Investigation of schemes for incorporating generator Q limits in the fast decoupled load flow method

Abstract

Fast Decoupled Load Flow (FDLF) is a very popular and widely used power flow analysis method because of its simplicity and efficiency. Even though the basic FDLF algorithm is well investigated, the same is not true in the case of additional schemes/modifications required to obtain adjusted load flow solutions using the FDLF method. Handling generator Q limits is one such important feature needed in any practical load flow method. This paper presents a comprehensive investigation of two classes of schemes intended to handle this aspect i.e. the bus type switching scheme and the sensitivity scheme. We propose two new sensitivity based schemes and assess their performance in comparison with the existing schemes. In addition, a new scheme to avoid the possibility of anomalous solutions encountered while using the conventional schemes is also proposed and evaluated. Results from extensive simulation studies are provided to highlight the strengths and weaknesses of these existing and proposed schemes, especially from the point of view of reliability.

Appendix

1.1 A. Computation of the new sensitivities

Obtaining the new sensitivities requires finding a few elements of the inverse of the matrix $B^{\prime \prime }_{new}$ in (11). Equation (11) is reproduced here as follows:

$$ B^{\prime\prime}_{new}= \left[ \begin{array}{ll} B^{\prime\prime}&C\\ C^{T}&H\\ \end{array} \right], $$

(13)

where the augmented columns of C and H correspond to the PV buses violating their Q limits. The computational problem is to obtain the elements of $B_{new}^{''-1}$ corresponding to the diagonal positions of H (the first approach proposed) or all the elements in the positions of H (the second approach proposed). The U ^T D ⁻¹ U factors of B ^″ are known as they are computed at the beginning of the FDLF iterations. The columns of the factors of $B^{\prime \prime }_{new}$ corresponding to the columns of C can be obtained (one by one) as explained in Section 3, using the computed factors of B ^″. Once computed they can be stored till the final solution is obtained because they are sparse columns and this way the need to compute them repeatedly in every iteration can be avoided.

An alternate approach would be to augment the columns of B ^″ with columns corresponding to all PV buses during the factorisation of B ^″ itself (before the beginning of the iterations) and going through the factorisation process even for the augmented columns. It may be noted that before the beginning of the iteration, the PV buses which violate their limits are not known. Hence, we augment B ^″ with $\hat {C}$ consisting of columns corresponding to all the PV buses. The C matrix in (13) would change from iteration to iteration depending on the PV buses that violate their limits in that iteration but always will be made up of the columns of $\hat {C}$. Denoting the modified elements (due to factorisation) of $\hat {C}$ as $\hat {C^{\prime }}$ in every iteration we build a matrix C ^′ by choosing the columns of $\hat {C^{\prime }}$ corresponding to the buses that violate their limits in that iteration. This way, at the end of the factorisation of B ^″, all the columns of the factors of $B^{\prime \prime }_{new}$ (corresponding to positions of C in various iterations) will be available with only a marginal increase in computational effort. In each iteration one has to compute a matrix H ^′

$$ H^{\prime}=H-C^{\prime T}\times D^{-1}\times C^{\prime}, $$

(14)

where the columns of C ^′ and H correspond to PV buses that violate their limit. This computation in (14) corresponds to a partial Gaussian elimination process of making elements in the rows of C ^T matrix in (13) zero. The computation in (14) is not demanding as C ^′ is a sparse matrix. As the interest here is in the elements of the inverse of H ^′, at this point one can find the triangular factors of H ^′ (exploiting sparsity of H ^′ if it is advantageous).

If only the diagonal elements of the inverse of H ^′ are needed, they can be obtained efficiently using the sparse Z bus approach proposed by Takahashi et al (1973) using the factors of H ^′. If one is implementing the second scheme, (using all the elements of H ^′ inverse) the required changes in the voltages at all violating PV buses can be computed by operating on the column of ΔQ/V using the factors of H ^′ to obtain all ΔVs as a repeat solution of a system of Eq. in (12) with the computed factors of the coefficient matrix H ^′.

1.2 B. Implementation of the proposed modifications for avoiding anomalous solutions

The proposed modifications can be applied to both the conventional adjustment schemes. It is easy to see that these schemes (bus type switching and sensitivity) are different in only how the changes in voltages of violating PV buses are calculated. But the method of identifying the violating buses and choosing the Q specified for these buses for that iteration is common for both. The proposed modifications are intended to change the conventional algorithms, in how the Q specified is chosen for the violating PV buses. Hence, only these critical steps for the proposed modified algorithms are given below. These steps can be incorporated into bus type switching as well as all the sensitivity schemes.

1.2.1 B.1 Modified scheme 1

• Case (1)  The PV bus has been treated as a PV bus in the previous iteration.

  Calculate the Reactive power (Q ^cal) at the bus

  • If this Q ^cal at the bus is within the specified limits, the bus is treated as a PV bus.

  • Else, this bus is switched to a PQ bus with the Q specified for this bus set at the violated limit.

• Case (ii)  The PV bus has been treated as a PQ bus in the previous iteration. In this case, there are two voltages corresponding to this bus at this stage. Its original specified voltage and the computed voltage after treating it as PQ bus. Calculate the reactive power (Q ₁) at the bus with the voltage at this bus assumed to be at the specified value.

  • If this Q ₁ is within limits, consider it as a PV bus with the voltage set at the specified value.

  Otherwise,

  • (***) If the present voltage value is lower than the specified value, the maximum limit of Q is considered as the value of the Q specified value at this bus.

  • (***) If the present voltage value is higher than the specified value, the minimum limit of Q is considered as the value of the Q specified value at this bus.

1.2.2 B.2 Modified scheme 2

This scheme is same as the previous scheme up to the last two steps (shown as ***) . The last two steps (shown as ***) are to be replaced by the following steps.

• Calculate Q ₂ at this bus corresponding to its present modified voltage

• If Q ₂ is outside the specified limits, this bus is considered as a PQ bus with the Q specified at the bus set at this violated limit.

• In case Q ₂ is within limits, this bus is considered as a PQ bus and the value of Q specified value at this bus is chosen as follows:

  • If the present voltage value is lower than the specified value, the maximum limit of Q is considered as the value of the Q specified value at this bus.

  • If the present voltage value is higher than the specified value, the minimum limit of Q is considered as the value of the Q specified value at this bus.