# A Bounded Exhaustive Search Technique for Optimal Phasor Measurement Unit Placement in Power Grids

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## Abstract

**:**

## 1. Introduction

## 2. Concept of System Observability

- All buses that are installed with PMU can be directly observed. The voltage phasor of the bus and the current phasor of all connected branches to the installed bus can be obtained directly without additional calculation, which is known as direct measurement.
- Buses adjacent to installed buses are also observable. Referring to Figure 1, the voltage phasor of the adjacent bus can be calculated using the interconnected line impedance, the voltage and current phasors at the installed buses based on Kirchhoff’s voltage law (KVL). The leaving or entering current phasor (i.e., load or generator) at the adjacent bus can also be obtained from the measured current phasors of the connected branches using Kirchhoff’s current law (KCL) which is known as pseudo measurement.
- All buses that do not comply the two conditions above are considered unobservable.

## 3. Bounded Search Technique

#### 3.1. Connectivity Constraints

- Select a bus location for PMU installation from AL. Assume that bus 1 is selected.
- Identify the direct and pseudo-measurement buses with respect to the selected bus. This step can be done by finding entry “1” from the corresponding row of matrix A. Store all pseudo-measurement buses in a vector denoted as SL. Bus 1 becomes a direct measurement bus, and only bus 2 can be identified as a pseudo-measurement bus (SL = [2]).
- Update RC by adding 1 to its elements that correspond to the identified buses. RC becomes [1 1 0 0 0].
- Check values at the direct measurement bus in RC. Remove the corresponding element of RC if the redundancy requirement is fulfilled, and update AL accordingly. Otherwise, select another bus to be installed with PMU for the next subsequence bit from the identified pseudo-measurement buses, SL. Repeat steps (2) to (4) until the direct measurement bus can be removed. AL = [2 3 4 5], and RC = [1 0 0 0] because $b=1$ is used.
- Check for other values in RC that are equal to b and eliminate the corresponding element of AL and RC. Skip this step if the condition is not satisfied. Now, AL = [3 4 5], and RC = [0 0 0].
- Repeat steps (1) to (5) for the next bits until AL is empty.

#### 3.2. Symmetry Constraints

## 4. Optimal PMU Placement Formulation

## 5. Results and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

BES | Bounded exhaustive search |

GPS | Global positioning system |

IEEE | Institute of Electrical and Electronics Engineers |

ILP | Integer linear programming |

KCL | Kirchhoff’s current law |

KVL | Kirchhoff’s voltage law |

MILP | Mixed integer linear programming |

NLP | Nonlinear programming |

PMU | Phasor measurement unit |

SORI | System observability redundancy index |

## Appendix A

**Table A1.**Voltage magnitude deviation index values of the used IEEE test systems, as described in [19].

Bus Location | Test System | |||
---|---|---|---|---|

IEEE 9-Bus | IEEE 14-Bus | IEEE 24-Bus | IEEE 30-Bus | |

1 | 0.9574 | 1.7201 | 1.2856 | 1.6339 |

2 | 1.0280 | 1.7026 | 1.2708 | 1.6345 |

3 | 1.0604 | 1.3472 | 1.4298 | 2.0849 |

4 | 1.1045 | 1.7973 | 1.4923 | 2.0914 |

5 | 1.3454 | 1.8608 | 1.5318 | 1.4099 |

6 | 1.3862 | 1.5202 | 1.3631 | 1.9985 |

7 | 1.1810 | 1.4788 | 0.7674 | 1.9058 |

8 | 1.3865 | 0.7764 | 1.2237 | 1.8178 |

9 | 1.2138 | 1.7900 | 1.3404 | 1.8489 |

10 | 1.9512 | 1.3782 | 2.4288 | |

11 | 1.9013 | 1.4247 | 0.8952 | |

12 | 1.9123 | 1.4038 | 1.9487 | |

13 | 1.8802 | 1.2670 | 1.1501 | |

14 | 1.9578 | 1.4119 | 2.3108 | |

15 | 1.5945 | 2.3790 | ||

16 | 1.6859 | 2.3363 | ||

17 | 1.7591 | 2.4512 | ||

18 | 1.6134 | 2.6522 | ||

19 | 1.8373 | 2.6629 | ||

20 | 1.7918 | 2.6585 | ||

21 | 1.5473 | 2.6141 | ||

22 | 1.2101 | 2.6332 | ||

23 | 1.4293 | 2.5362 | ||

24 | 1.5766 | 2.6065 | ||

25 | 2.5690 | |||

26 | 2.6244 | |||

27 | 2.4705 | |||

28 | 2.1352 | |||

29 | 2.6190 | |||

30 | 2.5620 |

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Item | Test System | |||
---|---|---|---|---|

IEEE 9-Bus | IEEE 14-Bus | IEEE 24-Bus | IEEE 30-Bus | |

Number of generator buses | 3 | 5 | 11 | 6 |

Number of load buses | 3 | 8 | 13 | 24 |

Number of transmission lines | 9 | 20 | 38 | 41 |

Maximum number of | 3 | 5 | 5 | 7 |

connected lines to a bus | ||||

Reference bus location | 1 | 1 | 13 | 1 |

Level of complexity | Simple mesh | Weakly mesh | Heavily mesh | Heavily mesh |

Test System | Number of Infeasible Combinations | Probability for Infeasible Combination (%) | ||
---|---|---|---|---|

Conventional | Bounded Search | Conventional | Bounded Search | |

IEEE 9-bus | 31,715 | 0 | 63.43 | 0 |

IEEE 14-bus | 31,193 | 0 | 62.39 | 0 |

IEEE 24-bus | 39,625 | 0 | 79.25 | 0 |

IEEE 30-bus | 45,118 | 0 | 90.24 | 0 |

Test System | Original Size | Standard Numbering | Alternative Numbering | Reduced Size without Duplications | ||
---|---|---|---|---|---|---|

Reduced Size | Number of Duplications | Reduced Size | Number of Duplications | |||

IEEE 9-bus | 511 | 92 | 84 | 12 | 4 | 8 |

IEEE 14-bus | 16,383 | 93 | 55 | 59 | 21 | 38 |

IEEE 24-bus | 16,777,215 | 140,500 | 140,108 | 3560 | 3168 | 392 |

IEEE 30-bus | 1,073,741,823 | 36,022 | 33,846 | 18,060 | 15,884 | 2176 |

Test System | Bus Re-Numbering as Referred to the Standard Numbering | ||||||||||||||

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |

IEEE 9-bus | 1 | 9 | 8 | 2 | 3 | 4 | 5 | 6 | 7 | ||||||

IEEE 14-bus | 1 | 2 | 3 | 4 | 5 | 6 | 13 | 14 | 10 | 11 | 12 | 7 | 8 | 9 | |

IEEE 24-bus | 1 | 5 | 8 | 6 | 2 | 4 | 23 | 22 | 7 | 3 | 16 | 24 | 15 | 17 | 10 |

IEEE 30-bus | 1 | 2 | 13 | 12 | 22 | 3 | 23 | 14 | 24 | 4 | 25 | 11 | 30 | 10 | 9 |

Test System | Bus Re-Numbering as Referred to the Standard Numbering | ||||||||||||||

16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | |

IEEE 24-bus | 11 | 18 | 19 | 12 | 13 | 20 | 21 | 14 | 9 | ||||||

IEEE 30-bus | 29 | 28 | 19 | 20 | 21 | 5 | 6 | 8 | 7 | 26 | 27 | 16 | 15 | 17 | 18 |

Test System | Optimum Number | Computational Time (s) | ||
---|---|---|---|---|

Exhaustive Search [8] | Proposed BES | Exhaustive Search [8] | Proposed BES | |

IEEE 9-bus | 3 | 3 | 0.108 | 0.024 |

IEEE 14-bus | 4 | 4 | 1.781 | 0.031 |

IEEE 24-bus | 7 | 7 | 1862.65 | 0.963 |

IEEE 30-bus | 10 | 10 | N/A | 7.547 |

Test System | Number of PMUs | PMU Placement (Bus) | Indicator $\mathit{\eta}$ |
---|---|---|---|

IEEE 9-bus | 3 | 1, 7, 9 | 3.3522 |

2, 4, 9 | 3.3463 | ||

3, 4, 7 | 3.3459 | ||

4, 7, 9 | 3.4993 | ||

IEEE 14-bus | 4 | 2, 6, 8, 9 | 5.7892 |

2, 7, 10, 13 | 7.0128 | ||

2, 7, 11, 13 | 6.9629 | ||

2, 8, 10, 13 | 6.3104 | ||

IEEE 24-bus | 7 | 2, 3, 7, 10, 16, 21, 23 | 9.5087 |

3, 4, 7, 10, 16, 21, 23 | 9.7302 |

Test System | Optimal PMU Placement (Bus) |
---|---|

IEEE 9-bus | 1, 7, 9 |

IEEE 14-bus | 2, 7, 10, 13 |

IEEE 24-bus | 3, 4, 7, 10, 16, 21, 23 |

IEEE 30-bus | 1, 7, 10, 11, 12, 19, 24, 26, 28, 29 |

Test System | Number of PMUs | Optimal PMU Placement (Bus) |
---|---|---|

IEEE 9-bus | 6 | 1,2,3,4,7,9 |

IEEE 14-bus | 9 | 2,3,5,6,7,8,9,10,13 |

IEEE 24-bus | 14 | 1,2,4,7,8,10,11,15,16,17,20,21,23,24 |

IEEE 30-bus | 21 | 1,3,5,7,8,9,10,11,12,13,15,17,18,19,22,24,25,26,28,29,30 |

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**MDPI and ACS Style**

Ibrahim, A.A.; Khalid, K.; Shareef, H.; Kamari, N.A.M.
A Bounded Exhaustive Search Technique for Optimal Phasor Measurement Unit Placement in Power Grids. *Symmetry* **2020**, *12*, 2021.
https://doi.org/10.3390/sym12122021

**AMA Style**

Ibrahim AA, Khalid K, Shareef H, Kamari NAM.
A Bounded Exhaustive Search Technique for Optimal Phasor Measurement Unit Placement in Power Grids. *Symmetry*. 2020; 12(12):2021.
https://doi.org/10.3390/sym12122021

**Chicago/Turabian Style**

Ibrahim, Ahmad Asrul, Khairuddin Khalid, Hussain Shareef, and Nor Azwan Mohamed Kamari.
2020. "A Bounded Exhaustive Search Technique for Optimal Phasor Measurement Unit Placement in Power Grids" *Symmetry* 12, no. 12: 2021.
https://doi.org/10.3390/sym12122021