# New Performance Indices for Voltage Stability Analysis in a Power System

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Power Voltage Performance Indexes

#### 2.1. Mathematical Formulation of the VCPI [17]

- ${I}_{m}$ = Current injected at node $m$
- ${V}_{m}$ = Voltage at m
^{th}node - ${V}_{k}$ = Voltage at k
^{th}node - ${Y}_{mk}$ = Mutual admittance between m
^{th}and k^{th}nodes

^{th}node is given as:

#### 2.2. Proposed Performance Voltage Stability Index (PVSI)

- ${U}_{nom}$ is the nominal voltage magnitude;
- ${U}_{m}$ is the voltage magnitude at $m$ node;
- $k$ is the reactive power loading condition.

_{max}) of a load node $m$.

#### 2.3. Conventional Modal Analysis Technique (CMAT)

#### 2.4. Improved Modal Analysis Technique (IMAT)

_{QV}for voltage stability analysis against the use of reduced Jacobian matrix as proposed by [14].

_{QV}as:

_{QV}.

#### 2.4.1. Determination of the Modes of a Power Network

- $\xi $ is the right eigenvector of the Jacobian matrix ${J}_{QV}$;
- $\tau $ represents the left eigenvector of the Jacobian matrix ${J}_{QV}$ and
- $\lambda $ represents the diagonal eigenvalues of the Jacobian matrix ${J}_{QV}$.

#### 2.4.2 Bus Participation Factor

## 3. Simulation Results and Discussion

#### Simulation Results of the VCPI, CMAT and the Proposed PVBI and IMAT

**Test Case A**: Results of the VCPI, PVBI, Voltage magnitude, CMAT and IMAT for the WSCC 9-bus test system.

**Test Case B**: Results of the IEEE 30-bus power system.

**Test Case C**: Results of the IEEE 57-bus power system.

## 4. Discussion of Results

#### 4.1. Results of the Traditional VCPI

#### 4.2. Proposed Performance Voltage Bus Index (PVBI)

^{−4}), minimum allowable load (40 MVar) and least voltage magnitude (0.5677 p.u.) as shown in Table 2 and Figure 3. We also observed a total voltage collapse on this bus, as the power flow solution did not converge for any additional load beyond the minimum permissible reactive power load of 40 MVar. Thus, of all the load buses of the WSCC 9 bus system, bus 5 is the critical bus liable to voltage collapse. The total computational time taken to identify the critical load bus 5 of the 9 bus system is 40.774147 s.

^{−4}, least sustainable load of 25 MVar, lowest total number of step size of 14 and lowest voltage magnitude (0.5677 p.u.). The voltage stability analysis of the IEEE 30 bus system took up to the total computational time of 90.984727 s to attain a solution. This amounts to 8.2% of time saving compared with the conventional modal analysis technique. Also, for a large scale IEEE 57 bus system, bus 31 has the least permissible reactive power load of 17 MVar, maximum PVBI value of 51.960784, lowest total number of step size of 6 and least voltage magnitude (0.5069 p.u.) at the minimum permissible load of 17 MVar. Thus, with the proposed PVBI, bus 31 is considered the weakest load bus of the IEEE 57 bus test system. It takes the total computational time of 198.559777 s to attain this solution. This amounts to 12.95% of time saving compared with the traditional approach of VCPI.

#### 4.3. Conventional Modal Analysis (CMAT) and the Proposed IMAT

#### 4.4. A Brief Comparison of all the Techniques Presented

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Kundur, P.; Paserba, J.; Ajjarapu, V.; Andersson, G.; Bose, A.; Canizares, C.; Hatziargyriou, N.; Hill, D.; Stankovic, A.; Taylor, C.; et al. Definition and classification of power system stability IEEE/CIGRE joint task force on stability terms and definitions. IEEE Trans. Power Syst.
**2014**, 19, 1387–1401. [Google Scholar] - Thukaram, D.; Vyjayanthi, C. Evaluation of suitable locations for generation expansion in restructured power systems: A novel concept of T-index. Int. J. Emerg. Electr. Power Syst.
**2009**, 10. [Google Scholar] [CrossRef] - Hong, Y.Y.; Gau, C.H. Voltage stability indicator for identification of the weakest bus/area in power systems. IEE Proc. Gener. Transm. Distrib.
**1994**, 141, 305–309. [Google Scholar] [CrossRef] - Su, H.Y. An Efficient Approach for Fast and Accurate Voltage Stability Margin Computation in Large Power Grids. Appl. Sci.
**2016**, 6, 335. [Google Scholar] [CrossRef] - Ellithy, K.; Shaheen, M.; Al-Athba, M. Voltage Stability Evaluation of Real Power Transmission System Using Singular Value Decomposition Technique. In Proceedings of the 2nd IEEE International Conference on Power and Energy (PECon 08), Johor Baharu, Malaysia, 1–3 December 2008. [Google Scholar]
- Su, H.Y.; Liu, C.W. Estimating the Voltage Stability Margin Using PMU Measurements. IEEE Trans. Power Syst.
**2016**, 31, 3221–3229. [Google Scholar] [CrossRef] - Ngo, V.D.; Le, D.D.; Le, K.H.; Pham, V.K.; Berizzi, A. A Methodology for Determining Permissible Operating Region of Power Systems According to Conditions of Static Stability Limit. Energies
**2017**, 10, 1163. [Google Scholar] [CrossRef] - Kessel, P.; Glavitsch, H. Estimating the Voltage Stability of a Power System. IEEE Trans. Power Deliv.
**1986**, 1, 346–354. [Google Scholar] [CrossRef] - Claudia, R. A Comparison of Voltage Stability Indices. In Proceedings of the IEEE MELECON 2006, Malaga, Spain, 16–19 May 2006. [Google Scholar]
- Moghawemi, M.; Omar, F.M. A Line Outage Study for Prediction of Static Voltage Collapse. IEEE Power Eng. Rev.
**1998**, 18, 52–54. [Google Scholar] [CrossRef] - Taylor, C.W. Power System Voltage Stability; McGraw-Hill Companies: New York, NY, USA, 1994. [Google Scholar]
- Ajjarapu, V.; Christy, C. The continuation power flow: A tool for steady state voltage stability analysis. IEEE Trans. Power Syst.
**1992**, 7, 416–423. [Google Scholar] [CrossRef] - Tamura, Y.; Mori, H.; Iwamoto, S. Relationship between voltage instability and multiple load flow solutions in electric power systems. IEEE Trans. Power Appar. Syst.
**1983**, 5, 1115–1125. [Google Scholar] [CrossRef] - Gao, B.; Morison, G.K.; Kundur, P. Voltage stability evaluation using modal analysis. IEEE Trans. Power Syst.
**1922**, 7, 1529–1542. [Google Scholar] [CrossRef] - Acharjee, P. Identification of voltage collapse points and weak buses under security constraints using hybrid particle swarm optimization technique. Int. Trans. Electr. Energy Syst.
**2013**, 23, 230–248. [Google Scholar] [CrossRef] - Moger, T.; Dhadbanjan, T. A novel index for identification of weak nodes for reactive compensation to improve voltage stability. IET Gener. Transm. Distrib.
**2015**, 9, 1826–1834. [Google Scholar] [CrossRef] - Balamourougan, V.; Sidhu, T.S.; Sachdev, M.S. Technique for online prediction of voltage collapse. IEE Proc. Gener. Transm. Distrib.
**2004**, 151, 453–460. [Google Scholar] [CrossRef] - Musirin, I.; Rahman, T.K. Estimating Maximum Loadability for Weak Bus Identification Using FVSI. IEEE Power Eng. Rev.
**2002**, 22, 50–52. [Google Scholar] [CrossRef] - Andrade, A.C.; Barbosa, F.P.M. Voltage Collapse Preventive Control—A New method. In Proceedings of the 12th IEEE Mediterranean Electrotechnical Conference, MELECON 2004, Dubrovnik, Croatia, 12–15 May 2004. [Google Scholar]
- Saadat, H. Power Systems Analysis; McGraw-Hill: New York, NY, USA, 1999. [Google Scholar]
- Sharma, C.; Ganness, M.G. Determination of the applicability of using modal analysis for the prediction of voltage stability. In Proceedings of the 2008 IEEE/PES Transmission and Distribution Conference and Exposition, Chicago, IL, USA, 21–24 April 2008; pp. 1–7. [Google Scholar]
- Asija, D.; Choudekar, P.; Soni, K.M.; Sinha, S.K. Power Flow Study and Contingency status of WSCC 9 Bus Test System using MATLAB. In Proceedings of the 2015 International Conference on Recent Developments in Control, Automation and Power Engineering (RDCAPE), Noida, India, 12–13 March 2015; pp. 338–342. [Google Scholar]
- Kumar, V.; Gupta, M.; Sharma, N.K.; Banerjee, G.K. Comparative analysis of different Power delivery System using Voltage Stability Index. In Proceedings of the 2016 Second International Innovative Applications of Computational Intelligence on Power, Energy and Controls with their Impact on Humanity (CIPECH), Ghaziabad, India, 18–19 November 2016. [Google Scholar]

Qmax (MVar) | Bus No | Traditional VCPI Method | Total Computation Time (s) | Ranking Order |
---|---|---|---|---|

40 | 5 | 0.5235 | 3.396875 | 1st |

256 | 6 | 0.4752 | 23.886528 | 3rd |

240 | 8 | 0.5063 | 22.758630 | 2nd |

**Table 2.**Results of the proposed performance voltage bus index (PVBI) for the 9-bus test system. PVDBI: performance voltage deviation index.

Bus No | Qmax (MVar) | Number of Step Size | Proposed PVDBI | Proposed PVBI 10^{−4} | Total Computation Time (s) | Ranking Order |
---|---|---|---|---|---|---|

5 | 40 | 5 | 0.4060 | 20.3000 | 2.569802 | 1st |

6 | 256 | 32 | 0.7933 | 0.9684 | 20.109783 | 3rd |

8 | 240 | 30 | 0.8286 | 1.1508 | 18.094562 | 2nd |

Load Bus | Eigenvalues | Mode | CMAT | Rank | Computational Time (s) |
---|---|---|---|---|---|

4 | 38.3118 | 1 | 0.0002 | 6th | |

5 | 34.7552 | 2 | 0.8535 | 1st | |

6 | 29.8090 | 3 | 0.0016 | 5th | 0.559172 |

7 | 9.1099 | 4 | 0.0631 | 3rd | |

8 | 5.2841 | 5 | 0.0731 | 2nd | |

9 | 0.1919 | 6 | 0.0067 | 4th |

**Table 4.**Results of the Proposed improved modal analysis technique (IMAT) of the WSCC 9-bus power system.

Load Bus | Eigenvalues | Mode | Proposed IMAT | Rank | Computational Time (s) |
---|---|---|---|---|---|

4 | 37.7712 | 1 | 0.0004 | 6th | |

5 | 34.3277 | 2 | 0.8558 | 1st | |

6 | 29.7026 | 3 | 0.0024 | 5th | 0.490854 |

7 | 9.0171 | 4 | 0.0533 | 3rd | |

8 | 5.3754 | 5 | 0.0806 | 2nd | |

9 | 1.4946 | 6 | 0.0067 | 4th |

Qmax (MVar) | Bus No | Traditional VCPI Method | Total Computation Time (s) | Ranking Order |
---|---|---|---|---|

102 | 23 | 0.2470 | 21.549021 | 4th |

108 | 24 | 0.0826 | 24.209167 | 5th |

25 | 27 | 0.4030 | 15.709843 | 1st |

35 | 29 | 0.2779 | 19.089312 | 3rd |

31 | 30 | 0.3225 | 18.563290 | 2nd |

Bus No | Qmax (MVar) | Number of Step Size | Proposed PVDBI | Proposed PVBI 10^{−4} | Total Computation Time (s) | Ranking Order |
---|---|---|---|---|---|---|

23 | 102 | 55 | 1.253418 | 2.234257 | 20.870932 | 4th |

24 | 108 | 58 | 0.234238 | 0.373943 | 22.760321 | 5th |

27 | 25 | 14 | 0.751936 | 21.48388 | 13.039880 | 1st |

29 | 35 | 21 | 0.800705 | 10.893945 | 17.903762 | 3rd |

30 | 31 | 18 | 0.61050 | 10.940860 | 16.409832 | 2nd |

Load Bus | Eigenvalues | Mode | CMAT | Rank | Computational Time (s) |
---|---|---|---|---|---|

7 | 109.7610 | 1 | 0.0003 | 23rd | |

8 | 98.2275 | 2 | 0.0004 | 22nd | |

9 | 65.8089 | 3 | 0.0005 | 21st | 0.923476 |

10 | 48.9036 | 4 | 0.0002 | 24th | |

11 | 37.9613 | 5 | 0.0026 | 20th | |

12 | 35.1885 | 6 | 0.0040 | 17th | |

13 | 33.3827 | 7 | 0.0107 | 10th | |

14 | 23.0648 | 8 | 0.0072 | 13th | |

15 | 22.9861 | 9 | 0.0041 | 16th | |

16 | 18.8202 | 10 | 0.0159 | 9th | |

17 | 17.0175 | 11 | 0.0172 | 8th | |

18 | 15.5751 | 12 | 0.0022 | 19th | |

19 | 13.6245 | 13 | 0.0050 | 15th | |

20 | 12.8474 | 14 | 0.0071 | 14th | |

21 | 11.2777 | 15 | 0.0034 | 18th | |

22 | 0.4921 | 16 | 0.0083 | 11th | |

23 | 1.1468 | 17 | 0.0193 | 7th | |

24 | 1.6633 | 18 | 0.0080 | 12th | |

25 | 2.8717 | 19 | 0.0376 | 6th | |

26 | 7.9392 | 20 | 0.1140 | 5th | |

27 | 4.0336 | 21 | 0.2045 | 2nd | |

28 | 6.9975 | 22 | 0.1099 | 4th | |

29 | 5.1742 | 23 | 0.1997 | 3rd | |

30 | 6.0380 | 24 | 0.2178 | 1st |

Load Bus | Eigenvalues | Mode | Proposed IMAT | Rank | Computational Time (s) |
---|---|---|---|---|---|

7 | 101.5484 | 1 | 0.0002 | 23rd | |

8 | 79.0143 | 2 | 0.0003 | 22nd | |

9 | 59.9206 | 3 | 0.0005 | 21st | 0.809821 |

10 | 44.7144 | 4 | 0.0001 | 24th | |

11 | 30.8457 | 5 | 0.0027 | 18th | |

12 | 29.7287 | 6 | 0.0030 | 17th | |

13 | 28.1175 | 7 | 0.0080 | 10th | |

14 | 21.2441 | 8 | 0.0061 | 14th | |

15 | 17.7133 | 9 | 0.0031 | 16th | |

16 | 16.8427 | 10 | 0.0125 | 9th | |

17 | 13.5606 | 11 | 0.0137 | 8th | |

18 | 13.1216 | 12 | 0.0016 | 20th | |

19 | 11.3142 | 13 | 0.0037 | 15th | |

20 | 10.2906 | 14 | 0.0062 | 13th | |

21 | 9.7484 | 15 | 0.0026 | 19th | |

22 | 0.4481 | 16 | 0.0072 | 11th | |

23 | 1.0261 | 17 | 0.0180 | 7th | |

24 | 1.2531 | 18 | 0.0068 | 12th | |

25 | 2.4551 | 19 | 0.0354 | 6th | |

26 | 3.3246 | 20 | 0.1133 | 4th | |

27 | 4.1938 | 21 | 0.2458 | 1st | |

28 | 4.7525 | 22 | 0.0971 | 5th | |

29 | 5.9044 | 23 | 0.1960 | 3rd | |

30 | 5.5165 | 24 | 0.2161 | 2nd |

Qmax (MVar) | Bus No | Traditional VCPI Method | Voltage Mag. (p.u.) | Total Computation Time (s) | Ranking Order |
---|---|---|---|---|---|

180 | 12 | 0.3584 | 0.6734 | 83.525160 | 8th |

24 | 25 | 0.4826 | 0.5601 | 14.947784 | 5th |

114 | 27 | 0.4004 | 0.5794 | 68.497318 | 7th |

21 | 30 | 0.5002 | 0.5491 | 12.975753 | 4th |

17 | 31 | 0.7400 | 0.5069 | 7.010215 | 1st |

20 | 32 | 0.5021 | 0.5226 | 10.208482 | 3rd |

19 | 33 | 0.6327 | 0.5176 | 9.462271 | 2nd |

36 | 57 | 0.4203 | 0.5758 | 21.478248 | 6th |

Bus No | Qmax (MVar) | Number of Step Size | Proposed PVDBI | Proposed PVBI 10^{−4} | Voltage Mag. (p.u.) | Total Computation Time (s) | Ranking Order |
---|---|---|---|---|---|---|---|

12 | 180 | 60 | 1.9853 | 1.83824 | 0.6734 | 76.987023 | 8th |

25 | 24 | 11 | 0.6402 | 24.25000 | 0.5601 | 12.690832 | 5th |

27 | 114 | 41 | 1.7494 | 3.74283 | 0.5794 | 56.89705 | 7th |

30 | 21 | 10 | 0.8453 | 40.25238 | 0.5491 | 11.098412 | 4th |

31 | 17 | 6 | 0.5300 | 51.960784 | 0.5069 | 5.809732 | 1st |

32 | 20 | 9 | 0.8774 | 48.744444 | 0.5226 | 8.892109 | 3rd |

33 | 19 | 8 | 0.7607 | 50.046053 | 0.5176 | 7.091207 | 2nd |

Load Bus | Eigenvalues | Mode | CMAT | Rank | Computational Time (s) |
---|---|---|---|---|---|

8 | 167.0587 | 1 | 0.00028 | 46th | |

9 | 117.2839 | 2 | 0.00012 | 49th | 1.709213 |

10 | 100.1632 | 3 | 0.00049 | 43rd | |

11 | 96.9489 | 4 | 0.00090 | 36th | |

12 | 83.4685 | 5 | 0.00089 | 37th | |

13 | 81.8242 | 6 | 0.00082 | 38th | |

14 | 63.5906 | 7 | 0.0010 | 35th | |

15 | 59.2406 | 8 | 0.00038 | 45th | |

16 | 57.7710 | 9 | 0.00026 | 47th | |

17 | 52.9183 | 10 | 0.000071 | 50th | |

18 | 51.8406 | 11 | 0.00018 | 48th | |

19 | 43.5320 | 12 | 0.0025 | 30th | |

20 | 43.1314 | 13 | 0.0048 | 24th | |

21 | 36.4075 | 14 | 0.0083 | 19th | |

22 | 35.8421 | 15 | 0.0084 | 18th | |

23 | 32.3515 | 16 | 0.0089 | 17th | |

24 | 32.4130 | 17 | 0.0173 | 11th | |

25 | 28.7774 | 18 | 0.0940 | 5th | |

26 | 25.5944 | 19 | 0.0144 | 14th | |

27 | 25.0823 | 20 | 0.0041 | 26th | |

28 | 21.7844 | 21 | 0.0017 | 33rd | |

29 | 17.7898 | 22 | 0.00073 | 41st | |

30 | 16.4373 | 23 | 0.1233 | 4th | |

31 | 15.4034 | 24 | 0.1687 | 1st | |

32 | 15.1964 | 25 | 0.1582 | 3rd | |

33 | 14.0792 | 26 | 0.1618 | 2nd | |

34 | 0.2184 | 27 | 0.0328 | 6th | |

35 | 0.5586 | 28 | 0.0255 | 7th | |

36 | 0.8936 | 29 | 0.0202 | 10th | |

37 | 1.0088 | 30 | 0.0165 | 13th | |

38 | 1.1925 | 31 | 0.0074 | 20th | |

39 | 1.5095 | 32 | 0.0167 | 12th | |

40 | 2.2811 | 33 | 0.0204 | 9th | |

41 | 2.5580 | 34 | 0.0063 | 21st | |

42 | 3.3895 | 35 | 0.0105 | 16th | |

43 | 3.7053 | 36 | 0.0020 | 32nd | |

44 | 4.1351 | 37 | 0.0054 | 22nd | |

45 | 4.5815 | 38 | 0.0021 | 31st | |

46 | 5.4765 | 39 | 0.0026 | 29th | |

47 | 5.8164 | 40 | 0.0046 | 25th | |

48 | 6.6397 | 41 | 0.0053 | 23rd | |

49 | 7.3517 | 42 | 0.0039 | 27th | |

50 | 7.8873 | 43 | 0.0031 | 28th | |

51 | 8.6792 | 44 | 0.0013 | 34th | |

52 | 9.1597 | 45 | 0.00081 | 39th | |

53 | 12.3263 | 46 | 0.0008 | 40th | |

54 | 11.5016 | 47 | 0.00060 | 42nd | |

55 | 10.7510 | 48 | 0.00039 | 44th | |

56 | 10.9852 | 49 | 0.0129 | 15th | |

57 | 10.9553 | 50 | 0.0254 | 8th |

Load Bus | Eigenvalues | Mode | Proposed IMAT | Rank | Computational Time (s) |
---|---|---|---|---|---|

8 | 117.6642 | 1 | 0.00021 | 47th | |

9 | 90.2499 | 2 | 0.00085 | 35th | 1.035109 |

10 | 79.5325 | 3 | 0.00037 | 44th | |

11 | 76.8631 | 4 | 0.00073 | 37th | |

12 | 66.0945 | 5 | 0.00063 | 39th | |

13 | 60.5387 | 6 | 0.00066 | 38th | |

14 | 58.9733 | 7 | 0.00082 | 36th | |

15 | 50.1497 | 8 | 0.00031 | 45th | |

16 | 52.2588 | 9 | 0.00019 | 48th | |

17 | 42.3857 | 10 | 0.000053 | 50th | |

18 | 40.9070 | 11 | 0.00012 | 49th | |

19 | 39.8208 | 12 | 0.0029 | 28th | |

20 | 35.3923 | 13 | 0.0055 | 21st | |

21 | 28.2427 | 14 | 0.0083 | 18th | |

22 | 27.6232 | 15 | 0.0082 | 19th | |

23 | 23.6718 | 16 | 0.0088 | 17th | |

24 | 24.2669 | 17 | 0.0191 | 11th | |

25 | 22.1195 | 18 | 0.0814 | 5th | |

26 | 21.0441 | 19 | 0.0164 | 14th | |

27 | 18.7207 | 20 | 0.0042 | 25th | |

28 | 16.0007 | 21 | 0.0014 | 33rd | |

29 | 14.5588 | 22 | 0.0005 | 42nd | |

30 | 13.1264 | 23 | 0.1143 | 4th | |

31 | 12.5574 | 24 | 0.1669 | 1st | |

32 | 11.2184 | 25 | 0.1539 | 3rd | |

33 | 11.0863 | 26 | 0.1602 | 2nd | |

34 | 10.6553 | 27 | 0.0423 | 6th | |

35 | 9.8133 | 28 | 0.0320 | 7th | |

36 | 9.3805 | 29 | 0.0237 | 10th | |

37 | 0.1920 | 30 | 0.0184 | 13th | |

38 | 0.5041 | 31 | 0.0071 | 20th | |

39 | 0.7385 | 32 | 0.0187 | 12th | |

40 | 0.8344 | 33 | 0.0238 | 9th | |

41 | 1.1019 | 34 | 0.0051 | 22nd | |

42 | 1.2206 | 35 | 0.0109 | 16th | |

43 | 1.8570 | 36 | 0.0016 | 31st | |

44 | 2.0165 | 37 | 0.0050 | 23rd | |

45 | 7.6179 | 38 | 0.0015 | 32nd | |

46 | 8.1450 | 39 | 0.0020 | 30th | |

47 | 7.9238 | 40 | 0.0037 | 26th | |

48 | 2.6906 | 41 | 0.0046 | 24th | |

49 | 2.8162 | 42 | 0.0031 | 27th | |

50 | 3.3008 | 43 | 0.0023 | 29th | |

51 | 3.9684 | 44 | 0.00094 | 34th | |

52 | 4.3339 | 45 | 0.0006 | 41st | |

53 | 4.8112 | 46 | 0.00061 | 40th | |

54 | 6.3361 | 47 | 0.00046 | 43rd | |

55 | 6.1087 | 48 | 0.00028 | 46th | |

56 | 5.3311 | 49 | 0.0157 | 15th | |

57 | 5.4760 | 50 | 0.0291 | 8th |

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## Share and Cite

**MDPI and ACS Style**

Adebayo, I.; Sun, Y. New Performance Indices for Voltage Stability Analysis in a Power System. *Energies* **2017**, *10*, 2042.
https://doi.org/10.3390/en10122042

**AMA Style**

Adebayo I, Sun Y. New Performance Indices for Voltage Stability Analysis in a Power System. *Energies*. 2017; 10(12):2042.
https://doi.org/10.3390/en10122042

**Chicago/Turabian Style**

Adebayo, Isaiah, and Yanxia Sun. 2017. "New Performance Indices for Voltage Stability Analysis in a Power System" *Energies* 10, no. 12: 2042.
https://doi.org/10.3390/en10122042