# Analytical Methods of Voltage Stability in Renewable Dominated Power Systems: A Review

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## Abstract

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## 1. Introduction

- Investigate the analysis and verification of voltage stability studies based on different renewable energy generation types;
- Classify and compare voltage stability analysis methods based on different microgrid operation modes and types of DGs; and
- Evaluate voltage stability techniques and conduct a simulation verification to demonstrate the most suitable simulation platform with different microgrid settings.

## 2. Voltage Stability Methods of Analysis

#### 2.1. Static Voltage Analysis Techniques

#### 2.1.1. Continuation Load Flow Method Using P–V and V–Q Curves

#### 2.1.2. Modal Analysis of the Jacobian Matrix Based on V–Q Sensitivity

#### 2.1.3. Singular Value Decomposition Using Network-Load Admittance Ratio

#### 2.1.4. Transfer Capability Evaluation Using Static Analysis Methods

#### 2.2. Dynamic Voltage Analysis Techniques

#### 2.2.1. Small Signal Analysis Method

#### 2.2.2. Large Signal Analysis Method

## 3. Voltage Stability Analysis Indices

#### 3.1. VSI Classification

#### 3.2. Voltage Stability Indices Review

#### 3.2.1. Jacobian-Matrix-Based VSIs

#### 3.2.2. System-Variable-Based VSIs

## 4. Verification Case Studies for the Voltage Stability Analysis

#### 4.1. Analysis and Verification Case Studies with Integrated PV Generation Only

#### 4.2. Analysis and Verification Case Studies with Integrated Wind Generation Only

#### 4.3. Analysis and Verification Cases with Hybrid Distributed Generation

- When a sampling method uses the standard error of the mean (SEM), the fitting probability ratio may be negative, while sampling methods using CMEM have greater effectiveness and accuracy;
- The computational speed of the method based on CMEM is significantly higher than that of the Monte Carlo method, resulting in a time saving of 99.95%;
- The higher the penetration rate of renewable energy, the greater the load margin fluctuation, leading to a more unstable system;
- As the correlation degree of external weather factors, such as the wind speed and solar irradiation rate, increases, the mean value of the load margin is almost unchanged, but the fluctuation degree increases.

#### 4.4. Examples of Simulation Validation under Different Scenarios

- Basic load condition;
- Different load models;
- The model works under the critical state.

- A two-node power system model with a 90-degree initial voltage angle for a flat start;
- A 1900 MW pure active load connected at the receiving end of the power system.

- Bus 8–9 outage;
- G3 outage;
- Bus 12 load increment.

## 5. Conclusions

- Systematic development of dynamic voltage stability analysis methods: Although several dynamic methods to evaluate the voltage profile of a system are available, additional work needs to be performed to improve their accuracy and efficacy levels.
- Online real-time techniques for assessing the state of the system’s voltage and the threshold of instability: It can be anticipated that power systems can be further optimized in an efficient and timely manner if the voltage collapse is detected at an early stage.
- Coping with increasing asynchronous generation from renewables: The increasing complexity of the network due to the higher level of renewable penetration may lead to more stability issues. Increasing the integration of DGs may exponentially increase the risk of large disturbance instability. Therefore, it may become important to coordinate the expanding asynchronous power supplies with the current synchronous generation.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**(

**a**) P–V and (

**b**) V–Q curves obtained for the voltage stability analysis of a power system showing stable and unstable regions.

**Figure 7.**(

**a**) Two-bus system; (

**b**) equivalent circuit after removal of the transformer (elements transferred from the primary side to the secondary side) in the presence of tap chargers and DGs.

Ref | VSI/Method | Analytical Foundation | Index Type | Equation | Stability Threshold |
---|---|---|---|---|---|

[78] | ${J}^{C}\in {\complement}^{2n\times 2n}$ | Jacobian matrix singular point | Static | ${J}^{C}=\left[\begin{array}{cc}{}_{}{}^{-}I{}^{n}& {V}^{n}{}_{}{}^{-}Y{}^{b}\\ {}_{}{}^{-}V{}^{n}{Y}^{b}& {I}^{n}\end{array}\right]=\phantom{\rule{0ex}{0ex}}\left[\begin{array}{cc}{V}^{n}& \\ & {}_{}{}^{-}V{}^{n}\end{array}\right]\left[\begin{array}{cc}{\left({V}^{n}\right)}^{-1}{}_{}{}^{-}I{{}^{n}}^{}& {}_{}{}^{-}Y{}^{b}\\ {Y}^{b}& {\left({}_{}{}^{-}V{}^{n}\right)}^{-1}{I}^{n}\end{array}\right]$ $\mathsf{\Lambda}={\left({V}^{n}\right)}^{-1}{Y}^{b/n}{V}^{n}$ | ${\lambda}^{cr}=\underset{{\lambda}_{i}}{\mathrm{min}}\left|{\lambda}_{i}-1\right|=0$, |

[79] | ${L}_{s}$-index | Load flow equation | Static | ${L}_{i}=\frac{1}{{V}_{i}}\sqrt{{f}^{2}+{g}^{2}}$ | ${L}_{s}\le 1$ |

[75] | TVAI | Approximated step function | Dynamic | $F={F}_{1}+{F}_{2}={{\displaystyle \sum}}_{i=0}^{n}{{\displaystyle \sum}}_{j=1}^{m}{K}_{j}{g}_{j}\left(V\left|{t}_{i}\right|\right)\left|V\left[{t}_{i}\right]-{V}_{N}\right|\Delta t+{{\displaystyle \sum}}_{i=1}^{n}\frac{k}{1+{e}^{-\frac{\left(V\left|{t}_{i}\right|-{V}_{N}-a\right)c}{b}}}$ | ${g}_{j}\left(V\left|{t}_{i}\right|\right)=\left\{\begin{array}{c}1\left({V}_{cr.j+1}\le V\left[{t}_{i}\right]\le {V}_{cr.j}\right)\\ 0(V\left[{t}_{i}\right]\le {V}_{cr.j+1}orV\left[{t}_{i}\right]{V}_{cr.j})\end{array}\right.$ |

[73] | VSI | Two-bus equivalent circuit | Static | $\mathrm{VSI}={V}_{1}{}^{2}-4\left(PR+XQ\right)$ | $\mathrm{VSI}\ge 0$ |

[80] | VSI | Two-bus equivalent circuit | Static | $\mathrm{VSI}=\frac{4{R}_{i}\times {P}_{i}{}^{2}}{{\left({Q}_{i}{V}_{i-1}sin{\delta}_{i}+{P}_{i}{V}_{i-1}cos{\delta}_{i}\right)}^{2}}\times \left({P}_{i}+\frac{{Q}_{i}{}^{2}}{{P}_{i}}\right)$ | $\mathrm{VSI}\le 1$ |

[60] | $v\left({X}^{S},{X}^{u}\right)$ | Energy function | Static | $v\left({X}^{S},{X}^{u}\right)={\displaystyle \sum _{i=1}^{n}}\left[{{\displaystyle \int}}_{{\theta}_{i}{}^{S}}^{{\theta}_{i}^{u}}{f}_{i}\left(\theta ,V\right)d{\theta}_{i}+{{\displaystyle \int}}_{{V}_{i}^{S}}^{{V}_{i}^{u}}{g}_{i}\left(\theta ,V\right)d{V}_{i}\right]$ | $v\left({X}^{S},{X}^{u}\right)\ge 0$ |

[43] | ${M}_{n/d}$ | Load flow Jacobian matrix | Static | ${M}_{n/d}=1-\frac{{P}_{L1}}{{P}_{L1}^{Lim}}=1-\frac{{R}_{n/d}{\left|1\angle {\alpha}_{loss}+1\angle {\alpha}_{d}\right|}^{2}}{{\left|1\angle {\alpha}_{loss}+{R}_{n/d}\angle {\alpha}_{d}\right|}^{2}}$ | Ranges from 0 to 1 (Stability limit point to no load) |

[81] | $ca{t}_{VSI\left(n\right)}$ | Saddle-node and finite induced bifurcation | Static | $ca{t}_{VSI\left(n\right)}={\left({P}_{n}{R}_{mn}+{Q}_{n}{X}_{mn}-0.5{\left|{V}_{m}\right|}^{2}\right)}^{2}-{Z}_{mn}^{2}\left({P}_{m}{}^{2}+{Q}_{m}{}^{2}\right)$ | Ranges from 0.25 to 0 (No load to collapse point) |

[82] | Sensitivity matrix | Linearized load flow equation | Static | $\left[\begin{array}{c}\Delta P\\ \Delta Q\end{array}\right]=\left[\begin{array}{cc}{J}_{P\theta}& {J}_{PV}\\ {J}_{Q\theta}& {J}_{QV}\end{array}\right]\left[\begin{array}{c}\Delta \theta \\ \Delta V\end{array}\right]$ $\Delta Q=\left({J}_{QV}-{J}_{Q\theta}{J}_{P\theta}{}^{-1}{J}_{PV}\right)\Delta V$ | If the sensitivity measure is positive, the system is stable; if not, the system is unstable. |

[83] | GSA | Optimal load flow and probabilistic model | Static | $P{L}_{ij}=-{t}_{ij}{G}_{ij}{V}_{i}{}^{2}+{V}_{i}{V}_{j}\left({G}_{ij}cos{\theta}_{ij}+{B}_{ij}sin{\theta}_{ij}\right)$ $Q{L}_{ij}={t}_{ij}{B}_{ij}{V}_{i}{}^{2}-\frac{{B}_{ij}{V}_{i}{}^{2}}{2}+{V}_{i}{V}_{j}\left({G}_{ij}sin{\theta}_{ij}-{B}_{ij}cos{\theta}_{ij}\right)$ | ${P}_{2}{L}_{ij}+{Q}_{2}{L}_{ij}\le {S}_{2}{L}_{ij,max}$ |

[84] | IB index | Traditional IB index | Dynamic | $I{B}_{i}=\frac{\left|{{Z}^{\prime}}_{eq,i}\right|}{\left|{{Z}^{\prime}}_{Li}\right|}=\frac{\left|{{Z}^{\prime}}_{LLii}+{Z}_{coupled,i}\right|}{\left|{r}_{i}^{2}{Z}_{Li}\right|}$ | If the load impedance ${{Z}^{\prime}}_{Li}$ is located inside the circle with a radius $\left|{{Z}^{\prime}}_{eq,i}\right|$, the system is unstable. |

[85] | MSV(Minimum Singular Value) | Singular point of Jacobian matrix | Dynamic | $diag\left[\Delta \sum \right]=diag\left|{\left[SU\right]}^{T}\xb7\left[HP,V\right]\xb7\left[\Delta V\right]\xb7\left[SV\right]\right|$ $\sum Case=\sum BaseCase+\Delta \sum Casel$ | ΔΣ is the change in singular value due to the uncertainty of wind power. MSV is used to assess whether the added wind turbine generator has a positive or negative effect on the voltage stability of the power system. |

[86] | V–Q modal analysis, V–Q curve analysis | V–Q modal analysis, V–Q curve analysis | Static | $\Delta U=\xi \xb7u$ $\Delta Q=\xi \xb7q$ | For modal analysis: A positive value means the system is stable. A negative value means the system is unstable. For the V–Q curve, the reactive power margin can show the voltage collapse margin. |

[87] | $VS{I}_{ij}$ | P–V Curve theory | Static | $VS{I}_{ij}={V}_{i}^{4}-4\left({P}_{j}{R}_{ij}+{Q}_{j}{X}_{ij}\right){V}_{i}^{2}-4\left({P}_{j}{X}_{ij}-{Q}_{j}{R}_{ij}\right)$ ${f}_{2}=\mathrm{max}\left(\mathrm{min}\left(VS{I}_{ij}\right)\right)$ | This essay uses the combined method to conduct the voltage stability analysis for the P–V curve; the active power margin can show the voltage collapse margin. For VSI, the larger the voltage stability index, the more stable the system. |

[88] | Monte Carlo based voltage stability analysis | Eigenvalue, reactive power margin, real and reactive power loss Monte Carlo simulation | Static | $\Delta Q=\lambda \varphi \zeta \Delta V$ $\Delta V=\varphi \lambda -1\zeta \Delta Q\Delta V=\sum i\varphi i\lambda -1i\zeta i\Delta Q$ $Pki=\varphi ki\zeta ki$ ${P}_{L,i}^{hr,k}=\mathbb{R}\left({\mathbb{P}}_{L,i}^{hr,k}~\mathcal{N}\left({\mu}_{L,i}^{hr,k},{\sigma}_{L,i}^{hr,{k}^{2}}\right)\right)$ ${Q}_{L,i}^{hr,k}={P}_{L,i}^{hr,k}\mathrm{tan}\left(co{s}^{-1}\left(pf\right)\right)$ | For the modal analysis: A positive value means the system is stable. A negative value means the system is unstable. For the V–Q curve, the reactive power margin can show the voltage collapse margin. |

[89] | LILO | Integral-integral estimate theory, LIOS properties | Dynamic | $\theta \circ \left[{\alpha}_{0}^{IOS}\left(\left|{x}_{0}\right|\right)+{\phi}^{IOS}\circ {{\displaystyle \int}}_{0}^{t}\left(\left|\omega \left(s\right)\right|\right)ds\left]\le \mathrm{min}\right[{\mathsf{\Theta}}_{\tilde{{x}_{0}}},Y\right]$ | The system outputs satisfy the equation |

[90] | VPS | P–V and V–Q curve | Static | $VPS=\Vert \frac{dV}{dP}\Vert $ | The active power margin can show the margin of voltage collapse |

[74] | $\mathrm{FVSI},\text{}{L}_{mn}$ | Line stability index | Static | $FVS{I}_{ij}=\frac{4{Z}^{2}{Q}_{j}}{{\left|{V}_{i}\right|}^{2}{X}^{\prime}}$ ${L}_{mn}=\frac{4X{Q}_{j}}{{{\left[{V}_{i}sin\left(\theta -\delta \right)\right]}^{2}}^{\prime}}$ | ${L}_{mn}\le 1$, the system is stable $FVSI\le 1$, the system is stable FVSI is close to 1, and the system is close to instability. |

[91] | Voltage Stability Condition | Steady-state load properties, Lyapunov stability theory | Static | $\left({\beta}_{{s}_{i}}-2\right){Q}_{{s}_{i}}\left({V}_{i}\right)+{\displaystyle \sum _{j\in {N}_{G}}}{w}_{{\u03f5}_{{j}_{i}}^{gl}}^{Q}$ | Assuming that $\left|{\theta}_{i}-{\theta}_{j}\right|\le \pi /2$ for any branch (i,j), the power system is at a QV regular operating point, if the following condition is satisfied:$\left({\beta}_{{s}_{i}}-2\right){Q}_{{s}_{i}}\left({V}_{i}\right)+{\displaystyle \sum _{j\in {N}_{G}}}{w}_{{\u03f5}_{{j}_{i}}^{gl}}^{Q}>0\left(i=1,\dots ,n\right)$ |

[92] | P–V and V–Q curve | P–V and V–Q curve | Static | $VPS=\Vert \frac{dV}{dP}\Vert $ | The active power margin can show the margin of voltage collapse. |

[5] | PV analysis | Continuation load flow algorithm | Static | ${P}_{Di}\left(\lambda \right)={P}_{Di0}+{k}_{Di}\lambda {P}_{Di0}=\left(1+{k}_{Di}\lambda \right){P}_{Di0}$ ${Q}_{Gi}\left(\lambda \right)={Q}_{Gi0}+{k}_{Gi}\lambda {Q}_{Gi0}=\left(1+{k}_{Gi}\lambda \right){Q}_{Gi0}$ | The active power margin can show the margin of voltage collapse. |

[93] | PV analysis | Continuation load flow algorithm | Static | $\left[\begin{array}{c}\Delta {V}_{Re,k}\\ \Delta {V}_{Im,k}\\ \vdots \\ \Delta {V}_{Re,m}\\ \Delta {V}_{Im,m}\\ \Delta \gamma \end{array}\right]$$={\left[\begin{array}{cccc}& & & \\ & & & \\ & J& & {J}_{I,\gamma}\\ & & & \\ & 0& & 1\end{array}\right]}^{-1}\left[\begin{array}{c}\Delta {I}_{Im,k}\\ \Delta {I}_{Re,k}\\ \vdots \\ \Delta {I}_{Im,m}\\ \Delta {I}_{Re,m}\\ \Delta \gamma \end{array}\right]$ | The active power margin can show the margin of voltage collapse. |

[94] | Software-based Simulation method | Software function | Static | N/A | Compare the system voltage plots with the voltage sag or UCAP between simulation software packages. |

[95] | VSI | Optimal load flow | Static | $S{I}_{k+1}=\frac{\begin{array}{c}4\xb7{\left[{P}_{k+1}\xb7{X}_{k}-{R}_{k}\xb7{Q}_{k+1}\right]}^{2}\xb7\\ 4\xb7\left[{P}_{k+1}\xb7{R}_{k}+{X}_{k}\xb7{Q}_{k+1}\right]\xb7{V}_{M1}^{2}\end{array}}{{V}_{M1}^{4}}$ $\mathrm{VSI}=\mathrm{MAX}(\mathrm{S}{I}_{k+1})\mathrm{for}\text{}k=1,2,3,\dots N$ | $\mathrm{VSI}\le 1$ |

[96] | Simulation Software-based method | Modal Analysis | Static | N/A | Determined using the General Algebraic Modeling System (GAMS) optimization software and analyzed with the CONOPT4 solver. |

[97] | P–V and V–Q curve | P–V and V–Q curve | Static | ${P}_{L}=VRIcos{\theta}_{L}={V}_{S}^{2}{Z}_{L}cos{\theta}_{L}{Z}_{TL}{}^{2}+{Z}_{L}{}^{2}+2{Z}_{TL}{Z}_{L}cos\left({\theta}_{TL}-{\theta}_{L}\right)$ ${Q}_{L}=VRIsin{\theta}_{L}={V}_{S}^{2}{Z}_{L}sin{\theta}_{L}{Z}_{TL}{}^{2}+{Z}_{L}{}^{2}+2{Z}_{TL}{Z}_{L}cos\left({\theta}_{TL}-{\theta}_{L}\right)$ | The active power margin can show the margin of voltage collapse. |

[98] | ${L}_{k}$ | P–V curve | Static | ${L}_{k}=\frac{\sqrt{\left(\left({R}_{S}^{2}+{X}_{S}^{2}\right)\left({\left({P}_{r}-{P}_{DG}\right)}^{2}+{Q}_{r}^{2}\right)\right)}}{\left|{\left|{V}_{S}\right|}^{2}-2\left({R}_{S}\left({P}_{r}-{P}_{DG}\right)+{X}_{S}{Q}_{r}\right)\right|}$ | ${L}_{k}\le 1$ |

[54] | $f\left(\left|\overrightarrow{U}\right|\right)$ | Topological model | Static | $f\left(\left|\overrightarrow{U}\right|\right)=\frac{\left|\overrightarrow{E}\right|\xb7{Z}_{3}}{\left|\overrightarrow{U}\right|\xb7\left({Z}_{1}+{Z}_{3}\right)-[\frac{P}{\left|\overrightarrow{U}\right|}+k\xb7[{U}_{0}-\left|\overrightarrow{U}\right|\xb7j]\xb7\mathbb{Z}]}$ | The number of intersection points between the unit circle and the function’s curve can show stability. The presence of zero intersection points indicates instability, and the presence of two intersection points indicates stability. The presence of one intersection point indicates a stable margin. |

[34] | VSI | Time-synchronized measurements | Dynamic | $VSI=\mathrm{min}\left(\frac{{P}_{max}-P}{{P}_{max}},\frac{{Q}_{max}-Q}{{Q}_{max}},\frac{{S}_{max}-S}{{S}_{max}}\right)$ ${P}_{max}=\frac{QR}{X}-\frac{{V}_{S}^{2}R}{2{X}^{2}}+\frac{\left|{Z}_{th}\right|{V}_{s}\sqrt{{V}_{s}^{2}-4QX}}{2{X}^{2}}$ ${Q}_{max}=\frac{PX}{R}-\frac{{V}_{S}^{2}X}{4{R}^{2}}+\frac{\left|{Z}_{th}\right|{V}_{s}\sqrt{{V}_{s}^{2}-4PR}}{2{R}^{2}}$ ${S}_{max}=\frac{{V}_{S}^{2}\left[\left|{Z}_{th}\right|-\left(sin\left(\theta \right)X+cos\left(\theta \right)R\right)\right]}{2{\left(cos\left(\theta \right)X-sin\left(\theta \right)R\right)}^{2}}$ | The system is stable if the VSI is 1. The system is unstable if the VSI is 0. |

[70] | ${S}_{\lambda \omega}$ | Jacobian matrix singular point, PDF | Static | ${S}_{\lambda \omega}=\frac{\Delta \lambda}{\Delta \omega}=-\frac{M{\left.F\right|}_{\omega}}{M{\left.F\right|}_{\lambda}}$ $\Delta {\lambda}_{i}={\displaystyle \sum _{r}}{s}_{ir0}\Delta {\omega}_{r}$ | The formulation can measure the loading margin. |

Operation Mode | Type of DG(s) | References |
---|---|---|

Grid-Connected | PV | [43,78,88,90,97,98] |

Wind | [5,54,74,84,85,89,94,96] | |

PV, Wind | [70,79,80,83,92] | |

PV, Hydro | [75] | |

PV, Wind, Hydro | [86] | |

Islanded | PV | [60] |

Wind | [81] | |

PV, Wind | [34,82] |

Voltage Stability Index | Formulation | Calculation Runtime (Units) |
---|---|---|

Index 2003 | $SI=0.5{\left|{V}_{2}\right|}^{2}-PR-QX$ | 0.8171 |

Index 2014 | $VSI={V}_{1}^{2}-2PR-2QX-R{P}_{loss}-X{Q}_{loss}$ | 0.8172 |

Novel Index | $VSI={V}_{1}^{2}-4\left(PR+XQ\right)$ | 0.7997 |

Voltage Stability Analysis Method | Simulation Result |
---|---|

L-index method | This method requires the least amount of calculation and has a good level of consistency with most other methods. |

Modal analysis | The method is most suitable for determining the strongest and weakest buses in the system. |

V–Q sensitivity analysis | This scheme has difficulty distinguishing different stability modes in the system and may be misleading when applied to large systems with multiple regions. |

Power flow based methods | Too many system parameters are considered in the calculation, and the accuracy is relatively low. |

Dynamic voltage stability analysis | Cannot accurately calculate the stability margin for each bus. Overlapped time-domain actions in the interconnected networks may exist, leading to the wrong analysis result. |

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**MDPI and ACS Style**

Liang, X.; Chai, H.; Ravishankar, J.
Analytical Methods of Voltage Stability in Renewable Dominated Power Systems: A Review. *Electricity* **2022**, *3*, 75-107.
https://doi.org/10.3390/electricity3010006

**AMA Style**

Liang X, Chai H, Ravishankar J.
Analytical Methods of Voltage Stability in Renewable Dominated Power Systems: A Review. *Electricity*. 2022; 3(1):75-107.
https://doi.org/10.3390/electricity3010006

**Chicago/Turabian Style**

Liang, Xinyu, Hua Chai, and Jayashri Ravishankar.
2022. "Analytical Methods of Voltage Stability in Renewable Dominated Power Systems: A Review" *Electricity* 3, no. 1: 75-107.
https://doi.org/10.3390/electricity3010006