Decoupled Load Flow Methods (DLF):

Real powers and bus voltage angles, as well as reactive powers and voltage magnitudes, exhibit a strong interdependence, which is a crucial feature of any practical electric power transmission system operating in steady state. The Fast Decoupled Load Flow method (DLF), in which P-δ and Q-V problems are solved separately, were developed as a result of this interesting property of weak coupling between P-δ and Q-V variables.

Decoupled Newton Methods

The weak coupling mentioned above is represented by half of the Jacobian matrix’s elements, which can be ignored in any conventional Newton method. Any such approximation transforms the real quadratic convergence into a geometric one, but there are computational advantages to make up for this. In the literature, a lot of decoupled algorithms have been developed. However, we only present the most widely used decoupled Newton version here.
Submatrices [N] and [J] are the components in Eq. (6.67) that should be ignored.

The resulting decoupled linear Newton equations become

where it can be shown that

and

Equations (6.76) and (6.77) can be constructed and solved simultaneously with each other at each iteration, updating the [H] and [L] matrices in each iteration using Eqs (6.78) to (6.80). A better approach is to conduct each iteration by first solving Eq. (6.76) for Δδ, and use the updated δ in constructing and then solving Eq. (6.77) for Δ|V|. This will result in faster convergence than in the simultaneous mode.

The main benefit of the Decoupled Load Flow Methods (DLF) over the NR method is the Jacobean’s need for less memory to store. Speed-wise, there is not much of an advantage because the DLF takes almost as long per iteration as the NR method does, and because of approximation, it always requires more iterations to converge.

Fast Decoupled Load Flow (FDLF):

Using the DLF model mentioned in the previous subsection, additional physically justifiable simplifications may be made to gain some speed advantage without suffering significantly from accuracy loss in the solution. This work culminated in 1974 with B. Stott’s invention of the Fast Decoupled Load Flow (FDLF) method. The following are the presumptions that are true for regular power system operation:

and

With these assumptions, the entries of the [H] and [L] submatrices will become considerably simplified and are given by

and

Matrices [H] and [L] are square matrics with dimension (n_PQ+n_PV) and n_PQ respectively.
Equations (6.76) and (6.77) can now be written as

where B′_ij, B″_ij are elements of [- B] matrix.

The Fast Decoupled Load Flow FDLF algorithm is further decoupled and logically simplified by:

1. Excluding from [B′] the representation of the network components, such as shunt reactances and transformer off-nominal in-phase taps, that primarily influence reactive power flows;
2. Neglecting from [B”] the angle shifting effects of phase shifters;
3. Dividing each of the Eqs. (6.84) and (6.85) by |V_i| and setting |V_j| = 1 pu in the equations;
4. calculating the components of [B′], which becomes the dc approximation power flow matrix, without taking into account series resistance.

With the above modifications, the resultant simplified FDLF equations become

Both [B’] and [B”] are real, sparse, and have the [H] and [L] structures, respectively, in Eqs. (6.86) and (6.87). They are constant and only need to be inverted once at the start of the study because they only contain admittances. Both [B’] and [B”] are always symmetrical in the absence of phase shifters, and their constant sparse upper triangular factors are calculated and stored just once at the start of the solution.
Equations (6.86) and (6.87) are solved alternatively always employing the most recent voltage values. One iteration implies one solution for [Δδ], to update [δ] and then one solution for [Δ|V|] to update [|V|] to be called 1-δ and 1-V iteration. Separate convergence tests are applied for the real and reactive power mismatches as follows:

where ε_P and ε_Q are the tolerances.

A flow chart giving FDLF algorithm is presented in Fig. 6.13.