NEWTON RAPHSON’S METHOD

The key idea behind Newton-Raphson is to use sequential linearization. Newton’s method is a successive approximation procedure based on an initial estimate of the unknown and the use of Taylor series expansion. Because of its quadratic convergence, Newton’s method is mathematically superior to the Gauss-Seidel model and is less prone to divergence with ill-conditioned problems.

Consider the solution of a one-dimensional equation given by

f(x) = c

If x(0) is an initial estimate of the solution, and ∆x(0) is a small deviation from the correct solution, we must have

Expanding the left- hand side of the above equation in Taylor’s series about x(0) yields

Assuming the error ∆x(0) is very small , the higher – order terms can neglected which results in

Where

Adding ∆x(0) to the initial estimate will result in the second approximation

Successive use of this procedure leads to Newton-Raphson algorithm

It can be rearranged as

Where

1. Formulate the YBUS

2. Assume initial voltages as follows:

(at all PV buses)

( at all PQ buses)3. At (r+1)st iteration, calculate Pi

(r+1) at all the PV and PQ buses and Qi(r+1) at all the PQ

buses, using voltages from previous iteration, Vi(r). The formulae to be used are

4. Calculate the power mismatches (power residues)

5. Calculate the Jacobian [J(r)] using Vi(r) and its elements spread over H, N, M, Lsub-

matrices using the relations already derived.

6. Compute

7. Update the variables as follows:

8. Go to step 3 and iterate till the power mismatches are within acceptable tolerance.

ADVANTAGES AND DISADVANTAGES OF NEWTON RAPHSON’S METHOD

Advantages:

• Faster,

• more reliable and

• results are accurate,

• require less number of iterations;

Disadvantages:

• Program is more complex,

• memory is more complex.

GAUSS-SEIDEL METHOD

The Gauss-Seidel method is also known as the method of successive displacements

Consider a non – linear equation given by

F(x) = 0

The above function can be rearranged and written as

x = g(x)

If x(k) is an initial estimate of the variable x, the following iterative sequence is formed.

x(k+1) = g(x(k))

A solution is obtained when the difference between the absolute value of the successive iteration is less than s specified accuracy, i.e.,

|x(k+1) – x(k)| ≤ ϵ

Where ϵ is the desired accuracy.

GAUSS- SEIDEL METHOD AS A SOLUTION TO POWER FLOW

ALGORITHM FOR G.S METHOD1. 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 < 00. This is normally referred to as the flat start solution.4. Update the voltages. In any (k+1)st iteration, the voltages are given by

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.5. Continue iterations till

Where, e is the tolerance value. Generally it is customary to use a value of 0.0001 pu.6. Compute slack bus power after voltages have converged [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.

ADVANTAGES AND DISADVANTAGES OF GAUSS SIEDEL METHOD

Advantages:

• Each iteration is relatively fast

• Calculations are simple and so the programming task is less.

• The memory requirement is less.

• Useful for small systems;

Disadvantages:

• Need to program using complex numbers.

• Requires large no. of iterations to reach convergence .

• Not suitable for large systems because it has tendency to fail to find solutions .

• Convergence time increases with size of the system

• Tends to diverge on cases with negative branch reactances

DECOUPLED SYSTEMS

Strategy-1(i) Calculate ∆P(r), ∆Q(r), J(r)

(ii) Compute

(iii) Update d and |V|(iv) Go to step (i) and iterate till convergence is reached

Strategy-2

1. Compute ∆P(r) and Sub-matrix H(r). Find2. Update d using d(r+1) = d(r) + ∆d(r)

3. Use d(r+1) to calculate ∆Q(r) and L(r)

4. Compute

5. Update

6. Go to step 1 and iterate until convergence is reached.

In the first strategy, the variables are solved simultaneously. In the second strategy the iteration is conducted by first solving for ∆d and using updated values of d to calculate ∆|V|. Hence, the second strategy results in faster convergence, compared to the first strategy.

FAST DECOUPLED METHOD

Power system transmission lines have a very high X/R ratio. For such a system, real power changes are less sensitive to changes in the voltage magnitude and are most sensitive to changes in the phase angle. Similarly, reactive power is less sensitive to changes in angle and are mainly dependent on changes in voltage magnitude. Therefore it is reasonable to set elements J2 an J3 of the Jacobian matrix to zero i.e

The equations above show that the matrix equation is separated into two decoupled equations requiring considerably fast time to solve

In a fast decoupled power flow algorithm, the successive voltage magnitude and phase angle changes are

This method ids very useful in contingency analysis where numerous outages are to be simulated or a power flow soluyion is required for the output.

Advantages

• Computation time per iteration is less

• The number of iterations are does not dependent of the size of the system

• Less memory requirements than N.R. method.

Disadvantage

• Require more number of iterations than N.R method.

COMPARISON OF LOAD FLOW METHODS