Uncertainty Modelling for Probabilistic Power System Stability Analysis: Comparison
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Please note this is a comparison between Version 2 by Sirius Huang and Version 1 by Ali M. Hakami.

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 [118,119][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 [119][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 [1][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 [1,3][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 [1,3][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 [120,121,122,123,124][5][6][7][8][9], system loads [120,124,125][5][9][10], and PV generation [120,125][5][10]. Rate of change of frequency (ROCOF) [120,123,124,126][5][8][9][11], frequency nadir [120,123,124,125,126][5][8][9][10][11], frequency excursion [123,126][8][11], and frequency response inadequacy (FRI) [120,121,122,123][5][6][7][8]. MC [14,123,124,]125,[126,127,11][12][13]128][8][9][10[14], SMC [22][15], and cumulant-based method [39][16].
Probabilistic voltage stability Wind speed [12,129,130][17][18][19], PV generation [129,130][18][19], generation scenarios [12,131][17][20], and system loads [4,129,130,132,133,134][18][19][21][22][23][24]. System loadability [129,135][18][25], active load margin [129,131][18][20], reactive power margin [12,[131,17136]][20][26], frequency of voltage instability [133][23], probability of voltage instability [133][23], expected voltage stability margin [133][23], pdf of the load increase limit [41][27] and probabilistic critical eigenvalue [132][22]. MC [4,15,120,129,131,132,137,138][5][18][20][21][22][28][29][30], QMC [4,120,139][5][21][31], Sobol [1,4][3][21], Halton [1,4][3][21], Latin hypercube [4][21], MCMC [28][32], PEM [33], probabilistic collocation method [41,120][5][27], and cumulant-based method [36,120,140,141][5][34][35][36].
Probabilistic transient stability Wind speed [12,129][17][18], PV generation [12,129][17][18], wind power [12,129][17][18], automatic reclosing [142][37], load demand [32][38], loading level [143][39], fault type [143,144][39][40], fault clearing time [32,144,145][38][40][41], and fault location [143,144][39][40]. Generating rejection requirement [142][37], transfer limit calculation [142][37], probability of instability of different lines [144][40], probability of system instability [144,146,147,148,149,150,151,152][40][42][43][44][45][46][47][48], most critical lines [148][44], maximum rotor angle deviation [153][49], transient instability [152][48], maximum relative rotor angle deviation (MRRAD) [121][6], and probability of transient instability [154][50]. MC [4,12,13,129,142,153,155,156,157,158,159][17][18][21][37][49][51][52][53][54][55][56], QMC [18][57], SMC [21,160,161,162][58][59][60][61], MCMC [4,27][21][62], PEM [32][38], Physics-informed Sparse Gaussian Process (SGP) [50][63], and probabilistic collocation method [40][64].
Probabilistic small-disturbance stability Wind-hydro-thermal system [163][65], wind power [15,18,164][28][57][66], different levels of wind penetration [145][41], the uncertainty of generation [163][65], load demand [163][65], generation or transmission outages [163][65], and PEV (plug-in electric vehicle) [18][57]. Damping ratios [18,163,165][57][65][67], damping of oscillations [18,163,164,166,167,168][57][65][66][68][69][70], critical eigenvalues of the system [145][41], participation factors [163,165][65][67], and the sensitivity of critical eigenvalues to the variation of wind power penetration into the system [15][28]. MC [9,145,163,165,166,167,168,169,170][41][65][67][68][69][70][71][72][73], QMC [18,171][57][74], PEM [9,172,173][71][75][76], cumulant-based method [9,164][66][71], probabilistic collocation method [9,41,42,174,175][27][71][77][78][79], and essential sampling technique [4][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 [119,124][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 [124][9].
The rate of change of frequency (ROCOF in Hz/s) and frequency nadir (in Hz) is the most widely used frequency stability indicators [124,176][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 [124][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 [14[8][9][10][11][12][13][14],123,124,125,126,127,128], SMC [22][15], and Cumulant-based method [39][16]. Input variables highly influence the probabilistic analysis of power system frequency stability, which is modelled by considering their appropriate probability distributions. Wind speed [120,121[5][6][7][8][9],122,123,124], system loads [120[5][9][10],124,125], and PV generation [120,125][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) [120,123[5][8][9][11],124,126], frequency nadir (i.e., minimum system frequency) [120[5][8][9][10][11],123,124,125,126], frequency excursion [123[8][11],126], and frequency response inadequacy (FRI) [120[5][6][7][8],121,122,123], 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 [119][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 [12][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 [4][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 [1][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 [136][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 [129][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 [177][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 [4[5][18][20][21][22][28][29][30],15,120,129,131,132,137,138], QMC [4[5][21][31],120,139], Sobol [1[3][21],4], Halton [1[3][21],4], Latin hypercube [4][21], MCMC [28][32], PEM [33], Cumulant-based Method [36[5][34][35][36],120,140,141], and Probabilistic Collocation method [41,120][5][27]. Various input uncertain system parameters are considered in probabilistic voltage stability assessment; the uncertain variability input is considered in generation scenarios [12,131][17][20], system loads [4,129[18][19][21][22][23][24],130,132,133,134], wind speed [12,129[17][18][19],130], and PV generation [129,130][18][19].
Probabilistic output variables are presented as system loadability [129[18][25],135], active load margin [129[18][20],131], reactive power margin [12[17][20][26],131,136], frequency of voltage instability [133][23], probability of voltage instability [133][23], expected voltage stability margin [133][23], pdf of the load increase limit [41][27] and probabilistic critical eigenvalue [132][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) [1,119][2][3]. The main factors that can affect power system transient stability are the disturbance’s severity and the system’s initial operating state [1][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) [129][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 [4[17][18][21][37][49][51][52][53][54][55][56],12,13,129,142,153,155,156,157,158,159], SMC [21[58][59][60][61],160,161,162], MCMC [4,27][21][62], PEM [32][38], Physics-informed Sparse Gaussian Process (SGP) [50][63], and Probabilistic collocation method [40][64]. Typically, in the probabilistic transient stability, the analysis of system uncertainties accounts for the automatic reclosing [142][37], wind speed [12[17][18],129], PV generation [12,129][17][18], wind power [12[17][18],129], load demand [32][38], loading level [143][39], fault type [143[39][40],144], fault clearing time [32[38][40][41],144,145], and fault location [143,144][39][40].
The output results are presented by: making a dismissal request to maintain system stability [142][37], showing the transfer limit calculation [142][37], probability of instability of different lines [144][40], probability of system instability [144[40][42][43][44][45][46][47][48],146,147,148,149,150,151,152], the most critical lines [148][44], transient stability index (TSI) based on the maximum rotor angle deviation [153][49], expected frequency of occurrence of transient instability [152][48], maximum relative rotor angle deviation (MRRAD) [121][6], and probability of transient instability [154][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 [119][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 [1][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 [129][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 [9[41][65][67][68][69][70][71][72][73],145,163,165,166,167,168,169,170], QMC [18[57][74],171], PEM PEM [9[71][75][76],172,173], a cumulant-based method [9[66][71],164], probabilistic collocation method [9,41,42,174[27][71][77][78][79],175], and important sampling technique [4][21]. In the probabilistic small-disturbance stability assessment, uncertain system input variables are considered as wind-hydro-thermal system [163][65], different levels of wind penetration [145][41], wind power [15[28][57][66],18,164], the uncertainty of generation [163][65], load demand [163][65], disturbance uncertainty concerning element (generation, transmission) outages [163][65], as well as PEV (plug-in electric vehicle) [18][57].
The probabilistic output results are the real part of the critical eigenvalue, i.e., damping ratios [18[57][65][67],163,165], damping of oscillations [18,163,164,166,167,168][57][65][66][68][69][70], critical eigenvalues of the system [145][41], participation factors [163[65][67],165], and the sensitivity of critical eigenvalues to the variation of wind power penetration into the system [15][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)

[4,18][21][57]		 None Low Low None
Point Estimate Method (PEM)

[9,32,33][33][38][71]		 None Low Low Low
Important Sampling (IS)

[4,178][21][82]		 None Low Low Low
Cumulant-based method

[9,36,39][16][34][71]		 Low Low Medium Low
Polynomial Chaos Expansion (PCE) [81,179][83][84] None Low None Low
Sparse Gaussian Process (SGP)

[50,180][63][85]		 None None Low Low
Spatio-temporal Kriging and Analog models [4][21] None Low Low None
Artificial Neural Networks (ANN)

[86,181,182][86][87][88]		 Low Low Low Low

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