Power System Analysis

POWER
SYSTEM
ANALYSIS
III B.Tech II Semester
Course Teacher:
Dr. MAHABOOB SHAREEF SYED

Students should able to
 Necessity of power flow studies.
 Derivation of static power flow equations.
 Power flow solution using
 Gauss-Seidel Method
 Newton Raphson Method
(Rectangular and polar coordinates form)
 Decoupled
 Fast Decoupled methods
Power Flow Studies

Power flow studies are undertaken for various reasons, some of
which are the following:
1.The line flows
2. The bus voltages and system voltage profile
3. The effect of change in configuration and incorporating new
circuits on system loading
4. The effect of temporary loss of transmission capacity and/or
generation on system loading and accompanied effects
5. The effect of in-phase and quadrative boost voltages on system
loading
6. Economic system operation
7. System loss minimization
8. Transformer tap setting for economic operation
9. Possible improvements to an existing system by change of
conductor sizes and system voltages.
NECESSITY OF POWER
FLOW STUDIES

 Load flow studies are one of the most important
aspects of power system planning and operation.
 The load flow gives us the sinusoidal steady state of the
entire system - voltages, real and reactive power
generated and absorbed and line losses.
 Since the load is a static quantity and it is the power
that flows through transmission lines, it is prefer to
call this Power Flow studies rather than load flow
studies.
NECESSITY OF POWER
FLOW STUDIES

 Through the load flow studies it can be obtained the voltage
magnitudes and angles at each bus in the steady state.
 it is required to held the bus voltages within a specified
limits.
 Once the bus voltage magnitudes and their angles are
computed using the load flow, the real and reactive power
flow through each line can be computed.
 Also based on the difference between power flow in the
sending and receiving ends, the losses in a particular line
can also be computed.
 Furthermore, from the line flow we can also determine the
over and under load conditions.
NECESSITY OF POWER
FLOW STUDIES

 The power system network is a such a large interconnected
network, where various buses are connected through a
transmission lines.
 At any bus, complex power is injected into the bus by the
generators and complex power is drawn by the loads.
 The complex power Si at any bus i is given by,
 Where Vi is voltage at bus i with angle δi.
Vi = |Vi| ∟δi = |Vi| (cos δi + j sin δi)
Yik = Gik + j Bik
POWER FLOW
EQUATIONS

POWER FLOW
EQUATIONS

 The general practice in power-flow studies is to
identify types of buses in the network.
 The load flow problem consists of determining the
magnitudes and phase angles of Voltages at each
bus and active and reactive power flow in each line.
 While solving power flow problem, the system is
assumed to be operating under balanced condition.
 The quantities associated with each bus are: δi,
|Vi|, Pi and Qi.
TYPES OF BUSES

 Depending upon which two variables are
specified a priori, the buses are classified into
three categories.
Load bus: A non generator bus is called load
bus. At this bus, the real power demand and
reactive power demands are specified. The
parameters is to be determined are δi and |Vi|. A
load bus is often called as P-Q bus.
TYPES OF BUSES

Voltage controlled Bus: Any bus of the system
at which the voltage magnitude is kept constant
is said to be voltage controlled.
At each bus to which there is a generator
connected, the megawatt generation can be
controlled by adjusting the prime mover, and the
voltage magnitude can be controlled by adjusting
the generator excitation .
These buses are the generator buses.
TYPES OF BUSES

 At these buses, the real power generation and
voltage magnitudes are specified. The other two
parameters δi and reactive powers has to be
determined.
 These buses are also called P-V bus.
 Some load buses with continuous reactive
power variation capability are also called
voltage controlled bus at which the real power
generation is simply zero.
TYPES OF BUSES

Slack Bus: This bus is also called slack bus or swing
bus, which is taken as reference to the entire system.
 The voltage angle of the slack bus serves as
reference for the angles of all other bus voltages.
The real and reactive powers are not specified at
this bus.
 Generally Bus 1 is considered to be a slack bus.
 The voltage magnitude and phase angles are
specified at this bus.
TYPES OF BUSES

 In any load flow study, the losses in the system
cannot be known a priori, without the solution of
voltages at all the buses.
 The slack bus supplies the difference between the
total system load plus losses and the sum of the
complex powers at the remaining buses.
 Slack bus is a generator bus as it needs to supply
the losses.
 The bus connected to the largest generating station
is normally selected as slack bus.
TYPES OF BUSES

TYPES OF BUSES

 The GS method is most popular iterative
algorithm for solving non linear algebraic
equations.
 At every subsequent iteration, the solution is
updated till convergence is reached.
 It is applied to power flow problem as described
in subsequent sections.
GAUSS-SEIDEL
(GS) METHOD

GS METHOD

 Starting from an initial estimate of all the bus
voltages, the most recent values of the bus voltages
are substituted.
 One iteration of the method involves computation
of all bus voltages.
 The values of the updated voltages are used in the
computation of subsequent voltages in the same
iteration.
 Iterations are carried out till the magnitudes of all
bus voltages do not change more than the
tolerance value.
GS METHOD

 In GS method, number of iterations increases
with increase of size of the system.
 It can be reduced, if the corrections in voltage
at each bus is accelerated.
 Therefore, a multiplication factor called
acceleration factor (α) is introduced.
 Generally, α is taken between 1.6 to 2.0.
 A wrong value of α may lead to divergence.
GS METHOD

Students should able to Problem
GS METHOD

 In the first step formulate Ybus
Problem
GS METHOD

Case (ii):
In case of any PV Bus in a system:
1. Calculate Q at that bus.
2. Calculate V by using above Q value.
3. Since, it is a PV bus |V| = V specified,
but angle only has to be updated.
GS METHOD

Case (iii):
In case Q limits specified at PV Bus in a system:
1. Calculate Q at that bus.
2. Check for Q limits, If the limits are not satisfied,
treat that PV bus as PQ bus with Q as follows:
GS METHOD

The most widely used method for solving
simultaneous non linear algebraic equations.
NEWTON – RAPHSON
METHOD

NEWTON – RAPHSON
METHOD

NEWTON – RAPHSON
METHOD (Polar Co-ordinates)

In the above equation, bus 1 is assumed to be
slack bus. The Jacobian matrix gives the
linearized relation between small changes in
with small changes in real and
reactive power Elements of jacobian
matrix are the partial derivatives of power flow
equations. It can be written as:
NEWTON – RAPHSON
METHOD (Polar Co-ordinates)

Students should able to NEWTON – RAPHSON
METHOD (Rectangular Co-
ordinates)

Students should able to Problem
NR METHOD

 When solving large scale power system, an alternative strategy for
improving computational efficiency and reducing computer
storage is introduced called Decoupled power flow method.
 This makes use of an approximate version of NR method.
 The basic principle underlying the decoupled approach is based
on two observations:
 Change in the voltage angle δ at a bus primarily affects the flow of
real power P in the transmission lines and leaves the flow of
reactive power Q relatively unchanged.
 Change in the voltage magnitude IVI at a bus primarily affects the
flow of reactive power Q in the transmission lines and leaves the
flow of real power P relatively unchanged.
Decoupled Load
flow

 The first observation states essentially that is
dP/dδ much larger than dQ/dδ, which we now
consider to be approximately zero.
 The second observation states that dQ/d|v| is
much larger than dP/d|V| , which is also
considered to be approximately zero.
 Incorporation of these approximations into the
jacobian:
 J2=0 and also J3=0;
Decoupled Load
flow

Students should able to Decoupled Load
flow

 These equations are decoupled in the sense that the
voltage-angle corrections ∆δ are calculated using only
real power mismatches ∆P, while the voltage-magnitude
corrections are calculated using only ∆ Q mismatches.
 However, these two interdependent.
 But this scheme would still require evaluation and
factoring of the two coefficient matrices at each
iteration.
 To avoid such computations, we introduce further
simplifications, which are justified by the physics of
transmission line power flow.
Decoupled Load
flow

 In a well-designed and properly operated power
transmission system:
 The angular differences (δi - δj) between typical
buses of the system are usually so small that
Cos(δi - δj) = 1; Sin(δi - δj) = (δi - δj);
 The line susceptances Bij are many times larger
than the line conductances, Gij so that,
Gij Sin(δi - δj) << Bij Cos(δi - δj) << Bij
Fast Decoupled
Load flow

Students should able to Fast Decoupled
Load flow

Students should able to Fast Decoupled
Load flow - Problem

Students should able to Limitations of GS
method

Students should able to Limitations & Merits
of NR method

The Bus impedance matrix (ZBUS) can be
formulated by two methods.
1. Formulating YBUS and taking the inverse.
2. Based on algorithm.
 By using system parameters and coded bus
numbers.
 Construction of the network is carried out by
adding one element at a time.
Z–BUS FORMULATION

Consider the following partial network with
‘m’ no. of buses and ‘0’ as the reference
node.
Algorithm for
Z–BUS FORMULATION
The performance equation in the
bus frame of reference:
EBUS = ZBUS IBUS
EBUS is the vector of bus voltages of size
mx1 measured w.r.to reference node.
IBUS is the vector of bus currents of size
mx1.

The voltage equations can be written as:
E1= Z11I1+Z12I2+…..+Z1kIk+….+Z1mIm
.
.
Ek= Zk1I1+Zk2I2+…..+ZkkIk+….+ZkmIm
.
.
Em= Zm1I1+Zm2I2+…..+ZmkIk+….+ZmmIm
By injecting the currents at Kth bus, by keeping all
other bus current injections as ‘0’.
Ii = 0 , i≠k
Algorithm for
Z–BUS FORMULATION

Ek= Zkk Ik
Ei= Zik Ik
Now Ik= 1.0 pu
Ek= Zkk
Ei= Zik
The ZBus formulation can be carried out in
two aspects.
When an element p-q is added to the partial
network, it may be branch or a link as
shown in figures.
Algorithm for
Z–BUS FORMULATION

Algorithm for
Z–BUS FORMULATION
P is an existing bus in
a partial network and
q is a new bus, this
results into p-q
branch.
Both p and q are
existing in the partial
network in this case
p-q is a link.

Let us assume, the added branch p-q is mutually
coupled with some elements of the partial network.
The performance equation with the added branch
is:
Addition of a
Branch

If the elements of the network are bilateral passive
elements: Zqi = Ziq
Addition of a
Branch
Where, Zqi is the voltage
at qth bus by injecting
1.0 pu current at ith
bus.
The voltage across the
added element is:
Vpq= Ep - Eq

The current through the element pq is:
Ipq : Current through the element pq
Irs : Current through the elements of partial network.
Vpq : Voltage across the element pq
Vrs : Voltage across the elements of partial network.
Ypq pq = Self admittance of added element.
Ypq rs = Vector of mutual admittance between pq-rs of
partial networks.
Yrs rs = Primitive admittance of partial network
Yrs pq = [Ypq rs ]T
Addition of a
Branch

Since pq is a branch, ipq = 0 but Vpq ≠ 0. Vrs= Er – Es
Addition of a
Branch

To obtain Zqq, inject 1 p.u current at qth node and at
remaining ‘0’
Addition of a
Branch

Students should able to Addition of a
Branch

The link is an element added in between two
existing buses.
Addition of a
Link
Firstly, an voltage
source ‘el’ is
connected in series
with the added
element.
This creates a
fictitious node 'l'.
The el is selected
such that Ipq = 0.
So that the p-l is
treated as an
addition of branch.

The performance equations are given by:
Addition of a
Link

Students should able to Addition of a
Link

Since, the node l is added which is a fictitious, its effect is to
be eliminated. Mathematically, it can be done by short
circuiting the series voltage source.
Addition of a
Link

Form ZBUS for the network shown.
Problem

Problem

The change of bus impedance matrix includes in following
cases:
 Removal of elements:
It can be done by modifying the already existing Zbus.
By adding an element in parallel, whose impedance is equal
to negative of the impedance of the element to be removed, if
the element is not mutually coupled to any of the element in
the partial network.
This is nothing but addition of a link.
Modification of Z-Bus

 Changes in the impedance of elements:
BY adding an link in parallel with the element such that the
equivalent impedance of the two elements is the desired
value.
Modification of Z-Bus

• Transients on a Transmission line-Short circuit
of synchronous machine(on no-load)
• 3–Phase short circuit currents and reactances
of synchronous machine
• Short circuit MVA calculations
• Series reactors and their Selection of reactors.
Symmetrical Fault
Analysis

• A number of undesirable but unavaoidable
incidents can temporarily disrupt this
condition. Such incidents can be said as a
fault.
• A fault in a circuit is an event, which causes
a deviation from the normal flow of current.
• This deviates the power system behavior.
Symmetrical Fault
Analysis

A fault may occur on a power system due to
number of reasons.
Some of them are listed below:
• Insulation failure of equipment
• Flashover of lines initiated by lightning
stroke.
• Falling of a tree along a line.
• Overloading of underground cables.
• Wind and Ice loading on the transmission
line.
• Accidental faulty operation.
Reasons for fault

• When a fault occurs on a system, the system get short
circuited.
• The current flowing into the fault depends on the path
met by the current, on the severity and nearness of
fault to the sources of power.
• The system must be protected against flow of heavy
short circuit currents, other wise it leads to the damage
of electrical equipment.
• This can be done by separating the faulty part of the
system from the healthy part by properly selecting
Faults

Basic two main types of faults are:
Series faults: The fault occurs through a high impedance
in series with the line.
Ex: Open circuit, This occurs when a circuit breaker or a
line is opened.
Shunt faults: In this type a low impedance is connected
between the line and ground.
Ex: Short circuit of lines.
Protective relays are employed to trip the circuit breaker
under faulty condition.
Types of Faults

The shunt faults are again classified into:
1. Symmetrical faults:
• In this all the three phases are short circuited to
each other and to the earth also.
• These are balanced and symmetrical.
• The voltages and currents remains balanced after
the occurrence of fault also.
• It is sufficient to consider one phase for the fault
analysis. Ex: 3- Phase short circuit faults.
Types of Faults

2. UnSymmetrical faults:
In this type only one or two phases only involves in
a fault.
The voltages and currents becomes unbalanced
after the occurrence of the fault.
Each phase has to be analyzed separately, for the
fault current calculations.
Ex: Line to Ground (LG) , Line to Line (LL/2L),
Double line to ground (LLG/2L-G).
Types of Faults

• The computation of fault currents for
unsymmetrical faults involves method of
symmetrical components.
• Frequency of faults in decreasing severity
Types of Faults
Type of fault Frequency of
occurrence
Three Phase faults 5 %
LLG 10 %
LL 15 %
LG 70 %

Students should able to SYNCHRONOUS
MACHINE (ON NO LOAD)

Students should able to Short Circuit
Capacity (SCC)

Problems

Students should able to TRANSIENT ON A
TRANSMISSION LINE

The current limiting reactor is an inductive coil having a
large inductive reactances in comparison to their
resistance and is used for
 limiting short circuit currents during fault conditions.
 Reduce the voltage disturbances on the rest of the
system.
It is installed in feeders and ties, in generators leads, and
between bus sections, for reducing the magnitude of
short circuit currents and the effect of the respective
voltage disturbance.
Current Limiting
Reactor

Location of Reactors
 Reactors are located at different location in a power
system for reducing the short circuit current. These
reactors may be connected in series with the
generators, feeders or in bus-bars as explained below.
• Generators Reactors
• Feeders Reactors
• Bus-Bar Reactor
Current Limiting
Reactor

Generators Reactors
Generator reactors are inserted
between the generator and the
generator bus. Such reactors
protect the machines individually.
In power station generator, reactors
are installed along with the
generators. The magnitude of
reactors is approximately about
0.05 per unit. The main
disadvantages of such type of
reactors are that if the fault occurs
on one feeder, then the whole of the
system will be adversely affected by

Feeder Reactors
Reactors, which is connected in
series with the feeder is called
feeders reactor. When the fault
occurs on any one feeder, then
the voltage drops occur only in its
reactors and the bus bar is not
affected much. Hence the
machines continue to supply the
load. The other advantage is that
the fault occurs on a feeder will
not affect the others feeders, and
thus the effects of fault are
localized
The disadvantage of such type
of reactors is that it does not
provide any protection to the
generators against short circuit
faults occurs across the bus
bars. Also, there is a constant
voltage drop and constant
power loss in reactors during
normal operating conditions.

Bus-Bar Reactors (Ring System)
 Bus-bar reactors are used to tie together the
separate bus sections. In this system sections
are made of generators and feeders and these
sections are connected to each other to a
common bus bar.
 In such type of system normally one feeder is fed
from one generator. In normal operating
conditions a small amount of power flows
through the reactors. Therefore voltage drop and
the power loss in the reactor is low. The bus bar
reactor, therefore, made with high ohmic
resistance so that there is not much voltage drop
across it.
Bus-Bar Reactors

Bus-Bar Reactors
When the fault occurs on any
one feeders, only one
generator feeds the fault while
the current of the other
generator is limited because
of the presence of the bus-bar
reactors.
The heavy current and voltage
disturbances caused by a
short circuit on a bus section
are reduced and restricted to
that faulty section only. The
only drawback of such type of
reactor is that it does not
protect the generators
connected to the faulty
sections.

Bus-bar Reactors (Tie-
Bus System)
This is the modification of the above
system. In tie-bus system, the
generator is connected to the
common bus-bar through the
reactors, and the feeder is fed from
generator side.
The operation of the system is
similar to the ring system, but it has
got additional advantages.
In this system, if the number of
sections is increased, the fault
current will not exceed a certain
value, which is fixed by the size of
the individual reactors

Stability
• Definition (IEEE / CIGRE ):
Power system stability is the ability of an electric
power system, for a given initial operating condition,
to regain a state of operating equilibrium after being
subjected to a physical disturbance, with most system
variables bounded so that practically the entire
system remains intact.
• The disturbances mentioned in the definition could be
faults, load changes, generator outages, line outages,
voltage collapse or some combination of these.

Classification of Power system
stability

• The study of steady state stability is basically concerned with
the determination of the upper limit of machine loadings
before losing synchronism, provided the loading is increased
gradually.
• The system is said to be dynamically stable if the oscillations
do not acquire more than certain amplitude and die out
quickly.
– Dynamic stability can be significantly improved through the use of
power system stabilizers.
• For a large disturbances in angular differences may be so
large as to cause the machines to falling out of step. This
type of instability is known as transient instability and is a
fast phenomenon.

Dynamics of a synchronous Machines
• Inertia constant: It is defined as the ratio of
energy stored in mega joules to the rating of the
machines in MVA.
• Energy stored = H x G
• Kinetic energy stored by the rotating body is
given by:

• On comparing above 2 expressions,

Swing Equation
• The Swing Equation of a
generator describes the
relative motion between
the rotor axis and the
synchronously rotating stator
filed axis with respect to time.
• it describes the rotor dynamics for a synchronous machine.
• It is a second-order differential equation.
• This equation is very helpful in analyzing the stability of
connected machines.

It is called the swing equation.

Swing Equation for coherent Machines:

Swing Equation for Non - coherent
Machines:

Steady state stability – Small Disturbances

Transient Stability
• It involves to find whether the system retained its synchronism
after machine has been subjected to severe disturbances.
• The disturbances may be sudden application of load, loss of
generation, loss of large load or occurrence of fault on a
system.
• In these instants the swing equation becomes highly non
linear and difficult to solve.
• A method known as Equal Area Criterion is used for a quickly
prediction of stability.
• This method is applicable to one machine connected to infinite
bus or two machine system.

Applications of Equal Area Criterion
The Equal Area Criterion can be imply in the following
cases and the transient stability analysis can be
performed.
• Sudden change in mechanical input.
• Sudden loss of one of the parallel lines.
• Sudden short circuit on one of the parallel lines.
 Short circuit at one end
 Short circuit away from the line ends

Sudden change in mechanical input
• Consider the following system SMIB.
• The electrical power transmitted is given by

Critical clearing time and Critical clearing angle
• Consider a system operating with mechanical input Pm
at steady angle of δ0.
• If a 3 phase fault occurs at point p of the outgoing
radial line, the electrical output of the generator
instantly reduces to zero. i.e. Pe = 0.

Methods to improve Steady State Stability
1. Reduction of transfer reactance
• A power system which has a lower value of transfer reactance can
have better steady-state stability limit. This can be achieved by:
i) use of parallel lines
ii) use of series capacitors
• If the power has to be transferred through long distance
transmission lines, use of parallel lines reduce transfer reactance as
well as improve voltage regulations.
• Similarly series capacitors are sometimes employed in lines to get
the same features.
2. Increase in the magnitudes of E and V.
Higher and fast field excitation system enhances steady-state
power limits

Methods to improve Transient stability
• Transient stability of the system can be improved by increasing the system voltage.
• Increase in the X/R ratio in the power system increases the power limit of the line. Thus
helps to improve the stability.
• High speed circuit breakers helps to clear the fault as quick as possible. The quicker the
breaker operates, the faster the fault cleared and better the system restores to normal
operating conditions.
• By Turbine fast valving: One of the main reason for the instability in the power system is
due to the excess energy supplied by the turbine during the fault period. Fast Valving helps
in reducing the mechanical input power when the generator is under acceleration during
the fault and hence improves the stability of the system.
• Use of Auto Re-closing: Majority of the faults in the power system will be momentary and
can be self cleared. Hence circuit breakers employed for fault clearance opens in sensing
the fault with time delay of 2 cycles and re-closes after particular time to determine whether
the fault is cleared.
• Some of the other ways to improve the transient stability are by
employing lightning arresters, high neutral grounding impedance, single pole switching,
quick Automatic Voltage Regulators.

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