This lecture covers load flow analysis in power systems, focusing on the bus impedance matrix, load flow equations, and various calculation methods such as Gauss-Seidel and Newton-Raphson. It emphasizes the creation of the bus admittance matrix and offers guidance on carrying out power flow studies to determine voltage magnitudes and phase angles at each bus. The document includes details on node equations, bus type classification, and examples of load flow calculations to aid in understanding the subject matter.

1. Power System Analysis
Lecture on
Load Flow Analysis
Presenter
Kazi Firoz Ahmed
Senior Assistant Professor, Dept. of EEE
American International University-Bangladesh (AIUB)
Lecture-5

2. Lecture Outcomes
By the end of this lecture, the students should be able to:
1) Identify the bus impedance matrix of a power system model.
2) Deduce the load flow equation for different methods such as
Gauss Seidal, Newton Raphson etc.
3) Solve complex engineering problems related to load flow
analysis using different methods.
4) Distinguish between different methods of load flow analysis.

3. Source Conversion
Equivalence of Sources
A constant emf and series impedance can be replaced by a constant current source and
shunt impedance if the shunt impedance is identical to the series impedance and if the
constant current equal to the value of the emf divided by its series impedance.
Conversely, a constant-current source and shunt impedance can be replaced by a constant
emf and series impedance if the emf equal to the product of the constant current and the
shunt impedance and if the series impedance equals the shunt impedance.

4. Node Equation
When two or more pure elements (R, L, C, or an ideal source of voltage or
current) are connected to each other at their terminals are called nodes or
junctions.
The current relation of a system
can be written as follows:
The node equations for a four buses system can
be written as follows:

5. Node Equation…
The diagonal admittances (Y11, Y22, Y33, and Y44) are called self-admittances at the nodes,
and each equals the sum of all the admittances terminating on the node identified by the
repeated subscripts.
The other admittances are the mutual admittances of the nodes, and each equals the
negative of the sum of all admittances connected directly between the nodes identified by
the double subscripts.
The above equation can be
written as:
YBus is called the bus admittance matrix.

6. Formulation of Admittance Matrix [Y]

7. Formulation of Admittance Matrix [Y]
As an example for a four bus system as shown in Fig. 7.2. The reactance diagram is shown
in Fig. 7.3. The Fig. 7.4 illustrated the equivalent circuit of Fig. 7.3 by replacing all voltage
source by current sources and impedance by admittance.
Circuit of Fig. 7.3 with current sources replacing the equivalent voltage
sources. Values shown are admittances in per unit.

8. Formulation of Admittance Matrix [Y]

9. Formulation of Admittance Matrix [Y]
For node 1 we can write the current equation
using KCL as:
According to this figure:

10. Formulation of Admittance Matrix [Y]
Similarly, for node 2 we can write the current
equation using KCL as:

11. Formulation of Admittance Matrix [Y]

12. Formulation of Admittance Matrix [Y]

13. Formulation of Admittance Matrix [Y]

14. Bus Impedance Matrix [Z]
The bus admittance matrix Ybus matrix can be inverted and called the resultant matrix
the bus impedance matrix Zbus.

15. Homework
Make the admittance matrix , Ybus, for the following networks.

16. Homework
Make the admittance matrix , Ybus, for the following networks.

17. Load/Power Flow Studies
From a load flow study the following information are obtained:
✓ the magnitude (|V|), and phase angle (δ) of voltage at each bus,
✓ the real (P) and reactive (Q) power flowing in each bus.
The purpose of power flow studies is to plan ahead and account for various
hypothetical situations.
The analysis in normal steady-state operation is called a power-flow
study or load-flow study.
The purpose of power flow studies is to plan ahead and account for various
hypothetical situations.
For instance, what if a transmission line within the power system properly supplying
loads must be taken off line for maintenance. Can the remaining lines in the system
handle the required loads without exceeding their rated parameters?

18. Basic Steps of Load Flow Calculation
The simplest way to perform power-flow calculations is by iteration:
1. Create a bus admittance matrix Ybus for the power system;
2. Make an initial estimate for the voltages at each bus in the system;
3. Update the voltage estimate for each bus (one at a time), based on the
estimates for the voltages
and power flows at every other bus and the values of the bus admittance
matrix:
4. Repeat this process to make the voltages at each bus approaching the
correct answers closer
and closer

19. Basic Steps of Load Flow Calculation
The equations used to update the estimates differ for different types of
busses.
At each node or bus two of the variables are specified as input and two
are unknowns
Voltage magnitude: │V│
Phase angle: δ
Net real power injected: P
Net reactive power injected: Q

20. Bus Type Classification
Load / PQ
Bus
Generator / PV
Bus
Slack / Swing/
Infinite / Vδ
Bus

21. Bus Type Classification
Load / PQ
Bus
✓ real and reactive power are specified, and for which the
bus voltage will be calculated.
✓ Power supplied > positive
✓ Power consumed > negative
✓ All busses having no generators are load busses.

22. Bus Type Classification
Generator / PV
Bus
✓ the voltage is kept constant by adjusting the field current
of a synchronous generator on the bus.
✓ we can control and specify the magnitude of the bus
voltage and real power supplied.

23. Bus Type Classification
Slack / Swing/ Infinite / Vδ
Bus
✓ A special generator bus serving as the reference bus for the
power system. Its voltage is assumed to be fixed in both magnitude
and phase (for instance, 1∠0˚ pu).
✓ The real and reactive powers are uncontrolled: the bus supplies
whatever real or reactive power is necessary to make the power
flows in the system balance.
✓ The controls on the swing generator will be set up to maintain a
constant voltage and frequency, allowing P and Q to increase or
decrease as loads change.

24. Load/Power Flow Studies

25. Load Flow Calculations

26. Load Flow Calculations
The above two equations of real and reactive power constitute the polar form of the
power-flow equations; they provide calculated values for the net real power Pk and
reactive power Qk entering the network at typical bus k.

27. Load Flow Calculations

28. Example of Load Flow Calculations
Example 6.9 [Ref. 3, p. 307] Fig, 6.2 shows a single-line diagram of a five bus power
system. Input Data are given in Table 6.1, 6.2 and 6.3. As shown in Table 6.1, bus 1,
to which a generator is connected, is the swing bus. Bus 3, to which a generator and a
load are connected, is a voltage cotrolled (or PV) bus. Buses 2, 4, and 5 are load
buses. Note that the loads at bus 2 and 3 are inductive since Q2= - QL2= - 2.8 and -
QL3= - 0.4 are negative. For each bus k, determine which of the variables Vk, δk, Pk,
and Qk are input data and which are unknowns. Also, compute the elements of the
second row of Ybus.

29. Example of Load Flow Calculations

30. Example of Load Flow Calculations

31. Example of Load Flow Calculations

32. Example of Load Flow Calculations

33. Gauss Seidel Method
The basic procedure of Gauss-Siedel iterative method is
1) Calculate the bus admittance matrix Ybus including the admittances of all
transmission lines, transformers, etc., between busses but exclude the admittances
of the loads or generators themselves.
2) Select a slack bus: one of the busses in the power system, whose voltage will
arbitrarily be assumed as 1.0∠0˚.
3) Select initial estimates for all bus voltages: usually, the voltage at every load bus
assumed as 1.0∠0˚ (flat start) lead to good convergence.
4) Write voltage equations for every other bus in the system. The generic form is:

34. Gauss Seidel Method
The basic procedure of Gauss-Siedel iterative method is
5) Calculate an updated estimate of the voltage at each load bus in succession using
(8.5) except for the slack bus.
6) Compare the differences between the old and new voltage estimates: if the differences
are less than some specified tolerance for all busses, stop. Otherwise, repeat step 5

35. Gauss Seidel Method
Acceleration Factor: the rate of convergence is increased by applying an
acceleration factor to the approx. solution obtained from each iteration. For kth
bus, the acceleration value of the voltage at the (i+1)th iteration is given by,

36. Gauss Seidel Method
7) Confirm that the resulting solution is reasonable: a valid solution typically has bus
voltages, whose phases range in less than 45˚ and the difference between nearby
buses less than about 10˚ and often very small.
At a bus where voltage magnitude rather than reactive power is specified [PV or
generator bus], the real and imaginary components of the voltage for each
iteration are found by first computing a value for the reactive power from Eq.
(8.5)

37. Example of Gauss Seidel Method
Follow the Lecture Sheet # 3
(Load Flow Analysis Lecture Sheet)

38. 38
References
[1] Willaim D. Stevenson, Elements of Power System Analysis, Fouth Edition,
McGraw-Hill International Editions, Civil Engineering Series, McGraw-Hill Inc.
[2] John J. Grainger, William D. Steevnson, Jr., Power System Analysis, McGraw-
Hill Series in Electrical and Conputer Engineering, McGraw-Hill Inc.
[3] J. Duncan Glover, Mulukutla S. Sharma, Thomas J. Overbye, Power System
Analysis and Design, Fouth Edition (India Edition), Course Technology Cengage
Learning
[4] Hadi Saadat, Power System Analysis, Tata McGraw-Hill Publishing Company
Limited [5] I J Nagrath, D P Lothari, Modern Power System Analysis, Second
Edition, Tata McGraw-Hill Publishing Company Liited
[6] V. K. Mehta, Rohit Mehta, Principles of Power System, Multicolor Illustrative
Edition, S. Chand and Company Limited