Advanced Techniques For Power Flow Solution

The purposes of these new techniques are:

· Reduction (decreasing) of running time.

· Reduction of memory (storage) space.

1-Advanced Forms Of Decoupled And Decomposed Power Flow Solution Method

A decoupled and decomposed power flow solution method is derived and discussed. In the derived method, the active power flow model is decomposed into the mutually connected generator active power flow model and the load active power & low model. Also, the difference of the coefficient matrices of the load active and the load reactive power flow models is equal to zero, which enables the same matrix triangulation to be used in solving the load active and load reactive power flow models. In some cases, the total number of iterations may be reduced by dividing the power mismatch vectors by a constant reference bus voltage instead of by the variable voltage vector. For power system with a relatively small number of generators, the proposed method has a greater computational speed, lower storage requirements and better or equal convergence when compared with stott and Alsac's fast decoupled power flow.

Notation:

The well-known fast decoupled power flow :

is taken as a starting point for the derivation whether the power mismatches and voltages and angle corrections for the slack bus are not included in the elements of matrices [B'] and [B''] which do not equal the elements of matrix B, because [B'] does not include shunt reactance and off-nominal, in phase, transformer taps and series resistances, while [B''] neglects the angle-shifting effects of phase shifters.

As a result of the decomposition of the set of all system buses into the set generator (i.e. P, V) buses and the set of load (i.e. P, Q) buses, the model of FDLF can be presented in the decomposition form as :

Because the voltage magnitudes are specified in the generator buses, the vector is equal to zero, and the following equation can be written as

The matrices

are not equal. However, usually they differ only slightly

and this will lead to the following :

After the process of triangulation, the model becomes :

Active power flow model is in the decomposition form, because the vectors

and the triangular factors of B' are decomposed in this model. Active and reactive power flow models are mutually decoupled. It is important to note that the triangular factor D and A are identical to the upper triangular sub factors in the previous figure .Also, it is clear that:

The decomposition active power flow model in :

and the reactive power flow model :

which are mutually decoupled can be written in the following way:

Thus, decomposition of the active power flow model enables the representation of this model in the form of two decomposed, mutually connected sub models, the generator active power flow sub model (2), (3) and the load active power flow sub model (1), (4)

In some cases, for the purpose of obtaining better convergence, the following assumptions may be introduced:

Where the reference bus voltage Vr is constant during the iterations (e.g. Vr = 1.05).

Substituting the previous eq into eq (1) (2) and (5) gives:

The pevious Equations make an iterative decomposed and decoupled power flow model. It is important to note that the equations in this model have a uniquely defined order, which is given by number from 1 to 6.

The decomposition active power flow model gives the possibility of exploiting the different convergence speeds of the generator active and the load active power flow models. The following decomposition active power flow model:

Where A, B, C, D, E and F were defined previously. In the previous model , the reactors

are not decomposed, but the triangular factors of B' are decomposed on sub matrices A, B, C, D, E and F

Until now, little attention has been paid to power flow solution methods with variables expressed in rectangular form. These methods could have better computational time and convergence characteristics than methods in polar form. In general several rectangular decoupled power flows are given, one of them has very good convergence and computation time characteristics, and can be written in the following way:

-The matrix B'''' is taken to equal to the matrix B''' (B''''=B'''). Analogously the following rectangular decoupled and decomposed power flow can be obtained from the previous eqs:

The matrices A1, B1, C1, D1, E1 and F1 are obtained by the triangulation and the decomposition or the matrix B'''.The previous equations make an iterative rectangular decoupled and decomposed power flow model.

2-Modification Of Fast Decoupled Load Flow

For solving power flow problems, the fast decoupled method is probably the most popular because of its efficiency, it's relatively for most power systems is very high but it does have difficulties in convergence for system with high ratio r/x.

The power flow have steadily improved:

1) By fast decoupled which most commonly used because it is computationally the fastest and simple to use however convergence difficulties.

2) On systems with branches that have large r/x ratio.

3) The modification using series and parallel compensation.

This property makes this modification reliable for use in those cases where the r/x ratio are marginal.

In fact, the basic Newton's method is quite insensitive to branch has high r/x ratios. Since the FDM is still the best suited method for most systems, a simple modification of the same program to handle high r/x ratios would be easier to use for these exceptional systems. For this method, the compare will be between modification proposed and fast decoupled method

Short Comparison Of The Power Flow Solution Methods

Acceleration ,Speed Computation time (Running rime),Memory and Convergence ]

The Gauss-Seidel method is the oldest of the power flow solution methods. It's simple, reliable and usually tolerant of poor voltage and reactive power conditions. In addition, it has low computer memory requirements. This method has a slow coverage rate.

The Newton-Raphson method has a good convergence rate. The computation-time increases only linearly with system size. This method has convergence problems when the initial voltages are significantly different from their true values. Once the voltage solution is near the true solution, the convergence is very rapid. Also it is suited for applications involving large systems requiring very accurate solutions.

The Fast decoupled load flow methods are basically approximations to the NR method. The approximations made in the FDLF methods generally results in a small increase in the no. of iterations. However, the computation effort is significantly reduced since the Jacobin doesn't have to be replaced in each iteration. In addition, the computer memory requirements are reduced. The convergence rate of the FDLF methods is linear as compared to quadratic rate of NR methods.

Ahmad Atiia

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Advanced Techniques For Power Flow Solution