Transient stability assessment in large-scale power systems using sparse logistic classifiers
Introduction
Power systems nowadays are more and more operating close to their physical limits. On one hand, the world electricity demand is always increasing; on the other hand, physically expanding existing power network facilities is usually slow and costly. As a result, dynamic security assessment is becoming more and more important for power system planning and control, as well as for power engineering researchers. Among various types of power system instability problems, the transient stability is the most important one [1]. It refers to the capability of power systems to stay synchornism when an incident of large disturbance happens. These incidents include loss of a load, loss of a generator, or a fault of a certain type on the transmission line, just to name a few.
So far, the most accurate method for predicting post-fault transient behavior is the time-domain simulations [2]. In this approach, a large number of differential and algebraic equations (DAE) describing the network operation conditions are solved through step-by-step numerical methods. However, when an transient stability event occur in real scenarios, power engineering users only have very limited time to judge whether the contingency leads to stable or unstable operating conditions, whereas in the later case, network controlling devices such as a relay should be activated immediately to separate the contingency regions from further harming the entire network. Time-domain simulation methods in a large real power network usually take long time for calculation, therefore become infeasible in practice. On the other hand, with the advent of the big data age, researches have realized that the transient stability problem can be more and more accurately solved by the well-tuned machine learning/ artificial intelligence (AI) methods. In this approach, the power engineering users generate a large number of network operational conditions, and for each types of contingencies, use time-domain simulations to determine whether these operational conditions are transiently stable or not. Then these off-line data are used to train AI models/classifiers, and these models/classifiers are used in on-line scenarios to determine whether field data condition leads to transiently stable or unstable operation conditions. In this approach, it usually takes a long time to train the models/classifiers offline; whereas when these models are applied, it takes almost no time to predict and to make instant decisions in real-time.
In the previous studies, various artificial intelligence methods have been applied. These works include regression methods, as well as pattern recognition (pattern classification) ones. The regression approach includes methods such as artificial neural networks (ANN) [3], kernel ridge regression [4], Lasso regression [5], additive models [6], [7], single index models (SIM) [8], etc; whereas the pattern classification approach includes works that implemented neural networks (NN) [9], [10], [11], support vector machines (SVM) [12], decision trees [13], k-nearest neighbor (kNN) classifier [14], etc. It is worth mentioning that our approach in this paper belongs to the pattern classification category.
In contrast to the statistical methodologies such as Lasso or kernel ridge regression, the emerging of the non-statistical approaches in computer science in the past decades have been accompanied by a even wider rage of techniques applied in the dynamic security assessment in the past years. These techniques include the fuzzy knowledge-based systems [15], short-term memory (LSTM) [16], [17], extreme learning machines [18], deep belief networks (DBN) [19], [20], convolutional neural networks (CNN) [21], and the aforementioned neural network (NN) method. Apart from those data-driven point forecasting techniques, the probablistic forecasting has been more and more applied in the transient stability research [22], [23], [24], [25], [26], [27], [28], [29]. These methods provide a probabilistic reference to the forecasting values and some of these work allow an ambiguous class apart from the standard transient stable and unstable ones.
Apart from the time-domain simulations and the aforementioned AI approach, there are generally another two types of works [30] in the transient stability research. Namely, they are the 1) direct methods including the transient energy function (TEF) methods [31], [32], and 2) Lyanunov exponents methods [33]. However, those two approaches have limitations [30] which make them explored not as much as the AI approach in transient stability analysis.
Furthermore, the work [17] has taken into consideration of the measurement data transmission delay. In [34], the frequency-domain features were investigated and proved very useful. The work [35] implemented a hybrid fault cluster and Thévenin equivalent based framework to identify the stability of faults.
In our research, we use a large number of features to predict the transient stability problem. These features include
- 1.
the voltage magnitudes of all the buses in the power network,
- 2.
the voltage angles of all the buses in the power network,
- 3.
the active power and reactive power of all the generators in the entire power network,
- 4.
the active power and reactive power of all the loads in the entire power network.
In large power networks, the inclusion of all these features makes the problem very high-dimensional. The so-called “curse of dimensionality” phenomenon [36], [37] refers to the fact that in high dimension problems, the traditional pattern recognition or system modeling techniques converge so slowly that they hardly work, unless they are equipped with proper dimensionality-reduction techniques.
One way to circumvent this problem is through the design of an extra criterion [36], [38]. We use this approach and combine it with the classical logistic classifiers. In the case studies we examine this approach using a 470-bus network. By comparing our methodology with several other competing methods recently applied in the field, we demonstrate that our approach provides good prediction performance in large power networks.
The rest of this paper is organized as follows. Section 2 gives a description about the transient stability problem. Section 3 describes the logistic classifier and its extension in the high-dimensional design, as well as the algorithms to numerically solve the sparse logistic classification. Section 4 discusses the generation of data, and compares the sparse logistic modeling with two competing methods recently applied in the field. Section 5 concludes the paper.
Section snippets
The transient stability classification problem
The power systems will come to a new equilibrium state after the occurrence of a transient disturbance. If the disturbance is unstable, the network will undergo cascading outage, and sometimes a major proportion of the systems may be shut down. On the other hand, if the disturbance is stable, then the network will reach stable operating conditions after the fault is cleared. After a transient stability event occurs, the behavior of a power system is fully characterized by the nature of the
Maximum likelihood classification and the logistic classifier
In pattern recognition problems, the researchers are often given the data set of the formwhere is a p-dimensional vector representing the input observations, is a label representing the class. The pattern recognition researcher aims to find the mapping function associated with the classification rule [41]so that for future observations, the classification error would be as small as possible.
In our
Test data generation
Our proposed sparse machine learning algorithms for transient stability analysis is demonstrated by a 470-bus power network. The single-line diagram of this system is shown in Fig. 4. This network consists of 470 buses, 214 loads, 45 generating units, 482 transmission lines, 374 adjustable transformers and 152 fixed shunts. We refer to [53] for a detailed description of this power network.
The training data are generated using the following way. Different load and generation patterns are
Concluding remarks
This paper presents the application of sparse logistic classification in the transient stability classification problem. The high-dimensional nature of the transient stability problem fits the sparse design of the logistic classification using the criterion. Our modeling achieves better classification accuracy compared to two competing methods recently applied in the field. The future work would be to explore how other machine learning techniques can be extended to the high-dimensional
CRediT authorship contribution statement
Jiaqing Lv: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data Curation, Writing – original draft, Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
The author was supported by the Polish National Center of Science under Grant DEC-2018/31/N/ST7/03977.
The author would like to thank Dr. Jayasekara for providing the transient stability data used in this paper.
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