Power Systems : Load Flow Analysis

By Aman Agrawal|Updated : July 6th, 2021

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Load flow studies deals with the study of the real and reactive power flow, and the magnitude and phase angle of the voltage at each bus of a given transmission system.

In load flow analysis, it is possible to define the bus power in terms of generated power, load power, and transmitted power at a given bus.

For example, the bus power (i.e., the net power) of the ith bus of an ‘n’ bus power system can be expressed as

Si = Pi + jQi = (PGi – PLi) + j(QGi –  QLi)


Si = three-phase complex bus power at ith bus

Pi = three-phase real bus power at ith bus

Qi = three-phase reactive bus power at ith bus

PGi = three-phase real generated power flowing into ith bus

PLi = three-phase real load power flowing out of ith bus 

QGi = three-phase reactive generated power flowing into ith bus

QLi = three-phase reactive load power flowing out of ith bus

In load flow studies, the basic assumption is that the given power system is a balanced three-phase system operating in its steady state with a constant frequency.

Therefore, the system can be represented by its single-phase positive sequence network with a lumped series and shunt branches.

The power flow problem can be solved either by using the bus admittance matrix (YBUS) or the bus impedance matrix (ZBUS) representation of the given network.

Thus, if the bus voltages are known, the bus currents is expressed as

[Ibus] = [Ybus][Vbus]

or in its inverse form,

[Vbus] = [Ybus]–1[Ibus] = [Zbus][Ibus]


Classification of Bus:

Bus Type

Known Quantities

Unknown Quantities


|V| = 1.0, θ = 0

P, Q

Generator (PV bus)

P, |V|

Q, θ

Load (PQ bus)

P, Q

|V|, θ

Each bus of a network has four variable quantities associated with it, i.e., the real and reactive power, the (line-to-ground) voltage magnitude, and voltage phase angle. Any two of the four may be the independent variables and are specified, whereas the other two remain to be determined.

Sign of Real & Reactive Powers:

The lagging reactive power is a positive reactive power due to the inductive current and the leading reactive power is a negative power due to the capacitive current and the positive bus current is in the direction that flows toward the bus. Since the generator current flows toward the bus and the load current flows away from the bus, the sign of power is positive for the generator bus and negative for the load bus.

Therefore, the following observations can be made:

  1. The real and reactive powers associated with the inductive load bus (i.e., the lagging power factor load bus) are both negative.
  2. The real and reactive powers associated with the capacitive load bus (i.e., the leading power factor load bus) are negative and positive, respectively.
  3. The real and reactive powers associated with the inductive generator bus (i.e., the bus with a generator operating in lagging power factor mode) are both positive.
  4. The real and reactive powers associated with the capacitive generator bus (i.e., the bus with a generator operating in leading power factor mode) are positive and negative, respectively.
  5. The reactive power of a shunt capacitive compensation located at a bus is positive.

Since transmission losses in a given system are associated with the bus voltage profile, until a solution is obtained, the total power generation requirement cannot be determined. Therefore, the generator at the slack bus is used to supply the additional active and reactive power necessary owing to the transmission losses. Thus, at the slack bus, the magnitude and phase angle of the voltage are known, and the real and reactive power generated are the quantities to be determined.

The generator bus is also known as the PV bus, since an overexcited synchronous generator supplies current at a lagging power factor, the reactive power ‘Q’ of a generator is not required to be specified.

The load bus is also known as the PQ bus, because the real and reactive powers are specified at a given load bus.

Note that, the slack bus voltage is set to 1.0 p.u. with a phase angle of 0° (i.e., the slack bus voltage is used as a reference voltage).


Comparison of Load Flow Methods

  • Gauss-Seidel Method and Newton Raphson methods are compared when both use YBUS as the network model. It is experienced that the Gauss-Seidel Method works well when programmed using rectangular coordinates, whereas Newton Raphson requires more memory when rectangular coordinates are used. Hence, polar coordinates are preferred for the Newton Raphson methods.
  • The Gauss-Seidel Method requires the fewest number of arithmetic operations to complete an iteration. This is because of the sparsity of the network matrix and the simplicity of the solution techniques. Consequently, this method requires less time per iteration.
  • With the NR method, the elements of the Jacobian are to be computed in each iteration, so the time is considerably longer. For typical large systems, the time per iteration in the NR method is roughly equivalent to 7 times that of the Gauss-Seidel Method. The time per iteration in both these methods increases almost directly as the number of buses of the network.

  • The rate of convergence of the Gauss-Seidel Method is slow (linear convergence characteristic), requiring a considerably greater number of iterations to obtain a solution than the NR method which has quadratic convergence characteristics and is the best among all methods from the standpoint of convergence.

  • In addition, the number of iterations for the Gauss-Seidel Method increases directly as the number of buses of the network, whereas the number of iterations for the Newton Raphson methods remains practically constant, independent of system size.

  • The Newton Raphson methods needs 3 to 5 iterations to reach an acceptable solution for a large system. In the Gauss-Seidel Method and other methods, convergence is affected by the choice of slack bus and the presence of series capacitor, but the sensitivity of the Newton Raphson methods is minimal to these factors which cause poor convergence.

  • The chief advantage of the Gauss-Seidel Method is the ease of programming and most efficient utilization of core memory. It is, however, restricted in use of small size system because of its doubtful convergence and longer time needed for the solution of large power networks.

  • Thus the Newton Raphson methods is decidedly more suitable than the Gauss-Seidel Method for all but very small systems.

  • The FDLF can be employed in optimization studies and is specially used for accurate information of both real and reactive power for multiple load flow studies, as in contingency evaluation for system security assessment and enhancement analysis
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