A Review of Uncertainty Modelling Techniques for Probabilistic Stability Analysis of Renewable-Rich Power Systems

Abstract

In pursuit of identifying the most accurate and efficient uncertainty modelling (UM) techniques, this paper provides an extensive review and classification of the available UM techniques for probabilistic power system stability analysis. 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 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). This review recommends the most accurate and efficient UM techniques that could be used for probabilistic stability analysis of renewable-rich power systems.

1. Introduction

The significant integration of renewable energy resources (RESs) and new types of electrical loads, including electric vehicles and energy storage, present substantial power system uncertainties. These uncertainties are mainly related to the intermittency of RESs’ power generation and the fluctuation of the system loads, leading to strict requirements on the power networks to operate within acceptable operational boundaries [1,2]. Uncertainty modelling with appropriate probabilistic stability analysis can help the network operators to ensure that the system meets all the operational requirements. However, by concentrating on the worst-case scenarios, many stability studies neglect the system uncertainties; hence they do not appropriately reflect the actual performance [3]. Therefore, uncertainty modelling (UM) techniques, also known as sampling techniques, are needed to model and assess the system uncertainties to present the actual system behaviour.

A wide range of UM techniques is available to evaluate system uncertainties, which must be carefully analysed and selected for their suitability in probabilistic power system analysis [4]. The UM techniques are the statistical methods that include generating random samples to describe and infer some probability for the uncertain system parameters [5,6]. Typically, the UM techniques can be categorised into two groups: (i) probabilistic UM techniques and (ii) non-probabilistic UM techniques [7,8]. Among these two groups, the probabilistic UM techniques are mainly used to model uncertain system parameters in probabilistic power system analysis [1,4]. By considering the theory of probability, samples are selected in probabilistic UM techniques, which means that each possible set of samples is assigned a probability of selection. A random procedure is used to choose the samples, and the evaluated confidence interval is known. On the other hand, the random selection of samples is not used in non-probabilistic uncertainty modelling techniques [8].

Different UM techniques are implemented in the literature to model the system uncertainties. For example, the Monte Carlo (MC) technique has been applied to model the wind and solar power and load uncertainties to investigate small-disturbance, voltage, and transient stability analysis [3]. Six different types of sampling techniques, including MC, three kinds of Quasi-Monte Carlo (QMC), i.e., Latin Hypercube, Halton, and Sobol, Markov Chain Monte Carlo (MCMC) and important sampling (IS) techniques have been used to analyse power system voltage stability in [4]. The point-estimated method, probabilistic collocation method, and cumulant-based method have been presented and compared for small-disturbance stability analysis in [9]. A review of uncertainty modelling techniques has been presented in [10], emphasizing various categories of modelling approaches for their application in power system studies, including interval analysis, robust optimisation, information gap theory, probabilistic, and combined probabilistic approaches. The most widely used probabilistic techniques in power system stability assessment and their advantages, disadvantages and application areas have been presented in [1]. The non-probabilistic sampling techniques (i.e., sequential and snowball sampling) are presented and compared in [11]. However, the majority of the previous literature has not presented a wide range of uncertainty modelling techniques, which can be accurate and efficient and can be applied in different aspects of probabilistic power system analysis, which requires the inclusion of all the system uncertainties.

Based on the previous discussions and published literature, the main contributions of this paper can be highlighted as follows:

• Altogether, 40 UM techniques (probabilistic and non-probabilistic techniques) have been collated with their characteristics, advantages, disadvantages, and application areas, particularly highlighting their accuracy and efficiency, for the first time, to determine their suitability for probabilistic power system stability assessments.
• Selection of the most accurate and efficient UM techniques that could be used for probabilistic stability analysis, including frequency, voltage, transient, and small disturbance stability analyses.
• Highlight the applications of different UM techniques for specific power system stability analysis and identification of uncertain input and probabilistic output parameters that are necessary for various probabilistic power system stability studies.

The rest of the paper is organised as follows: uncertainty modelling techniques are presented in Section 2. The accurate and efficient sampling techniques are described in Section 3. Section 4 outlines the UM techniques for probabilistic analysis of power system stability studies. The conclusions and future works are summarised in Section 5.

2. Uncertainty Modelling Techniques

2.1. Benchmark Uncertainty Modelling Technique (Monte Carlo Method)

The MC method is the most commonly used UM technique, which is also the benchmark for UM techniques. With the increasing number of simulations, the accuracy of MC is improved. The MC simulation involves defining an input domain that could generate random input samples from the input probability distribution across the domain. Then it performs precise simulations, analyses for every sample, and empirically analyses the results [1,4].

The required number of MC simulations can be determined using the MC stopping criteria formula, which is presented in (1) [4].

$$\epsilon =[\{{\varnothing }^{-1}(1-(\delta /2)·\sqrt{{\alpha }^2(x)/N)\}}c/\overline{\chi }]$$

(1)

In (1), $\epsilon$ is the error of the sample, ${\varnothing }^{-1}$ is the inverted Gaussian standard probability distribution with a zero mean and one standard deviation, $\overline{\chi }$ is the mean of the sample, ${\alpha }^2\left(x\right)$ is the variance of the sample, and $\delta$ is the confidence level.

However, the application of the MC technique is somewhat restricted, particularly in large-scale power system studies with many uncertain system variables, due to its low efficiency, which requires a large number of generated random samples to provide an accurate result [1]. Figure 1 shows an example of generating random samples used in the MC technique with different numbers of samples, such as (a) 100 samples, (b) 500 samples, (c) 1000 samples, and (d) 10,000 samples, respectively. This demonstrates that the simulation gradually starts covering the whole search space with more samples.

2.2. Alternative Uncertainty Modelling Techniques

So many different UM techniques have been reported in the published literature to model uncertain random parameters. Forty of the important UM techniques are summarised in

Table 1. Uncertainty modelling techniques with their description, advantages, disadvantages, and applications.

Model	Description	Advantages	Disadvantages	Applications
Monte Carlo [1,3,4,9,12,13,14,15,16]	MC works by compiling a large data set of uncertain parameters from a given system [1,4,12].	High flexibility. Easy to extend and modify [1].	Needs a high number of simulations. Time-consuming [1,3].	Transient stability [13], small-disturbance stability [9], frequency stability [14], voltage stability [15], and probability distribution [16].
QMC—Latin Hypercube Sampling [17,18,19,20]	QMC divides the variable’s space into subspaces to be ensured that each sample is generated for each subspace [17].	Fills subspace uniformly. Better for numerical integration. Need less number of simulations [17].	Difficult to add samples after the higher computational cost [17].	Small-disturbance stability [18], probabilistic optimal power flow [19], load flow, and generation planning [20].
QMC—Sobol Sequences [9,17,18,19,20]	It is an example of low discrepancy sequences, which are deterministic sequences of numbers that merge fast to a uniform distribution [9].	Show low error. Efficient sampling scheme with partial rank correlation coefficients (PRCC) and predict high value [17].	Cannot guarantee superior theoretical properties for non-uniform multivariate distributions [17].	Small-disturbance stability [18], probabilistic optimal power flow [19], load flow and generation planning [20].
Sequential Monte Carlo (SMC) [1,3,21,22,23,24,25,26]	It can accurately capture the time-dependent occurrences for the generated samples and simulate the system with the generation of chronological data [1].	Provide accurate frequency. Provide time for several actions [1,3].	Needs very high mathematical effort [3].	Transient stability [21], frequency stability [22], wind generation [23], risk analysis [24], reliability [25], and transmission reinforcement planning [26].
Markov Chain Monte Carlo (MCMC) [1,3,4,27,28,29,30]	The random samples are generated based on the probability distribution, which is constructed by the Markov Chain from a target distribution [4].	Efficient for simulating rare failure events. Prior information is not required [4].	Complex implementation. Need a large number of simulations. Need many burn-in samples [1,3].	Transient stability [27], voltage stability [28], composite system reliability evaluation [29], and load modelling [30].
Point Estimate Method (PEM) [1,3,9,31,32,33,34,35]	Generating a small number of selected sample points and associated weights to estimate the system statistics output [3,9,31].	Provides low error. Require 2n calculations for n uncertain parameters [9].	Need many simulations for more accuracy. Cannot be used with a correlated system [1,3,9].	Transient stability [32], small-disturbance stability [9], voltage stability [33], power transfer capability [34], and probabilistic power flow [35].
Cumulant-based Method [1,3,9,36,37,38,39]	It presents an alternative to the moment of the probability distribution to obtain the variation in the output [9].	High efficiency. Used with the non-parametric distribution [9].	Depends on the correct derivation of the input-output sensitivity [1,3].	Small-disturbance stability [9], voltage stability [36], probabilistic power flow [37], probabilistic OPF [38], and frequency stability [39].
Probabilistic Collocation Method [3,9,40,41,42,43]	It produces a simple approximate model that estimates the system’s output as a function of the uncertain system parameter [3,9].	Computationally efficient. Reduce computational time [3,9].	Limited accuracy compared to other methods [3,9].	Transient stability [40], small-disturbance stability [9], voltage stability [41], probabilistic controller tuning [42], and power flow [43].
Probabilistic Game Theory [44,45,46,47]	It is a mathematical tool to model competitive conditions to imply rational decision-makers interaction [44,45,46].	Predicate the opponents’ decisions and present accuracy. Transform incomplete data into complete data [44,45,46].	Complex and cannot handle imprecise data. Normalization issues [47].	Congestion management [44], transmission cost allocation [45], and strategic power-market bidding [46].
Spatio-temporal Kriging and Analog [48]	It is used as the input value to solve an economic dispatch issue [48].	Low operating costs. Provide accuracy [48].	Limited accuracy with less number of samples [48].	Forecast wind power generation [48], power planning studies [48], and economic dispatch problems [48].
Trigonometric Direction Diurnal Model [49]	It generalizes the Regime-Switching Spatio-temporal model by considering the wind directions [49].	Reduce system-wide generation, ancillary system services and operating system cost [49].	Cannot decrease the error in the forecasting [49].	Power system economic dispatch and short-term wind speed forecasting [49].
Importance Sampling [4,6]	Biased sampling technique, which uses a variance reduction strategy to raise the MC method efficiency [4].	High accuracy. High efficiency [6].	Biased. Need a large amount of sampling [6].	Small-disturbance stability [4], power system security [4], and power system reliability [4].
Sparse Gaussian Process (SGP) [50,51,52,53]	It is a stochastic process (random variables indexed are selected randomly by time or space) [50].	Require a small number of samples and improve efficiency. Provide high accuracy with limited data [50].	Insufficient to model stable and unstable dynamic reactions [50].	Probabilistic transient stability [50], distribution grid analysis [51,52], transmission system power flow [53], and optimal power flow [53].
Compressive Spatio-temporal Forecasting [48,54,55]	It is inspired by compressive sensing and structured-sparse recovery [55].	Effectiveness in making short-term predictions. Low error metrics [48].	Needs more effort and time. Challenging and tricky to implement, and expensive [48].	Wind speed forecasting [48] and photovoltaic power forecasting [54].
Spatio-temporal Markov Chain [56,57]	It describes the sequence of probable events, which is dependent on the state of the previous event [56,57].	More accurate and stable for forecasting errors [56].	The accuracy depends on the prediction output accuracy [57].	Wind power forecasting and electricity markets [56].
Copula Function [58,59,60]	Copula function is a multivariate distribution function [58].	Enable various unusual shapes. Allow the modelling of multivariate data in a natural way [58].	Fail to represent asymptotic tail dependence. Become unreliable conventional risk measures to statistical dependence [59].	Large-scale integration of wind power [59], statistics [60], finance [60], economics [60], image processing [60], and engineering applications [60].
Spatio-temporal Kriging [62,63]	An optimal spatio-temporal prediction approach. It depends on the process of the statistical spatio-temporal dependencies [62].	Reduce the average absolute errors. Use a small number of sampled data points to get the value [62,63].	The merit goes to their ability to perform forecasting at unobserved locations. Not appropriate for real-world wind speed [63].	Very short-term solar irradiance forecasting [62], forecast wind power generation [63], power planning studies [63], and economic dispatch problems [63].
Low-cost Spatio-Temporal Adaptive Filter [64]	Used for predicting both wind speed and wind direction [64].	Minimise the mean square error. Produce wind speed and direction predictions [64].	Less accuracy with high gradient noise. Negatively impacting the predictor’s performance [64].	Wind speed and direction [64].
Cluster Sampling [8]	Naturally occurring groups are chosen as samples [8].	Need less time and effort. Reduces variability [8].	It is a biased sampling. Less accuracy [8].	Geographic studies [8].
Systematic Sampling [65,66]	It takes the samples at regular intervals (time, place or order) [65,66].	Spreads the sample more evenly. Easy to conduct [65,66].	Expensive and time-consuming. Require more effort and a specific amount or size [65,66].	Quality control [65].
Kernel Sampling [67,68]	Use Mercer kernels to create nonlinear versions of traditional algorithms [67,68].	Dealing with sparse data happens when the amount of data is limited [67].	Calculation complexity with a new sample. Over-fitting [67,68].	Bioinformatics [68], image analysis [68], 3D reconstruction [68], and geostatistics [68].
Stratified Sampling [6,66,69]	It divides the population member into groups, same or different probability places (strata) before selection [6].	Perform a high representative sample. Allow generalizing [66].	Require high effort. Expensive. Time consumption [69].	Regional distribution of cyanobacteria [69].
Trajectory Sensitivity Analysis [4,70,71,72,73,74]	It can determine the change in a trajectory due to (small) changes in initial conditions [70].	Require fewer numbers of simulations [71].	High cost. Difficult to implement [4].	Parameter and control estimation [72], stability assessment [73,74], and dynamic security [74].
Decision Tree [75,76,77,78]	This sequential model logically combines a series of straightforward tests, each of which compares nominal or numeric features with a range of potential values [75,76].	Easy to understand. Require a small amount of data preparation. Used numerical and categorical data. Easy to interpret [75,76].	Can be unstable Can create a biased tree. Any data change can change the overall look of the techniques [75,76].	Medicine [75,76], image processing [75], industry [76], intelligent vehicles [76], business [76], e-commerce [76], energy modelling [77], remote sensing [77], and power flow [78].
Polynomial Chaos Expansion (PCE) [79,80,81,82,83,84]	It can be used to identify uncertain quantities, which it represents as expansions [79].	Facilitates the mathematically optimal construction. Originate from the Wiener chaos expansion [79].	Has scalability problems. Fails to manage large numbers of uncertain parameters [80].	Stability and control [81], solid mechanics [82], electronic circuits [83], and computational fluid dynamics [84].
Artificial Neural Networks (ANN) [85,86,87]	It is a data processing system modelled after biological neurons [85].	Provide significant progress. Can detect trends and patterns to be classified [85].	Using a variety of sensors can cause issues for real systems [85].	Robotics [85], signal processing [85], transient stability [86], solar energy [87], wind power [87], and hydropower [87].
Unscented Transformation [88,89,90,91,92,93,94,95,96]	It is used for calculating the statistics of a random variable [88].	Can calculate mean and covariance. Numerically accurate and efficient [88].	Influenced by the non-global sampling problem [89].	Probabilistic load flow [90], power system dynamic state estimation [91,92], phasor estimation [93], and state estimation [88,96].
Convenience Sampling [97,98,99]	Useful where the target population is defined in terms of a very broad category [97].	Needs less effort and is inexpensive. Collect data quickly. Easy to do research [97].	Sampling biases. Systematic errors [98].	Science [98], and medicine [99].
Logistic Regression (LR) [78]	Used for binary classification tasks, where an output belongs to one class or another [78].	Easy to implement and predict the class of new inputs. Better performance in binary classification tasks [78].	Cannot solve non-linear problems [78].	State estimation of distribution network [78].
Predictive Deep Convolutional Neural Network [61]	Predicting wind speeds by employing the CNNs and the multi-layer perceptron in order to represent the spatial and temporal correlations [61].	Excellent performance. Able to capture the spatial correlation of wind speed. Individual error control capacity [61].	Shallow architectures. Computationally complex. Lacks a targeted processing mechanism for spatial details [61].	Wind speed prediction and operational planning of power systems [61].
K-Nearest Neighbors (KNN) [78]	It is used to investigate data with similar characteristics and group them in the same class [78].	Simple to understand. Easy to implement for a multi-class problem. Used for classification and regression [78].	Low accuracy. Sensitive to noise. Does not perform sufficiently [78].	Power quality detection [78], voltage and reactive power control [78], and power factor compensator [78].
Naïve Bayes (NB) [78]	This technique compares the probability of a current event happening to a past event that has already happened [78].	Easy to implement. Predict labels of new inputs [78].	Difficult to have data with independent features [78].	Classification and spam filtering [78].
Support Vector Machine (SVM) [100,101]	It is used in classification tasks and implemented in regression problems [100].	Perfect classification accuracy [101].	Mathematically complex. Computationally expensive [100].	Power quality [78], power flow [78], biological [101], and biomedical [101].
Random Forest (RF) [78,100]	This technique is similar to the decision tree, but RF uses several decision trees rather than one [78,100].	Applied with complex data. Provide high accuracy and multiple trained decision tree classifiers [78,100].	Difficult to implement. Slower than other techniques [78].	Power flow [78], power quality [78], transportation [78], and science [100].
Linear Regression (LR) [78]	It is used with a linear relationship between variables [78].	Prevent overfitting [78].	Not flexible for non-linear variable relationships and complex patterns [78].	Power flow [78].
Regression Tree (RT) [78,100,102]	It helps to predict a numerical value and take it as input [100].	Easy to implement. Robust to outliers [78].	This technique disposed to overfitting problems [102].	Remote sensing, vegetation mapping and predicting species invasions [102].
Extreme Learning Machine [78,103,104]	It is an algorithm training for a single hidden layer feed-forward neural network [78,103].	Faster training phase. Best interpolation results [78,103].	Requires a large number of hidden neurons. Not all the samples will be classified. Slow technique [104].	Medical [103], chemistry [103], economy [103], transportation [103], geography [103], control systems and robotics [103].
Deep Neural Network (DNN) [105]	It can be used as a regression algorithm and classifier [105].	Interpreted the network performance without any influence [105].	Computationally expensive. Difficult to apply [105].	Language identification [105], dialogue state [105], and transferring cross-language knowledge [105].
Purposive Sampling [106,107]	In this technique, objects will be selected carefully—not randomly chosen. It is used to select useful information [106].	Justifies researchers to make generalisations. Offers a wide range of non-probability sampling [106].	Prone to researcher bias. Challenging to convince the reader. Manipulate the data being collected [106].	Redesign of stroke services [107], child and family health nurses [107], and safety [107].
Snowball Sampling [106,108]	The objects are selected depending on their relationship to the previously selected objects [106].	Identify objects that are not in the sampling frame [106].	Sampling biases results and systematic errors due to network connection [106].	Economics and medical [108].

, highlighting their characteristics, advantages and disadvantages, and application areas. These modelling techniques include MC [1,3,4,9,12,13,14,15,16], QMC [9,17,18,19,20], Sequential MC (SMC) [1,3,21,22,23,24,25,26], MCMC [1,3,4,27,28,29,30], point estimate method (PEM) [1,3,9,31,32,33,34,35], cumulant-based method [1,3,9,36,37,38,39], probabilistic collocation method [3,9,40,41,42,43], probabilistic game theory [44,45,46,47], spatio-temporal kriging and analogue models [48], trigonometric direction diurnal model [49], importance sampling [4,6], physics-informed sparse gaussian process (SGP) [50,51,52,53], compressive spatio-temporal forecasting [48,54,55], spatio-temporal Markov Chain [56,57], copula function [58,59,60], predictive deep convolutional neural network [61], spatio-temporal kriging [62,63], low-cost spatio-temporal adaptive filter [64], cluster sampling [8], systematic sampling [65,66], kernel sampling [67,68], stratified sampling [6,66,69], trajectory sensitivity analysis (TSA) [4,70,71,72,73,74], decision tree [75,76,77,78], polynomial chaos expansion (PCE) [79,80,81,82,83,84], artificial neural networks (ANN) [85,86,87], unscented transformation [88,89,90,91,92,93,94,95,96], convenience sampling [97,98,99], logistic regression (LR) [78], k-nearest neighbours (KNN) [78], naïve bayes [78], support vector machine (SVM) [100,101], random forest (RF) [78,100], linear regression (LR) [78], regression tree (RT) [78,100,102], extreme learning machine (ELM) [78,103,104], deep neural network (DNN) [105], purposive sampling [106,107], and snowball sampling [106,108].

These UM techniques can be classified into numerical or analytical based on their fundamental sample generation process. The numerical technique is a set of simulation-based techniques, e.g., MC, QMC, MCMC, and IS. On the other hand, the analytical techniques calculate the probability density function (PDF) based on mathematical expressions, e.g., PEM, cumulant-based method, and probabilistic collocation method [1,10]. These UM techniques have different aspects when accuracy and efficiency are considered. In terms of accuracy and efficiency, these UM techniques can be categorised into three groups:

• Accurate and efficient: this includes QMC [9,17,18,19,20], point estimate method (PEM) [1,3,9,32,33,34,35], cumulant-based method [1,3,9,36,37,38,39], importance sampling technique [4,6], polynomial chaos expansion (PCE) [79,80,81,82,83,84], physics-informed sparse gaussian process (SGP) [50], Spatio-temporal kriging and analogue models [48], copula function [58,59,60], and artificial neural networks (ANN) [85,86,87];
• Accurate but not much efficient: this includes MC [1,3,4,9,12,13,14,15,16], sequential MC (SMC) [1,3,21,22,23,24,25,26], MCMC [1,3,4,27,28,29,30], probabilistic game theory [44,45,46,47], Spatio-temporal Markov Chain [56,57], support vector machine (SVM) [100,101], random forest (RF) [78,100], purposive sampling [106,107], snowball sampling [106,108], predictive deep convolutional neural network [61], and deep neural network (DNN) [105], and;
• Efficient but not much accurate: this includes probabilistic collocation [3,9,40,41,42,43], compressive Spatio-temporal forecasting [48,54,55], Spatio-temporal kriging [62,63], trajectory sensitivity analysis [4,70,71,72,73,74], decision tree [75,76,77,78], convenience sampling [97,98,99], k-nearest neighbours (KNN) [78], naïve Bayes [78], and extreme learning machine (ELM) [78,103,104].

Regarding the UM techniques, mostly mentioned advantages in Table 1 are: more accurate and efficient [17], need less simulation [17], and need fewer efforts [8], increase the actual wind resources utilization [49], lower operating costs [48], reduced the average of absolute errors [62,63], computationally efficient, reduced computational time [3,9], provide high accuracy with limited data [50], and allow the modelling of multivariate data in a natural way [58].

In Table 1, the most highlighted disadvantages are the need for a high number of simulations to be more accurate which leads to time consumption [1,3], accuracy is limited in comparison to other methods with a similar computational burden [9], difficulty to accommodate the unpredictable influence of variables [57], may be very costly [65,66], needs high efforts [69], not flexible for non-linear variable relationships and complex patterns [78], and mathematically complex [100].

The most relevant applications of the UM techniques that have been discussed in Table 1 are transient stability [13,21,27,32], small-disturbance stability [4,9,18], frequency stability [14,22], voltage stability [15,28,33,36,41], forecasting wind power generation [48], probabilistic power flow [35], power flow [43], and very short-term solar irradiance forecasting [62].

3. Accurate and Efficient UM Techniques

Some UM techniques have high accuracy and efficiency for probabilistic power system stability analysis [1,4,9,50] and can be applied to a large-scale power system with higher uncertainty. These sampling techniques are QMC, point estimate method (PEM), important sampling (IS) technique, cumulant-based method, physics-informed sparse Gaussian process (SGP), polynomial chaos expansion (PCE), Spatio-temporal kriging and analogue models, and artificial neural networks (ANN). These accurate and efficient sampling techniques are described in the following subsections.

Figure 2 shows a graphical representation of the working principle of the various UM techniques. It can be seen that the QMC technique generates its samples using low-discrepancy sequences, which have a uniform demeanour and are based on equal distribution compared to the MC, which generates its samples randomly. In addition, the PEM technique generates small sample points and calculates them on the input probability distributions. The IS technique generates its samples by calculating the importance weight of the probability distribution of the uncertain parameter. Furthermore, the cumulant-based method provides analytical solutions to obtain the variation. The PCE technique analysed uncertainty propagation to get accurate statistical information efficiently. The SGP follows differential and algebraic equations when using uncertain input to generate samples. Spatio-temporal kriging and analogue models calculate the values of points of interest as a weighted sum of values at other points by analysing and modelling the measured locations’ Spatio-temporal distribution. Finally, the ANN technique is a data processing system modelled after biological neurons.

3.1. Quasi-Monte Carlo (QMC) [1,3,4,109,110]

The QMC method works similarly to the standard MC approach, although a different technique is applied to generate the sample sets. While the MC approach actively modifies sampling to ensure that it effectively covers the intended section of the input domain, QMC creates samples using a pseudorandom sequence, which generates an accurate sample from the input distributions. This is often accomplished by producing equidistant samples in the input domain rather than equiprobable using quasi-random (low-discrepancy) sequences like Halton or Sobol sequences. However, such sample biassing is more frequently referred to as importance sampling. Additionally, more significant changes can be made to the sample to completely ignore portions of the input domain if necessary (for instance, this usually applies to extreme event samples that occur in the tails of distributions) [1,3,4].

While using the QMC method, system inputs must be defined with relation to expectations, standard deviation values, the correlation matrix, 3rd and 4th moments, and the length of the desired low-discrepancy sequences (n). This ultimately produces a Halton/Sobol sequence that follows a uniform distribution in [0, 1]. Meanwhile, the performance times of deterministic simulations must be established to derive numerical characteristics and calculate probability distributions. Moreover, the sample points are produced to evenly fill the input domain, while post-processing can be applied to weigh the outputs based on the probability of the input samples. Thus, the results encapsulate the input distribution information. Nonetheless, it is essential to note that the raw results of the QMC simulations are insignificant if post-processing is not performed [1,3,4].

The Sobol and Halton (low-discrepancy techniques) sequences are two of the most popular techniques to generate QMC samples due to their accuracy, efficiency, and implementation simplicity. Firstly, to generate the Sobol sequence for the jth component of the points, it is required to select a primitive polynomial of some degree ($s_j$) in the field (Z₂), which is in a polynomial of the form as shown in Equation (2) [109].

$$x^{s_j}+a_{1,j}{x}^{s_j-1}+\dots +a_{s_j-1,j^x}+1$$

(2)

where the coefficients $a_{1,j,}$, …, $a_{s_j-1,j}$ are either 0 or 1. These coefficients will be used to determine a sequence {$m_1$, $j$, $m_2$, $j$, …} of positive integers by the recurrence relation [108].

$$m_{k,j}=2a_{1,j}{m}_{k-1,j}\oplus 2^2{a}_{2,j}{m}_{k-2,j}\oplus \dots \oplus 2^{s_j-1}{a}_{s_j-1,j}{m}_{k-s_j+1,j}\oplus 2^{s_j}{m}_{k-s_j,j}\oplus m_{k-s_j,j},$$

(3)

For k ≥ $s_j$ _{+ 1}, where ⊕ is the bit-by-bit exclusive-OR operator. The initial values $m_{1,j}$, $m_{2,j}$, …, $m_{s_j,j}$ can be selected and freely presented that each $m_{k,j}$, 1 ≤ k ≤ $s_j$, is odd and less than 2^k. The direction numbers {$v_{1,j}$, $v_{2,j}$, …} are determined by Equation (4) [109].

$$v_{k,j}≔\frac{m_{k,j}}{2^k}$$

(4)

Then ${\mathrm{X}}_{i,j}$, the jth components of the $i$th points in the Sobol samples, is calculated by Equation (5) [109].

$${\mathrm{X}}_{i,j}=b_1{v}_{1,j}\oplus b_2{v}_{2,j} \oplus \dots ,$$

(5)

In addition, the low-discrepancy Halton sequence in relatively prime bases b₁, b₂ … b_s is defined as a sequence as shown in Equation (6) [110].

$${\mathrm{X}}_n={\mathsf{\Phi }}_{b1}\left(n\right),\dots {\mathsf{\Phi }}_{bj}\left(n\right),\dots {\mathsf{\Phi }}_{bs}\left(n\right)$$

(6)

In (6), ${\mathsf{\Phi }}_{bj}\left(n\right)$ is the jth radical inverse function which can be calculated by Equation (7) [109].

$${\mathsf{\Phi }}_{bj}\left(n\right)={\displaystyle \sum }_k{a}_{ij}\left(k\right)b_j^{-k-1}$$

(7)

where the b_j is the jth prime and the $a_{ij}$(k) values are in the base of b_j digits of the integer $i-1$, which is calculated as follows in Equation (8) [109]:

$$i-1={\displaystyle \sum }_{k=0}{a}_{ij}\left(k\right)b_j^k$$

(8)

3.2. Point Estimate Method (PEM) [1,3,31]

A small number of specified sample points and relevant weights are required to apply PE methods, and these are calculated based on the input probability distributions, which can be used to determine the estimated system output values. Several types of PE methods are available, each of which has different requirements in terms of the number of samples required (this is often associated with the number of uncertainties n, with methods such as 2n + 1 and 4n + 1). Moreover, the computational efficiencies of different PE methods vary significantly, with each approach having a different capacity to manage correlated and asymmetrically distributed variables (it is vital to consult original references before applying such models) [1,3].

Many steps are required when using PE methods, including calculating input concentrations about uncertainty (i.e., sample points and associated weights), carrying out deterministic studies for each concentration, using deterministic simulation weights and outputs to compute raw moments, working out the standard and output central moments, and creating pdfs using expansions if necessary [1,3].

Let us assume a function y = f(x) to briefly describe PEM, including the n-dimensional input parameter x with mean $x_m$. Then, the estimated points are ($s_i$ + $x_m$), and the weight is $w_j$, which a certain PEM can calculate. Then the statistical moments of y are estimated by l sigma points. Thus, the estimated mean of y can be calculated as given in (9), and the kth order central moments of y can be calculated as presented in (10) [31].

$$E\left(y\right)\approx {\displaystyle \sum }_{i=1}^l{w}_{i}f(s_i+x_m)$$

(9)

$$E[y-E(y){]}^K\approx {\displaystyle \sum }_{i=1}^l{w}_i [f{w}_i(s_i+x_m)-{\displaystyle \sum }_{j=1}^l{w}_{j}f(s_i+x_m){]}^k$$

(10)

3.3. Important Sampling (IS) [4,6,111]

Calculation times are significantly faster with the IS when a sampling distribution is selected that produces interesting samples for the quantity that needs to be calculated. Additionally, the IS technique is an MC method that employs proposal distribution to determine the distribution weight’s statistical importance. This helps to resample the random samples [4,6].

The following steps can be used to generate an IS sample when taking $f(x_i$) as the target distribution, $w\left(x_i\right)$ as the importance weight, and g(x_i) as the proposal distribution. Firstly, a random sample must be selected from a proposed distribution (g(x_i)). Secondly, the sample size can be determined, and the probability for each sample can be computed. Thirdly, the values of the importance weights can be calculated, as this is equal to $f(x_i)$/g(x_i). Finally, the dataset can be resampled according to the importance weights [4,6].

The IS technique’s sample generation process is presented in Equation (11) [111].

$${\widehat{P}}_{IS}=\frac{1}{n}{\displaystyle \sum }_{i=1}^n{1}_E\left(x_i\right)w\left(x_i\right)$$

(11)

In (11), the $1_E\left(x_i\right)$ denotes the indicator function of the set E and the $w(x_i)$ is the importance weight which can be calculated using Equation (12) [111].

$$w(x_i)=\frac{f\left(x_i\right)}{g\left(x_i\right)}$$

(12)

where g(x_i) is the PDF’s importance weight, and $f(x_i)$ is the target distribution.

3.4. Cumulant-Based Method [1,3,10]

The cumulative probability distribution is one of the alternatives to modelling uncertainty which refers to a simple sum of cumulants from the separate input uncertainties and can be used to characterise the system output’s cumulants [9]. The output cumulants can then be utilised to determine the output moments. In general, it is easier to determine the cumulant than the distribution moment. Thus, this viable method can be used to analytically calculate output varies depending on input uncertainty [1,3].

The cumulant-based technique requires the following steps to be performed: Calculate uncertain input cumulants using input means and central moment values, Develop system input-output sensitivity (this can be achieved through numerical simulations or analytical methods), Compute the change cumulants in system output, Work out the system output’s central moments, Acquire output standard moments and Formulate pdf if necessary [1,3].

In the cumulants-based method, it is possible to define a probability distribution of a linear combination of several uncertain parameters using a simple arithmetic process rather than convolution. The moments and cumulants technique performs feature extraction from a probability distribution to avoid the complex convolution computation. The γth moment corresponding to a continuous uncertain parameter x is determined by Equation (13) [10].

$${\alpha }_{\gamma }={{\displaystyle \int }}_{-\infty }^{+\infty }{x}^{\gamma }dF\left(x\right)$$

(13)

In (13), the $\gamma$ denotes the moment order. The F(x) presents the cumulative probability distribution of uncertain parameter (x), and the moments about the mean ($\mu \mathrm{of} x$) are known as the central moments, which can be calculated by Equation (14) [10].

$${\beta }_{\gamma }=E[\left(x-\mu {)}^{\gamma }\right]={{\displaystyle \int }}_{-\infty }^{+\infty }{(x-\mu )}^{\gamma }dF\left(x\right)$$

(14)

If x represents a discrete uncertain parameter and there is a probability ($p_c$) for a corresponding element x_c of x; then the γth moment of x can be represented by Equation (15) [10].

$${\alpha }_{\gamma }={\displaystyle \sum }_{c=1}^{\infty }{p}_c{x}_c^{\gamma }$$

(15)

3.5. Polynomial Chaos Expansion (PCE) [79,80,112,113]

The PCE examines uncertainty propagation to achieve accurate statistical information efficiently. It is a spectral method that can be considered a Fourier analysis for random variables. By finding a polynomial chaos expansion for a random variable, it creates a deterministic and finitely parameterised approximation. This approximation can then be used to perform model evaluations and get a PCE representation of the model output. Moreover, the PCE approximates the initially complicated model using orthogonal polynomials and the corresponding coefficients [79,80].

The orthogonality ensures that the first two moments of model output are encoded in the coefficients [112]. The PCE combines mathematical results from measure theory, probability theory, and Hilbert space theory and finds application in a broad range of modelling tasks under uncertainty [79,80].

Suppose an uncertain vector with independent elements x ∈ RM is determined by the combined probability distribution f(x). If Y = M(x), Y ∈ R is assumed as a limited friction computational model as a map such as Equation (16) [113]:

$$E \left[Y^2\right]={{\displaystyle \int }}_{D_x}{M}^2\left(x\right)fx\left(x\right)dx<\infty$$

(16)

Then, the PCE of M(x) is calculated by Equation (17) [113].

$$Y=M\left(x\right)={\displaystyle \sum }_{\propto \in N^M}{y}_{\alpha }{\Psi }_{\alpha }\left(x\right)$$

(17)

In (17), the ${\Psi }_{\alpha }\left(x\right)$ denotes the multivariate polynomials regarding f(x), $\propto \in N^M$ is a multi-index function that identifies the components of the multivariate polynomials ${\Psi }_{\alpha }$ and the $y_{\alpha }$ ∈ R is the corresponding coefficient.

3.6. Physics-Informed Sparse Gaussian Process (SGP) [50,53]

The SGP method uses differential and algebraic equations for uncertain inputs to create samples. However, only sampled data and data that have been disseminated using differential and algebraic equations should be employed to build the SGP model. The SGP has a significant advantage over other sample-intensive or data-hungry deep learning algorithms as only a small number of samples (i.e., a few hundred) are needed [50].

SGP can be applied to large systems with high-dimensional data and unknown input distributions. This method’s computational complexity is significantly reduced while accuracy is maintained [53]. Additionally, it can record the peaks and troughs of uncertain system parameters, improving awareness of system risks and providing important information for preventive control [50].

3.7. Spatio-Temporal Kriging and Analog Models [48]

In this method, any variations in the value are based on assumptions relating to the data distribution’s underlying mean function. Moreover, it calculates the values of points of interest as a weighted sum of values at other points. Through the analysis and modelling of the measured locations’ spatio-temporal distribution, it is possible to reduce the interpolation error. An experimental variogram can also be used to present this spatiotemporal distribution or spatiotemporal variability visually. In essence, the use of kriging is based on the variogram. Typically, kriging models are geographical models, although they can be extended to solve spatial-temporal problems by taking time into account as a separate dimension [48].

This approach has several benefits, including that it provides the mean squared error for the estimation and an estimate of a function’s value. Moreover, it fully uses temporal and spatial information without making any assumptions regarding the probability distribution of the data. Finally, it can be used to estimate a variable’s value across a continuous spatial field using a small set of sampled data points [48].

Suppose a spatio-temporal distribution of a set of available datasets is Z (s_i, t_i) = z₁(s₁, t₂), …, Z_n (s_n, t_n), where Z_i is a value of the parameter (Z) in time (t) and at location (s). Then, the kriging weight is calculated using the means of a covariance function C (s, t) or a semivariogram g (s, t). The semivariogram is more advantageous than the covariance function, which can be calculated with real-case scenarios as it is a graphical representation of the covariance between each pair of its samples. For each pair of samples, the semivariance is plotted over the distance between each pair of samples. The practical semivariogram $\widehat{\gamma }$ (s, t) is expressed as Equation (18) [48].

$$\widehat{\gamma }\left(s,t\right)=\frac{1}{2|N(h_s,h_t)|}{\displaystyle \sum }_{N\left(h_s,h_t\right)}{[Z\left(s_i,t_j\right)-Z\left(s_i+h_s,t_j+h_t\right)]}^2$$

(18)

In (18), the N ($h_s,t_j$) contains the samples that are in the same spatial distance $\left(h_s\right)$ and time ($h_t$). The relation between the semivariogram and the covariance function is given by Equation (19) [48].

$$\gamma \left(s,t\right)=C\left(0,0\right)-C\left(s,t\right)$$

(19)

3.8. Artificial Neural Networks (ANN) [87,114,115,116,117]

The nonlinear behaviours of uncertain systems can be assessed using ANN [114]. These networks offer improved output precision, shorter analysis times, and less computing work. It excels at extracting valuable data from complex datasets via training and learning. The output obtained in the output layer is then compared against the target output to calculate the errors. Then these errors are used as feedback or feedforward to change bias weights until the predicted output results are good enough to reduce the error to an acceptable level [87].

The ANN has high prediction capabilities and is gaining popularity compared to traditional forecasting techniques like time series and regression. It is a well-known method used for precise prediction of renewable energy generation [115], where the trained networks have a correlation coefficient of more than 80% while also becoming more accurate every day with each cycle of real-time data training [116]. The input dataset, the number of hidden layers and their neurons, and the learning process affect the ANN’s prediction performance. The ANN approach can be used to predict the daily, weekly, monthly and yearly mean values for uncertain parameters that are vital for making energy predictions in renewable energy systems. Such parameters include global solar irradiance, solar azimuth angle, temperature, and solar elevation for solar power systems. Meanwhile, wind speed/direction, air pressure, and temperature are critical for predicting wind energy, whilst rainfall levels, temperature, and water pressure are crucial in hydropower generation [87].

Some standard ANN training algorithms are the back-propagation and Levenberg–Marquardt algorithms. The objective of the training is to reduce the error $E$, which can be calculated by Equation (20) [117].

$$E=\frac{1}{P}{\displaystyle \sum }_{P=1}^P{E}_P$$

(20)

In (20), the $E_P$ is the error for training pattern P, where P is the total number of training patterns. The $E_P$ can be calculated using Equation (21) [117].

$$E_P=\frac{1}{2}{\displaystyle \sum }_{k=1}^N{(o_k-t_k)}^2$$

(21)

In (21), the N is the total nodes of the output, the $o_k$ is the output of the system at the kth (output node), and the $t_k$ is the target output at the kth.

The straightforward implementation of the standard back-propagation learning is to update the system biases and weights in the direction of the system function to decrease rapidly to the negative of the gradient. Thus, an iteration of the back-propagation learning algorithm can be expressed as Equation (22) [117].

$$x_{k+1}=x_k-{\alpha }_k{\mathrm{ǥ}}_k$$

(22)

where $x_k$ is the vector of biases and current weights and ${\alpha }_k$ is the current gradient, and ${\mathrm{ǥ}}_k$ is the learning rate.

In the Levenberg–Marquardt algorithm, when the performance function contains a sum of squares (usual in training feedforward networks), then the Hessian matrix is approximated as given in Equation (23), and the gradient can be calculated as in Equation (24) [117].

$$H=J^{T}J$$

(23)

$$\mathrm{ǥ}=J^{T}e$$

(24)

where $J$ is the Jacobian matrix, which has the first derivative of the system error concerning the weights and biases and $e$ is the vector of the system error. Furthermore, the Levenberg–Marquardt algorithm employs this approximation to the Hessian matrix as presented in Equation (25) and is called a Newton-like update [117]:

$$x_{k+1}=x_k-{[J^{T}J+\mu I]}^{-1} J^{T}e$$

(25)

In the case of the scalar $\mu$ being zero, this is only Newton’s technique employing the approximation of the Hessian matrix, whereas if the $\mu$ is large, it evolves gradient descent with a small step size.

4. Uncertainty Modelling (UM) Techniques for Probabilistic Power System Stability Analysis

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]. 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]. 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]. The following subsections 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 UM techniques that have been applied to the different aspects of power system stability analysis.

4.1. Applications of UM Techniques for RESs Modelling

The increased penetration of RESs, particularly the variable generation of solar photovoltaic (PV) plants and wind turbines, has brought significant challenges in power system operation [1,3]. 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].

Table 2. 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], system loads [120,124,125], and PV generation [120,125].	Rate of change of frequency (ROCOF) [120,123,124,126], frequency nadir [120,123,124,125,126], frequency excursion [123,126], and frequency response inadequacy (FRI) [120,121,122,123].	MC [14,123,124,125,126,127,128], SMC [22], and cumulant-based method [39].
Probabilistic voltage stability	Wind speed [12,129,130], PV generation [129,130], generation scenarios [12,131], and system loads [4,129,130,132,133,134].	System loadability [129,135], active load margin [129,131], reactive power margin [12,131,136], frequency of voltage instability [133], probability of voltage instability [133], expected voltage stability margin [133], pdf of the load increase limit [41] and probabilistic critical eigenvalue [132].	MC [4,15,120,129,131,132,137,138], QMC [4,120,139], Sobol [1,4], Halton [1,4], Latin hypercube [4], MCMC [28], PEM [33], probabilistic collocation method [41,120], and cumulant-based method [36,120,140,141].
Probabilistic transient stability	Wind speed [12,129], PV generation [12,129], wind power [12,129], automatic reclosing [142], load demand [32], loading level [143], fault type [143,144], fault clearing time [32,144,145], and fault location [143,144].	Generating rejection requirement [142], transfer limit calculation [142], probability of instability of different lines [144], probability of system instability [144,146,147,148,149,150,151,152], most critical lines [148], maximum rotor angle deviation [153], transient instability [152], maximum relative rotor angle deviation (MRRAD) [121], and probability of transient instability [154].	MC [4,12,13,129,142,153,155,156,157,158,159], QMC [18], SMC [21,160,161,162], MCMC [4,27], PEM [32], Physics-informed Sparse Gaussian Process (SGP) [50], and probabilistic collocation method [40].
Probabilistic small-disturbance stability	Wind-hydro-thermal system [163], wind power [15,18,164], different levels of wind penetration [145], the uncertainty of generation [163], load demand [163], generation or transmission outages [163], and PEV (plug-in electric vehicle) [18].	Damping ratios [18,163,165], damping of oscillations [18,163,164,166,167,168], critical eigenvalues of the system [145], participation factors [163,165], and the sensitivity of critical eigenvalues to the variation of wind power penetration into the system [15].	MC [9,145,163,165,166,167,168,169,170], QMC [18,171], PEM [9,172,173], cumulant-based method [9,164], probabilistic collocation method [9,41,42,174,175], and essential sampling technique [4].

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.

4.2. 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]. 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].

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]. 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 (26).

$$f_{N,n}=f_{0,n}-\Delta f_n$$

(26)

In (26), f_N donates frequency nadir, f₀ is the initial frequency, and $\Delta f$ 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 (27). The lowest ROCOF value means a better system response for a stable and robust power system [124].

$$ROCO{F}_n={\left(\frac{df\left(t\right)}{dt}\right)}_n$$

(27)

In (27), f stands for frequency (Hz), and the $df\left(t\right)$/$dt$ can be calculated based on the per-unit formulation of the swing equation in the probabilistic analysis, as shown in Equation (28). The n is the number of samples/simulations based on the generated datasets using the UM techniques.

$${\left(\frac{df\left(t\right)}{dt}\right)}_n={\left(\frac{\frac{\Delta P\left(t\right)}{S_{b-i}}}{2H_i}{f}_0\right)}_n={\left(\frac{\frac{\Delta P\left(t\right)}{S_{b-i}}}{T_{N-1}}\right)}_n$$

(28)

where the $\Delta P$ is the variation of the active power (MW), ($S_{b-i}$) is the nominal value of apparent power of the generator (MVA), and ($T_{N-1}$) is the acceleration time constant (sec.). The $H_i$ is the system inertia constant, and $f_0$ 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 2, these sampling techniques are MC [14,123,124,125,126,127,128], SMC [22], and Cumulant-based method [39]. Input variables highly influence the probabilistic analysis of power system frequency stability, which is modelled by considering their appropriate probability distributions. Wind speed [120,121,122,123,124], system loads [120,124,125], and PV generation [120,125] 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) [120,123,124,126], frequency nadir (i.e., minimum system frequency) [120,123,124,125,126], frequency excursion [123,126], and frequency response inadequacy (FRI) [120,121,122,123], as presented in Table 2.

4.3. 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]. 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] 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]. 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].

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]. 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]. The load margin can be calculated in the probabilistic analysis based on Equation (29).

$$P_{margin,n}=P_{max,n}-P_{0,n}$$

(29)

In (29), $P_{margin}$ is the load margin (system loadability), $P_{max}$ is the maximum active power, and $P_0$ 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 (VSF_i), which can be calculated in the probabilistic analysis as presented in Equation (30), and the stability measure is VSF_i > 0 [177].

$$VS{F}_{i,n}=\frac{\Delta V_{i,n}}{\Delta Q_{i,n}}$$

(30)

In (30), $\Delta V_i$ represents the voltage variation in a load busbar between the operating point and voltage critical (collapse) point, whereas $\Delta Q_i$ 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 2, different UM techniques have been employed for voltage stability analysis, including MC [4,15,120,129,131,132,137,138], QMC [4,120,139], Sobol [1,4], Halton [1,4], Latin hypercube [4], MCMC [28], PEM [33], Cumulant-based Method [36,120,140,141], and Probabilistic Collocation method [41,120]. Various input uncertain system parameters are considered in probabilistic voltage stability assessment; the uncertain variability input is considered in generation scenarios [12,131], system loads [4,129,130,132,133,134], wind speed [12,129,130], and PV generation [129,130].

Probabilistic output variables are presented as system loadability [129,135], active load margin [129,131], reactive power margin [12,131,136], frequency of voltage instability [133], probability of voltage instability [133], expected voltage stability margin [133], pdf of the load increase limit [41] and probabilistic critical eigenvalue [132], as presented in Table 2.

4.4. 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]. The main factors that can affect power system transient stability are the disturbance’s severity and the system’s initial operating state [1].

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 ($TS{I}_n$) for a significant disturbance rotor angle stability can be calculated in the probabilistic analysis as shown in Equation (31) [129].

$$TS{I}_n=\frac{360-{\delta }_{max,n}}{360+{\delta }_{max,n}}\times 100$$

(31)

In (31), $TSI$ stands for the transient stability index and ${\delta }_{max}$ 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 2, several types of UM techniques have been used for probabilistic transient stability analysis, which are Monte Carlo [4,12,13,129,142,153,155,156,157,158,159], SMC [21,160,161,162], MCMC [4,27], PEM [32], Physics-informed Sparse Gaussian Process (SGP) [50], and Probabilistic collocation method [40]. Typically, in the probabilistic transient stability, the analysis of system uncertainties accounts for the automatic reclosing [142], wind speed [12,129], PV generation [12,129], wind power [12,129], load demand [32], loading level [143], fault type [143,144], fault clearing time [32,144,145], and fault location [143,144].

The output results are presented by: making a dismissal request to maintain system stability [142], showing the transfer limit calculation [142], probability of instability of different lines [144], probability of system instability [144,146,147,148,149,150,151,152], the most critical lines [148], transient stability index (TSI) based on the maximum rotor angle deviation [153], expected frequency of occurrence of transient instability [152], maximum relative rotor angle deviation (MRRAD) [121], and probability of transient instability [154].

4.5. 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]. 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].

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 (32) in the probabilistic analysis [129].

$${\xi }_{i,n}=\frac{-{\sigma }_{i,n}}{\sqrt{{\sigma }_{i,n}^2+{\omega }_{i,n}^2}}$$

(32)

In (32), ${\xi }_i$ is the damping ratio ${\sigma }_i$ denotes the damping of the critical eigenvalue and ${\omega }_i$ 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 2. These UM techniques are MC [9,145,163,165,166,167,168,169,170], QMC [18,171], PEM PEM [9,172,173], a cumulant-based method [9,164], probabilistic collocation method [9,41,42,174,175], and important sampling technique [4]. In the probabilistic small-disturbance stability assessment, uncertain system input variables are considered as wind-hydro-thermal system [163], different levels of wind penetration [145], wind power [15,18,164], the uncertainty of generation [163], load demand [163], disturbance uncertainty concerning element (generation, transmission) outages [163], as well as PEV (plug-in electric vehicle) [18].

The probabilistic output results are the real part of the critical eigenvalue, i.e., damping ratios [18,163,165], damping of oscillations [18,163,164,166,167,168], critical eigenvalues of the system [145], participation factors [163,165], and the sensitivity of critical eigenvalues to the variation of wind power penetration into the system [15].

4.6. Summary of Accurate and Efficient Uncertainty Modelling 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 3. 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]		None	Low	Low	None
Point Estimate Method (PEM) [9,32,33]		None	Low	Low	Low
Important Sampling (IS) [4,178]		None	Low	Low	Low
Cumulant-based method [9,36,39]		Low	Low	Medium	Low
Polynomial Chaos Expansion (PCE) [81,179]		None	Low	None	Low
Sparse Gaussian Process (SGP) [50,180]		None	None	Low	Low
Spatio-temporal Kriging and Analog models [4]		None	Low	Low	None
Artificial Neural Networks (ANN) [86,181,182]		Low	Low	Low	Low

. Interestingly, most UM techniques have not been used for frequency stability studies.

5. Conclusions and Future Works

This paper presents an extensive review and classification of a wide range of sampling techniques that have been employed in various aspects of probabilistic power system analysis. The most widely used sampling techniques are collated and summarised with their characteristics, advantages, disadvantages, and application areas. This review can assist in determining the most accurate and appropriate sampling techniques for probabilistic power system studies.

Among the UM techniques that can be applied to model uncertain random variables for different aspects of power system analysis, the most accurate and efficient UM techniques are the QMC, IS technique, cumulant-based method, PEM, and physics-informed sparse Gaussian process (SGP). These techniques show high accuracy and efficiency for various applications.

Identifying the proper UM techniques for probabilistic power system analysis is vital for assuring future power grids’ operational efficiency and accuracy. The sampling techniques in the probabilistic assessment are more critical than ever with the increased penetration of RESs, new types of system loads, and increased system volatility. Some of the current challenges in the probabilistic studies of the future power network are identified as follows and can be the motivations for further research:

• Implementation, development, or proposition of novel, accurate, and efficient uncertainty modelling techniques for probabilistic analysis of system input parameters to be used in a large-scale power system, particularly their application in performing high-resolution and dynamic stability studies;
• Determination of the power system and its equipment operating boundaries under uncertain circumstances;
• Applications of probabilistic analysis for power electronic-based (inverters, converters) generators, loads, control strategies and their impacts under uncertain circumstances;
• Applications of accurate and efficient UM techniques that have not been applied before for stability analysis in large-scale power networks.