Iterative Approaches for Solving Power Flow Equations, Gauss‐Seidal  Method, Newton‐ Rapshon Methods and their Comparison

Gauss Seidal methods:

The complexity of obtaining a formal solution for power flow in a power system arises because of the differences in the type of data specined for the different kinds of buses. Although the formulation of sufficient equations to match the number of unknown state variables is not diHicult, as we have seen, the closed form of solution is not practical. Digital solutions of the power-flow problems follow an iterative process by assigning estimated values to the unknown bus voltages and by calculating a new value for each bus voltage from the estimated values at the other buses and the real and reactive power specifieed. A new set of values for the voltage at each bus is thus obtained and used to calculate still another set of bus voltages. Each calculation of a new set of voltages if called an iteration. The iterative process is repeated until the changes at each bus are less than a specified minimum value.

Stepwise Solution of Gauss Seidal method:

1)Read the Parameters to specified variables:

• Y_bus
• Number of buses in system;N
• Injected Real & reactive power (P&Q) at load buses.
• Injected real power (P) ,voltage magnitude V and maximum and minimum reactive power limit generator.
• Specified voltage magnitude V and reference power angle ð at reference buses.

2) Assume Suitable value voltage magnitude V and ð at load buses.

3)Start iteration count k=0

4)Start bus count p=1

5)Check is it a reference bus?

YES:Goto step 9

NO:Goto next step

6)Check is it a generator bus?

NO:Goto Next step(It is neither reference bus nor generator bus)

YES:Calculate:

We have,

Q_gp=Q_p+Q_dp

Check Q_gp >Q_gp min?

NO: Q_p= (Q_gp min - Q_dp)

Goto next step,treat this as load bus.

YES: Q_gp< Q_gp max?

NO: Q_p =Q_gp max - Q_dp

Goto next step,treat this as load bus.

YES:Confirmed that this is generator bus.

7)Solve the voltage equation;

Here in equation, Pth bus is treated as ith bus.

8)Calculate the change in voltage

That can be calculated by subtracting the fresher value to new one.

9)Advance bus count; p=p+1;

10)Check; (P <N+1)?

YES:Goto Step 5.

NO:Goto next step.

11)Determine the largest absolute value of change in voltage,

i.e.

Check; is it less than error that we can accept?

YES:Goto next step.

NO:k=k+1; GOto step 4.

12)Evaluate line flows & slack bus Pp & Qp.

13)Stop

14)End.

Newton Rapshons methods:

Newton Rapshon's method is mathematically superior to the Gauss Seidal method and less prone to divergence problem.For the large power system,it is more efficient and practical.The number of iteration required to obtain a solution is independent of system size,but more functional evaluations are required at each iteration.Taylor's series expansion for a function of two or more variables is the basis for the N ewton-Raphson method of solving the power-flow problem. Our study of the method begins by a discussion of the solution of a problem involving only two equations and two variables. Then, we see how to extend the analysis to the solution of power-flow equations.

The steps to solve the power system problem by Newton rapshons methosd is described below:

1)

Read the Parameters to specified variables:

• Y_bus
• Number of buses in system;n
• Injected Real & reactive power (P&Q) at load buses.
• Injected real power (P) ,voltage magnitude V and maximum and minimum reactive power limit generator.
• Specified voltage magnitude V and reference power angle ð at reference buses.

2)Assume Suitable value voltage magnitude V and ð at load buses.

3)Start iteration count 0;

4)Start bus count i=1

5)Compute P_cal at all buses except slack bus & compute Q_cal at load buses.

Calculate the error vector;

6)Compute the jacobian matrix.

Or, E=JC

Where, J is jacobian matrix.

6)compute the correction vectors;

7)Compute the updated value of ð & V.

8)compute Q_i at each generator buses.

9)Compare Q_gp as,

Q_gp=Q_i + Q_dp

Check, Q_gp >Q_gp min?

NO: Q_p= (Q_gp min - Q_dp)

Goto next step,treat this as load bus.

YES: Q_gp< Q_gp max?

NO: Q_p =Q_gp max - Q_dp

Goto next step,treat this as load bus.

YES:Confirmed that this is generator bus.

10) Check the convergence,

YES:Next step

NO:increase iteration count & goto step 4.

11)Compute P & Q at slack bus.

12)Compute network looses(P_loss & Q_loss).Compute line flows if required.

13)Print result

14)End

Q. Analyse the error and jacobian matrix when

I) Total number of buses=3,number of load buses=2

II)Total number of buses=3,number of load buses=1,number of generator bus=1

Comparison between Gauss Seidal and Newton Rapshon methods: