A Practical Security Assessment Methodology for Power System Operations Considering Uncertainty

Abstract

Today, renewable energy sources (RESs) are increasingly being integrated into power systems. This means adding more sources of uncertainty to the power system. To deal with the uncertainty of input random variables (RVs) in power system calculation and analysis problems, probabilistic power flow (PPF) techniques have been introduced and proven to be effective. Currently, although there are many techniques proposed for solving the PPF problem, the Monte Carlo simulation (MCS) method is still considered as the method with the highest accuracy and its results are used as a reference for the evaluation of other methods. However, MCS often requires very high computational intensity, and this makes practical application difficult, especially with large-scale power systems. In the current paper, an advanced data clustering technique is proposed to process input RV data in order to the decrease computational burden of solving the PPF problem while upholding an acceptable level of accuracy. The proposed method can be effectively applied to solve practical problems in the operating time horizon of power systems. The developed approach is tested on the modified IEEE-300 bus system, indicating good performance in reducing computation time.

1. Introduction

During the operation of a power system, operating mode parameters such as the voltage at the buses, power transmitted through the branches, etc., need to be regularly calculated to assess the security of the system by comparing the parameters with their allowable limits. In the event of a safety risk, reasonable solutions should be proposed to resolve it. Deterministic power flow (DPF) is one of the essential tools for power system operation and planning. Nevertheless, during the computation process, the traditional approach uses fixed values of nodal power injections (from power generation, load, etc.) and the known grid structure so that sources of uncertainty from these factors are not considered. This is the main limitation of the traditional power flow (PF) method [1].

To overcome the above-mentioned disadvantage, PPF was proposed and has become a very effective calculation tool. The load, the power generation from a plant, and the operation of an element such as a line, transformer, etc., can follow certain rules of probability. In particular, for today’s power systems, when additional RESs such as solar and wind power, etc., are connected to the system, modeling renewables’ intermittency is very challenging. The intermittency often changes very quickly and stochastically, increasing the uncertainty level in the system. Therefore, a calculation method is required to be able to integrate uncertainties into the calculation process. By using PPF methods, the outputs, i.e., the voltage at buses, PF transmitted on branches, etc., also change randomly according to a law of probability distribution [1]. The PPF analysis allows us to appraise the probability of line overloading, the probability of over-/under-voltage, etc. From there, depending on the characteristics of the system and the severity of the violation, the operator could consider and suggest appropriate solutions to improve the system security.

The PPF approach was first introduced by Borkowska in 1974 [2] and, since then, several research works for PPF have been proposed around the world. Generally, methods for calculating the PF using the PPF technique can be classified into three main categories, i.e., numerical, analytical, and approximation approaches.

The analytical approach [3,4,5,6] makes use of algorithms and analytical techniques such as the convolution and cumulant techniques. Applying these analytical techniques combined with the relationship of the input and output of a PPF problem allows for determination of the distribution function of the output RVs such as the power transmission on the line, nodal voltage, phase angle, etc., according to the system parameters, e.g., the total line impedance, total transformer impedance, etc., and probability distributions of the input RVs of the load and power generation from traditional generators and RESs, as well as the operating status of the devices. The relationship between the input and output of the PF calculation problem is non-linear. Nevertheless, the analytical method works well with a linear relationship between the input and output of the problem. Therefore, the relationship, firstly, needs to be linearized using an expansion technique, e.g., Taylor expansion. One of the outstanding advantages of the analytical approach is that it can give very fast results. Among the cumulant and convolution approaches, the convolution approach is more computationally intensive than the cumulant one. Hence, currently, the cumulant approach is more popular than the convolution approach. To achieve the distribution functions for the output RVs, the cumulant approach is often used simultaneously with expansion techniques such as Gram–Charlier or Cornish–Fisher expansion [4]. Owing to the advantage of fast computation, the analytical approach could be applied for a large-scale power system in practice. Nevertheless, the analytical approach has some drawbacks. Firstly, the accuracy of the analytical approach is significantly affected by the use of techniques that linearize the input and output relationship, especially when the input RV changes over a wide range, for example, in the case of RESs. Secondly, the analytical approach uses expansion techniques that can perform well in the case of the distribution functions of the input RVs being either Gausian distribution or close to Gausian distribution. In fact, the distribution functions of the input RVs of the PPF problem for a power system, in practice, often follow non-Gausian distribution, so the achieved results will be limited. To be able to integrate discrete distribution functions of input RVs into the calculation process, the Von Mises method is proposed [1].

The typical approach for the group of approximation ones in calculating PPF is the point estimate approach [7,8]. In this approach, the input RV is decomposed into a sequence of value and weight pairs. Next, the moment of the output RV is computed as a function of the input RV and then the output RV distribution function is obtained. The point estimate approach can provide relatively fast results. Moreover, different from the analytical approach, this approach uses the non-linear relationship between the input and output of the PF computation problem. However, the main limitation of the point estimate approach is that its accuracy decreases as the order of the moment increases. Another drawback is that the computation time required increases significantly as the number of input RVs increases.

A typical numerical approach is MCS [9,10,11,12,13,14]. In MCS, the input RVs are sampled and then the DPF calculation is carried out for all samples. It repeats the simulation with a very large number of samples to obtain a highly accurate result. MCS uses the non-linear relationship between the input and output of the PF problem, like the traditional approach. The main advantage of the MCS approach is that it gives very accurate and reliable results. Moreover, the probability distributions of the input RVs in MCS are easy to represent. However, the biggest drawback is that the calculation volume is heavy and the calculation time is relatively long, thus making it difficult to apply for a practical large-scale power network. To reduce the computational burden of the MCS method, several clustering algorithms are proposed to reduce the number of samples, and then DPF is run for each cluster instead of running it for all samples. In [15], a PSO algorithm is proposed to use for the clustering task, while K-means is used in [16,17]. Each clustering algorithm has its own weaknesses. For PSO, its iterative convergence rate is modest, and it often becomes stuck at local optimums in dealing with high-dimensional datasets. For the K-means algorithm, it is sensitive to the choice of k and it is difficult to find the optimal k for a given dataset. Moreover, K-means does not scale well to large problems, which is why, in the present study, this problem is focused on.

From the above analysis, it can be seen that each PPF approach has its own characteristics, advantages, and disadvantages.

In addition to the above overview of PPF, to solve various problems related to uncertainty, recent advancements have also been found. In [18], a two-stage robust coordinated dispatch method for multi-energy microgrids is developed to alleviate all of the negative effects of diverse uncertainties from wind power and loads. The interval method is employed in [18] to characterize the uncertainties. A committed carbon emission operation region (CCEOR) of integrated energy systems (IESs) is proposed in [19]. The developed method converts the proposed uncertain non-linear CCEOR model into a deterministic mixed-integer convex CCEOR model. In [20], to take into account different types of uncertainties from the outcomes of disasters, extreme events, loads, and renewable generation, both the pre-restoration and real-time stage measures are coordinated via a two-stage stochastic programming method.

The main contributions of this paper are summarized as follows: (1) The core objective of this study is to develop an approach to calculating PPF that ensures a certain level of accuracy compared to MCS but which must give very fast results close to “real time” operation of the power system. In order to exploit the advantages of the accuracy of the MCS method while reducing the time-consuming, and volume of, calculation, a real time clustering technique combined with MCS in PPF calculation is proposed. The clustering technique applied is simple but effective and suitable for practical application. The large number of samples of input RVs of the PPF calculation problem using MCS is significantly and effectively reduced, so the PPF calculation gives fast results. Thanks to this outstanding feature, the PPF approach can be applied to large power systems in practice and to the operational time frame. (2) In addition, the discussion on the application of PPF methods in power system calculation and analysis is also presented in detail in this article. This provides a clearer and more intuitive picture of the application of PPF analysis in both the planning and operation of power systems. The current limitations of PPF methods in general are also pointed out to provide topics for future research.

The rest of this paper is structured as follows. Section 2 presents the developed methodology, while the results obtained by the developed approach are discussed in Section 3. In Section 4, further discussion on the applicability of various PPF methods in power system analysis problems is given. Concluding remarks are provided in Section 5.

2. Methodology

2.1. Real Time Clustering Technique

Clustering is the division of data into a number of groups so that data points in the same group have similar characteristics to each other and are dissimilar to data points in other groups. It is basically the task of dividing data in a dataset on the basis of similarities and differences between them. Among clustering techniques, K-means is known as the most popular one and is applied in all fields due to its simplicity, efficiency, scalability, and ease of implementation. It can handle large datasets effectively, making it a practical choice for numerous applications. A comprehensive review of the application of K-means clustering in modern power systems is presented in [21]. The K-means clustering algorithm is a type unsupervised machine learning that divides the unlabeled dataset into k different clusters by an iterative algorithm.

The K-means algorithm is implemented as follows:

Step 1: Randomly choose k points or centroids from considered data to initialize the groups or clusters;

Step 2: For each point in the dataset, compute the distance between the point and each of the k centroids; assign each point to its closest centroid to form k clusters;

Step 3: Replace the centroid for each cluster with the mean of all data points assigned to the cluster;

Step 4: Repeat Steps 2 and 3 until the centroids no longer change significantly or after a pre-selected maximum number of iterations. The outputs obtained are the last cluster centroids and the data points assigned to clusters.

Although the K-means method has many advantages, it has some disadvantages that can affect its applicability and performance. The K-means algorithm is not considered to have good scalability for large problems. It is sensitive to the choice of k and it is difficult to find the optimal k for a given dataset. K-means converges to a local minimum, so different initializations will result in different results.

K-means can be very time-consuming with large datasets. For a dataset including n data points, it needs to be run O(nkT) times to calculate the distances between the n data points and each of the k centroids (T is the number of iterations) [22]. In the K-means algorithm, each iteration takes a time proportional to k and n. This explains why the K-means algorithm has poor scalability. The running time will increase with an increasing n or k, or both. Therefore, its efficiency can be significantly enhanced by decreasing the runtime related to n. To deal with the problem of poor scalability, in [22], the authors develop a K-means-lite approach that can obtain the aimed-for centroids in O(1) time with respect to n and exhibits and improved speed-up factor as k and n increase. The accuracy has also been shown to increase. The statistical inference technique is used, in which the k centroids are calculated using a few small samples, instead of repeated exhaustive comparison between centroids and data points. This idea comes from an intuitive extension of the classical central limit theorem. In particular, its use does not need special data structures, does not need to keep distances computed in memory, and does not require repeated exhaustive assignments. It is demonstrated that the use of K-means-lite obtains a drastic efficiency gain and can solve large datasets in real time; it is called advanced data clustering (ADC) in this paper [22].

2.2. Representation of Input Uncertainties

In this paper, to account for the uncertainties regarding the power outputs of generators, loads, etc., and the parameters of components, they are represented by probabilistic distributions. Based on their historical data, the distributions can be estimated. They can also be provided by a forecast technique, especially in solving operational problems.

• Wind generation

For wind speed modeling, the Weibull distribution [23] is commonly used. Its probability density function (PDF) is represented as:

$$f(v)={\displaystyle \frac{h}{c}}·{\left({\displaystyle \frac{v}{c}}\right)}^{h-1}·\exp \left[-{\left({\displaystyle \frac{v}{c}}\right)}^h\right]$$

(1)

where h: the shape parameter; c: the scale parameter; v: wind speed.

The wind turbine characteristic curve can be estimated by wind power–wind speed pairs measurement data [24]. It can also be modeled by a piecewise function as follows:

$$P_{wo}(v)=\{\begin{array}{ll}0\hfill & v\le v_{ci}orv>v_{co}\hfill \\ P_w{\displaystyle \frac{v-v_{ci}}{v_r-v_{ci}}}\hfill & v_{ci}<v\le v_r\hfill \\ P_{wr}\hfill & v_r<v\le v_{co}\hfill \end{array}$$

(2)

where, v_ci, v_co, and v_r are the cut-in, cut-out, and rated wind speed, respectively; P_wr and P_wo are the rated power and output of the wind generation, respectively.

• Solar generation

The solar radiation distribution can be estimated by its observed data. It is also usually represented by a Beta distribution [25], as:

$$f(r)={\displaystyle \frac{\mathrm{\Gamma }(\alpha +\beta )}{\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(\beta )}}·{\left({\displaystyle \frac{r}{r_{\max }}}\right)}^{\alpha -1}·{\left(1-{\displaystyle \frac{r}{r_{\max }}}\right)}^{\beta -1}$$

(3)

where r and r_max are the real and maximum solar radiations, respectively; α and β are two main parameters of the distribution; Γ(·) is the well-known Gamma function.

$$P_{vo}(r)=\{\begin{array}{ll}{P}_{vr}{\displaystyle \frac{r^2}{r_c{r}_{std}}}\hfill & r<r_c\hfill \\ P_v{\displaystyle \frac{r}{r_{std}}}\hfill & r_c\le r\le r_{std}\hfill \\ P_{vr}\hfill & r>r_{std}\hfill \end{array}$$

(4)

where r_c is the radiation at a certain point; r_std is the standard radiation (corresponding to the standard environment); P_vr and P_vo are the rated and output powers of the photovoltaic unit, respectively. Solar generation is commonly required to operate in the unity power factor mode, i.e., its reactive power is equal to zero.

• Load

The uncertainty of each load is usually represented by a Gaussian or normal distribution [1]. The normal distribution function is a continuous function and one of the most commonly used functions in most fields.

The PDF of a normal distribution is as follows:

$$f\left(x\right)={\displaystyle \frac{1}{\sqrt{2\pi }\sigma }}{e}^{-{\displaystyle \frac{{(x-\mu )}^2}{2{\sigma }^2}}}$$

(5)

in which μ and σ are the expectation (average value) and the standard deviation, respectively.

The cumulative distribution function (CDF) of the normal distribution function is calculated as follows:

$$F\left(x\right)={\displaystyle \frac{1}{\sqrt{2\pi }\sigma }}{\displaystyle \underset{-\infty }{\overset{x}{\int }}}{e}^{-{\displaystyle \frac{{(t-\mu )}^2}{2{\sigma }^2}}}dt$$

(6)

For modeling a load, the expected value (mean) is its base power while the standard deviation is assumed to be equal to a certain percentage, e.g., 10%, of the mean.

In addition to the popular probability distributions, shown above, which are very suitable for representing uncertainties from RESs and loads in power systems, currently, in the fields of probability and statistics, there are several other probability distributions and density functions used to incorporate uncertainties into the PPF problem. In other words, the fact that we assume the above distribution functions for RESs and loads does not lose the generality of the use of the PPF method developed in this study.

2.3. Advanced Data Clustering-Based Probabilistic Power Flow

The flowchart of the proposed approach, i.e., the advanced data clustering-based probabilistic power flow (ADCPPF), used for probabilistic security assessment is shown in Figure 1. In the flowchart in Figure 1, the main part that helps significantly improve the calculation time is in the ADC block.

3. Tests and Results

The application of the developed approach is illustrated on a modified IEEE 300-bus system. The information needed for DPF analysis of the system, i.e., the electrical network diagram, bus, branch, generator data, is given in [26]. The system includes 300 buses, 409 branches, 195 loads, and 69 generators. In this test, uncertainties from both loads and RESs are considered. The system is modified by adding 10 solar photovoltaic and 8 wind power plants to buses, as shown in

Table 1. Information on Beta distributions of solar photovoltaic power.

Bus	Rated Power (MW)	α Parameter	β Parameter
196	50	2.5	8
198	80	1.6	9
203	40	1.2	6
204	60	2.2	8
215	70	3.2	7
217	50	3.5	10
221	35	4.2	11
229	90	2.8	8
245	65	1.9	7
246	95	3.1	8

and

Table 2. Information on Weibull distributions of wind power.

Bus	Rated Power (MW)	Scale Parameter	Shape Parameter
118	90	10	2.4
121	80	15	1.6
126	100	11	2.4
142	50	14	1.5
154	40	20	2.2
156	60	28	1.8
159	70	16	1.7
161	95	12	2.3

, respectively.

For the sake of simplicity, but without the loss of generality, the uncertainties of loads and RESs are assumed to be known by forecast techniques. The uncertainty of each load is modeled by a normal distribution with an expected value equal to the base value and standard deviation equal to 10% of the expected value. The solar photovoltaic power uncertainty at each plant is assumed to follow Beta distribution, with its parameters given in Table 1. The distributions are also assumed to be correlated with a correlation coefficient of 0.7. In this study, it is assumed that the simulation scenario occurs during daylight hours, with potential conditions for some solar photovoltaic power generation. Additionally, we assume that the loads and solar and wind power plants are not under abnormal weather conditions (e.g., extreme heat wave, large-scale winter storm, and hurricanes), extremely rare phenomena (e.g., solar eclipse), and catastrophes (e.g., war, earthquakes, and tsunamis). Some of these abnormal events are extremely challenging to forecast with low standard deviation. These abnormal and rare events can also be described by appropriate distribution functions and included in the PPF problem. However, this issue is out of the scope of the current study and is intended to be considered in future studies. Similarly, the uncertainty of the power output at each wind farm is assumed to have Weibull distributions, with the parameters shown in Table 2. The distributions are correlated with a correlation coefficient of 0.8.

In the current test, the base power of 100 MVA is used. The MCS results are used as the reference to evaluate the results obtained by other methods. All tests are executed in Matlab (R2015b) on an Intel Core i5 CPU 2.53 GHz and 4.00 GB RAM PC.

PPF computation is performed to achieve all results of interest of the output RVs in terms of PDFs and/or CDFs. For the purpose of illustration, the distributions of a number of selected output RVs are shown. Figure 2 plots the CDFs of real PF through branch 2–8 (i.e., denoted as P_2–8), while that of the voltage at bus 89 (i.e., denoted as V₈₉) is given in Figure 3. It can be seen from the figures that the developed ADCPPF approach can match well with the curves from MCS, indicating the good performance of the ADCPPF approach.

Because it is difficult to observe clearly when plotting all the results on the same figure, only the results corresponding to the ADCPPF method are depicted. However, to demonstrate its effectiveness, we also compare the results of PPF using ADC with K-means clustering.

As discussed in Section 2.1, K-means clustering can be very time-consuming with large datasets due to the poor scalability problem. Different from K-means, the K-means-lite approach was developed in [16] and can give the results very quickly. Its accuracy has also been shown to increase in comparison with the K-means technique. Therefore, in this test, we do not focus on proving the accuracy of the ADC method compared to the K-means method. Instead, the processing time performance is focused on, thereby indicating that the proposed approach has good applicability for solving problems in the time frame of power system operation. It is clearly shown in

Table 3. Execution time comparison.

Method	Time (s)
MCS	725
ADCPPF with 5 clusters	2.64
ADCPPF with 10 clusters	2.79
ADCPPF with 20 clusters	2.98
ADCPPF with 30 clusters	3.26
ADCPPF with 40 clusters	3.47
ADCPPF with 50 clusters	3.72
ADCPPF with 70 clusters	4.21
K-means based PPF with 5 clusters	140
K-means based PPF with 10 clusters	420

that the ADCPPF approach can give the result in a few seconds, in comparison to the hundreds of seconds needed by MCS. In particular, as previously mentioned, for a dataset including n data point, K-means clustering needs an amount of time equal to O(nkT) to run (in which T is the number of iterations). The running time of K-means will sharply increase with an increasing n and/or k. The modified IEEE 300-bus system is a large-scale system, so K-means-based PPF runs very hard and takes a long time.

From Figure 2 and Figure 3 and Table 3, the accuracy of ADCPPF increases (i.e., intuitively, the corresponding curve in Figure 2 and Figure 3 follows the MCS curve more closely) and the time required to execute the method also increases with an increasing number of clusters. Comparing MCS using K-means and ADCPPF, K-means is challenging in this case and causes the calculation time to increase by much more than when using ADCPPF. Through the above analysis, it is shown that the ADCPPF method has the advantage of both relatively high accuracy and a significantly reduced execution time.

PPF can provide distributions for output RVs that are good for power system security assessment. The probability of under-/over-voltage, line overloading, and so on can be judged. For instance, the upper limit of the real PF of branch 2–8 is supposed to be equal to 445 MW, i.e., corresponding to the vertical line in Figure 2, and the probability that power transmitted through the branch is over its limit can be computed as:

$$\mathbf{P}\left\{P_{2-8}>445\right\}=1.4\%$$

(7)

Similarly, the probability that voltage at a considered bus is out of the operating range can be assessed. However, in this test, the voltages at all buses in the system (for example, V₈₉ in Figure 3) are within the range, i.e., [0.9, 1.1] p.u.

The results obtained by the ADCPPF method can help the operator of the system to evaluate the operating states in order to make suitable decisions and provide solutions for the system.

4. Further Discussion on Applicability of Various PPF Methods

PPF methods can be selected for application in both planning and operation problems of power systems.

• Applying PPF in planning problems: The MCS method is suitable for solving planning problems with long time frames (such as years, seasons, months, weeks) or operational planning problems in the time frame of a few days. In such cases, the time to achieve results does not need to be very fast. In addition to network configuration data, data on sources (especially RES) and loads collected over long periods, i.e., months, a year, or several years, are used to estimate PDFs. These data can also be used by a forecasting technique to provide results for operating the system. If the forecasting technique follows the point forecast approach, the forecast results are provided as a set value at each forecast time point and a corresponding error. These values are considered the expectation and standard deviation of the normal distribution function. These functions are the input information of the MCS problem. If the forecasting technique follows an uncertainty forecast approach such as the probabilistic forecast or scenario forecast approaches [24], then the probability distribution function will be more useful. In fact, when applying MCS in practice, if the system is too large, with many input variables, the processing of MCS is very difficult, taking up an extremely large amount of memory, making it very difficult to process, and, in many cases, it may not be possible. In such cases, clustering (as in the current paper) and dimensionality reduction techniques should be used.
• Application in operational problems: For operational problems with extremely short time frames ranging from a few minutes (i.e., very short-term frame) to a few hours (i.e., short-term frame) and within 24 h (day-ahead), analytical methods and approximation methods with a fast processing time can be applied. The MCS method can also be used in the time frame of several hours or more when the power system is small in scale and the number of input RVs is also small. The MCS method combined with clustering techniques, such as the one proposed in this paper, can be implemented with short or even extremely short operating time frames. Based on data collected from RESs or loads (in addition to other network configuration data), probability distribution functions at the time points of the operational problem are built. These functions can also be provided by a forecast technique.

The next part is an illustrative example. Suppose a technique for forecasting the load or power generation of RESs provides forecast information at different time points in the operational time frame. At that time, the results obtained from PPF methods will be very useful for the system operator.

Figure 4 illustrates the PDF of the current flowing on a line of interest at consecutive times (resolution of 1 h) in the 24 h time horizon of day-ahead operation of the system. The system operator will have a very clear “picture” of the risk of system insecurity so that they can propose appropriate solutions. On the basis of comparing the PDF curve with the maximum limit Imax (corresponding to the limit of the power) of the line of interest, it is possible to determine the point at which there is a risk of the current passing through that line exceeding the allowable value.

For example, in Figure 4, from 16:00 to 18:00, the current gradually increases and, near 19:00, the upper boundary of the PDF line begins to touch and surpass the limit Imax and the calculated probability that corresponds to 19:00 is p1. It should be noted that, as the evening approaches, the solar power source gradually decreases, and in this example, by 19:00, this source no longer generates power. However, this is the peak period in the system where the load increases rapidly until reaching the peak. At this time, the uncertainty in the system comes from the loads and other sources, if any, and not from the solar power source. Then, at 20:00, the current tends to decrease and the level of intrusion has a value of p2. Thus, in the above case, it can be roughly considered that the risk of overloading lasts about 1 h. The overload time can be estimated more accurately if the calculation time-step is smaller (30 min, 15 min, etc.), and this depends on the capabilities of the forecasting technique as well as the requirements of the system operator. If the period of risk of overloading lasts long but is still within the allowable limit, then the overload is considered a temporary overload and does not require any intervention. On the contrary, when there is a risk of overloading for a long period of time and the level of overload is severe, the operator must come up with suitable solutions to ensure security for the system.

It should be noted that, in Figure 4, if we only care about the expected value (similar to the results obtained from the traditional DPF), then all calculated current values are smaller than the limit Imax and safety risks due to the overloading are “not seen”. This is a very new point in the view of system security assessment.

Figure 5 is an illustrative example of assessing the risk of over-/under-voltage at a bus of interest. In this example, the risk of under-voltage at times 19:00 and 20:00 is calculated as p1 and p2, respectively.

The method of calculating and analyzing the security of the system is based on information and data obtained from random input quantities of the problem such as the loads and output powers of RESs. The result is in terms of probability distributions of output quantities such as the node voltage, or the power or current transmitted on branches. This process is performed before the real operation to find the probability distributions of the quantities of interest.

For actual power systems with a SCADA EMS system, this system will provide information about mode parameters and be updated regularly in close to real-time, so monitoring of the operation of mode parameters degrees is performed continuously. In addition, when there is a SCADA EMS system, the data collected for random factors will be more convenient and continuously updated. When the dataset of random factors is more complete, the information obtained is clearer. That is also another benefit when using a SCADA EMS system combined with the PPF method.

The main goal of this paper is to develop an approach to calculating PPF that ensures a certain acceptable accuracy but that also provides very fast results to meet the desired application in the time-frame of almost “real time” in operation of the power system. As mentioned above, the MCS method faces many challenges and is even impossible when applied to real power systems, especially large-scale systems with very short operating time-frames. The clustering algorithm proposed to be applied for solving the PPF problem in this paper is simple, easy to implement, and helps effectively deal with the poor scalability of the K-means algorithm. Therefore, ADCPPF can give quick results with large data that can be applied to solve PPF problems for large-scale power systems.

However, in addition to the advantages and contributions of the proposed method in practical applications, it also has a limitation that needs to be addressed: predetermining the value of k like in the K-means algorithm. In addition, the inherent limitations of the PPF, which does not cover electromagnetic transients, and extreme abnormal operating conditions are unresolved issues. Power system planning and operation should also focus on worst case future and abnormal scenarios (e.g., heat waves, winter storms, etc.). The extreme weather events exacerbated by climate change and global warming pose a high level of uncertainty. Therefore, more diversity in types of uncertainty should be considered for integration into the PPF problem. Energy storage systems (e.g., batteries, pumped storage hydropower, and electric vehicle-to-grid schemes) can mitigate the uncertainty related to RESs that is also not considered in this study. Hardware is one of the important factors affecting PPF analysis, especially in relation to the calculation time. Future research can also focus on processing the algorithm in high-performance computers, overcoming the time processing concerns, and allowing more focus on the model accuracy. These limitations open up topics for consideration in future research.

5. Conclusions

PPF is an effective tool in calculating and analyzing power systems and considering uncertainties existing in the system. It can help the operator of the system in assessing the security. Among the various techniques for PPF, MCS gives highly accurate results but is often very computationally intensive. This article focuses on solving the problem of reducing the computation time for Monte Carlo simulation to achieve a practical tool with high accuracy that gives fast calculation results to be used to solve problems in the time horizon of power system operations. To achieve that goal, we make use of an advanced data clustering technique called K-means-lite. The developed approach, ADCPPF, is extensively tested on a modified IEEE-300 bus system, showing good performance in reducing computation time.