Matrices – System of Linear Equations (Part 1) | Don’t Memorise
Matrices – System of Linear Equations (Part 1) | Don’t Memorise

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



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


Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Conflicts of Interest


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Ref VSI/Method Analytical


Index Type Equation Stability Threshold
[78] Jacobian matrix singular point Static ,
[79] -index Load flow equation Static
[75] TVAI Approximated step function Dynamic
[73] VSI Two-bus equivalent circuit Static
[80] VSI Two-bus equivalent circuit Static
[60] Energy function Static
[43] Load flow Jacobian matrix Static Ranges from 0 to 1

(Stability limit point to no load)

[81] Saddle-node and finite induced bifurcation Static Ranges from 0.25 to 0

(No load to collapse point)

[82] Sensitivity matrix Linearized load flow equation Static 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
[84] IB index Traditional IB index Dynamic If the load impedance is located inside the circle with a radius , the system is unstable.
[85] MSV(Minimum Singular Value) Singular point of Jacobian matrix Dynamic 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 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] P–V Curve theory Static 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 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 The system outputs satisfy the equation
[90] VPS P–V and V–Q curve Static The active power margin can show the margin of voltage collapse
[74] Line stability index Static , the system is stable

, 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 Assuming that for any branch (i,j), the power system is at a QV regular operating point, if the following condition is satisfied:
[92] P–V and V–Q curve P–V and V–Q curve Static The active power margin can show the margin of voltage collapse.
[5] PV analysis Continuation load flow algorithm Static The active power margin can show the margin of voltage collapse.
[93] PV analysis Continuation load flow algorithm Static 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
[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 The active power margin can show the margin of voltage collapse.
[98] P–V curve Static
[54] Topological model Static 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 The system is stable if the VSI is 1. The system is unstable if the VSI is 0.
[70] Jacobian matrix singular point, PDF Static 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 0.8171
Index 2014 0.8172
Novel Index 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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Liang, X.; Chai, H.; Ravishankar, J. Analytical Methods of Voltage Stability in Renewable Dominated Power Systems: A Review. Electricity 2022, 3, 75-107.

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

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.

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