Power-flow study

The load flow calculation is in the electrical industry a method of numerical analysis of power grids. In contrast to the traditional circuit analysis simplified representations of how the single phase diagram or per- unit system (pu) are used with respect to various forms of electrical power as the power factor, active power and apparent power instead of current and voltage. The power grid is analyzed in the normal state ( steady state ). There are several software solutions for load flow calculations.

In addition to the load flow calculation have many software solutions over other methods of analysis, such as the short-circuit analysis and cost-effectiveness analysis. Many programs use linear programming to determine the optimal power flow to find the state with the lowest cost per generated kilowatt.

The great importance of load flow calculation is in the planning of future expansions of energy supply systems, and to identify the optimal operating condition of existing systems. The basic information which are obtained, are voltage magnitude and phase angle of each distribution rail or active and reactive power on each line.

Problem formulation

The aim of load flow calculation is relative to get complete information (voltage / phase angle ) for each busbar load and generator real power. If this information is known, active and reactive power flow in each branch as well as the generator output can be determined analytically. Due to non-linear nature of this problem, numerical methods are used to obtain solutions within acceptable tolerances.

The solution begins with identification of known and unknown variables of the system. These variables depend on the type of distribution rail. A rail without generator is called load rail. Rails with at least one generator generator rails. The exception is a randomly selected track with generator. Such rails are called balance node (English Slack bus).

In the problem-solving is believed that the active power and reactive power PD QD are known for each load bar. For this reason, load rails are called PQ - rails. Generator for rails, it is assumed that the active power PG and generated voltage | V | are known. For the Record node is assumed voltage | V | and phase angle Θ are known. For each load bus voltage and phase angle are unknown and must be calculated; for each generator track the phase angle must be calculated; there are no unknown variables for the balance node. In a system with N tracks and R generators there are unknowns.

To solve for the unknowns, equations must be set up which use no other unknowns. The possible equations use the power balance, which can be placed for each bar with respect to the active and reactive power.

The equation of power balance is:

Here is the power fed into the track i, the active component of the element YBUS is corresponding to the row i and column k, the imaginary part of the element in YBUS is corresponding to the row i and column k and the difference in phase angle between i and k rail.

The reactive power equation is:

Here, the reactive power, which is fed to bar i.

The equations contain the active and reactive power for each load bus and the active power balance for each generator rail. For the generator only, the rail active power balance is set up, because it is assumed that the injected reactive power is known. For the same reason, no equations for the balance nodes are placed.

Solution methods

There are different methods for solving systems of nonlinear equations such as Newton's method. This method begins with the estimation of all unknown variables (voltage and phase angle of the load rails, and phase angle of the generator rails ). Thereafter, a Taylor series are prepared, the result is a linear equation system:

For replacement and equations are set up:

And is a matrix of partial derivatives, known as the Jacobian matrix:

The linearized system of equations is solved by determining the next estimate ( m 1) of voltage magnitude and phase angle based on:

The process is repeated until a stop condition occurs. The stop condition occurs usually when the solution of the equations replacement is within a certain tolerance.

A crude solution of the load flow problem is: