# A Recap of Voltage Stability Indices in the Past Three Decades

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## Abstract

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## 1. Introduction

## 2. Voltage Stability Indices

#### 2.1. L Index

#### 2.2. Power Stability Index (PSI)

#### 2.3. Voltage Deviation Index (VDI)

#### 2.4. Stability Index (SI)

#### 2.5. Voltage Collapse Prediction Index (VCPI)

- Prediction of voltage collapse in a power system for each bus.
- This index needs a modest amount of calculations for estimating the VCPI.
- This index can be used for the recognition of weak buses.
- This index can be used for both online and offline applications.

#### 2.6. Sensitivity Analysis (SA)

#### 2.7. Bus Participation Factor (BPF)

#### 2.8. Voltage Stability Index (VSI)

#### 2.9. Equivalent Node Voltage Collapse Index (ENVCI)

- Accuracy in index modeling; this index covers the influence of both local and outside networks.
- Analysis of internal and external impedances.
- Easy calculation with less computation time compared to the customary power flow-based methods.

#### 2.10. Voltage Collapse Index (VCI)

#### 2.11. Improved Voltage Stability Index (IVSI)

#### 2.12. Voltage Stability Factor (VSF)

#### 2.13. Line Stability Index (${L}_{mn}$)

#### 2.14. Line Stability Factor (LQP)

#### 2.15. L Index

#### 2.16. Voltage Collapse Proximity Indicator (VCPI)

#### 2.17. Voltage Instability Proximity Index (VIPI)

#### 2.18. Integral Steady-State Margin (ISSM)

#### 2.19. Novel Line Stability Index (NLSI)

#### 2.20. Fast Voltage Stability Index (FVSI)

#### 2.21. Critical Voltage (${V}_{cr}$)

#### 2.22. Power Transfer Stability Index (PTSI)

#### 2.23. Line Voltage Stability Index (LVSI)

#### 2.24. Impedance Ration Indicator

#### 2.25. Minimum Eigenvalue and Right Eigenvector (RE) Method

#### 2.26. Singular Value Indicator

- The smallest singular value (${\mathit{\sigma}}_{\mathit{n}}$) can be used as a steady-state stability limit indicator;
- The right singular vector (${\mathit{v}}_{\mathit{n}}$) corresponding to the smallest singular value (${\mathit{\sigma}}_{\mathit{n}}$) indicates sensitive voltage and angles;
- The left singular vector (${\mathit{u}}_{\mathit{n}}$) corresponding to the smallest singular value (${\mathit{\sigma}}_{\mathit{n}}$) indicates the most sensitive direction for changes of active and reactive power injections.

#### 2.27. Predicting the Voltage Collapse Index ($V/{V}_{o}$)

#### 2.28. Test Function

#### 2.29. Tangent Vector Index ($TV{I}_{i}$)

#### 2.30. Second-Order Index ($i$ Index)

#### 2.31. Critical Boundary Index (CBI)

#### 2.32. Line Voltage Stability Index (LVSI)

#### 2.33. Integrated Transmission Line Transfer Index (ITLTI)

#### 2.34. Miscellaneous Indices

## 3. Voltage Stability Indices Categorization

## 4. Results and Discussion

^{®}and PowerWorld

^{®}education and business simulation tools. Simulation results were obtained based on the indices’ foundation, performance, application, merit, demerit, and overall behavior. Although the analysis and simulation of more than 40 voltage stability indices were not feasible at the same time, therefore indices were generalized in consensus groups considering their wide applicability. Also, due to the limitation of accessibility, the simulation of some indices for real-time online monitoring was ignored. Verification and testing of these indices require special hardware and software tools.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Type | Index | Abbreviation | Calculation | Stability Threshold | Reference | |
---|---|---|---|---|---|---|

System parameters (variables)-based | For Bus | L Index | $L$ | $L=\begin{array}{c}MAX\\ j\in {\alpha}_{L}\end{array}\left|1-\frac{{{\displaystyle \sum}}_{i\in {\alpha}_{G}}{\overline{F}}_{ji}{\overline{V}}_{i}}{{\overline{V}}_{j}}\right|$ | $L<1$ | [27] |

Power Stability Index | $PSI$ | $PSI=\frac{4{r}_{ij}\left({P}_{L}-{P}_{G}\right)}{{\left[\left|{V}_{i}\right|\mathrm{cos}\left(\theta -\delta \right)\right]}^{2}}$ | $PSI\le 1$ | [50] | ||

Voltage Deviation Index | $VDI$ | $VD{I}_{j}=\left|1-{V}_{j}\right|$ | Details are given in the reference | [51] | ||

Stability Index | $SI$ | $\begin{array}{cc}SI\left(m2\right)=\{{\left|V\left(m1\right)\right|}^{4}-4.0\{P& \left(m2\right)x\left(jj\right)\\ & -Q\left(m2\right)r\left(jj\right)\}{\}}^{2}\\ & -4.0\{P\left(m2\right)r\left(jj\right)\\ & +Q\left(m2\right)x\left(jj\right)\}{\left|V\left(m1\right)\right|}^{2}\end{array}$ | The smallest magnitude is the most sensitive to voltage collapse | [54] | ||

Voltage Collapse Prediction Index | $VCP{I}_{kth\text{}bus}$ | $VCP{I}_{kth\text{}bus}=1-\frac{{{\displaystyle \sum}}_{\begin{array}{c}m=1\\ m\ne k\end{array}}^{N}\left|{V}_{m}^{\prime}\right|}{{V}_{k}}$ | $VCP{I}_{kth\text{}bus}1$ | [67] | ||

Sensitivity Analysis | $SA$ | $\Delta {V}_{i}/\Delta {Q}_{i}$ $\Delta {V}_{i}/\Delta {P}_{i}$ | Details are given in the reference | [39] | ||

Bus Participation Factor | $BPF$ | Details are given in [40] | Using a power system simulation tool | [40] | ||

Voltage Stability Index | $VSI$ | $VS{I}_{i}={\left[1+\left(\frac{{I}_{i}}{{V}_{i}}\right)\left(\frac{\Delta {V}_{i}}{\Delta {I}_{i}}\right)\right]}^{\alpha}$ | $VS{I}_{i}\ge 0$ | [58] | ||

Equivalent Node Voltage Collapse Index | $ENVCI$ | $ENVCI=2\left({e}_{k}{e}_{n}+{f}_{k}{f}_{n}\right)-\left({e}_{k}^{2}+{e}_{k}^{2}\right)$ | $ENVCI>0$ | [59] | ||

Voltage Collapse Index | $VCI$ | $VC{I}_{i}={\left[1+\left(\frac{{I}_{i}\Delta {V}_{i}}{{V}_{i}\Delta {I}_{i}}\right)\right]}^{\alpha}$ | $VC{I}_{i}\ge 0$ | [58] | ||

Improved Voltage Stability Index | $IVSI$ | $\frac{-4{{\displaystyle \sum}}_{j=0}^{n}\left({G}_{ij}-{B}_{ij}\right)\left({P}_{i}+{Q}_{i}\right)\left(IVSI\le 1\right)}{{\left[{{\displaystyle \sum}}_{j=1}^{n}\left|{V}_{j}\right|\left[{G}_{ij}\left(\mathrm{cos}{\delta}_{ij}+\mathrm{sin}{\delta}_{ij}\right)-{B}_{ij}\left(\mathrm{cos}{\delta}_{ij}+\mathrm{sin}{\delta}_{ij}\right)\right]\right]}^{2}}$ | [51] | |||

Voltage Stability Factor | $VSF$ | $VS{F}_{total}={\displaystyle {\displaystyle \sum}_{m=1}^{k-1}}\left(2{V}_{m+1}-{V}_{m}\right)$ | The greatest magnitude is more stable | [25] | ||

Voltage Instability Proximity Index | VIPI | $VIPI=\theta =co{s}^{-1}\frac{{Y}_{s}^{T}\text{\hspace{1em}}Y\left(a\right)}{\Vert {Y}_{s}\Vert \text{}\Vert Y\left(a\right)\Vert}$ | Value is between the operating and critical load conditions | [20] | ||

For Line | L_{mn} Index | ${L}_{mn}$ | ${L}_{mn}=\frac{4Qrx}{{\left[\left|{V}_{s}\right|\mathrm{sin}\left(\theta -\delta \right)\right]}^{2}}$ | ${L}_{mn}<1$ | [60] | |

Line Voltage Factor | $LQP$ | $LQP=4\left(\frac{X}{{V}_{i}^{2}}\right)\left(\frac{X}{{V}_{i}^{2}}{P}_{i}^{2}+{Q}_{j}\right)$ | $LQP<1$ | [45] | ||

Line Index | $L$ | $L=4\left[{\left({x}_{eg}{P}_{leg}-{r}_{eg}{Q}_{leg}\right)}^{2}+{x}_{eg}{Q}_{L}+{r}_{eg}{P}_{leg}\right]$ | $L<1$ | [68] | ||

Voltage Collapse Proximity Indicator | $VCPI$ | $VCPI\left(1\right)=\frac{{P}_{r}}{{P}_{r}\left(\mathrm{max}\right)}$ $VCPI\left(2\right)=\frac{{Q}_{r}}{{Q}_{r}\left(\mathrm{max}\right)}$ $VCPI\left(3\right)=\frac{{P}_{l}}{{P}_{l}\left(\mathrm{max}\right)}$ $VCPI\left(4\right)=\frac{{Q}_{l}}{{Q}_{l}\left(\mathrm{max}\right)}$ | $VCPI<1$ | [69] | ||

Novel Line Stability Index | $NLSI$ | $NLS{I}_{ij}=\frac{{R}_{ij}{P}_{j}+{X}_{ij}{Q}_{j}}{0.25{V}_{i}^{2}}$ | $NLS{I}_{ij}<1$ | [70] | ||

Fast Voltage Stability Index | $FVSI$ | $FVS{I}_{ij}=\frac{4{Z}^{2}{Q}_{j}}{{V}_{i}^{2}x}$ | $FVS{I}_{ij}<1$ | [71] | ||

Critical Voltage | ${V}_{cr}$ | ${V}_{cr}=\frac{E}{2\mathrm{cos}\theta}$ | The critical voltage value | [52] | ||

Power Transfer Stability Index | $PTSI$ | $PTSI=\frac{2{S}_{L}{Z}_{Thev}\left(1+\mathrm{cos}\left(\beta -\alpha \right)\right)}{{E}_{Thev}^{2}}$ | $PTSI<1$ | [53] | ||

Line Voltage Stability Index | $LVSI$ | $LVSI=\frac{4r{P}_{r}}{{V}_{s}\mathrm{cos}{\left(\theta -\delta \right)}^{2}}$ | $LVSI\le 1$ | [1] | ||

Critical Boundary Index | $CBI$ | $CB{I}_{ik}=\sqrt{\Delta {P}_{ik}^{2}+\Delta {Q}_{ik}^{2}}$ | $CBI>1$ | [61] | ||

Line Voltage Stability Index | $LVSI$ | $LVSI=max\left(LVS{I}_{j}\right)\text{\hspace{1em}}\forall \text{}j=1,2,3,\dots l$ | $LVSI>1$ | [62] | ||

Integrated Transmission Line Transfer Index | $ITLTI$ | ${P}_{R}=-\frac{A{V}_{R}^{2}}{B}\mathrm{cos}\left(\beta -\alpha \right)+\frac{{V}_{S}{V}_{R}}{B}\mathrm{cos}\left(\beta -\alpha \right)$ | Details are given in the reference | [26] | ||

Jacobian matrix-based | Impedance Ratio Indicator | $\frac{{Z}_{ii}}{{Z}_{i}}$ | $\frac{{Z}_{ii}}{{Z}_{i}}\le 1$ | [55] | ||

Minimum Eigenvalue and Right eigenvector method | RE | $\Delta V={\displaystyle \sum}_{i}\frac{{\xi}_{i}\text{}{\eta}_{i}}{{\lambda}_{i}}\Delta Q$ | All eigenvalues should be positive | [21] | ||

Minimum Singular value | $\left[\begin{array}{c}\Delta \theta \\ \Delta V\end{array}\right]=V{{\displaystyle \sum}}^{-1}{U}^{T}\left[\begin{array}{c}\Delta F\\ \Delta G\end{array}\right]$ | Details are given in the reference | [18] | |||

Predicting Voltage Collapse | $\frac{V}{{V}_{0}}$ | The smallest index value | [1] | |||

Test Function | ${t}_{cc}=\left|{e}_{c}^{T}J{J}_{cc}^{-1}{e}_{c}\right|$ | Details are given in the reference | [58] | |||

Tangent Vector Index | $TVI$ | $TV{I}_{i}={\left|\frac{d{V}_{i}}{d\lambda}\right|}^{-1}$ | Depends on load increase | [59] | ||

Second-Order Index | $i$ | $i=\frac{1}{{i}_{0}}\text{}\frac{{\sigma}_{max}}{d{\sigma}_{max}/d{\lambda}_{total}}$ | $i>0$ | [60] | ||

Integral Steady-State Margin | ISSM | $ISSM=\left|\frac{{J}_{c}}{{J}_{o}}\right|$ | Between 0 and 1 | [20] | ||

Phasor Measurement Units (PMU)-based | Local Measurement-based | Recursive Least Square | $RLS$ | ${x}_{k}={x}_{k-1}+{G}_{k}\left({y}_{k}-{H}_{k}^{T}{x}_{k-1}\right)$ ${G}_{k}={P}_{k-1}{H}_{k}{\left(\lambda I+{H}_{k}^{T}{P}_{k-1}{H}_{k}\right)}^{-1}$ ${P}_{k}=\frac{1}{\lambda}\left(I-{G}_{k}{H}_{k}^{T}\right){P}_{k-1}$ | Details are given in the reference | [69] |

Voltage Instability Predictor | $VIP$ | $\Delta S=\frac{{\left({V}_{k}-{Z}_{Th}{I}_{k}\right)}^{2}}{4{Z}_{Th}}$ | Details are given in the reference | [68] | ||

Voltage Stability Load Bus Index | $VSLBI$ | $VSLB{I}_{k}=\frac{\left|{V}_{i}\left(k\right)\right|}{\left|\Delta {V}_{i}\left(k\right)\right|}$ | Details are given in the reference | [63] | ||

Approximate Approach | ${V}_{Li}={E}_{eq,i}-{Z}_{eq}{I}_{Li}$ ${Z}_{eq}={Z}_{LLii}$ | Details are given in the reference | [70] | |||

Simplified Voltage Stability Index | $SVSI$ | $SVS{I}_{i}=\frac{\Delta {V}_{i}}{\beta {V}_{i}}$ | $SVS{I}_{i}<1$ | [17] | ||

Observability-based | Voltage Collapse Proximity Indicator | $VCPI$ | $VCP{I}_{kth\text{}bus}=\left|1-\frac{{{\displaystyle \sum}}_{\begin{array}{c}m=1\\ m\ne k\end{array}}^{N}{V}_{m}^{\prime}}{{V}_{k}}\right|$ | $VCP{I}_{kth\text{}bus}1$ | [31] | |

Margin Voltage Stability Index | $MVSI$ | $VSI=min$ $\left(\frac{{P}_{margine}}{{P}_{max}}\frac{{Q}_{margine}}{{Q}_{max}}\frac{{S}_{margine}}{{S}_{max}}\right)$ | Details are given in the reference | [71] | ||

Sensitivity Related Eigenvalue | ${S}_{Qgq}=-{g}_{q}^{T}{\left({g}_{x}^{T}\right)}^{-1}{\Delta}_{x}{Q}_{g}$ | Details are given in the reference | [72] |

**Table 2.**The obtained indices’ magnitude for critical branch identification by each index (IEEE 14-bus system).

Branch | NLSI [48] | VCPI [47] | FVSI [49] | L_{mn} [59] | LQP [45] | L [46] | V_{cr} [50] | LVSI [52] | ||
---|---|---|---|---|---|---|---|---|---|---|

From | To | P | Q | |||||||

1 | 2 | 0.041723 | 0.05211165 | 0.052112 | 0.029621675 | 0.031626 | 0.026752 | 0.527999 | 0.527999 | 0.097140945 |

1 | 5 | 0.027323 | 0.053664748 | 0.053665 | 0.013449821 | 0.0129961 | 0.012704 | 0.527983 | 0.527983 | 0.651156176 |

2 | 3 | 0.299916 | 0.603222227 | 0.603222 | 0.145540502 | 0.158298 | 0.137778 | 0.509657 | 0.509657 | 1.250791857 |

2 | 4 | 0.076555 | 0.256248432 | 0.256248 | −0.027923844 | −0.029982 | −0.025188 | 0.514064 | 0.514064 | 0.636275 |

2 | 5 | 0.026044 | 0.045083448 | 0.045083 | 0.0112837 | 0.0118415 | 0.010191 | 0.516391 | 0.516391 | 0.1137122 |

3 | 4 | 0.099444 | 0.277198618 | 0.277199 | −0.030169995 | −0.029247 | −0.026155 | 0.496846 | 0.496846 | 1.187916261 |

4 | 5 | 0.006521 | 0.011445692 | 0.011446 | 0.002863807 | 0.0028176 | 0.002602 | 0.502886 | 0.502886 | 0.051265482 |

4 | 7 | 0 | 0 | 0 | 0 | 0 | 0.495065 | 0.495065 | 0 | |

4 | 9 | 0.356591 | 0.627703555 | 0.627704 | 0.356590282 | 0.3589291 | 0.35659 | 0.491635 | 0.491635 | 0.000174855 |

5 | 6 | 0.07274 | 0.115325447 | 0.115325 | 0.072739998 | 0.0734022 | 0.07274 | 0.494136 | 0.494136 | 4.77773E-05 |

6 | 11 | 0.024123 | 0.028720079 | 0.02872 | 0.015360637 | 0.0155092 | 0.012508 | 0.517274 | 0.517274 | 0.060026116 |

6 | 12 | 0.040494 | 0.056754618 | 0.056755 | 0.01760098 | 0.0178617 | 0.0143 | 0.516576 | 0.516576 | 0.131357137 |

6 | 13 | 0.057598 | 0.070516198 | 0.070516 | 0.03320431 | 0.0337737 | 0.026398 | 0.516382 | 0.516382 | 0.14285981 |

7 | 8 | 0 | 0 | 0 | 0 | 0 | 0.516396 | 0.516396 | 0 | |

7 | 9 | 0.064826 | 0.114111626 | 0.114112 | 0.064825202 | 0.0648746 | 0.064825 | 0.512818 | 0.512818 | 0.001376514 |

9 | 10 | 0.027853 | 0.032929845 | 0.03293 | 0.020073907 | 0.0201163 | 0.017582 | 0.509736 | 0.509736 | 0.081531112 |

9 | 14 | 0.116444 | 0.154906644 | 0.154907 | 0.059217931 | 0.0603135 | 0.048499 | 0.507436 | 0.507436 | 0.346865422 |

10 | 11 | 0.022919 | 0.028003345 | 0.028003 | 0.014804447 | 0.0147367 | 0.01252 | 0.508084 | 0.508084 | 0.069127813 |

12 | 13 | 0.148792 | 0.15512846 | 0.155128 | 0.092525866 | 0.0928123 | 0.041648 | 0.509234 | 0.509234 | 0.194360274 |

13 | 14 | 0.155423 | 0.203896609 | 0.203897 | 0.078305626 | 0.0795052 | 0.063087 | 0.504769 | 0.504769 | 0.447143333 |

**Table 3.**The obtained indices’ magnitude for critical branch identification by each index (IEEE 30-bus system).

Branch | NLSI [48] | VCPI [47] | FVSI [49] | L_{mn} [59] | LQP [45] | L [46] | V_{cr} [50] | LVSI [52] | ||
---|---|---|---|---|---|---|---|---|---|---|

From | To | P | Q | |||||||

1 | 2 | 0.040829 | 0.050792427 | 0.050792 | 0.028895385 | 0.0310744 | 0.025997 | 0.098337212 | 0.52767 | 0.090772386 |

1 | 3 | 0.010919 | 0.014938477 | 0.014938 | 0.007585635 | 0.0083082 | 0.007057 | 0.012527058 | 0.52543 | 0.025650757 |

2 | 4 | 0.026048 | 0.045052501 | 0.045053 | 0.011276206 | 0.0118532 | 0.01018 | 0.028476133 | 0.51566 | 0.112409942 |

2 | 5 | 0.30087 | 0.604439542 | 0.60444 | 0.145826669 | 0.160899 | 0.138008 | 0.328598596 | 0.50665 | 1.144735735 |

2 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.5128 | 0 |

3 | 4 | 0.006176 | 0.01041434 | 0.010414 | 0.002609098 | 0.002668 | 0.002327 | 0.007126896 | 0.50382 | 0.03010354 |

4 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.49661 | 0 |

4 | 12 | 0.07499 | 0.118891841 | 0.118892 | 0.074989455 | 0.0757245 | 0.074989 | 0.085385582 | 0.48891 | 4.50637E-05 |

5 | 7 | 0.090705 | 0.115125046 | 0.115125 | 0.057375984 | 0.0563883 | 0.049579 | 0.119787331 | 0.49235 | 0.340584256 |

6 | 7 | 0.058802 | 0.07838679 | 0.078387 | 0.0386867 | 0.0395283 | 0.034978 | 0.09272165 | 0.49284 | 0.206965597 |

6 | 8 | 0.063398 | 0.0702055 | 0.070206 | 0.053334474 | 0.0537404 | 0.049309 | 0.39519504 | 0.49482 | 0.170861069 |

6 | 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.49027 | 0 |

6 | 10 | 0.043518 | 0.108695808 | 0.108696 | 0.043517353 | 9.9869167 | 0.043517 | 0.045080286 | 0.06397 | 2.27972E-07 |

6 | 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.49504 | 0 |

8 | 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.49455 | 0 |

9 | 10 | 0.007967 | 0.019898883 | 0.019899 | 0.007966678 | 0.5637133 | 0.007967 | 0.00954225 | 0.0665 | 2.13041E-07 |

9 | 11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.50967 | 0 |

10 | 17 | 0.028633 | 0.033734145 | 0.033734 | 0.02059127 | 0.0737685 | 0.017952 | 0.041426624 | 0.50263 | 0.01411637 |

10 | 20 | 0.012902 | 0.01767585 | 0.017676 | 0.00643366 | 0.0065182 | 0.005359 | 0.014111878 | 0.50097 | 0.04240821 |

10 | 21 | 0.053035 | 0.060349918 | 0.06035 | 0.037360687 | 0.0376328 | 0.030728 | 0.097034106 | 0.50193 | 0.121570373 |

10 | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.50197 | 0 |

12 | 13 | 0 | 0 | 0 | 0 | 0 | 0 | 0.51065 | 0 | |

12 | 14 | 0.041984 | 0.059037913 | 0.059038 | 0.018051019 | 0.0183289 | 0.014659 | 0.046906472 | 0.50847 | 0.136465701 |

12 | 15 | 0.031106 | 0.041296468 | 0.041296 | 0.014679599 | 0.0149431 | 0.011672 | 0.035572384 | 0.50824 | 0.088769537 |

12 | 16 | 0.024647 | 0.02937138 | 0.029371 | 0.015701355 | 0.0158558 | 0.012805 | 0.028024412 | 0.50924 | 0.061544138 |

14 | 15 | 0.084992 | 0.09163833 | 0.091638 | 0.040839478 | 0.0409835 | 0.018357 | 0.092755535 | 0.50151 | 0.120696655 |

15 | 18 | 0.020048 | 0.027428989 | 0.027429 | 0.009061192 | 0.0091584 | 0.007301 | 0.021616528 | 0.49755 | 0.062840994 |

15 | 23 | 0.023879 | 0.028382608 | 0.028383 | 0.014939354 | 0.0150415 | 0.011999 | 0.026091741 | 0.49812 | 0.058727501 |

16 | 17 | 0.058128 | 0.072986174 | 0.072986 | 0.043887473 | 0.0440292 | 0.040854 | 0.070101281 | 0.50263 | 0.239539698 |

18 | 19 | 0.039604 | 0.051041895 | 0.051042 | 0.020694215 | 0.0207567 | 0.016627 | 0.043741276 | 0.49231 | 0.115489427 |

19 | 20 | 0.004651 | 0.006144183 | 0.006144 | 0.002260905 | 0.0022532 | 0.001809 | 0.005017176 | 0.49186 | 0.014409058 |

21 | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0.4962 | 0 | |

22 | 24 | 0.0823 | 0.086120355 | 0.08612 | 0.06338884 | 0.0639164 | 0.044869 | 0.098570945 | 0.49575 | 0.125606794 |

23 | 24 | 0.112158 | 0.121864537 | 0.121865 | 0.085002814 | 0.0852595 | 0.068605 | 0.12586848 | 0.4924 | 0.222962198 |

24 | 25 | 0 | 0 | 0 | 0 | 0 | 0 | 0.49107 | 0 | |

25 | 26 | 0.068102 | 0.072452792 | 0.072453 | 0.048854314 | 0.0493389 | 0.033735 | 0.07095236 | 0.4881 | 0.108666623 |

25 | 27 | 0 | 0 | 0 | 0 | 0 | 0 | 0.49042 | 0 | |

28 | 27 | 0 | 0 | 0 | 0 | 0 | 0 | 0.48512 | 0 | |

27 | 29 | 0.034381 | 0.042956211 | 0.042956 | 0.018252083 | 0.0186823 | 0.014258 | 0.036055342 | 0.49025 | 0.084977094 |

27 | 30 | 0.173159 | 0.252141728 | 0.252142 | 0.056013018 | 0.0583534 | 0.043683 | 0.189629084 | 0.48792 | 0.515014545 |

29 | 30 | 0.135086 | 0.197026775 | 0.197027 | 0.043749241 | 0.0444823 | 0.034177 | 0.139543964 | 0.47839 | 0.435559311 |

**Table 4.**The obtained indices’ magnitude for weak bus identification by each index (IEEE 14-bus system).

Bus | Branch | VSF | PSI | Vj/Vo | BPF | RE | S | |
---|---|---|---|---|---|---|---|---|

From | To | |||||||

4 | 2 | 4 | 1.080872 | 1.93368064 | 0.94908 | 0.0139 | 0.119854 | 0.044 |

3 | 4 | 1.004157 | 5.533106601 | 0.94908 | ||||

5 | 1 | 5 | 1.107938 | 0.263729008 | 0.954419 | 0.0064 | 0.080149 | 0.0427 |

2 | 5 | 1.074834 | 0.306090238 | 0.954419 | ||||

4 | 5 | 1.016568 | 0.244039621 | 0.954419 | ||||

7 | 4 | 7 | 0.976958 | 0 | 0.949303 | 0.1616 | 0.401572 | 0.1417 |

9 | 4 | 9 | 0.986545 | 3.55711E-05 | 0.929802 | 0.2256 | 0.476716 | 0.1377 |

7 | 9 | 1.067908 | 1.96145E-05 | 0.929802 | ||||

10 | 9 | 10 | 1.060878 | 1.045255773 | 0.925681 | 0.2333 | 0.48392 | 0.1621 |

11 | 6 | 11 | 1.083193 | 0.302194356 | 0.93142 | 0.93142 | 0.0926 | 0.48392 |

10 | 11 | 1.045101 | 0.532718621 | 0.93142 | ||||

12 | 6 | 12 | 1.085044 | 0.665506183 | 0.930355 | 0.0095 | 0.096489 | 0.1377 |

13 | 6 | 13 | 1.089898 | 0.645425171 | 0.926024 | 0.0198 | 0.138994 | 0.0872 |

12 | 13 | 1.060002 | 0.36546909 | 0.926024 | ||||

14 | 9 | 14 | 1.076698 | 2.150384317 | 0.912321 | 0.2374 | 0.48619 | 0.2233 |

13 | 14 | 1.065465 | 2.537073855 | 0.912321 |

**Table 5.**The obtained indices’ magnitude for three weakest buses identification by each index (IEEE 30-bus system).

Bus | From | To | VSF | PSI | Vj/Vo | BPF | RE | S |
---|---|---|---|---|---|---|---|---|

26 | 25 | 26 | 1.036053 | 0.369924625 | 0.93985 | 0.174031 | 0.414541 | 0.7299 |

29 | 27 | 29 | 1.044453 | 0.439171371 | 0.945386 | 0.157802 | 0.399068 | 0.6733 |

30 | 29 | 30 | 1.016233 | 2.707961113 | 0.934087 | 0.156804 | 0.394602 | 0.6024 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Danish, M.S.S.; Senjyu, T.; Danish, S.M.S.; Sabory, N.R.; K, N.; Mandal, P. A Recap of Voltage Stability Indices in the Past Three Decades. *Energies* **2019**, *12*, 1544.
https://doi.org/10.3390/en12081544

**AMA Style**

Danish MSS, Senjyu T, Danish SMS, Sabory NR, K N, Mandal P. A Recap of Voltage Stability Indices in the Past Three Decades. *Energies*. 2019; 12(8):1544.
https://doi.org/10.3390/en12081544

**Chicago/Turabian Style**

Danish, Mir Sayed Shah, Tomonobu Senjyu, Sayed Mir Shah Danish, Najib Rahman Sabory, Narayanan K, and Paras Mandal. 2019. "A Recap of Voltage Stability Indices in the Past Three Decades" *Energies* 12, no. 8: 1544.
https://doi.org/10.3390/en12081544