Power flow solution

Prepared by Balaram Das, Dept. of Electrical Science, GIET, Gunupur Page 3.1
Chapter-03
Power Flow Solutions
Introduction:
In a three phase ac power system active and reactive power flows from the generating
station to the load through different networks buses and branches. The flow of active
and reactive power is called power flow or load flow.
Power flow studies is a systematic mathematical approach for determination of various
bus voltages, phase angle, active and reactive power flows through different branches,
generators and loads under steady state condition.
Power flow analysis is used to determine the steady state operating condition of a
power system. Power flow analysis is also required for planning and designing the
future expansion of power system as well as determining the best operation of existing
system.
Objective of Load Flow Study
Power flow analysis is very important in planning stages of new networks or addition
to existing ones like adding new generator sites, meeting increase load demand and
locating new transmission sites.
 The load flow solution gives the nodal voltages and phase angles and hence the
power injection at all the buses and power flows through interconnecting power
channels.
 It is helpful in determining the best location as well as optimal capacity of
proposed generating station, substation and new lines.
 It determines the voltage of the buses. The voltage level at the certain buses
must be kept within the closed tolerances.
 System transmission loss minimizes.
 Economic system operation with respect to fuel cost to generate all the power
needed.
 The line flows can be known. The line should not be overloaded, it means, we
should not operate the close to their stability or thermal limits.
Bus Classification
A bus is a node at which one or many lines, one or many loads and generators are
connected. In a power system each node or bus is associated with 4 quantities, such
as magnitude of voltage, phage angle of voltage, active power and reactive power. In
load flow problem two out of these 4 quantities are specified and remaining 2 are
required to be determined through the solution of equation. Depending on the
quantities that have been specified, the buses are classified into 3 categories.

a. Load bus: No generator is connected to the bus. At this bus the real and reactive
power are specified. It is desired to find out the voltage magnitude and phase angle
through load flow solutions.
b. Generator bus or voltage controlled bus: Here the voltage magnitude
corresponding to the generator voltage and real power Pg corresponds to its rating are
specified. It is required to find out the reactive power generation Qg and phase angle of
the bus voltage.
c. Slack (swing) bus: For the Slack Bus, it is assumed that the voltage magnitude |V|
and voltage phase Φ are known, whereas real and reactive powers Pg and Qg are
obtained through the load flow solution.
Bus type
Specified
quantities
Unknown
quantities
Load bus/ PQ bus P, Q ІVІ, δ
Generator bus/
Voltage controlled
bus/regulated
bus/PV bus
P, ІVІ Q, δ
Slack bus/
Reference bus/
Swing bus
ІVІ, δ P, Q
The power flow problem:
The complex power injected by the generating source into the ith bus of a power
system is given as
Where, Vi – voltage at the ith bus w.r.to ground
Ii* - Complex conjugate of source current Ii injected into the bus.
It is convenient to handle load flow problem by using Ii rather than Ii*. So taking the
complex conjugate of equation(1)
Substituting equation(3) in (2)

Equating real and imaginary parts we have
So,
Above two equations are known as Static Load Flow Equation (SLFE).
The Gauss-Seidal Method:
It is an iterative technique used to solve non linear algebric equations. This method
can be discussed in two cases.
(i) When PV buses are absent
(ii) When PV buses are present
(i) When PV buses are absent:
Total no. of bus = n
One bus is selected as slack bus
Remaining buses = n-1 = PQ bus
For slack bus,
For PQ bus,
Complex power injected in system by ith bus

Above equation used to calculate the value of voltage of buses and it starts from i=
2,3,4…..n
First of all we need to choose the initial values for bus voltages and angles.
………………………..
………………………..
If convergence occurs then we calculate the power for slack bus otherwise we will do
new iteration.
Acceleration of Convergence
To speed up the convergence we use an acceleration factor, . For an ith bus the value
of accelerated voltage is given by
The value of lies between 1 to 2 but usually its value is 1.6

When PV buses are present
Total no. of bus = n
i.e 1,2,3……..m-1, m, m+1….n
Number of PQ buses = m-1
Number of PV buses = n-m
One bus is selected as slack bus
Given data,
We need to found
S1 for slack bus
Advantages:
 Simplicity of technique.
 Small computer memory requirement
 Less computational time per iteration
Disadvantages:
 Slow rate of convergence resulting in larger number of iterations.
 Increase in number of iterations directly with the increase in the number of
buses.
 Effect on convergence due to choice of slack bus.
Due to above disadvantages, GS method is limited to systems with smaller number of
buses.
Example:01
Using GS method, find the bus voltage at the end of one iteration for the following 2 bus
system. Line reactances are shown in the Fig. Ignore resistance and line charging.
Assume initial voltage at all buses to 1.0∟0. Use 1.0 as acceleration factor. The bus
data is given below.

Bus No. Specified P(pu) Injection Q(pu) Specified voltage
1 - - 1.0
2 0.3 - 1.0
3 0.5 0.2 -
Solution:
Admittance of each line
The admittance matrix as
For generation bus-2
For Load bus-3

Example-02
The following is the system data for load flow solution
Bus code Admittance
1-2 2-j8
1-3 1-j4
2-3 0.666-j2.664
2-4 1-j4
3-4 2-j8
The schedule of active and Reactive powers
Bus code P Q V Remarks
1 - - 1.06 Slack
2 0.5 0.2 1+j0 PQ
3 0.4 0.3 1+j0 PQ
4 0.3 0.1 1+j0 PQ
Determine the voltages at the end of 1st iteration using GS method. Take
Solution:
The powers of generator bus are taken as positive while the powers for load buses are
taken as Negetive.

The Newton-Raphson Method:(using Polar Coordinates)
In polar form the results obtained in a smaller number of equations then the total
number of equations involved in rectangular form.
For any ith bus, we have
Where, δ- phase angle of the bus voltage and θik is an admittance angle.
Substituting the values of Vi*, Vk, Yik from equation
Now the linear equation in polar form becomes

Where J1, J2, J3 and J4 are the elements of Jacobian matrix and can be determined
from power equations.
The off diagonal and diagonal elements of J1 are
The formulation in the polar coordinates takes the less computational effort and also
less memory space.
Sparcity: In a power system, there may be large number of buses but each bus is
connected to only a small number of the remaining buses. It means that Ybus of a
large power system is very sparse( It has a large number of zero elements). In a large

system of 100 nodes, the non zero elements may be as small as 20% of the total
elements. The sparcity features Ybus minimizes the computer memory requirement
and results faster computations.
Comparison of GS and NR method:
GS Method NR Method
Rectangular coordinate method used. Both rectangular and Polar coordinate
method is used.
The time taken is less for computing one
iteration
It takes longer time to perform one
iteration.
Rate of convergence is slow. Rate of convergence is fast.
Convergence characteristic is linear. It has quadrature convergence
characteristic.
GS method takes more computer time NR method takes less computer time.
It is used to compute the solution of small
system
NR method is used with advantages for
large power system.
Fast Decoupled Power Flow method:
This is an extension of Newton-Raphson method formulated in polar coordinates with
certain approximation which results into a fast algorithm for load flow solution.
In this method both speed and sparcity are exploited. This method makes use of loose
coupling between MW flow-voltage angle and MVAR flow-voltage magnitude. In other
words a small change in the magnitude of the bus voltage does not affect the real
power flow at the bus and similarly a small change in phase angle of the bus voltage
has hardly any affect on reactive power flow.
Because of this loose physical interaction between MW and MVAR flows in a power
system, The MW-voltage angle and MVAR-V calculations can be decoupled. This
decoupling results in a very simple, fast and reliable algorithm.
Equation() can be written as
Where H,N,M and L are the elements of the Jacobian matrix.
Since Changes in real power are less sensitive to changes in voltage magnitude (are
mainly sensitive to angle) and changes in reactive power are less sensitive to change in
phase angle of voltage (but mainly sensitive to change in voltage magnitude), above
equation can be reduced to

The equation is decoupled equation and can be expanded as
Off diagonal element of H is
Similarly off diagonal elements of L is
The diagonal elements of H are given as
Similarly the diagonal elements of L are given as
In the case of fast decoupled load flow method following approximations are further
made for evaluating Jacobian element:

With the above assumption, the Jacobian elements becomes
With these Jacobian elements
Further decoupling is obtained as follows:
(i) Omit from B’ the angle shifting effects of phase shifters.
(ii) Omit from B’’ the representation of those network elements that affect MVAr flows
i.e., shunt reactors and off-nominal in phase transformer taps.
(iii) Dividing above two equation by Vi and assuming Vk=1.0 pu and also neglecting
series resistance in calculating the elements of B’
With the above assumption above equations becomes
It is to be noted that [B′] and [B″] are real and sparse and have similar structures as
those of H and L respectively. Since the two matrices are constant and do not change
during successive iterations for solution of the load flow problem, they need be
evaluated only once and inverted once during the first iteration and then used in all
successive iterations. It is because of the nature of Jacobian matrices [B′] and [Β″] and
the sparsity of these matrices that the method is fast.
Regulating Transformer
Definition: The transformer which changes the magnitude and phase angle at the
certain point in the power system is known as the regulating transformer. It is mainly
used for controlling the magnitude of bus voltage and for controlling the power flow,
which is controlled by the phase angle of the transformer. They provide the small
component of voltage between the line or phase voltage.

The main function of the regulating transformer is to control the magnitude of voltage
and power flow of the transmission line. The regulating transformer is of two types.
One is used for changing the magnitude of voltage which is called online tap changing
transformer and the other is called phase shifting transformer. The regulating
transformer compensates the fluctuation of voltage and current. The arrangement of
the regulating transformer is shown in the figure below.
The main application of power system is to reduce the circulating current and
minimise the losses in the power system. The regulating transformer reduces the
losses in the power system network, and it also controls the unwanted exchange of the
reactive power in the system.
Consider that,
The primary winding of the transformer is connected in delta, and the secondary
winding phase is connected in star. The voltage of the secondary phases is adjustable.
From the phase shift property in delta-star transformer we have

The magnitude of Vm can be controlled in a small range. It is used for adjusting the
three phase voltage Va, Vb, and Vc. The voltage of the tertiary winding is given below.
The magnitude of VΦ is adjustable and is used for control of the phase angle of the
voltage Va’, Vb’ and Vc’
The voltage Vkm, Vlk, Vml is derived from the voltage Vko, Vlo, Vmo as
The Vkm, Vkl and Vml are in phase with the system voltages Van, Vbn, and Vcn The voltage
Vrl, Vsm, and Vkl are 90º out of phase with the same voltage. The incremental voltage is
given by

The phasor diagram of the voltage is shown in the figure above.

Multiple Choice questions
1) Which among the following quantities are to be determined in voltage controlled bus?
a. P and Q
b. Q and |V|
c. |V| and δ
d. Q and δ
ANSWER: Q and δ
2) Which among theses quantities are to be determined in slack bus?
a. P and Q
b. Q and |V|
c. |V| and δ
d. Q and δ
ANSWER: P and Q
3) Which among the following buses constitute the maximum number in a power system?
a. Slack bus
b. P Q bus
c. P V bus
d. All of these
e. None of these
ANSWER: P Q bus
4) What percentage of buses in the power system are generator buses?
a. 5 %
b. 25 %

c. 70 %
d. 10 %
ANSWER: 10 %
5) Which among the following quantities are specified at the generator bus?
a. P and Q
b. P and |V|
c. Q and |V|
d. P and δ
ANSWER: P and |V|
6) Which among the following quantities are specified at the load bus?
a. P and Q
b. P and |V|
c. Q and |V|
d. P and δ
ANSWER: P and Q
7) Why are load flow studies carried out?
a. To study of stability of the system
b. For fault calculations
c. For planning the power system
d. All of these
ANSWER: For planning the power system
Short Questions and Answers
1. What is power flow study or load flow study?
The study of various methods of solution to power system network is referred to as load
study. The solution provides the voltages at various buses, power flowing in Various lines
and line losses.
2. What is the need for load flow study?
The load flow study of a power system is essential to decide the best operation of existing
system and for planning the future expansion of the system. It is also essential for
designing a new power system.
3. What are the different types of buses in a power system?
The buses of a power system can be classified into three types based on the Quantities
being specified for the buses, which are as follows:
a. Load bus or PQ bus (P and Q are specified)
b. Generator bus or voltage controlled bus or PV bus (P and V are specified)
c. Slack bus or swing bus or reference bus (|V| and δ are specified)
4. Define voltage controlled bus (generator bus/PV bus).

A bus is called voltage controlled bus if the magnitude of voltage |V| and real power (P) are
specified for it. In a voltage controlled bus, the magnitude of the voltage is not allowed to
change. Voltage controlled bus is also called as Generator bus and PV bus.
5. What is PQ bus (load bus)?
A bus is called PQ bus or load bus when real and reactive components of power are
specified for the bus. In a load bus, the voltage is allowed to vary within permissible limits.
6. What is swing bus(slack bus/reference bus)?
A bus is called swing bus when the magnitude and phase of bus voltage are specified for it.
The swing bus is the reference bus for load flow solution and it is required for accounting
for the line losses. Usually one of the generator bus is selected as the swing bus.
7. What is the need for slack bus?
The slack bus is needed to account for transmission line losses. In a power system, the
total power generated will be equal to sum of power consumed by loads and losses. In a
power system, only the generated power and load power are specified for the buses. The
slack bus is assumed to generate the power required for losses. Since the losses are
unknown, the real and reactive power are not specified for slack bus. They are estimated
through the solution of line flow equations.
8. List the quantities specified and the quantities to be determined from load flow study
for various types of buses.
The following table shows the quantities specified and the quantities to be obtained for
various types of buses.
9. Write the load flow equation of Gauss and Gauss-Seidel method.
The load flow equation of Gauss method is given by,

10.Write the load flow equation of Newton-Raphson method.
The load flow equation of Newton Raphson method is given by,
Note: Refer your class note
11.Discuss the effect of acceleration factor in the load flow solution algorithm.
In load flow solution by iterative methods, the number of iterations can be reduced if the
correction voltage at each bus is multiplied by some constant. The multiplication of the
constant will increase the amount of correction to bring the voltage closer to the value it is
approaching. The multipliers that accomplish this improved converged are called
acceleration factors. An acceleration factor of 1.6 is normally used in load flow problems.
12.What do you mean by a flat voltage start?
In iterative methods of load flow solution, the initial voltage of all buses except slack bus is
assumed as 1+j0 p.u. This is referred to as flat voltage start.
13.When the generator bus is treated as load bus? What will be the reactive power and
bus voltage when the generator bus is treated as load bus?
If the reactive power of a generator bus violates the specified limits, then the generator bus
is treated as load bus. The reactive power of that particular bus is equated to the limit it
has violated and the previous iteration value of bus voltage is used for calculating current
iteration value.
14.What are the advantages of Gauss-Seidel method?
The advantages of Gauss-Seidel method are,
a. Calculations are simple and so the programming task is less
b. The memory requirement is less
c. Useful for small systems.
15.What are the disadvantages of Gauss-Seidel method?

The disadvantages of Gauss-Seidel method are,
a. Requires large number of iterations to reach convergence.
b. Not suitable for large systems.
c. Convergence time increases with size of the system.
16.How approximation is performed in Newton-Raphson method?
In Newton-Raphson method, the set of non-linear simultaneous (load flow) equations are
approximated to a set of linear simultaneous equations using Taylor’s series expansion and
the terms are limited to first order approximation.
17.What is Jacobian matrix? How the elements of Jacobian matrix are computed?
The matrix formed from the derivates of load flow equations is called Jacobian matrix and it
is denoted by J. The elements of Jacobian matrix will change in every iteration. In each
iteration, the elements of the Jacobian matrix are obtained by partially differentiating the
load flow equations with respect o unknown variable and then evaluating the first derivates
using the solution of previous iteration.
18.What are the advantages of Newton-Raphson method?
The advantages of Newton-Raphson method are,
a. This load flow method is faster, more reliable and he results are accurate.
b. Requires less number of iterations for convergence.
c. The number of iterations are independent of the size of the system.
d. Suitable for large system.
19.What are the disadvantages of Newton-Raphson method?
The disadvantages of Newton-Raphson method are,
a. Programming is more complex.
b. The memory requirement is more.
c. Computational time per iteration is higher due to larger number of calculations per
iteration.
20.Mention (any) three advantages of N-R method over G-S method?
The three advantages of N-R method over G-S method are,
a. The N-R method has quadratic convergence characteristics and so converges faster than G-
S method.
b. The number of iterations for convergence is independent of the system in N-R method.
c. In N-R method, the convergence is not affected by the choice of slack bus.
21.Compare G-S method and N-R methods of load flow solutions.

22.How the convergence of N-R method is speeded up?
The convergence can be speeded up in N-R method by using Fast Decoupled Load Flow
(FDLF) algorithm. In FDLF method, the weak coupling between P-δ and Q-V are decoupled
and then the equations are further simplified using the knowledge of practical operating
conditions of a power system.
Long Questions
1. Explain the different methods of voltage control?
The voltage of the power system may vary with the change in load. The voltage is normally high
at light load and low at the heavy-load condition. For keeping the voltage of the system in
limits, some additional equipment requires which increase the system voltage when it is low
and reduces the voltage when it is too high. The following are the methods used in the power
system for controlling the voltage.
1. On – Load Tap Changing Transformer
2. Off – Load Tap Changing transformer
3. Shunt Reactors
4. Synchronous Phase Modifiers
5. Shunt Capacitor
6. Static VAR System (SVS)
Controlling the system voltage by the help of shunt inductive element is known as shunt
compensation. The shunt compensation is of two types, i.e., the static shunt compensation and
the synchronous compensation. In static shunt compensation, the shunt reactor, shunt
capacitor and static VAR system are used, whereas the shunt compensation uses the
synchronous phase modifier. The methods used for controlling the voltage are explained below
in details.

1. Off – Load Tap Changing Transformer – In this method, the voltage is controlled by
changing the turn ratio of the transformer. The transformer is disconnected from the supply
before changing the tap. The tap changing of the transformer mostly done manually.
2. On – Load Tap Changing Transformer – This arrangement is used for changing the turn
ratio of the transformer for regulating the system voltage when the transformer delivers the
load. Most of the power transformer is provided with on-load tap changer.
3. Shunt Reactor – The shunt reactor is the inductive current element which is connected
between the line and neutral. The shunt reactor compensates the inductive current from the
transmission line or underground cables. It is mainly used in the long distance EHV and UHV
transmission lines for reactive power control.
The shunt reactors are used in the sending end substation, receiving end substation and in the
intermediate substation of long EHV and UHV line. In the long transmission line, the shunt
reactor is connected at the distance of 300 Km to limit the voltage at an intermediate point.
4. Shunt Capacitors – The shunt capacitors are the capacitors connected in parallel with the
line. It is installed at the receiving end substation, distribution substations and in the
switching substations. The shunt capacitor injected the reactive volt-ampere to the line. It is
placed in the three phase bank.
5. Synchronous Phase Modifier – The synchronous phase modifier is the synchronous motor
running without a mechanical load. It is connected with the load at receiving the end of the
line. The synchronous phase modifier absorbs or generates the reactive power by varying the
excitation of the field winding. It keeps the voltage constant at any condition of the load and
also improves the power factor.
6. Series Var Systems (SVS) – The static VAR compensator inject or absorb the inductive VAR
to the system when the voltage becomes higher or lower than the reference value. In static VAR
compensator, the thyristor is used as switching device in place of circuit breakers. Nowadays,
the thyristor switching is used in the system in place of mechanical switching because
thyristor switching is faster and provides transient free operation by controlling the switching.

Power flow solution

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