POWER FLOW ANALYSIS

UNIT - II
 OVERVIEW

• 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.
• Through the load flow studies we can obtain the voltage magnitudes and angles at each bus in the steady
state.
• This is rather important as the magnitudes of the bus voltages are required to be held within a specified
limit.
• 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.
 OVERVIEW

• The steady state power and reactive power supplied by a bus in

a power network are expressed in terms of nonlinear algebraic
equations.
• We therefore would require iterative methods for solving these
equations.
• In this chapter we shall discuss two of the load flow methods.
 REAL AND REACTIVE POWER INJECTED IN A BUS

• For the formulation of the real and reactive power entering a bus, we need to define the following
quantities.
• Let the voltage at the i th bus be denoted by
• Also let us define the self admittance at bus- i as
• Similarly the mutual admittance between the buses i and j can be written as

• Let the power system contains a total number of n buses. The current injected at bus- i is given as
 REAL AND REACTIVE POWER INJECTED IN A BUS

• Assume the current entering a bus to be positive and that leaving the bus to be negative.
• As a consequence the power and reactive power entering a bus will also be assumed to be positive.
• The complex power at bus- i is then given by
 REAL AND REACTIVE POWER INJECTED IN A BUS

Therefore substituting in eqn. we get the real and reactive power as

 CLASSIFICATION OF BUSES

• For load flow studies - Assumed that the loads are constant and they are defined by their real
and reactive power consumption.
• Further assumed that the generator terminal voltages are tightly regulated and so constant.
• The main objective of the load flow is to find the voltage magnitude of each bus and its angle
when the powers generated and loads are pre-specified.
• To facilitate this we classify the different buses of the power system shown in the chart below.
 CLASSIFICATION OF BUSES

Load Buses :
• In these buses no generators are connected and hence the generated real power PGi and
reactive power QGi are taken as zero.
• The load drawn by these buses are defined by real power -PLi and reactive power -QLi in
which the negative sign accommodates for the power flowing out of the bus.
• This is why these buses are sometimes referred to as P-Q bus.
• The objective of the load flow is to find the bus voltage magnitude |Vi| and its angle δi.
 CLASSIFICATION OF BUSES

Voltage Controlled Buses :

• These are the buses where generators are connected.
• Therefore the power generation in such buses is controlled through a prime mover while
the terminal voltage is controlled through the generator excitation.
• Keeping the input power constant through turbine-governor control and keeping the bus
voltage constant using automatic voltage regulator, we can specify constant PGi and
| Vi | for these buses.
• This is why such buses are also referred to as P-V buses.
• It is to be noted that the reactive power supplied by the generator QGi depends on the
system configuration and cannot be specified in advance.
• Furthermore we have to find the unknown angle δi of the bus voltage.
 CLASSIFICATION OF BUSES

Slack or Swing Bus :

• Usually this bus is numbered 1 for the load flow studies.
• This bus sets the angular reference for all the other buses.
• It sets the reference against which angles of all the other bus voltages are measured. For
this reason the angle of this bus is usually chosen as 0° .
• Furthermore it is assumed that the magnitude of the voltage of this bus is known.
 SLACK BUS

• Consider a typical load flow problem in which all the load demands are known.
• Even if the generation matches the sum total of these demands exactly, the mismatch
between generation and load will persist because of the line I 2R losses.
• Since the I 2R loss of a line depends on the line current which, in turn, depends on the
magnitudes and angles of voltages of the two buses connected to the line, it is rather
difficult to estimate the loss without calculating the voltages and angles.
• For this reason a generator bus is usually chosen as the slack bus without specifying its
real power. It is assumed that the generator connected to this bus will supply the balance
of the real power required and the line losses.
 PREPARATION OF DATA FOR LOAD FLOW

• Let real and reactive power generated at bus- i be denoted by PGi and QGi respectively.
Let us denote the real and reactive power consumed at the i th bus
as PLi and QLi respectively. Then the net real power injected in bus- i is:

• Let the injected power calculated by the load flow program be Pi, calc . Then the
mismatch between the actual injected and calculated values is given by
 PREPARATION OF DATA FOR LOAD FLOW

• In a similar way the mismatch between the reactive power injected and calculated values is given by

• The purpose of the load flow is to minimize the above two mismatches.
• It is to be noted that

are used for the calculation of real and reactive power in Eqn.
• However since the magnitudes of all the voltages and their angles are not known a priori, an
iterative procedure must be used to estimate the bus voltages and their angles in order to calculate
the mismatches.
• It is expected that mismatches ΔPi and ΔQi reduce with each iteration and the load flow is said to
have converged when the mismatches of all the buses become less than a very small number.
 PREPARATION OF DATA FOR LOAD FLOW

• For the load flow studies we shall consider the system of Fig., which has 2 generator and 3
load buses.
• Define bus-1 as the slack bus while taking bus-5 as the P-V bus. Buses 2, 3 and 4 are P-Q
buses.
 PREPARATION OF DATA FOR LOAD FLOW

• The line impedances and the line charging admittances are given in Table.
• Based on this data the Y bus matrix is given in Table.
• The sources and their internal impedances are not considered while forming
the Ybus matrix for load flow studies which deal only with the bus voltages.
Table: Line impedance and line charging data of the system of Fig.
PREPARATION OF DATA FOR LOAD FLOW

Table: Ybus matrix of the system of Fig.

 PREPARATION OF DATA FOR LOAD FLOW

• The bus voltage magnitudes, their angles, the power generated and consumed at each bus are
given in Table.
• In this table some of the voltages and their angles are given in boldface letters. This indicates that
these are initial data used for starting the load flow program.
• The power and reactive power generated at the slack bus and the reactive power generated at the
P-V bus are unknown. Therefore each of these quantities are indicated by a dash ( - ). Since we do
not need these quantities for our load flow calculations, their initial estimates are not required.
• Also note from Fig. that the slack bus does not contain any load while the P-V bus 5 has a local load
and this is indicated in the load column.
 LOAD FLOW BY GAUSS-SEIDEL METHOD

• The basic power flow equations are nonlinear.

• In an n -bus power system, let the number of P-Q buses be np and the number of P-V
(generator) buses be ng such that n = np + ng + 1.
• Both voltage magnitudes and angles of the P-Q buses and voltage angles of the P-V
buses are unknown making a total number of 2np + ng quantities to be determined.
• Amongst, the known quantities are 2np numbers of real and reactive powers of the P-Q
buses, 2ng numbers of real powers and voltage magnitudes of the P-V buses and
voltage magnitude and angle of the slack bus.
• Therefore there are sufficient numbers of known quantities to obtain a solution of the
load flow problem. However, it is rather difficult to obtain a set of closed form equations
from PFE. We therefore have to resort to obtain iterative solutions of the load flow
problem.
 LOAD FLOW BY GAUSS-SEIDEL METHOD

• At the beginning of an iterative method, a set of values for the unknown quantities are
chosen. These are then updated at each iteration.
• The process continues till errors between all the known and actual quantities reduce
below a pre-specified value.
• In the Gauss-Seidel load flow we denote the initial voltage of the i th bus by Vi(0) ,
i = 2, ... , n . This should read as the voltage of the i th bus at the 0th iteration, or initial
guess. Similarly this voltage after the first iteration will be denoted by Vi(1) .
 LOAD FLOW BY GAUSS-SEIDEL METHOD

• In this Gauss-Seidel load flow, the load buses and voltage controlled buses are treated
differently.
• However in both these type of buses we use the complex power equation for updating the
voltages.

• Knowing the real and reactive power injected at any bus we can expand as
EXAMPLE
 CONVERGENCE OF THE ALGORITHM

• A total number of 4 real and 3 reactive powers are known to us.

• We must then calculate each of these powers (using PFEs) using the values of
the voltage magnitudes and their angle obtained after each iteration.
• The power mismatches are then calculated.
• The process is assumed to have converged when each of ΔP2 , ΔP3, ΔP4 , ΔP5 ,
ΔQ2 , ΔQ3 and ΔQ4 is below a small pre-specified value.
 ACCELERATION FACTOR

• To accelerate computation in the P-Q buses, the voltages obtained is multiplied by a

constant. The voltage update of bus- i is then given by

• where λ is a constant that is known as the acceleration factor .

• The value of λ has to be below 2.0 for the convergence to occur.
• Table lists the values of the bus voltages after the 1st iteration and number of
iterations required for the algorithm to converge for different values of λ.
• The algorithm converges in the least number of iterations when λ is 1.4 and the
maximum number of iterations are required when λ is 2.
• The algorithm will start to diverge if larger values of acceleration factor are chosen.
 GAUSS-SEIDEL METHOD: BUS VOLTAGES AFTER 1 ST
ITERATION AND NUMBER OF ITERATIONS REQUIRED FOR
CONVERGENCE FOR DIFFERENT VALUES OF
 SOLUTION OF A SET OF NONLINEAR EQUATIONS
BY NEWTON-RAPHSON METHOD
• Let us consider a set of n nonlinear equations of a total number of n variables x1 , x2 , ... , xn. Let
these equations be given by

where f1, ... , fn are functions of the variables x1 , x2 , ... , xn.

• We can define another set of functions g1 , ... , gn as given below
• Let us assume that the initial estimates of the n variables are x1(0) , x2(0) , ... , xn(0) . Let us add corrections
Δx1(0) , Δx2(0) , ... , Δxn(0) to these variables such that we get the correct solution of these variables defined by

• The functions then can be written in terms of the variables

• Expand the above equation in Taylor 's series around the nominal values of x1(0) , x2(0) , ... , xn(0) . Neglecting
the second and higher order terms of the series, the expansion of gk , k = 1, ... , n is given as
• Where is the partial derivative of gk evaluated at x2(0) , ... , xn(0) .
• Equation can be written in vector-matrix form as

• The square matrix of partial derivatives is called the Jacobian matrix J with J (0) indicating that the
matrix is evaluated for the initial values of x2(0) , ... , xn(0) .
• Since the Taylor 's series is truncated by neglecting the 2nd and higher order terms, we cannot
expect to find the correct solution at the end of first iteration.
• We shall then have

• These are then used to find J (1) and Δgk (1) , k = 1, ... , n . We can then find Δx2(1) , ... , Δxn(1)
• Calculate x1(2), x2(2) , ... , xn(2).
• The process continues till Δgk , k = 1, ... , n becomes less than a small quantity
 LOAD FLOW BY NEWTON-RAPHSON METHOD

• Let us assume that an n -bus power system contains a total np number of P-Q buses while the
number of P-V (generator) buses be ng such that n = np + ng + 1. Bus-1 is assumed to be the slack
bus.
• We shall further use the mismatch equations of ΔPi and ΔQi.
• The approach to Newton-Raphson load flow is similar to that of solving a system of nonlinear
equations using the Newton-Raphson method: At each iteration we have to form a Jacobian
matrix and solve for the corrections
• For the load flow problem, this equation is of the form
• where the Jacobian matrix is divided into submatrices as

• The size of the Jacobian matrix is ( n + np − 1) x ( n + np −1).

• For example, for the 5-bus problem, this matrix will be of the size (7 x 7). The dimensions of
the submatrices are as follows:
• J11: (n − 1)  (n − 1), J12: (n − 1)  np, J21: np  (n − 1) and J22: np  np
• The submatrices are
 LOAD FLOW ALGORITHM
• The Newton-Raphson procedure is as follows:
• Step-1: Choose the initial values of the voltage magnitudes |V| (0) of all np load buses
and n − 1 angles δ (0) of the voltages of all the buses except the slack bus.
• Step-2: Use the estimated |V|(0) and δ (0) to calculate a total n − 1 number of injected
real power Pcalc(0) and equal number of real power mismatch ΔP (0) .
• Step-3: Use the estimated |V| (0) and δ (0) to calculate a total np number of injected
reactive power Qcalc(0) and equal number of reactive power mismatch ΔQ (0) .
• Step-3: Use the estimated |V| (0) and δ (0) to formulate the Jacobian matrix J (0) .
• Step-4: Solve for δ (0) and Δ |V| (0) ÷ |V| (0).
• Step-5 : Obtain the updates from ; ;
• Step-6: Check if all the mismatches are below a small number. Terminate the
process if yes. Otherwise go back to step-1 to start the next iteration with the
updates
 FORMATION OF THE JACOBIAN MATRIX

• A. Formation of J11
Let us define J11 as

✓Lik's are the partial derivatives of Pi with respect to δk. The

derivative Pi (4.38) with respect to k for i ≠ k is given by
• Similarly the derivative Pi with respect to k for i = k is given by

• B. Formation of J21
Let us define J21 as

▪ The elements of J21 are the partial derivative of Q with respect to δ

▪ , i=k
C. Formation of J12
Let us define J12 as

J21 involve the derivatives of real power P with respect to magnitude of bus
voltage |V|

For i=k
• Formation of J22
• For the formation of J22 let us define

• For i=k
• Once the submatrices J11 and J21 are computed, the
formation of the submatrices J12 and J22 is fairly
straightforward.
• For large system this will result in considerable saving
in the computation time.
 SOLUTION OF NEWTON-RAPHSON LOAD FLOW

• The Newton-Raphson load flow is tested on the 5 bus system

For forming the off diagonal elements of J21

The real power injected at bus-2 is calculated as