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LOAD FLOW STUDIES
3.1 REVIEW OF NUMERICAL SOLUTION OF EQUATIONS
The numerical analysis involving the solution of algebraic simultaneous equations forms
the basis for solution of the performance equations in computer aided electrical power
system analyses, such as during linear graph analysis, load flow analysis (nonlinear
equations), transient stability studies (differential equations), etc. Hence, it is necessary to
review the general forms of the various solution methods with respect to all forms of
equations, as under:
1. Solution Linear equations:
* Direct methods:
- Cramer‟s (Determinant) Method,
- Gauss Elimination Method (only for smaller systems),
- LU Factorization (more preferred method), etc.
* Iterative methods:
- Gauss Method
- Gauss-Siedel Method (for diagonally dominant systems)
3. Solution of Nonlinear equations:
Iterative methods only:
- Gauss-Siedel Method (for smaller systems)
- Newton-Raphson Method (if corrections for variables are small)
4. Solution of differential equations:
Iterative methods only:
- Euler and Modified Euler method,
- RK IV-order method,
- Milne‟s predictor-corrector method, etc.
It is to be observed that the nonlinear and differential equations can be solved only by the
iterative methods. The iterative methods are characterized by the various performance
features as under:
_ Selection of initial solution/ estimates
_ Determination of fresh/ new estimates during each iteration
_ Selection of number of iterations as per tolerance limit
_ Time per iteration and total time of solution as per the solution method selected
_ Convergence and divergence criteria of the iterative solution
_ Choice of the Acceleration factor of convergence, etc.
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A comparison of the above solution methods is as under:
In general, the direct methods yield exact or accurate solutions. However, they are suited
for only the smaller systems, since otherwise, in large systems, the possible round-off
errors make the solution process inaccurate. The iterative methods are more useful when
the diagonal elements of the coefficient matrix are large in comparison with the off
diagonal elements. The round-off errors in these methods are corrected at the successive
steps of the iterative process.The Newton-Raphson method is very much useful for
solution of non –linear equations, if all the values of the corrections for the unknowns are
very small in magnitude and the initial values of unknowns are selected to be reasonably
closer to the exact solution.
3.2 LOAD FLOW STUDIES
Introduction: Load flow studies are important in planning and designing future expansion
of power systems. The study gives steady state solutions of the voltages at all the buses,
for a particular load condition. Different steady state solutions can be obtained, for
different operating conditions, to help in planning, design and operation of the power
system. Generally, load flow studies are limited to the transmission system, which
involves bulk power transmission. The load at the buses is assumed to be known. Load
flow studies throw light on some of the important aspects of the system operation, such as:
violation of voltage magnitudes at the buses, overloading of lines, overloading of
generators, stability margin reduction, indicated by power angle differences between buses
linked by a line, effect of contingencies like line voltages, emergency shutdown of
generators, etc. Load flow studies are required for deciding the economic operation of the
power system. They are also required in transient stability studies. Hence, load flow
studies play a vital role in power system studies. Thus the load flow problem consists of
finding the power flows (real and reactive) and voltages of a network for given bus
conditions. At each bus, there are four quantities of interest to be known for further
analysis: the real and reactive power, the voltage magnitude and its phase angle. Because
of the nonlinearity of the algebraic equations, describing the given power system, their
solutions are obviously, based on the iterative methods only. The constraints placed on the
load flow solutions could be:
_ The Kirchhoff‟s relations holding good,
_ Capability limits of reactive power sources,
_ Tap-setting range of tap-changing transformers,
_ Specified power interchange between interconnected systems,
_ Selection of initial values, acceleration factor, convergence limit, etc.
3.3 Classification of buses for LFA: Different types of buses are present based on
the specified and unspecified variables at a given bus as presented in the table below:
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Table 1. Classification of buses for LFA
Importance of swing bus: The slack or swing bus is usually a PV-bus with the largest
capacity generator of the given system connected to it. The generator at the swing bus
supplies the power difference between the “specified power into the system at the other
buses” and the “total system output plus losses”. Thus swing bus is needed to supply the
additional real and reactive power to meet the losses. Both the magnitude and phase angle
of voltage are specified at the swing bus, or otherwise, they are assumed to be equal to 1.0
p.u. and 00 , as per flat-start procedure of iterative
solutions. The real and reactive powers at the swing bus are found by the computer routine
as part of the load flow solution process. It is to be noted that the source at the swing bus is
a perfect one, called the swing machine, or slack machine. It is voltage regulated, i.e., the
magnitude of voltage fixed. The phase angle is the system reference phase and hence is
fixed. The generator at the swing bus has a torque angle and excitation which vary or
swing as the demand changes. This variation is such as to produce fixed voltage.
Importance of YBUS based LFA:
The majority of load flow programs employ methods using the bus admittance matrix, as
this method is found to be more economical. The bus admittance matrix plays a very
important role in load flow analysis. It is a complex, square and symmetric matrix and
hence only n(n+1)/2 elements of YBUS need to be stored for a n-bus system. Further, in
the YBUS matrix, Yij = 0, if an incident element is not present in the system connecting
the buses „i‟ and „j‟. since in a large power system, each bus is connected only to a fewer
buses through an incident element, (about 6-8), the coefficient matrix, YBUS of such
systems would be highly sparse, i.e., it will have many zero valued elements in it. This is
defined by the sparsity of the matrix, as under:
The percentage sparsity of YBUS, in practice, could be as high as 80-90%, especially
for very large, practical power systems. This sparsity feature of YBUS is extensively used
in reducing the load flow calculations and in minimizing the memory required to store the
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coefficient matrices. This is due to the fact that only the non-zero elements YBUS can be
stored during the computer based implementation of the schemes, by adopting the suitable
optimal storage schemes. While YBUS is thus highly sparse, it‟s inverse, ZBUS, the bus
impedance matrix is not so. It is a FULL matrix, unless the optimal bus ordering schemes
are followed before proceeding for load flow analysis.
3.4 THE LOAD FLOW PROBLEM
Here, the analysis is restricted to a balanced three-phase power system, so that the analysis
can be carried out on a single phase basis. The per unit quantities are used for all
quantities. The first step in the analysis is the formulation of suitable equations for the
power flows in the system. The power system is a large interconnected system, where
various buses are connected by transmission lines. At any bus, complex power is injected
into the bus by the generators and complex power is drawn by the loads. Of course at any
bus, either one of them may not be present. The power is transported from one bus to other
via the transmission lines. At any bus i, the complex power Si (injected), shown in figure
1, is defined as
where Si = net complex power injected into bus i, SGi = complex power injected by the
generator at bus i, and SDi = complex power drawn by the load at bus i. According to
conservation of complex power, at any bus i, the complex power injected into the bus must
be equal to the sum of complex power flows out of the bus via the transmission lines.
Hence,
Si = _Sij " i = 1, 2, ………..n
(3)
where Sij is the sum over all lines connected to the bus and n is the number of buses in the
system (excluding the ground). The bus current injected at the bus-i is defined as
Ii = IGi – IDi " i = 1, 2, ………..n (4)
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where IGi is the current injected by the generator at the bus and IDi is the current drawn
by the load (demand) at that bus. In the bus frame of reference
IBUS = YBUS VBUS
(5)
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Equations (9)-(10) and (13)-(14) are the „power flow equations‟ or the „load flow
equations‟ in two alternative forms, corresponding to the n-bus system, where each bus-i is
characterized by four variables, Pi, Qi, |Vi|, and di. Thus a total of 4n variables are
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involved in these equations. The load flow equations can be solved for any 2n unknowns,
if the other 2n variables are specified. This establishes the need for classification of buses
of the system for load flow analysis into: PV bus, PQ bus, etc.
3.4 DATA FOR LOAD FLOW
Irrespective of the method used for the solution, the data required is common for any load
flow. All data is normally in pu. The bus admittance matrix is formulated from these data.
The various data required are as under:
System data: It includes: number of buses-n, number of PV buses, number of loads,
number of transmission lines, number of transformers, number of shunt elements, the slack
bus number, voltage magnitude of slack bus (angle is generally taken as 0o), tolerance
limit, base MVA, and maximum permissible number of iterations.
Generator bus data: For every PV bus i, the data required includes the bus number,
active power generation PGi, the specified voltage magnitude i sp V , , minimum reactive
power limit Qi,min, and maximum reactive power limit Qi,max.
Load data: For all loads the data required includes the the bus number, active power
demand PDi, and the reactive power demand QDi.
Transmission line data: For every transmission line connected between buses i and k the
data includes the starting bus number i, ending bus number k,.resistance of the line,
reactance of the line and the half line charging admittance.
Transformer data:
For every transformer connected between buses i and k the data to be given includes: the
starting bus number i, ending bus number k, resistance of the transformer, reactance of the
transformer, and the off nominal turns-ratio a.
Shunt element data: The data needed for the shunt element includes the bus number
where element is connected, and the shunt admittance (Gsh + j Bsh).
GAUSS – SEIDEL (GS) METHOD
The GS method is an iterative algorithm for solving non linear algebraic equations. An
initial solution vector is assumed, chosen from past experiences, statistical data or from
practical considerations. At every subsequent iteration, the solution is updated till
convergence is reached. The GS method applied to power flow problem is as discussed
below.
Case (a): Systems with PQ buses only:
Initially assume all buses to be PQ type buses, except the slack bus. This means that (n–1)
complex bus voltages have to be determined. For ease of programming, the slack bus is
generally numbered as bus-1. PV buses are numbered in sequence and PQ buses are
ordered next in sequence. This makes programming easier, compared to random ordering
of buses. Consider the expression for the complex power at bus-i, given from (7), as:
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Equation (17) is an implicit equation since the unknown variable, appears on both sides of
the equation. Hence, it needs to be solved by an iterative technique. Starting from an initial
estimate of all bus voltages, in the RHS of (17) the most recent values of the bus voltages
is substituted. One iteration of the method involves computation of all the bus voltages. In
Gauss–Seidel method, the value of the updated voltages are used in the computation of
subsequent voltages in the same iteration, thus speeding up convergence. Iterations are
carried out till the magnitudes of all bus voltages do not change by more than the tolerance
value. Thus the algorithm for GS method is as under:
3.5 Algorithm for GS method
1. Prepare data for the given system as required.
2. Formulate the bus admittance matrix YBUS. This is generally done by the rule of
inspection.
3. Assume initial voltages for all buses, 2,3,…n. In practical power systems, the magnitude
of the bus voltages is close to 1.0 p.u. Hence, the complex bus voltages at all (n-1) buses
(except slack bus) are taken to be 1.0 0
0
. This is normally refered as the flat start solution.
4. Update the voltages. In any (k +1)st iteration, from (17) the voltages are given by
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Here note that when computation is carried out for bus-i, updated values are already
available for buses 2,3….(i-1) in the current (k+1)st iteration. Hence these values are used.
For buses (i+1)…..n, values from previous, kth iteration are used.
Where,e is the tolerance value. Generally it is customary to use a value of 0.0001 pu.
Compute slack bus power after voltages have converged using (15) [assuming bus 1 is
slack bus].
7. Compute all line flows.
8. The complex power loss in the line is given by Sik + Ski. The total loss in the system is
calculated by summing the loss over all the lines.
Case (b): Systems with PV buses also present:
At PV buses, the magnitude of voltage and not the reactive power is specified. Hence it is
needed to first make an estimate of Qi to be used in (18). From (15) we have
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Case (c): Systems with PV buses with reactive power generation limits specified:
In the previous algorithm if the Q limit at the voltage controlled bus is violated during any
iteration, i.e (k +1) i Q computed using (21) is either less than Qi, min or greater than
Qi,max, it means that the voltage cannot be maintained at the specified value due to lack
of reactive power support. This bus is then treated as a PQ bus in the (k+1)st iteration and
the voltage is calculated with the value of Qi set as follows:
If in the subsequent iteration, if Qi falls within the limits, then the bus can be switched
back to PV status.
Acceleration of convergence
It is found that in GS method of load flow, the number of iterations increase with increase
in the size of the system. The number of iterations required can be reduced if the
correction in voltage at each bus is accelerated, by multiplying with a constant α, called
the acceleration factor. In the (k+1)st iteration we can let
where is a real number. When =1, the value of (k +1) is the computed value. If 1<
<2 then the value computed is extrapolated. Generally _ is taken between 1.2 to 1.6, for
GS load flow procedure. At PQ buses (pure load buses) if the voltage magnitude violates
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the limit, it simply means that the specified reactive power demand cannot be supplied,
with the voltage maintained within acceptable limits.
3.6 Examples on GS load flow analysis:
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Since the difference in the voltage magnitudes is less than 10-6 pu, the iterations can be
stopped. To compute line flow
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The total loss in the line is given by S12 + S21 = j 0.133329 pu Obviously, it is observed
that there is no real power loss, since the line has no resistance.
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Example-2:
For the power system shown in fig. below, with the data as given in tables below, obtain
the bus voltages at the end of first iteration, by applying GS method.
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Example-3:
Obtain the load flow solution at the end of first iteration of the system with data as given
below. The solution is to be obtained for the following cases
(i) All buses except bus 1 are PQ Buses
(ii) Bus 2 is a PV bus whose voltage magnitude is specified as 1.04 pu
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(iii) Bus 2 is PV bus, with voltage magnitude specified as 1.04 and 0.25_Q2_1.0 pu.
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Limitations of GS load flow analysis
GS method is very useful for very small systems. It is easily adoptable, it can be
generalized and it is very efficient for systems having less number of buses. However, GS
LFA fails to converge in systems with one or more of the features as under:
• Systems having large number of radial lines
• Systems with short and long lines terminating on the same bus
• Systems having negative values of transfer admittances
• Systems with heavily loaded lines, etc.
GS method successfully converges in the absence of the above problems. However,
convergence also depends on various other set of factors such as: selection of slack bus,
initial solution, acceleration factor, tolerance limit, level of accuracy of results needed,
type and quality of computer/ software used, etc.
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3.7
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Here, the matrix [J] is called the Jacobian matrix. The vector of unknown variables is
updated using (30). The process is continued till the difference between two successive
iterations is less than the tolerance value.
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3. 8
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FINAL WORD
In this chapter, the load flow problem, also called as the power flow problem, has been
considered in detail. The load flow solution gives the complex voltages at all the buses and
the complex power flows in the lines. Though, algorithms are available using the
impedance form of the equations, the sparsity of the bus admittance matrix and the ease of
building the bus admittance matrix, have made algorithms using the admittance form of
equations more popular. The most popular methods are the Gauss-Seidel method, the
Newton-Raphson method and the Fast Decoupled Load Flow method. These methods
have been discussed in detail with illustrative examples. In smaller systems, the ease of
programming and the memory requirements, make GS method attractive. However, the
computation time increases with increase in the size of the system. Hence, in large systems
NR and FDLF methods are more popular. There is a trade off between various
requirements like speed, storage, reliability, computation time, convergence characteristics
etc. No single method has all the desirable features. However, NR method is most popular
because of its versatility, reliability and accuracy.
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UNIT-3&4
1. Using generalized algorithm expressions for each case of analysis,
explain the load flow studies procedure, as per the G-S method for
power system having PQ and PV buses, with reactive power
generations constraints.
2. Derive the expression in polar form for the typical diagonal
elements of the sub matrices of the Jacobian in NR method of load
flow analysis.
3. Compare NR and GS method for load flow analysis procedure in
respect of the following i) Time per iteration ii) total solution time
iii) acceleration factor iv)number of iterations
4. Explain briefly fast decoupled load flow solution method for
solving the non linear load flow equations.
5. Draw the flow chart of NR method for load flow analysis.
6. Explain the representation of transformer with fixed tap changing
during the load flow studies
7. What are the assumptions made in fast decoupled load flow
method? Explain the algorithm briefly, through a flow chart.
8. Explain the NR load flow method? Explain the algorithm briefly,
through a flow chart.
9. What is load flow analysis? What is the data required to conduct
load flow analysis? Explain how buses are classified to carry out
load flow analysis in power system. What is the significance of
slack bus.
10. The following is the system data for a load flow analysis. The line
data are shown and also real and reactive powers data is given.
Determine the voltage at the end of first iteration using GS
method. Take α=1.6
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