# State Estimation in Electric Power Systems Using an Approach Based on a Weighted Least Squares Non-Linear Programming Modeling

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## Abstract

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## 1. Introduction

- The first strategy consists of a mathematical programming approach labeled as classical non-linear programming model (classical NLPM). Such model emulates the formulation and iterative process of the WLS state estimator.
- The second strategy is a new mathematical model based on some modifications of the classical NLPM. This mathematical model is used within a new state estimation procedure that allows reducing the negative impact on the estimated results of conventional state estimation studies due to errors presented in the set of measurements. This strategy is labeled as New NLPM.

## 2. State Estimation in EPS

## 3. Representation of the State Estimation Problem as a Mathematical Programming Model

#### 3.1. Classical NLPM

- Active and reactive power injection measurements:$$\begin{array}{cc}\hfill \phantom{\rule{-14.45377pt}{0ex}}h{\left(\widehat{x}\right)}_{mij}={P}_{i}=\phantom{\rule{-6.50403pt}{0ex}}\sum _{k\phantom{\rule{0.72229pt}{0ex}}\in \phantom{\rule{0.72229pt}{0ex}}\mathsf{\Omega}B}\phantom{\rule{-6.50403pt}{0ex}}{V}_{i}{V}_{k}({G}_{ik}cos{\theta}_{ik}+{B}_{ik}sin{\theta}_{ik}),\hfill & {\forall}_{m,i,j}\phantom{\rule{0.72229pt}{0ex}}\in \phantom{\rule{0.72229pt}{0ex}}\mathsf{\Omega}M\phantom{\rule{-2.168pt}{0ex}}:m=1\end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{-14.45377pt}{0ex}}h{\left(\widehat{x}\right)}_{mij}={Q}_{i}=\phantom{\rule{-6.50403pt}{0ex}}\sum _{k\phantom{\rule{0.72229pt}{0ex}}\in \phantom{\rule{0.72229pt}{0ex}}\mathsf{\Omega}B}\phantom{\rule{-6.50403pt}{0ex}}{V}_{i}{V}_{k}({G}_{ik}sin{\theta}_{ik}-{B}_{ik}cos{\theta}_{ik}),\hfill & {\forall}_{m,i,j}\phantom{\rule{0.72229pt}{0ex}}\in \phantom{\rule{0.72229pt}{0ex}}\mathsf{\Omega}M\phantom{\rule{-2.168pt}{0ex}}:m=2\end{array}$$In this case, the elements of the ${Y}_{BUS}$ matrix in (4) and (5) are calculated using Equations (6) to (9). Please note that both conventional and phase-shifting transformers are taken into account.$$\begin{array}{c}\hfill {G}_{ii}=\sum _{(i,m)\phantom{\rule{0.72229pt}{0ex}}\in \phantom{\rule{0.72229pt}{0ex}}\mathsf{\Omega}L}{t}_{im}^{2}{g}_{im}\hfill \end{array}$$$$\begin{array}{c}\hfill {B}_{ii}={b}_{i}^{sh}+\sum _{(i,m)\phantom{\rule{0.72229pt}{0ex}}\in \phantom{\rule{0.72229pt}{0ex}}\mathsf{\Omega}L}\left({b}_{im}^{sh}+{t}_{im}^{2}{g}_{im}\right)\hfill \end{array}$$$$\begin{array}{c}\hfill {G}_{ik}=-tik\left({g}_{ik}cos{\phi}_{ik}+{b}_{ik}sin{\phi}_{ik}\right)\hfill \end{array}$$$$\begin{array}{c}\hfill {B}_{ik}=tik\left({g}_{ik}sin{\phi}_{ik}-{b}_{ik}cos{\phi}_{ik}\right)\hfill \end{array}$$
- Active power flow measurements:$$\begin{array}{c}\hfill h{\left(\widehat{x}\right)}_{mij}\phantom{\rule{-0.72229pt}{0ex}}=\phantom{\rule{-1.4457pt}{0ex}}{P}_{ij}\phantom{\rule{-1.4457pt}{0ex}}=\phantom{\rule{-0.72229pt}{0ex}}{g}_{ij}{\left({t}_{ij}{V}_{i}\right)}^{2}\phantom{\rule{-1.4457pt}{0ex}}-\phantom{\rule{-1.4457pt}{0ex}}{t}_{ij}{V}_{i}{V}_{j}({g}_{ij}cos({\theta}_{ij}\phantom{\rule{-1.4457pt}{0ex}}+\phantom{\rule{-1.4457pt}{0ex}}{\phi}_{ij})\phantom{\rule{-0.72229pt}{0ex}}+\phantom{\rule{-0.72229pt}{0ex}}{b}_{ij}sin({\theta}_{ij}\phantom{\rule{-1.4457pt}{0ex}}+\phantom{\rule{-1.4457pt}{0ex}}{\phi}_{ij}),\\ \hfill {\forall}_{m,i,j}\phantom{\rule{0.72229pt}{0ex}}\in \phantom{\rule{0.72229pt}{0ex}}\mathsf{\Omega}M\phantom{\rule{-2.168pt}{0ex}}:m=3\phantom{\rule{3.61371pt}{0ex}}\end{array}$$$$\begin{array}{c}\hfill h{\left(\widehat{x}\right)}_{mij}\phantom{\rule{-0.72229pt}{0ex}}=\phantom{\rule{-1.4457pt}{0ex}}{P}_{ji}\phantom{\rule{-1.4457pt}{0ex}}=\phantom{\rule{-0.72229pt}{0ex}}{g}_{ij}{V}_{j}^{2}\phantom{\rule{-1.4457pt}{0ex}}-\phantom{\rule{-1.4457pt}{0ex}}{t}_{ij}{V}_{i}{V}_{j}({g}_{ij}cos({\theta}_{ij}\phantom{\rule{-1.4457pt}{0ex}}+\phantom{\rule{-1.4457pt}{0ex}}{\phi}_{ij})\phantom{\rule{-0.72229pt}{0ex}}-\phantom{\rule{-0.72229pt}{0ex}}{b}_{ij}sin({\theta}_{ij}\phantom{\rule{-1.4457pt}{0ex}}+\phantom{\rule{-1.4457pt}{0ex}}{\phi}_{ij}),\\ \hfill {\forall}_{m,i,j}\phantom{\rule{0.72229pt}{0ex}}\in \phantom{\rule{0.72229pt}{0ex}}\mathsf{\Omega}M\phantom{\rule{-2.168pt}{0ex}}:m=3\phantom{\rule{3.61371pt}{0ex}}\end{array}$$
- Reactive power flow measurements:$$\begin{array}{c}h{\left(\widehat{x}\right)}_{mij}\phantom{\rule{-2.168pt}{0ex}}=\phantom{\rule{-2.168pt}{0ex}}{Q}_{ij}\phantom{\rule{-1.4457pt}{0ex}}=\phantom{\rule{-1.4457pt}{0ex}}-{V}_{i}^{2}({t}_{ij}^{2}{b}_{ij}\phantom{\rule{-1.4457pt}{0ex}}+\phantom{\rule{-1.4457pt}{0ex}}{b}_{ij}^{sh})\phantom{\rule{-1.4457pt}{0ex}}+\phantom{\rule{-1.4457pt}{0ex}}{t}_{ij}{V}_{i}{V}_{j}({b}_{ij}cos({\theta}_{ij}\phantom{\rule{-2.168pt}{0ex}}+\phantom{\rule{-2.168pt}{0ex}}{\phi}_{ij})\phantom{\rule{-0.72229pt}{0ex}}-\phantom{\rule{39.74872pt}{0ex}}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{g}_{ij}sin({\theta}_{ij}+{\phi}_{ij})),\phantom{\rule{57.81621pt}{0ex}}{\forall}_{m,i,j}\phantom{\rule{0.72229pt}{0ex}}\in \phantom{\rule{0.72229pt}{0ex}}\mathsf{\Omega}M\phantom{\rule{-2.168pt}{0ex}}:m=4\hfill \end{array}$$$$\begin{array}{c}\hspace{1em}h{\left(\widehat{x}\right)}_{mij}\phantom{\rule{-2.168pt}{0ex}}=\phantom{\rule{-2.168pt}{0ex}}{Q}_{ji}\phantom{\rule{-1.4457pt}{0ex}}=\phantom{\rule{-1.4457pt}{0ex}}-{V}_{j}^{2}({b}_{ij}\phantom{\rule{-1.4457pt}{0ex}}+\phantom{\rule{-1.4457pt}{0ex}}{b}_{ij}^{sh})\phantom{\rule{-1.4457pt}{0ex}}+\phantom{\rule{-1.4457pt}{0ex}}{t}_{ij}{V}_{i}{V}_{j}({b}_{ij}cos({\theta}_{ij}\phantom{\rule{-2.168pt}{0ex}}+\phantom{\rule{-2.168pt}{0ex}}{\phi}_{ij})\phantom{\rule{-0.72229pt}{0ex}}+\phantom{\rule{39.74872pt}{0ex}}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{g}_{ij}sin({\theta}_{ij}+{\phi}_{ij})),\phantom{\rule{57.81621pt}{0ex}}{\forall}_{m,i,j}\phantom{\rule{0.72229pt}{0ex}}\in \phantom{\rule{0.72229pt}{0ex}}\mathsf{\Omega}M\phantom{\rule{-2.168pt}{0ex}}:m=4\hfill \end{array}$$
- Voltage measurements:$$\begin{array}{c}\hfill \phantom{\rule{79.49744pt}{0ex}}h{\left(\widehat{x}\right)}_{mij}\phantom{\rule{-2.168pt}{0ex}}=\phantom{\rule{-2.168pt}{0ex}}{V}_{i},\phantom{\rule{57.81621pt}{0ex}}{\forall}_{m,i,j}\phantom{\rule{0.72229pt}{0ex}}\in \phantom{\rule{0.72229pt}{0ex}}\mathsf{\Omega}M\phantom{\rule{-2.168pt}{0ex}}:m=5\hfill \end{array}$$

#### 3.2. New NLPM

- Neglect the presence of errors in $\mathsf{\Omega}M$: For a specific measurement ${z}_{mij}\approx h{\left(\widehat{x}\right)}_{mij}$, $\Delta {z}_{mij}\approx 0$ and ${r}_{mij}\approx 0$, and the objective function $J\left(\widehat{x}\right)\approx 0$.
- Consider the presence of errors in $\mathsf{\Omega}M$: For a specific measurement ${z}_{mij}\ne h{\left(\widehat{x}\right)}_{mij}$, $\Delta {z}_{mij}\ne 0$ and ${r}_{mij}\ne 0$, and the objective function $J\left(\widehat{x}\right)>0$.

## 4. Numerical Results

#### 4.1. Preliminary Considerations

#### 4.2. Results with a 5-Bus Didactic Power System

#### 4.2.1. No Bad Data in the Measurement Set

#### 4.2.2. Multiple Bad Data Measurements in the Measurement Set

#### 4.3. IEEE Power Systems

#### 4.3.1. No Bad Data in the Measurement Set

#### 4.3.2. Multiple Bad Data Measurements in the Measurement Set

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Sets: | |

$\mathsf{\Omega}M$ | Set of system’s measurements. |

$\mathsf{\Omega}B$ | Set of system’s buses. |

$\mathsf{\Omega}L$ | Set of system’s transmission lines. |

Parameters: | |

${g}_{ij}$ | Line conductance at node i and j. |

${b}_{ij}$ | Line susceptance at node i and j. |

${b}_{ij}^{sh}$ | Line shunt susceptance at node i and j. |

${b}_{i}^{sh}$ | Shunt susceptance at node i. |

${t}_{ij}$ | Transformer tap ratio at node i and j. |

${\phi}_{ij}$ | Phase-shifting transformer angle at node i and j. |

${G}_{ii},{G}_{ij}$ | Conductance of the Ybus matrix. |

${B}_{ii},{B}_{ij}$ | Susceptance of the Ybus matrix. |

${z}_{mij}$ | Measurement m located at node i and j. |

${W}_{mij}$ | Weight of the measurement m at node i and j. |

p | Allowed mismatch percentage. |

$\u03f5$ | Tolerance of the iterative state estimator procedure. |

e | Random errors added in measurement set. |

Variables: | |

$\widehat{x}$ | State variables of the system. |

$J\left(\widehat{x}\right)$ | Least squares function. |

${r}_{mij}$ | Residual value of the measurement m located at node i and j. |

${h}_{mij}\left(\widehat{x}\right)$ | Non-linear measurement functions. |

$\Delta {z}_{mij}$ | Measurement mismatch. |

${V}_{i},{\theta}_{i}$ | Voltage module and phase angle at node i. |

${P}_{ij},{P}_{ji}$ | Active power flow of branch $ij$. |

${Q}_{ij},{Q}_{ji}$ | Reactive power flow of branch $ij$. |

${P}_{L},{Q}_{L}$ | Total active and reactive power loss. |

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Bus | Matpower Load Flow Analysis | Classical NLPM | New NLPM | |||
---|---|---|---|---|---|---|

${\mathit{V}}_{\mathit{i}}(\mathit{p}.\mathit{u}.)$ | ${\mathbf{\theta}}_{\mathit{i}}\left(\mathit{d}\mathit{e}\mathit{g}\right)$ | ${\mathit{V}}_{\mathit{i}}(\mathit{p}.\mathit{u}.)$ | ${\mathbf{\theta}}_{\mathit{i}}\left(\mathit{d}\mathit{e}\mathit{g}\right)$ | ${\mathit{V}}_{\mathit{i}}(\mathit{p}.\mathit{u}.)$ | ${\mathbf{\theta}}_{\mathit{i}}\left(\mathit{d}\mathit{e}\mathit{g}\right)$ | |

1 | 1.0600 | 0.0000 | 1.0600 | 0.0000 | 1.0600 | 0.0000 |

2 | 1.0474 | −2.8063 | 1.0474 | −2.8063 | 1.0474 | −2.8063 |

3 | 1.0242 | −4.9969 | 1.0242 | −4.9969 | 1.0242 | −4.9969 |

4 | 1.0236 | −5.3291 | 1.0236 | −5.3291 | 1.0236 | −5.3291 |

5 | 1.0179 | −6.1502 | 1.0179 | −6.1502 | 1.0179 | −6.1502 |

Matpower Load Flow Analysis | Classical NLPM | New NLPM | |
---|---|---|---|

${P}_{L}$ (MW) | 4.5868 | 4.5867 | 4.5867 |

${Q}_{L}$ (MVAr) | 13.7605 | 13.7600 | 13.7600 |

Bus | Matpower Load Flow Analysis | Classical NLPM | New NLPM | |||
---|---|---|---|---|---|---|

${\mathit{V}}_{\mathit{i}}(\mathit{p}.\mathit{u}.)$ | ${\mathbf{\theta}}_{\mathit{i}}\left(\mathit{d}\mathit{e}\mathit{g}\right)$ | ${\mathit{V}}_{\mathit{i}}(\mathit{p}.\mathit{u}.)$ | ${\mathbf{\theta}}_{\mathit{i}}\left(\mathit{d}\mathit{e}\mathit{g}\right)$ | ${\mathit{V}}_{\mathit{i}}(\mathit{p}.\mathit{u}.)$ | ${\mathbf{\theta}}_{\mathit{i}}\left(\mathit{d}\mathit{e}\mathit{g}\right)$ | |

1 | 1.0600 | 0.0000 | 1.0633 | 0.0000 | 1.0605 | 0.0000 |

2 | 1.0474 | −2.8063 | 1.0504 | −2.8036 | 1.0481 | −2.7945 |

3 | 1.0242 | −4.9969 | 1.0272 | −5.0240 | 1.0246 | −5.0096 |

4 | 1.0236 | −5.3291 | 1.0266 | −5.3584 | 1.0240 | −5.3426 |

5 | 1.0179 | −6.1502 | 1.0206 | −6.1902 | 1.0186 | −6.1699 |

Matpower Load Flow Analysis | Classical NLPM | New NLPM | |
---|---|---|---|

${P}_{L}$ (MW) | 4.5868 | 4.6718 | 4.6188 |

${Q}_{L}$ (MVAr) | 13.7605 | 14.0154 | 13.8564 |

Test System | Power Losses | Matpower Load Flow Analysis | Classical NLPM | New NLPM |
---|---|---|---|---|

IEEE-14 | ${P}_{L}$ (MW) | 13.3932 | 13.3929 | 13.3931 |

${Q}_{L}$(MVAr) | 54.5372 | 54.5358 | 54.5362 | |

IEEE-30 | ${P}_{L}$ (MW) | 17.5518 | 17.5488 | 17.5491 |

${Q}_{L}$ (MVAr) | 67.6996 | 67.6889 | 67.6900 | |

IEEE-57 | ${P}_{L}$ (MW) | 27.8611 | 27.8640 | 27.8635 |

${Q}_{L}$ (MVAr) | 121.6643 | 121.6698 | 121.6692 | |

IEEE-118 | ${P}_{L}$ (MW) | 132.4813 | 132.4820 | 132.4821 |

${Q}_{L}$ (MVAr) | 782.2786 | 782.2816 | 782.2830 |

Test System | Power Losses | Matpower Load Flow Analysis | Classical NLPM | New NLPM |
---|---|---|---|---|

IEEE-14 | ${P}_{L}$ (MW) | 13.3932 | 14.2413 | 13.6221 |

${Q}_{L}$ (MVAr) | 54.5372 | 58.4248 | 55.7560 | |

IEEE-30 | ${P}_{L}$ (MW) | 17.5518 | 16.3230 | 17.1545 |

${Q}_{L}$ (MVAr) | 67.6996 | 63.6821 | 66.331 | |

IEEE-57 | ${P}_{L}$ (MW) | 27.8611 | 27.0641 | 27.8068 |

${Q}_{L}$ (MVAr) | 121.6643 | 118.6022 | 121.4292 | |

IEEE-118 | ${P}_{L}$ (MW) | 132.4813 | 130.8553 | 130.8291 |

${Q}_{L}$ (MVAr) | 782.2786 | 766.1776 | 769.0600 |

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## Share and Cite

**MDPI and ACS Style**

Florez, H.A.R.; Marujo, D.; López, G.P.; López-Lezama, J.M.; Muñoz-Galeano, N.
State Estimation in Electric Power Systems Using an Approach Based on a Weighted Least Squares Non-Linear Programming Modeling. *Electronics* **2021**, *10*, 2560.
https://doi.org/10.3390/electronics10202560

**AMA Style**

Florez HAR, Marujo D, López GP, López-Lezama JM, Muñoz-Galeano N.
State Estimation in Electric Power Systems Using an Approach Based on a Weighted Least Squares Non-Linear Programming Modeling. *Electronics*. 2021; 10(20):2560.
https://doi.org/10.3390/electronics10202560

**Chicago/Turabian Style**

Florez, Hugo A. R., Diogo Marujo, Gloria P. López, Jesús M. López-Lezama, and Nicolás Muñoz-Galeano.
2021. "State Estimation in Electric Power Systems Using an Approach Based on a Weighted Least Squares Non-Linear Programming Modeling" *Electronics* 10, no. 20: 2560.
https://doi.org/10.3390/electronics10202560