Uncertainty Modelling for Probabilistic Power System Stability Analysis: History
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Contributor:
Ali M. Hakami
,
Kazi N. Hasan
,
Mohammed Alzubaidi
,
Manoj Datta

The increased penetration of system uncertainties related to renewable energy sources, new types of loads and their fluctuations, and deregulation of the electricity markets necessitates probabilistic power system analysis. The abovementioned factors significantly affect the power system stability, which requires computationally intensive simulation, including frequency, voltage, transient, and small disturbance stability. Altogether 40 uncertainty modelling (UM) techniques are collated with their characteristics, advantages, disadvantages, and application areas, particularly highlighting their accuracy and efficiency (as both are crucial for power system stability applications). 

  • power system stability
  • renewable energy
  • uncertainty modelling techniques

1. Introduction

The capability of a power system to continuously maintain its operating conditions within acceptable boundaries (with system integrity maintained) following small or large disturbances is known as power system stability [1][2]. The power system stability predominantly depends on the initial operating system conditions and the nature of the disturbances that occur in a system. The response of any power system to a contingency can involve one or more of the system equipment. The consequent changes can be observed in the system frequencies, system voltages, power flows, and rotor angle of the generators [2]. Therefore, the power system stability aspects can be classified into frequency, voltage, transient, and small disturbance based on the effect of the magnitude, type of the disturbances and the time (which is needed for evaluating the phenomenon), and devices involved during the system response to disturbances [3]. The following sections provide an overview of the classification of the power system stability, followed by a discussion on the probabilistic system inputs and system outputs related to each type of stability; then, the uncertainty modelling (UM) techniques that have been applied to the different aspects of power system stability analysis.

2. Applications of UM Techniques for RESs Modelling

The increased penetration of renewable energy resources (RESs), particularly the variable generation of solar photovoltaic (PV) plants and wind turbines, has brought significant challenges in power system operation [3][4]. Therefore, finding appropriate UM techniques that can accurately and efficiently model the PV and wind power generations as uncertain input parameters is essential for realistic power system stability assessment [3][4]. Table 1 presents the probabilistic input parameters, particularly the RESs and other significant uncertain parameters in power system stability applications, along with the probabilistic output indices and the UM techniques for each type of stability study.
Table 1. The probabilistic input parameters, output indices, and UM techniques and their applications in power system stability studies.
Power System Stability Probabilistic Input Parameters Probabilistic Output Indices UM Techniques
Probabilistic frequency stability Wind speed [5][6][7][8][9], system loads [5][9][10], and PV generation [5][10]. Rate of change of frequency (ROCOF) [5][8][9][11], frequency nadir [5][8][9][10][11], frequency excursion [8][11], and frequency response inadequacy (FRI) [5][6][7][8]. MC [8][9][10][11][12][13][14], SMC [15], and cumulant-based method [16].
Probabilistic voltage stability Wind speed [17][18][19], PV generation [18][19], generation scenarios [17][20], and system loads [18][19][21][22][23][24]. System loadability [18][25], active load margin [18][20], reactive power margin [17][20][26], frequency of voltage instability [23], probability of voltage instability [23], expected voltage stability margin [23], pdf of the load increase limit [27] and probabilistic critical eigenvalue [22]. MC [5][18][20][21][22][28][29][30], QMC [5][21][31], Sobol [3][21], Halton [3][21], Latin hypercube [21], MCMC [32], PEM [33], probabilistic collocation method [5][27], and cumulant-based method [5][34][35][36].
Probabilistic transient stability Wind speed [17][18], PV generation [17][18], wind power [17][18], automatic reclosing [37], load demand [38], loading level [39], fault type [39][40], fault clearing time [38][40][41], and fault location [39][40]. Generating rejection requirement [37], transfer limit calculation [37], probability of instability of different lines [40], probability of system instability [40][42][43][44][45][46][47][48], most critical lines [44], maximum rotor angle deviation [49], transient instability [48], maximum relative rotor angle deviation (MRRAD) [6], and probability of transient instability [50]. MC [17][18][21][37][49][51][52][53][54][55][56], QMC [57], SMC [58][59][60][61], MCMC [21][62], PEM [38], Physics-informed Sparse Gaussian Process (SGP) [63], and probabilistic collocation method [64].
Probabilistic small-disturbance stability Wind-hydro-thermal system [65], wind power [28][57][66], different levels of wind penetration [41], the uncertainty of generation [65], load demand [65], generation or transmission outages [65], and PEV (plug-in electric vehicle) [57]. Damping ratios [57][65][67], damping of oscillations [57][65][66][68][69][70], critical eigenvalues of the system [41], participation factors [65][67], and the sensitivity of critical eigenvalues to the variation of wind power penetration into the system [28]. MC [41][65][67][68][69][70][71][72][73], QMC [57][74], PEM [71][75][76], cumulant-based method [66][71], probabilistic collocation method [27][71][77][78][79], and essential sampling technique [21].

3. Probabilistic Frequency Stability Analysis

Frequency stability can be defined as the ability of the power network to keep and preserve a steady operating frequency condition following a severe disturbance resulting in losing the balance between the power system generation and load demand [2][9]. The imbalance or instability has commonly been global in the case of frequency stability in the form of supported frequency swings which may lead to tripping generators and systems loads. Frequency instability issues are related to inadequate system support response, insufficient generation supply, or poor coordination of protection and control devices [9].
The rate of change of frequency (ROCOF in Hz/s) and frequency nadir (in Hz) is the most widely used frequency stability indicators [9][80]. The frequency nadir is characterised by the lowest value of the system frequency obtained after any contingency, and it can be calculated in the probabilistic analysis as shown in Equation (1).
f N , n = f 0 , n − Δ f n
In (1), fN donates frequency nadir, f0 is the initial frequency, and Δf is the frequency deviation. The n is the number of samples/simulations based on the generated datasets using the UM techniques.
In addition, the rate of change of frequency (ROCOF) is the initial slope of the frequency difference instantly after a disturbance, and it can be calculated in the probabilistic analysis based on Equation (2). The lowest ROCOF value means a better system response for a stable and robust power system [9].
R O C O F n = ( d f ( t ) d t ) n
In (2), f stands for frequency (Hz), and the df(t)/dt can be calculated based on the per-unit formulation of the swing equation in the probabilistic analysis, as shown in Equation (3). The n is the number of samples/simulations based on the generated datasets using the UM techniques.
( d f ( t ) d t ) n = ( Δ P ( t ) S b − i 2 H i f 0 ) n = ( Δ P ( t ) S b − i T N − 1 ) n
where the ΔP is the variation of the active power (MW), (Sb−i) is the nominal value of apparent power of the generator (MVA), and (TN−1) is the acceleration time constant (sec.). The Hi is the system inertia constant, and f0 is the nominal value of frequency (Hz). The n is the number of samples/simulations based on the generated datasets using the UM techniques.

UM Techniques in Frequency Stability Analysis

Several types of UM techniques have been applied to frequency stability analysis. As shown in Table 1, these sampling techniques are MC [8][9][10][11][12][13][14], SMC [15], and Cumulant-based method [16]. Input variables highly influence the probabilistic analysis of power system frequency stability, which is modelled by considering their appropriate probability distributions. Wind speed [5][6][7][8][9], system loads [5][9][10], and PV generation [5][10] are considered uncertain input variables.
Generally, the probabilistic output indices are presented to assess the frequency stability, which are the rate of change of frequency (ROCOF) [5][8][9][11], frequency nadir (i.e., minimum system frequency) [5][8][9][10][11], frequency excursion [8][11], and frequency response inadequacy (FRI) [5][6][7][8], as presented in Table 1.

4. Probabilistic Voltage Stability Analysis

Voltage stability is defined as the capability of the power system to maintain or recover the voltages to an acceptable voltage range after being subjected to a contingency or system fault [2]. Voltage instability occurs in the form of voltage drops in one, some, or all busbars of the power system, and then it may rapidly decline and subsequently collapses [17] to zero. It may also result in a loss of load in a small area, or the tripping of power plants, resulting in cascading system outage or failure [21]. Voltage stability issues mainly occur in the heavily loaded system since the reactive power supplied to the system may not be sufficient to sustain and maintain the user-end voltage in its boundary. Hence, the power system’s system load is regarded as the driving force for voltage instability issues [3].
Voltage stability simulation typically involves the continuation of power flow, establishing the active power and voltage (PV)-curve and reactive power and voltage (QV)-curve for each busbar in the power systems [26]. In the PV-curve analysis, the stability index is the system’s load margin (also known as system loadability), which can be defined as the difference in the active power at the initial operating point and the critical (maximum) active power [18]. The load margin can be calculated in the probabilistic analysis based on Equation (4).
P m a r g i n , n = P m a x , n − P 0 , n
In (4), Pmargin is the load margin (system loadability), Pmax is the maximum active power, and P0 is the initial active power point. The n is the number of samples/simulations based on the generated datasets using the UM techniques.
In addition, another critical index for studying and assessing voltage stability is the voltage sensitivity factor (VSFi), which can be calculated in the probabilistic analysis as presented in Equation (5), and the stability measure is VSFi > 0 [81].
V S F i , n = Δ V i , n Δ Q i , n
In (5), ΔVi represents the voltage variation in a load busbar between the operating point and voltage critical (collapse) point, whereas ΔQi represents the reactive power variation. The n is the number of samples/simulations based on the generated datasets using the UM techniques. This stability index measures the busbar voltage’s sensitivity to reactive power variations.

UM Techniques in Voltage Stability Analysis

As shown in Table 1, different UM techniques have been employed for voltage stability analysis, including MC [5][18][20][21][22][28][29][30], QMC [5][21][31], Sobol [3][21], Halton [3][21], Latin hypercube [21], MCMC [32], PEM [33], Cumulant-based Method [5][34][35][36], and Probabilistic Collocation method [5][27]. Various input uncertain system parameters are considered in probabilistic voltage stability assessment; the uncertain variability input is considered in generation scenarios [17][20], system loads [18][19][21][22][23][24], wind speed [17][18][19], and PV generation [18][19].
Probabilistic output variables are presented as system loadability [18][25], active load margin [18][20], reactive power margin [17][20][26], frequency of voltage instability [23], probability of voltage instability [23], expected voltage stability margin [23], pdf of the load increase limit [27] and probabilistic critical eigenvalue [22], as presented in Table 1.

5. Probabilistic Transient Stability Analysis

Transient stability (also known as large-disturbance stability) refers to the change in the topology of a power system resulting from a severe disturbance such as a loss of a power plant, large loads, or a fault in transmission lines (i.e., a short circuit) [2][3]. The main factors that can affect power system transient stability are the disturbance’s severity and the system’s initial operating state [3].
The transient stability can be assessed by calculating the maximum relative rotor angle difference observed in a power network that follows a severe disturbance. The transient stability index (TSIn) for a significant disturbance rotor angle stability can be calculated in the probabilistic analysis as shown in Equation (6) [18].
T S I n = 360 − δ m a x , n 360 + δ m a x , n × 100
In (6), TSI stands for the transient stability index and δmax is the maximum rotor angle separation between any two generators during a post-fault response. The n is the number of samples/simulations based on the generated datasets using the UM techniques. A higher value of the TSI is better for the system and indicates that the system is stable, whereas a negative value of the TSI means the system is unstable.

UM Techniques in Transient Stability Analysis

As shown in Table 1, several types of UM techniques have been used for probabilistic transient stability analysis, which are Monte Carlo [17][18][21][37][49][51][52][53][54][55][56], SMC [58][59][60][61], MCMC [21][62], PEM [38], Physics-informed Sparse Gaussian Process (SGP) [63], and Probabilistic collocation method [64]. Typically, in the probabilistic transient stability, the analysis of system uncertainties accounts for the automatic reclosing [37], wind speed [17][18], PV generation [17][18], wind power [17][18], load demand [38], loading level [39], fault type [39][40], fault clearing time [38][40][41], and fault location [39][40].
The output results are presented by: making a dismissal request to maintain system stability [37], showing the transfer limit calculation [37], probability of instability of different lines [40], probability of system instability [40][42][43][44][45][46][47][48], the most critical lines [44], transient stability index (TSI) based on the maximum rotor angle deviation [49], expected frequency of occurrence of transient instability [48], maximum relative rotor angle deviation (MRRAD) [6], and probability of transient instability [50].

6. Probabilistic Small-Disturbance Stability Analysis

Small-disturbance stability is involved with the ability of synchronous machines of power systems to remain in synchronism after the network is subjected to a small perturbation, such as minor variations in generation and loads [2]. It is mainly concerned with insufficient damping of oscillation. Small disturbances occur in power networks where the rotor angle is presented in a linear variation to allow a linearisation of the system equation around the balance points for analysis [3].
For the small-disturbance stability assessment, calculating the damping of the critical oscillatory mode can be employed as the stability index, as given in Equation (7) in the probabilistic analysis [18].
ξ i , n = − σ i , n σ i , n 2 + ω i , n 2
In (7), ξi is the damping ratio σi denotes the damping of the critical eigenvalue and ωi is the angular frequency of the critical eigenvalue. The n is the number of samples/simulations based on the generated datasets using the UM techniques. Based on the damping ratio (ξ), when a complex eigenvalue has a negative real part, the oscillations decay and result in a stable system operation. Moreover, having a higher damping ratio (ξ) is desirable, which can lead to a faster system restoration after a small disturbance occurrence.

UM Techniques in Small-Disturbance Stability Analysis

Various UM techniques have been implemented for small-disturbance stability, as shown in Table 1. These UM techniques are MC [41][65][67][68][69][70][71][72][73], QMC [57][74], PEM PEM [71][75][76], a cumulant-based method [66][71], probabilistic collocation method [27][71][77][78][79], and important sampling technique [21]. In the probabilistic small-disturbance stability assessment, uncertain system input variables are considered as wind-hydro-thermal system [65], different levels of wind penetration [41], wind power [28][57][66], the uncertainty of generation [65], load demand [65], disturbance uncertainty concerning element (generation, transmission) outages [65], as well as PEV (plug-in electric vehicle) [57].
The probabilistic output results are the real part of the critical eigenvalue, i.e., damping ratios [57][65][67], damping of oscillations [57][65][66][68][69][70], critical eigenvalues of the system [41], participation factors [65][67], and the sensitivity of critical eigenvalues to the variation of wind power penetration into the system [28].

7. Summary of Accurate and Efficient UM Techniques and Their Applications in Power System Stability Studies

The accurate and efficient UM techniques and their applications in different power system stability studies are summarised in Table 2. Interestingly, most UM techniques have not been used for frequency stability studies.
Table 2. The application of accurate and efficient uncertainty modelling techniques in different power system stability studies.
  Applicability Frequency
Stability		 Voltage
Stability		 Small-
Disturbance Stability		 Transient
Stability
Method  
Quasi-Monte Carlo (QMC)
[21][57]		 None Low Low None
Point Estimate Method (PEM)
[33][38][71]		 None Low Low Low
Important Sampling (IS)
[21][82]		 None Low Low Low
Cumulant-based method
[16][34][71]		 Low Low Medium Low
Polynomial Chaos Expansion (PCE) [83][84] None Low None Low
Sparse Gaussian Process (SGP)
[63][85]		 None None Low Low
Spatio-temporal Kriging and Analog models [21] None Low Low None
Artificial Neural Networks (ANN)
[86][87][88]		 Low Low Low Low

This entry is adapted from the peer-reviewed paper 10.3390/en16010112

References

  1. Raymond, E.; Othmer, D.; Ed, F.; Kroschwitz, J.I.; Howe-Grant, M. Kirk-Othmer Encyclopedia of Chemical Technology, 3rd ed.; John Wiley & Sons: New York, NY, USA, 1978; Volume 2.
  2. 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. 2004, 19, 1387–1401.
  3. Hasan, K.N.; Preece, R.; Milanović, J.V. Existing approaches and trends in uncertainty modelling and probabilistic stability analysis of power systems with renewable generation. Renew. Sustain. Energy Rev. 2019, 101, 168–180.
  4. Milanović, J.V. Probabilistic stability analysis: The way forward for stability analysis of sustainable power systems. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 2017, 375, 20160296.
  5. Hasan, K.N.; Preece, R. Impact of stochastic dependence within load and non-synchronous generation on frequency stability. In Proceedings of the Bulk Power Systems Dynamics and Control Symposium-IREP’2017, Espinho, Portugal, 27 August–1 September 2017.
  6. Ahmadi, H.; Ghasemi, H. Maximum penetration level of wind generation considering power system security limits. IET Gener. Transm. Distrib. 2012, 6, 1164–1170.
  7. Negnevitsky, M.; Nguyen, D.H.; Piekutowski, M. Risk assessment for power system operation planning with high wind power penetration. IEEE Trans. Power Syst. 2014, 30, 1359–1368.
  8. Wu, L.; Infield, D. Power system frequency management challenges—A new approach to assessing the potential of wind capacity to aid system frequency stability. IET Renew. Power Gener. 2014, 8, 733–739.
  9. Alsharif, H.; Jalili, M.; Hasan, K.N. Power system frequency stability using optimal sizing and placement of Battery Energy Storage System under uncertainty. J. Energy Storage 2022, 50, 104610.
  10. Adrees, A.; Papadopoulos, P.N.; Milanovic, J.V. A framework to assess the effect of reduction in inertia on system frequency response. In Proceedings of the 2016 IEEE Power and Energy Society General Meeting (PESGM), Boston, MA, USA, 17–21 July 2016; pp. 1–5.
  11. Adrees, A.; Milanovic, J.V. Study of frequency response in power system with renewable generation and energy storage. In Proceedings of the 2016 Power Systems Computation Conference (PSCC), Genoa, Italy, 20–24 June 2016; pp. 1–7.
  12. Gevorgian, V.; Zhang, Y.; Ela, E. Investigating the impacts of wind generation participation in interconnection frequency response. IEEE Trans. Sustain. Energy 2014, 6, 1004–1012.
  13. O’Sullivan, J.; Rogers, A.; Flynn, D.; Smith, P.; Mullane, A.; O’Malley, M. Studying the Maximum Instantaneous Non-Synchronous Generation in an Island System—Frequency Stability Challenges in Ireland. IEEE Trans. Power Syst. 2014, 29, 2943–2951.
  14. Ruttledge, L.; Miller, N.W.; O’Sullivan, J.; Flynn, D. Frequency response of power systems with variable speed wind turbines. IEEE Trans. Sustain. Energy 2012, 3, 683–691.
  15. Issicaba, D.; Lopes, J.A.P.; da Rosa, M.A. Adequacy and security evaluation of distribution systems with distributed generation. IEEE Trans. Power Syst. 2012, 27, 1681–1689.
  16. Bu, S.; Wen, J.; Li, F. A generic framework for analytical probabilistic assessment of frequency stability in modern power system operational planning. IEEE Trans. Power Syst. 2019, 34, 3973–3976.
  17. Alzubaidi, M.; Hasan, K.N.; Meegahapola, L. Impact of Probabilistic Modelling of Wind Speed on Power System Voltage Profile and Voltage Stability Analysis. Electr. Power Syst. Res. 2022, 206, 107807.
  18. Qi, B.; Hasan, K.N.; Milanović, J.V. Identification of critical parameters affecting voltage and angular stability considering load-renewable generation correlations. IEEE Trans. Power Syst. 2019, 34, 2859–2869.
  19. Zhu, Y.; Qi, B.; Milanovic, J.V. Probabilistic ranking of power system loads for voltage stability studies in networks with renewable generation. In Proceedings of the 2016 IEEE PES Innovative Smart Grid Technologies Conference Europe (ISGT-Europe), Ljubljana, Slovenia, 9–12 October 2016; pp. 1–6.
  20. Almeida, A.B.; de Lorenci, E.V.; Leme, R.C.; de Souza, A.C.Z.; Lopes, B.I.L.; Lo, K. Probabilistic voltage stability assessment considering renewable sources with the help of the PV and QV curves. IET Renew. Power Gener. 2013, 7, 521–530.
  21. Alzubaidi, M.; Hasan, K.; Meegahapola, L.; Rahman, M. Identification of Efficient Sampling Techniques for Probabilistic Voltage Stability Analysis of Renewable-Rich Power Systems. Energies 2021, 14, 2328.
  22. Zhang, J.; Tse, C.; Wang, K.; Chung, C. Voltage stability analysis considering the uncertainties of dynamic load parameters. IET Gener. Transm. Distrib. 2009, 3, 941–948.
  23. Aboreshaid, S.; Billinton, R. Probabilistic evaluation of voltage stability. IEEE Trans. Power Syst. 1999, 14, 342–348.
  24. Zhang, J.; Tse, C.; Wang, W.; Chung, C. Voltage stability analysis based on probabilistic power flow and maximum entropy. IET Gener. Transm. Distrib. 2010, 4, 530–537.
  25. Alzubaidi, M.; Hasan, K.N.; Meegahapola, L.; Rahman, M.T. Probabilistic Voltage Stability Assessment Considering Load and Wind Uncertainties. In Proceedings of the 2021 IEEE PES Innovative Smart Grid Technologies-Asia (ISGT Asia), Brisbane, Australia, 5–8 December 2021; pp. 1–5.
  26. Alzubaidi, M.; Hasan, K.N.; Meegahapola, L.; Rahman, M.T. Probabilistic Voltage Stability Analysis Considering Variable Wind Generation and Different Control Modes. In Proceedings of the 2021 31st Australasian Universities Power Engineering Conference (AUPEC), Perth, Australia, 26–30 September 2021; pp. 1–6.
  27. Preece, R.; Woolley, N.C.; Milanović, J.V. The probabilistic collocation method for power-system damping and voltage collapse studies in the presence of uncertainties. IEEE Trans. Power Syst. 2012, 28, 2253–2262.
  28. Bu, S.; Du, W.; Wang, H. Probabilistic analysis of small-signal rotor angle/voltage stability of large-scale AC/DC power systems as affected by grid-connected offshore wind generation. IEEE Trans. Power Syst. 2013, 28, 3712–3719.
  29. Muñoz, J.; Cañizares, C.; Bhattacharya, K.; Vaccaro, A. An affine arithmetic-based method for voltage stability assessment of power systems with intermittent generation sources. IEEE Trans. Power Syst. 2013, 28, 4475–4487.
  30. Alzubaidi, M.; Hasan, K.N.; Meegahapola, L. Identification of Suitable Probability Density Function for Wind Speed Profiles in Power System Studies. In Proceedings of the 2020 Australasian Universities Power Engineering Conference (AUPEC), Hobart, Australia, 29 November–2 December 2020; pp. 1–6.
  31. Deng, W.; Zhang, B.; Ding, H.; Li, H. Risk-based probabilistic voltage stability assessment in uncertain power system. Energies 2017, 10, 180.
  32. Perninge, M.; Söder, L. Analysis of transfer capability by Markov chain Monte Carlo simulation. In Proceedings of the 2010 IEEE International Conference on Power and Energy, Kuala Lumpur, Malaysia, 29 November–1 December 2010; pp. 232–237.
  33. Liu, K.-Y.; Hu, L.; Sheng, W. Probabilistic evaluation of static voltage stability taking account of the variation of load and stochastic distributed generations. In Proceedings of the 2013 International Conference on Electrical Machines and Systems (ICEMS), Busan, Republic of Korea, 26–29 October 2013; pp. 418–421.
  34. Schellenberg, A.; Rosehart, W.; Aguado, J.A. Cumulant-based stochastic nonlinear programming for variance constrained voltage stability analysis of power systems. IEEE Trans. Power Syst. 2006, 21, 579–585.
  35. Ruiz-Rodríguez, F.J.; Hernández, J.C.; Jurado, F. Probabilistic load-flow analysis of biomass-fuelled gas engines with electrical vehicles in distribution systems. Energies 2017, 10, 1536.
  36. Hernández, J.; Ruiz-Rodriguez, F.; Jurado, F. Modelling and assessment of the combined technical impact of electric vehicles and photovoltaic generation in radial distribution systems. Energy 2017, 141, 316–332.
  37. Vaahedi, E.; Li, W.; Chia, T.; Dommel, H. Large scale probabilistic transient stability assessment using BC Hydro’s on-line tool. IEEE Trans. Power Syst. 2000, 15, 661–667.
  38. Karimishad, A.; Nguyen, T. Probabilistic transient stability assessment using two-point estimate method. In Proceedings of the 8th International Conference on Advances in Power System Control, Operation and Management (APSCOM 2009), London, UK, 8–11 November 2009; pp. 1–6.
  39. Yuan-Yih, H.; Chung-Liang, C. Probabilistic transient stability studies using the conditional probability approach. IEEE Trans. Power Syst. 1988, 3, 1565–1572.
  40. Aboreshaid, S.; Billinton, R.; Fotuhi-Firuzabad, M. Probabilistic transient stability studies using the method of bisection . IEEE Trans. Power Syst. 1996, 11, 1990–1995.
  41. Bu, S.; Du, W.; Wang, H.; Chen, Z.; Xiao, L.; Li, H. Probabilistic analysis of small-signal stability of large-scale power systems as affected by penetration of wind generation. IEEE Trans. Power Syst. 2011, 27, 762–770.
  42. Faried, S.; Billinton, R.; Aboreshaid, S. Probabilistic evaluation of transient stability of a power system incorporating wind farms. IET Renew. Power Gener. 2010, 4, 299–307.
  43. Faried, S.O.; Billinton, R.; Aboreshaid, S. Probabilistic evaluation of transient stability of a wind farm. IEEE Trans. Energy Convers. 2009, 24, 733–739.
  44. Billinton, R.; Kuruganty, P.; Carvalho, M. An approximate method for probabilistic assessment of transient stability. IEEE Trans. Reliab. 1979, 28, 255–258.
  45. Chiodo, E.; Gagliardi, F.; la Scala, M.; Lauria, D. Probabilistic on-line transient stability analysis. IEE Proc. Gener. Transm. Distrib. 1999, 146, 176–180.
  46. Billinton, R.; Kuruganty, P. A probabilistic index for transient stability. IEEE Trans. Power Appar. Syst. 1980, PAS-99, 195–206.
  47. Chiodo, E.; Lauria, D. Transient stability evaluation of multimachine power systems: A probabilistic approach based upon the extended equal area criterion. IEE Proc. Gener. Transm. Distrib. 1994, 141, 545–553.
  48. Billinton, R.; Kuruganty, P. Probabilistic assessment of transient stability in a practical multimachine system. IEEE Trans. Power Appar. Syst. 1981, PAS-100, 3634–3641.
  49. Papadopoulos, P.N.; Milanović, J.V. Probabilistic framework for transient stability assessment of power systems with high penetration of renewable generation. IEEE Trans. Power Syst. 2016, 32, 3078–3088.
  50. Zhao, X.; Zhou, J. Probabilistic transient stability assessment based on distributed DSA computation tool. In Proceedings of the 2010 IEEE 11th International Conference on Probabilistic Methods Applied to Power Systems, Singapore, 14–17 June 2010; pp. 685–690.
  51. Wu, W.; Wang, K.; Li, G.; Hu, Y. A stochastic model for power system transient stability with wind power. In Proceedings of the 2014 IEEE PES General Meeting|Conference & Exposition, National Harbor, MD, USA, 27–31 July 2014; pp. 1–5.
  52. Wang, K.; Crow, M.L. Numerical simulation of stochastic differential algebraic equations for power system transient stability with random loads. In Proceedings of the 2011 IEEE Power and Energy Society General Meeting, Detroit, MI, USA, 24–28 July 2011; pp. 1–8.
  53. Cepeda, J.C.; Rueda, J.L.; Colomé, D.G.; Echeverría, D.E. Real-time transient stability assessment based on centre-of-inertia estimation from phasor measurement unit records. IET Gener. Transm. Distrib. 2014, 8, 1363–1376.
  54. Shi, L.; Sun, S.; Yao, L.; Ni, Y.; Bazargan, M. Effects of wind generation intermittency and volatility on power system transient stability. IET Renew. Power Gener. 2014, 8, 509–521.
  55. Papadopoulos, P.N.; Milanović, J.V. Impact of penetration of non-synchronous generators on power system dynamics. In Proceedings of the 2015 IEEE Eindhoven PowerTech, Eindhoven, The Netherlands, 29 June–2 July 2015; pp. 1–6.
  56. Papadopoulos, P.N.; Adrees, A.; Milanovicć, J.V. Probabilistic assessment of transient stability in reduced inertia systems. In Proceedings of the 2016 IEEE Power and Energy Society General Meeting (PESGM), Boston, MA, USA, 17–21 July 2016; pp. 1–5.
  57. Huang, H.; Chung, C.; Chan, K.W.; Chen, H. Quasi-Monte Carlo based probabilistic small signal stability analysis for power systems with plug-in electric vehicle and wind power integration. IEEE Trans. Power Syst. 2013, 28, 3335–3343.
  58. Sankarakrishnan, A.; Billinton, R. Sequential Monte Carlo simulation for composite power system reliability analysis with time varying loads. IEEE Trans. Power Syst. 1995, 10, 1540–1545.
  59. Huang, G.M.; Li, Y. Power system reliability indices to measure impacts caused by transient stability crises. In Proceedings of the 2002 IEEE Power Engineering Society Winter Meeting. Conference Proceedings (Cat. No. 02CH37309), New York, NY, USA, 27–31 January 2002; Volume 2, pp. 766–771.
  60. Wangdee, W.; Billinton, R. Bulk electric system well-being analysis using sequential Monte Carlo simulation. IEEE Trans. Power Syst. 2006, 21, 188–193.
  61. Rei, A.M.; da Silva, A.L.; Jardim, J.L.; Mello, J. Static and dynamic aspects in bulk power system reliability evaluations. IEEE Trans. Power Syst. 2000, 15, 189–195.
  62. Fan, Y.; Zai, X.; Qian, H.; Yang, X.; Liu, L.; Zhu, Y. Transient stability analysis of power system based on bayesian networks and main electrical wiring. In Proceedings of the 2009 Asia-Pacific Power and Energy Engineering Conference, Wuhan, China, 27–31 March 2009; pp. 1–4.
  63. Ye, K.; Zhao, J.; Duan, N.; Zhang, Y. Physics-Informed Sparse Gaussian Process for Probabilistic Stability Analysis of Large-Scale Power System with Dynamic PVs and Loads. IEEE Trans. Power Syst. 2022.
  64. Han, D.; Ma, J.; Xue, A.; Lin, T.; Zhang, G. The uncertainty and its influence of wind generated power on power system transient stability under different penetration. In Proceedings of the 2014 International Conference on Power System Technology, Chengdu, China, 20–22 October 2014; pp. 675–680.
  65. Rueda, J.; Colome, D. Probabilistic performance indexes for small signal stability enhancement in weak wind-hydro-thermal power systems. IET Gener. Transm. Distrib. 2009, 3, 733–747.
  66. Bu, S.; Du, W.; Wang, H. Investigation on probabilistic small-signal stability of power systems as affected by offshore wind generation. IEEE Trans. Power Syst. 2014, 30, 2479–2486.
  67. Rueda, J.L.; Colome, D.G.; Erlich, I. Assessment and enhancement of small signal stability considering uncertainties. IEEE Trans. Power Syst. 2009, 24, 198–207.
  68. Hasan, K.N.; Preece, R.; Milanović, J.V. Priority ranking of critical uncertainties affecting small-disturbance stability using sensitivity analysis techniques. IEEE Trans. Power Syst. 2016, 32, 2629–2639.
  69. Hasan, K.N.; Preece, R.; Milanović, J.V. The influence of load on risk-based small-disturbance security profile of a power system. IEEE Trans. Power Syst. 2017, 33, 557–566.
  70. Hasan, K.; Preece, R.; Milanović, J. Efficient identification of critical parameters affecting the small-disturbance stability of power systems with variable uncertainty. In Proceedings of the 2016 IEEE Power and Energy Society General Meeting (PESGM), Boston, MA, USA, 17–21 July 2016; pp. 1–5.
  71. Preece, R.; Huang, K.; Milanovic, J.V. Probabilistic Small-Disturbance Stability Assessment of Uncertain Power Systems Using Efficient Estimation Methods. IEEE Trans. Power Syst. 2014, 29, 2509–2517.
  72. Wang, C.; Shi, L.; Yao, L.; Wang, L.; Ni, Y.; Bazargan, M. Modelling analysis in power system small signal stability considering uncertainty of wind generation. In Proceedings of the IEEE PES General Meeting, Providence, RI, USA, 25–29 July 2010; pp. 1–7.
  73. Preece, R.; Milanovic, J.V.; Almutairi, A.M.; Marjanovic, O. Probabilistic evaluation of damping controller in networks with multiple VSC-HVDC lines. IEEE Trans. Power Syst. 2012, 28, 367–376.
  74. Alabduljabbar, A.; Milanovic, J.; Al-Eid, E. Low discrepancy sequences based optimization algorithm for tuning PSSs. In Proceedings of the 10th International Conference on Probablistic Methods Applied to Power Systems, Rincon, PR, USA, 25–29 May 2008; pp. 1–9.
  75. Xu, X.; Lin, T.; Zha, X. Probabilistic analysis of small signal stability of microgrid using point estimate method. In Proceedings of the 2009 International Conference on Sustainable Power Generation and Supply, Nanjing, China, 6–7 April 2009; pp. 1–6.
  76. Yi, H.; Hou, Y.; Cheng, S.; Zhou, H.; Chen, G. Power system probabilistic small signal stability analysis using two point estimation method. In Proceedings of the 2007 42nd International Universities Power Engineering Conference, Brighton, UK, 4–6 September 2007; pp. 402–407.
  77. Preece, R.; Milanović, J.V. Tuning of a damping controller for multiterminal VSC-HVDC grids using the probabilistic collocation method. IEEE Trans. Power Deliv. 2013, 29, 318–326.
  78. Preece, R.; Milanović, J. The Probabilistic Collocation Method for dealing with uncertainties in power system small disturbance studies. In Proceedings of the 2012 IEEE Power and Energy Society General Meeting, San Diego, CA, USA, 22–26 July 2012; pp. 1–7.
  79. Meiyan, L.; Jin, M.; Dong, Z. Uncertainty analysis of load models in small signal stability. In Proceedings of the 2009 International Conference on Sustainable Power Generation and Supply, Nanjing, China, 6–7 April 2009; pp. 1–6.
  80. Mochamad, R.F.; Preece, R.; Hasan, K.N. Probabilistic multi-stability operational boundaries in power systems with high penetration of power electronics. Int. J. Electr. Power Energy Syst. 2022, 135, 107382.
  81. Hasan, K. Application of Probabilistic Methods in Power Systems Stability Assessment—A Review Identifying Future Research Needs; Faculty of Engineering and Physical Sciences, The University of Manchester: Manchester, UK, 2015.
  82. Preece, R.; Milanović, J.V. Efficient estimation of the probability of small-disturbance instability of large uncertain power systems. IEEE Trans. Power Syst. 2015, 31, 1063–1072.
  83. Hover, F.S.; Triantafyllou, M.S. Application of polynomial chaos in stability and control. Automatica 2006, 42, 789–795.
  84. Wang, Y.; Chiang, H.-D.; Wang, T. A two-stage method for assessment of voltage stability in power system with renewable energy. In Proceedings of the 2013 IEEE Electrical Power & Energy Conference, Halifax, NS, Canada, 21–23 August 2013; pp. 1–6.
  85. Pareek, P.; Nguyen, H.D. Probabilistic robust small-signal stability framework using Gaussian process learning. Electr. Power Syst. Res. 2020, 188, 106545.
  86. Sarajcev, P.; Kunac, A.; Petrovic, G.; Despalatovic, M. Artificial Intelligence Techniques for Power System Transient Stability Assessment. Energies 2022, 15, 507.
  87. Goh, H.; Chua, Q.; Lee, S.; Kok, B.; Goh, K.; Teo, K. Evaluation for voltage stability indices in power system using artificial neural network. Procedia Eng. 2015, 118, 1127–1136.
  88. Hooshmand, R.; Moazzami, M. Optimal design of adaptive under frequency load shedding using artificial neural networks in isolated power system. Int. J. Electr. Power Energy Syst. 2012, 42, 220–228.
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