can be varied only
by changing its “load reference point” which is integrated with the speed
governing mechanism. The block diagram of a governor with the governor
droop R, the time constant of hydraulic amplifier TG and the load reference set
point is shown in Fig .4
GG(s)
-
∆f 1/R 1/(1+sTG) ∆Xv
+
Load reference
Fig.4 Governor with Speed Load Reference Set Point
The adjustment of load reference set point is accomplished by operating the
“speed changer motor”. This in effect moves the speed droop characteristics up
and down.
Turbine model
For the purpose of load-frequency dynamics the turbine may be modelled by
an approximate model with a single time constant TT as given by equation
∆PT(s) = GT(s) ∆Xv(s) = (1/(1+sTT)) ∆Xv(s) (2)
The block diagram for single-area load-frequency control is assembled by
combining equation (2) and Fig. 4. The block diagram is given in Fig. 5.
Modelling and Analysis of Single-Area Load-Frequency Control
Fig.5 Block Diagram for Single-Area Load – Frequency Control
In the above diagram, all powers are in per unit to area rated capacity and
the frequency deviation is in hertz.
Kp = 1/D Hz / p.u.MW
Tp = (2H/f0
D) s e c
The load damping constant D is normally expressed in percent and typical
values of D are 1 to 2 percent. A value of D = 1.5 means that 1.0 percent
change in frequency would cause a 1.5 percent change in load.

The dashed portion of the diagram marked as the secondary loop
represents the integral controller whose gain is KI. This controller actuates the
load reference point until the frequency deviation becomes zero.
Steady-state analysis with governor control
Let the disturbance be a step increase in load, M p.u. MW. With only governor
control (integral controller deactivated) the frequency deviation will not be made
zero. The steady-state frequency deviation ∆fs can be determined by applying
final value theorem in s-domain
∆fs = Lim s{∆fs } (3)
s ->0
(4)
∆PD(s) = (M/s) (5)
Substituting equations (4) and (5) in equation (3) we obtain
∆fs = -(M / ß) Hz (6)
ß = Area Frequency Response Coefficient (AFRC)
ß = D +(1/R) Hz / p.u. MW
M is in p.u. MW

Example:
The data for a single-area power system is given below
Rated area capacity = Pr = 2000MW
Nominal operating load = P0
D = 1000MW
f0
= 50 Hz; D=1 %; R = 3%; H = 5 sec
Load increase = M= 20 MW. Compute steady-state frequency deviation.
Solution:
Steady-state analysis with integral control
By reducing the full block diagram Fig .5. and by applying final value theorem in
s-domain, one can show that the steady-state frequency deviation is made zero.
Transient analysis
The block diagram Fig. 5 can be used to derive the state variable model with the
following four states:
x1 =∆fs = frequency deviation
x2 =∆PT = Turbine power deviation
x3 =∆Xv = Steam valve/water gate position x4
∆Pref = Load-reference setting
Transient response for step change in load can be obtained by numerically
integrating the four state equations through Runge – Kutta fourth order method
or any other method. AU Power lab or any available software can be used for
this purpose.

EXERCISES
It is proposed to simulate using the software available the load-frequency
dynamics of a single-area power system whose data are given below:
Rated capacity of the area = ________ MVA
Normal operating load = ________ MW
Nominal frequency = 50 Hz
Inertia constant of the area = ________sec
Speed regulation (governor droop)
of all regulating generators = ________
Percent Governor Time Constant = ________sec
Turbine Time Constant = ________ sec
Assume linear load–frequency characteristics which means the connected
system load increases by one percent if the system frequency increases by one
percent.
The area has a governor control but not a load-frequency controller. The area
is subjected to a load increase of 20 MW.
(a) Simulate the load-frequency dynamics of this area using available software
and check the following:
(i) Steady – state frequency deviation ∆fs in Hz. compare it with the hand-
calculated value using “Area Frequency Response Coefficient”(AFRC).
(ii) Plot the time response of frequency deviation ∆f in Hz and change in
turbine power ∆PT in p.u MW upto 20 sec. What is value of the peak
overshoot in ∆f?
(b) Repeat the simulation with the following changes in operating condition,
plot the time response of ∆f and compare the steady-state error and peak
overshoot.
(i) Speed regulation = 3 percent
(ii) Normal operating load = 1500MW

Viva Questions
1. What is the function of load frequency control?
2. How is the real power in a power system controlled/
3. What is meant by fly ball governor?
4. Define inertia constant.
5. What is area control error?
6. Define per unit droop.
7. Write the principle of tie line bias control.
8. State the basic role of ALFC.
9. What is meant by control area?
10. Write the tie line power deviation equation in terms of frequency.
11. Differentiate static and dynamic response of an ALFC loop.
12. What are the assumptions made in dynamic response of uncontrolled case?
13. Define Speed regulation.
14. What is mean by AFRC?
15. Draw a block diagram for single area load frequency control.
16. What is a need of speed changer?
17. Why load frequency control is important in operation of power systems?
18. What is ALFC?
19. Define time constant.
20. What is meant by frequency deviation?

EXPERIMENT: Date:
ECONOMIC DISPATCH IN POWER SYSTEMS
AIM
To understand the basics of the problem of Economic Dispatch (ED) of optimally
adjusting the generation schedules of thermal generating units to meet the system
load which are required for unit commitment and economic operation of power
systems.
To understand the development of coordination equations (the mathematical
model for ED) without losses and operating constraints and solution of these
equations using direct and iterative methods
OBJECTIVES
i. To solving ED problem without transmission losses for a given load condition /
daily load cycle using
(a) Direct
(b) Lambda-iteration method
ii. To study the effect of reduction in operation cost resulting due to changing
from simple load dispatch to economic load dispatch.
iii. To study the effect of change in fuel cost on the economic dispatch for a
given load.
SOFTWARE REQUIRED
„ECONOMIC DISPATCH‟ module of AUPowerlab or equivalent.

Mathematical Model for Economic Dispatch of Thermal Units without
Transmission Loss
Statement of Economic Dispatch Problem
In a power system, with negligible transmission loss and with N number of
spinning thermal generating units the total system load PD at a particular interval
can be met by different sets of generation schedules
{PG1(k), PG1(k), PG1(k),…………………………………….. PG1(k) }; where k = 1,2,3,……..NS.
Out of these NS sets of generation schedules, the system operator has to choose
that set of schedule which minimizes the system operating cost which is essentially
the sum of the production costs of all the generating units. This economic dispatch
problem is mathematically stated as an optimization problem.
Given: the number of available generating units N, their production cost
functions, their operating limits and the system load PD,
To determine: the set of generation schedule,
PGi ; i = 1,2,……N (1)
Which minimizes the total production cost,
N
Min : FT =Σ Fi(PGi) (2)
i = 1
and satisfies the power balance constraint
(3)
and the operating limits
PGi, min ≤PGi ≤ PGi, max (4)
The unit production cost function is usually approximated by a quadratic function
Fi (PGi) = ai PG2
i + bi PGi +ci ;i =1,2, …….N (5)
Where, ai, bi and ci are constants.

Necessary conditions for the existence of a solution to ED problem
The ED problem is given by the equations (1) to (4). By omitting the inequality
constraints (4) tentatively, the reduced ED problem (1),(2) and (3) may be
restated as an unconstrained optimization problem by augmenting the objective
function (1) with the constraint function Φ multiplied by LaGrange Multiplier λ to
obtain LaGrange function L as,
(6)
The necessary conditions for the existence of solution to (6) are given by
(7)
(8)
The solution to ED problem can be obtained by solving simultaneously the
necessary conditions (7) and (8) which state that the economic generation
schedules not only satisfy the system power balance equation (8) but also
demand that the incremental cost rates of all the units be equal to λ which can be
interpreted as “incremental cost of received power”
When the inequality constraints (4) are included in the ED problem the
necessary condition (7) gets modified as
(9)
Methods of Solution for ED Without Loss
The solution to the ED problem with the production cost function assumed to be
a quadratic function, equation (5), can be obtained by simultaneously solving
equations (7) and (8) using a direct method as given below.
(10)
From equation (10) we obtain
(11)

Substituting equation (1) in equation (8) we obtain
(12)
The method of solution involves computing λ using equation (12) and then
computing the economic schedules PGi; i = 1, 2, . . . . . . . . . . . . . . N using
equation (11). In order to satisfy the operating limits (4) the following iterative
algorithm is to be used.
Algorithm for ED without loss (For quadratic production cost function)
Step 1: Compute λ using equation (12)
Step 2: Compute using equation (11) the economic schedules
PGi ; i=1,2,…..N
Step 3: If the computed PGi satisfy the operating limits
PGi, min ≤PGi ≤ PGi, max ; i=1,2,…..N (13)
then stop, the solution is reached. Otherwise proceed to step 4
Step 4: Fix the schedule of the NV number of violating units whose
generation PGi violates the operating limits (13) at the respective
limit, either PGi,max or PGi,min
Step 5: Distribute the remaining system load PD minus the sum of the fixed
generation schedules to the remaining units numbering NR (= N-NV)
by computing λ using equation (12) and the PGi;
P
G
i
i ∈α NR using equation (11) whereα NR is the set of remaining units.

Step 6: Check whether optimality condition (9) is satisfied.
If yes, stop the solution is reached. Otherwise, release the generation
schedule fixed at PGi,max or PGi,min of those generators not satisfying
optimality condition (9), include these units in the remaining units, modify
the sets αNV,αNR and the remaining load. Go to step 5
λ – Iteration Algorithm

Procedure
Using available software to solve the Economic Dispatch problem of a power
system with thermal units only for a given daily load cycle. Assume that the
production cost function of these units is quadratic and the transmission
loss of the system is negligible. Use the algorithm given in b e l o w
e x e r c i s e .
The program should have three sections: input section, compute section and
output section.
I. Input Section
The data to be read from an input file should contain the following:
(i) Number of thermal units in the system
(ii) Cost coefficients ai , bi , and ci , with cost in hundreds of rupees per
hour for all the units.
(iii) Maximum and minimum MW operating limits of all the units.
(iv) Daily load cycle in MW.
II. Compute Section
economic generation schedules for each one of the load levels in the load cycle
using the algorithm given in section 10.4.
III. Output Section
Create an output file in a report form comprising the following:
(i) Student information: as specified in exercise under experiment 3.
(ii) Input data: with proper headings
(iii) Results obtained: with proper headings for each load level
(a) Economic generation schedule of each unit
(b) Incremental fuel cost of each unit at economic schedule
(c) Incremental cost of received power.

Exercise
In a power system with negligible transmission loss, the system load varies
from a peak of 1200 MW to a valley of 500 MW. There are three thermal
generating units which can be committed to take the system load. The fuel
cost data and generation operation limit data are given below.
In hundreds of rupees per hour:
F1 = 392.7 + 5.544 P1 + 0.001093 P1
2
; P1 in MW
F2 = 217.0 + 5.495 P2 + 0.001358 P2
2
; P2 in MW
F3 = 65.5 + 6.695 P3 + 0.004049 P3
2
; P3 in MW
Generation limits:
150 ≤ P1 ≤ 600 MW
100 ≤ P2 ≤ 400 MW
50 ≤ P3 ≤ 200 MW
There are no other constraints on system operation. Obtain an optimum
(minimum fuel cost) unit commitment table for each load level taken in
steps of 100 MW from 1200 to 500. Adopt “brute force enumeration”
technique. For each load level obtain economic schedules using the
Economic Dispatch Program developed in exercise for each “feasible”
combination of units and choose the lowest fuel cost schedule among
these combinations.
Show the details of economic schedule and the component and total costs
of operation for each feasible combination of units for the load level of 900
MW.

Viva Questions
1. What is the purpose of economic dispatch?
2. In which condition, the transmission losses are negligible in economic dispatch
problem?
3. What is unit commitment?
4. Name the methods of finding economic dispatch.
5. What do you mean by base load method?
6. What is meant by total generator operating cost?
7. List the various constraints in the modern power systems.
8. What are the disadvantages of using participation factor?
9. What is the difference between load frequency controller and economic
dispatch controller?
10. What is Lagrangian multiplier?
11. Write the coordination equation neglecting losses.
12. What are the assumptions for deriving loss coefficients?
13. Draw incremental fuel cost curve.
14. Write the quadratic expression for fuel cost.
15. What is system incremental cost?
16. Write the relationship between λ and power demand when the cost curve is
given.
17. What is base load?
18. Define Lamda – iteration method
19. What is minimum fuel cost?
20. What are the difference between simple load dispatch and economic load
dispatch?

EXPERIMENT: Date:
ELECTROMAGNETIC TRANSIENTS IN POWER
SYSTEMS
AIM
To study and understand the electromagnetic transient a phenomenon in
power systems caused due to switching and faults by using Electromagnetic
Transients Program (EMTP).
OBJECTIVES
i. To study the transients due to energization of a single-phase and
three-phase load from a non- ideal source with the line represented by
Π model.
ii. To study the transients due to energization of a single-phase and
three-phase load from a non-ideal source and line represented by
distributed parameters.
SOFTWARE REQUIRED:
ELECTROMAGNETIC TRANSIENTS PROGRAM – UBC version module of
AU Power lab or equivalent
Solution Method for Electromagnetic Transients Analysis
Intentional and inadvertent switching operations in EHV systems initiate
over voltages, which might attain dangerous values resulting in destruction
of apparatus. Accurate computation of these over voltages is essential for
proper sizing, coordination of insulation of various equipment‟s and
specification of protective devices. Meaningful design of EHV systems is
dependent on modelling philosophy built into a computer program. The
models of equipment‟s must be detailed enough to reproduce actual
conditions successfully – an important aspect where a general purpose
digital computer program scores over transient network analysers.

The program employs a direct integration time-domain technique evolved
by Dommel [2]. The essence of this method is discretization of differential
equations associated with network elements using trapezoidal rule of
integration and solution of the resulting difference equations for the
unknown voltages. Any network which consists of interconnections of
resistances, inductances, capacitance, single & multiphase Π circuits,
distributed parameter lines, and certain other elements can be solved. To
keep the examinations simple, however, single phase network elements will
be used, rather than the more complex multiphase network elements.
Figure: Part of the Network around a Node of Large System

EXCERCISES
Prepare the data for the network given in the Annexure and run EMTP.
Obtain the plots of source voltage, load bus voltage and load current
following the energisation of a single-phase load. Comment on the results.
Double the source inductance and obtain the plots of the variables
mentioned earlier. Comment on the effect of doubling the source inductance.
ANNEXURE
Energization of a single phase 0.95 pf load from a non ideal source and a
more realistic line representation (lumped R, L, C)
Circuit Diagram
Figure: Energization of 0.95 pf load

Viva Questions
1. Define transient network analyser.
2. What is electromagnetic transient?
3. List out the advantage & disadvantage of EMT.
4. What are the differences between nodal and modified nodal analysis in
electromagnetic transient?
5. Develop the model for transmission system with lumped parameters.
6. What are the applications of EMT?
7. What is called as multi step method?
8. What is surge impedance?
9. Define Eigen value of power transients.
10. What is the importance of transients?
11. What are the requirements for transients in power system?
12. What is meant by thermal breakdown?
13. Mention the two components of voltage in power systems during transient period.
14. What is called as switching transients?
15. What are the sources of transients?
16. Define harmonics.
17. What is the need for harmonics and list out the types?
18. What is the significance of resistance in transients?
19. Define transient voltage & transient current.
20. What are the various types of power system transients?

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