# Exploiting the S-Iteration Process for Solving Power Flow Problems: Novel Algorithms and Comprehensive Analysis

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- Although the Jacobian updated mechanism proposed in [31] allows overcoming the slow-convergence issues in heavy loading systems, the whole iterative procedure remains linear. Consequently, many iterations are normally employed to achieve a feasible solution.
- The overall performance of the Newton-SIP technique studied in [31] strongly depends on the value of the parameters involved in the iterative procedure (s-parameters).

## 2. Newton-SIP Methods Applied to PF Analysis

#### 2.1. Background

#### 2.2. Newton-SIP Methods (Type 1)

_{0}to the methodology whose generic ${k}^{th}$ iteration for solving the PF is carried out as follows:

_{0}has been proposed in [32]. In this case, the generic ${k}^{th}$ iteration of SIP2-J

_{0}for solving the PF is given by:

_{0}and SIP2-J

_{0}are defined by only one s-parameter (namely $\alpha $). The main difference between SIP1-J

_{0}and SIP2-J

_{0}lies in the latter requiring two Jacobian evaluations. It is noteworthy that the iterative algorithms defined by (4) and (5) only evaluate the Jacobian at ${\mathit{x}}^{\left(0\right)}$ and ${\mathit{y}}^{\left(0\right)}$; hence, they are a priori more efficient than NR.

#### 2.3. Newton-SIP Methods (Type 2)

_{0}to that method whose generic ${k}^{th}$ iteration for solving the PF is carried out as follows:

_{0}proposed in [33] is carried out at its generic ${k}^{th}$ iteration for solving the PF problem as follows:

_{0}and SIP4-J

_{0}lies in the total number of Jacobian evaluations. While the latter requires three Jacobian evaluations, SIP3-J

_{0}only requires one Jacobian evaluation. An important difference between the methodologies proposed in [32] and those developed in [33] is the number of s-parameters involved. While SIP1-J

_{0}and SIP2-J

_{0}are defined by only one s-parameter ($\alpha $), SIP3-J

_{0}and SIP4-J

_{0}are characterized by a pair of s-parameters ($\alpha $, $\theta $). Finally, all studied Newton-SIP methods only evaluate the Jacobian matrix at the first iteration (in just one or various points); in this paper, we have considered alternative procedures in which the Jacobian matrices are updated each iteration (as in the standard NR). These alternative techniques have been called SIP1-J, SIP2-J, SIP3-J, and SIP4-J for the SIP1-J

_{0}, SIP2-J

_{0}, SIP3-J

_{0}, and SIP4-J

_{0}, respectively. With the aim to summarize, Table 1 collects the main characteristics of the studied PF solution techniques.

## 3. Convergence Analysis of Studied Newton-SIP Methods

**Theorem**

**1.**

**Proof**.

**Theorem**

**2.**

**Proof**.

**Theorem**

**3.**

**Proof**.

**Theorem**

**4.**

**Proof**.

_{0}, SIP2-J

_{0}, SIP3-J

_{0}, and SIP4-J

_{0}as a function of the iteration number. In this figure, the convergence rate of the conventional NR has been also included for comparison. From this figure, it can be deduced that SIP4-J

_{0}and SIP1-J

_{0}show the highest and the lowest convergence rate, respectively, while both SIP2-J

_{0}and SIP3-J

_{0}have the same convergence order. Anyway, for these techniques, the convergence rate is linear after the first iteration. Therefore, their convergence orders are always less than two, being so overcome by NR.

_{0}SIP2-J

_{0}, SIP3-J

_{0}, and SIP4-J

_{0}. To complete the section, Table 2 summarizes the convergence analysis of the considered techniques and the NR. As commented, the studied techniques achieve their maximum convergence rate when the s-parameters are equal to 1.

## 4. Comparison of the Efficiency of Different Iterative Algorithms

**Theorem**

**5.**

## 5. Numerical Experiments

#### 5.1. Well-Conditioned Cases

_{0}and SIP3-J

_{0}are the fastest methods, which is strongly linked with the number of factorizations required (one should note that the LU decomposition is the heaviest part of any PF calculation [13]). Among all the studied Newton-SIP techniques, only SIP2-J and SIP4-J are occasionally slower than NR. These results may look not coherent with the analysis performed in Section 4; however, Figure 4c provides a clear explanation about this issue. In this figure, it can be appreciated that these two techniques frequently required more factorizations than NR, which is reflected in a higher computational burden and therefore less competitive execution times. Regarding the total iterations required to attain the solution, the results are expected since the highest convergence rate has the lowest number of total iterations required for achieving the solution. Regarding those algorithms with linear convergence (Equations (4)–(7)), the following relations normally hold:

_{0}and SIP3-J

_{0}, since it can be observed in Figure 5 that clearly $i{t}_{\mathrm{SIP}3-{\mathrm{J}}_{0}}<i{t}_{\mathrm{SIP}2-{\mathrm{J}}_{0}}$.

#### 5.2. Ill-Conditioned Cases

_{0}, SIP2-J

_{0}, SIP3-J

_{0}, and SIP4-J

_{0,}since the latter normally showed wider convergence areas. There are some remarkable cases; for example, SIP1-J did not converge in the case3012wp and case3375wp; on the other hand, SIP1-J

_{0}and SIP3-J

_{0}frequently converged, regardless of the value of parameters. Finally, SIP4-J and SIP4-J

_{0}look very sensitive to the values of s-parameters. Their convergence rate precisely explains the superior robustness features of linear methods. In [43], it is said that the methods with high convergence rates normally show narrow Regions of Attraction; in other words, the highest convergence rate has the most sensitivity with respect to the initial guess. This fact can also be appreciated for other PF techniques such as [10,14], which introduce a discrete step size to reduce the convergence rate and obtain robust techniques properly.

_{0}, SIP2-J

_{0}, and SIP3-J

_{0}are the fastest techniques, while SIP1-J, SIP2-J, SIP3-J, and SIP4-J typically reached the solution employing less iterations. It is worth mentioning that SIP3-J is occasionally faster than SIP4-J

_{0}in the case13659pegase. This is because these two methods require the same number of factorizations in this system; however, SIP4-J

_{0}computes more calculations per iteration. SIP1-J only converged in the case13659pegase.

#### 5.3. Influence of the R/X Ratio

## 6. Conclusions and Future Works

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Consideration of Composite Loads

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**Figure 2.**Convergence degree of the error vector ${\mathit{e}}^{\left(0\right)}$ at different iteration counters.

**Figure 4.**Comparison of the results obtained in well-conditioned systems. (

**a**) Execution time (ms), (

**b**) total iterations, and (

**c**) total factorizations.

**Figure 5.**Comparison of the results obtained in well-conditioned systems in a limit load scenario. (

**a**) Execution time (ms), (

**b**) total iterations number and (

**c**) total factorizations.

**Figure 6.**(

**a**) Comparison of the total number of iterations of different linear Newton-SIP solvers, taking the s-parameters equal to 1. (

**b**) Considered s-parameters.

**Figure 7.**Convergence (green) and failure areas (red) in the s-parameter space for solving the case3012wp. Bold green areas indicate where the studied technique successfully converged, employing the least number of iterations.

**Figure 8.**Convergence (green) and failure areas (red) in the s-parameter space for solving the case3375wp. Bold green areas indicate where the studied technique successfully converged, employing the least number of iterations.

**Figure 9.**Convergence (green) and failure areas (red) in the s-parameter space for solving the case13659pegase. Bold green areas indicate where the studied technique successfully converged, employing the least number of iterations.

**Figure 10.**Comparison of the results obtained in ill-conditioned systems. (

**a**) Execution time (ms), (

**b**) total iterations number, and (

**c**) total factorizations. S-parameters have been tuned as all studied techniques employed the least number of iterations.

**Figure 11.**Total iterations employed in case3012wp with different solvers considering a flat start and the default starter provided in Matpower.

Method | Jacobian Evaluations | Function Evaluations | S-Parameters |
---|---|---|---|

NR | $K$ | $K$ | -- |

SIP1-J_{0} | 1 | $2\times K$ | $\alpha $ |

SIP1-J | $K$ | $2\times K$ | $\alpha $ |

SIP2-J_{0} | 2 | $2\times K$ | $\alpha $ |

SIP2-J | $2\times K$ | $2\times K$ | $\alpha $ |

SIP3-J_{0} | 1 | $3\times K$ | $\alpha ,\theta $ |

SIP3-J | $K$ | $3\times K$ | $\alpha ,\theta $ |

SIP4-J_{0} | 3 | $3\times K$ | $\alpha ,\theta $ |

SIP4-J | $3\times K$ | $3\times K$ | $\alpha ,\theta $ |

Method | Convergence Rate | |
---|---|---|

Minimum | Maximum | |

NR | 2 | 2 |

SIP1-J_{0} | Linear | Linear |

SIP1-J | 2 | 3 |

SIP2-J_{0} | Linear | Linear |

SIP2-J | 3 | 4 |

SIP3-J_{0} | Linear | Linear |

SIP3-J | 3 | 4 |

SIP4-J_{0} | Linear | Linear |

SIP4-J | 4 | 8 |

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**MDPI and ACS Style**

Tostado-Véliz, M.; Kamel, S.; Taha, I.B.M.; Jurado, F.
Exploiting the S-Iteration Process for Solving Power Flow Problems: Novel Algorithms and Comprehensive Analysis. *Electronics* **2021**, *10*, 3011.
https://doi.org/10.3390/electronics10233011

**AMA Style**

Tostado-Véliz M, Kamel S, Taha IBM, Jurado F.
Exploiting the S-Iteration Process for Solving Power Flow Problems: Novel Algorithms and Comprehensive Analysis. *Electronics*. 2021; 10(23):3011.
https://doi.org/10.3390/electronics10233011

**Chicago/Turabian Style**

Tostado-Véliz, Marcos, Salah Kamel, Ibrahim B. M. Taha, and Francisco Jurado.
2021. "Exploiting the S-Iteration Process for Solving Power Flow Problems: Novel Algorithms and Comprehensive Analysis" *Electronics* 10, no. 23: 3011.
https://doi.org/10.3390/electronics10233011