Real-time transient stability assessment in power system based on improved SVM
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Abstract
Due to the strict requirements of extremely high accuracy and fast computational speed, real-time transient stability assessment (TSA) has always been a tough problem in power system analysis. Fortunately, the development of artificial intelligence and big data technologies provide the new prospective methods to this issue, and there have been some successful trials on using intelligent method, such as support vector machine (SVM) method. However, the traditional SVM method cannot avoid false classification, and the interpretability of the results needs to be strengthened and clear. This paper proposes a new strategy to solve the shortcomings of traditional SVM, which can improve the interpretability of results, and avoid the problem of false alarms and missed alarms. In this strategy, two improved SVMs, which are called aggressive support vector machine (ASVM) and conservative support vector machine (CSVM), are proposed to improve the accuracy of the classification. And two improved SVMs can ensure the stability or instability of the power system in most cases. For the small amount of cases with undetermined stability, a new concept of grey region (GR) is built to measure the uncertainty of the results, and GR can assessment the instable probability of the power system. Cases studies on IEEE 39-bus system and realistic provincial power grid illustrate the effectiveness and practicability of the proposed strategy.
Keywords
Power system Transient stability assessment (TSA) Intelligent method Support vector machine Grey region1 Introduction
Real-time transient stability assessment (TSA) is one of the most important means for the power systems to prevent cascading failures, and avoid power system instability and large-area blackout. At the same time, it is also a necessary tool for power system security and stability analysis. However, until now TSA has not been widely used in practice, because each traditional analysis method has its own limitations and defects. The main research progress in TSA includes the following aspects. One aspect is the study of improving simulation speed with high-performance computing method such as parallel computing. The second aspect is the study of direct method which can determine the stability of a system directly according to the state of the system without numerical integration. The other aspects are the improving direct method research to adapt to large-scale grid, the combining direct method research with simulation method to form mixed method and method research of directly determining the system stability using response of post-fault system. It is a whole new research method on the TSA with data mining and big data technologies based on huge amount of data and intelligent algorithms. For the first four methods, a lot of research works are carried out, while there are still some bottle necks that cannot break. A time parallel algorithm is proposed [1], and the bottle neck is the contradiction between parallelism and convergence. There is literature that identifies the instability by the concavity and convexity of the plane trajectory of the post-fault equivalent system, while the recognition speed needs to be further improved [2].
In these years, the value of big data technology has been widely recognized all around the world. It is both opportunities and challenges to analyse big data and explore useful information or knowledge for various application [3]. In the era of big data, big data technology is used to mine useful information in massive data and to make correct decisions [4]. TSA based on data mining usually consists of two main phases, one is offline training and another is online application. Firstly the data mining method is used to train a model based on large amount of simulation data and historical data. Then the mapping from various physical variables to power systems’ steady states is find. At last the power systems’ stability rules are extracted. In online application phase, the pre-prepared stability assessment rules and the current state of the system are used to obtain a stability assessment result, while the stability assessment rules do not dependent on the system models, but rely on offline training. Early in 1989, [5] proposes an artificial neural network (ANN) based approach to predict the critical clearing time (CTT). It is one of the earliest attempts in this field, and since then relevant studies are underway through the world. The studies are categorized into two types, which are pre-fault type and post-fault type, and this paper belongs to the latter one. Generally speaking, the most commonly used methods are ANN [6, 7, 8], decision tree (DT) [9, 10] and support vector machine (SVM) [11, 12]. There are also researches about comparing these new methods, and SVM shows excellent performance [12, 13]. Apart from the traditional SVM, multi-SVM has been applied in the field of TSA [14]. SVM is also used to handle emergency control problem of power system [15] and forecast the wind power generation [16]. Since SVM has been widely used and has many advantages, this paper chooses to apply SVM algorithm in TSA study.
This paper addresses on problems of the data mining method with SVM. There are some deficiencies in data mining method. Firstly, the data mining method based on correlation does not focus on the specific physical model, while the traditional power system stability analysis based on the casual relationship has a mature physical model and theoretical method. Traditional methods have questioned the reliability of data mining method in TSA. Secondly, the classification rules established by data mining method cannot avoid false alarms and false dismissals. For the first problem, this paper integrates stability assessment rules with traditional theory of stability analysis of power systems, and establishes the stability assessment rules under the stability domain concept. To deal with the problems of false alarms and false dismissals, this paper proposes a new strategy with grey region (GR), and two novel SVMs, aggressive support vector machine (ASVM) and conservative support vector machine (CSVM). So that the data mining based on TSA can adapt to the special requirements of power system. In this strategy, most of the cases can be classified into stability or instability. For those small amounts of cases, on the one hand, they can be determined as instability to ensure the accuracy of classification; on the other hand, the probability of instability can be obtained.
2 Theoretical basis
In this section, theoretical basis is prepared for the following studies, which include basic concept of stability domain, a brief introduction of SVM algorithm, and the relationship between them.
2.1 Stability domain theory
The stability domain of the stable equilibrium point is its stable manifold, and is an open constant connected set. The boundary of \(A\left( {{\varvec{x}}_{s} } \right)\) is called the stable boundary of \({\varvec{x}}_{s}\), denoted by \(\partial {\varvec{x}}_{s}\). However, the greatest difficulty in assessing the stability domain is to solve the stability of the boundary, and for high-dimensional systems, the boundary of the actual system is difficult to obtain.
2.2 SVM algorithm
2.3 Theoretical basis of fitting stability boundary with SVM
According to the relevant theory of the stability of the power system, if the stability domain exits, the stable and unstable regions are nonlinearly separable, which means the stable boundary is linearly separable in an infinite dimensional space. This property provides a theoretical basis for the SVM to fit the stable boundary. The SVM realizes the transformation of low-dimensional space to high-dimensional space through mapping function, so that the two regions of nonlinear separable in low-dimensional space are linearly separable in high-dimensional space. Therefore, find the appropriate mapping function and get enough samples on both sides of stable boundary and then a stability assessment rule equivalent to the stable boundary can be established by training SVM. On the one hand, the stability domain theory provides theoretical support for the application of data mining methods and clarifies the physical meaning of the stability assessment rules; on the other hand, the data mining methods provide a new method for obtaining the stable boundary of the stable domain.
- 1)
In the real-time TSA, the stable equilibrium point refers to the one of the post-fault system. The different faults in the same operating mode will result in different post-fault systems, and these systems have different stable equilibrium points and also the stable boundaries are different. It is unrealistic to use a stable boundary as a stability assessment rule to fit all possible stable boundaries in the offline training phase.
- 2)
Real-time transient assessment requires measurement of input characteristics immediately after a fault, and then the results are obtained by substituting the measured values into the stabilization rules. Nevertheless, the states of an actual system are not easy to get through measurement.
- 1)
When the scale of system is large enough, it is considered that the stable domain corresponding to the post-fault equilibrium point obtained by different disturbances before the same fault equilibrium is approximately the same stable domain.
- 2)
In this study, only the state quantities which are easy to measure and the non-state quantities such as active and reactive power of the line and the phase angle of the bus voltage are taken as alternatives of the input features.
The most important approximation is built on the large scale of system, which means the bigger system is, the better result is. But it does not mean the algorithm can’t be used to a system not so big. From the case study in this paper, application in the provincial grid of China is well satisfied.
3 Real-time TSA strategy using ASVM and CSVM
Due to the factors, such as model error, stability domain approximation, and input space dimensionality constraints, the TSA rule cannot be exactly equivalent to the stability boundary of the system stability domain. Therefore, the stable region and unstable region cannot be completely separated in the regular input space, and there exits the GR. In this section, the GR is defined to ensure that the assessment results outside the GR are accurate and credible, rather than find the boundaries of the two regions.
3.1 GR
- 1)
The mapping function corresponding to the kernel function cannot map the input space exactly to the high dimensional space of which the stability boundary is linearly separable, so the mapping relationship error is one of the causes.
- 2)
Because of the two approximations in Section 2.3, it is impossible to find an interface to accurately separate the two types of samples. So, the accuracy rate of stability assessment rules cannot reach 100%.
When kernel function, pre-fault mode of operation and input features of stability rules are determined, the shape and size of GR will be determined. In this case, misclassification cannot be avoided if the accuracy is still focused on. Therefore, a new idea for stability assessment is needed.
3.2 ASVM and CSVM
In traditional SVM, the samples that fall above L1 are classified as unstable, while below are stable. It is obvious that misclassification exits. In this section, a new idea based on GR is proposed: region above L2 is determined as unstable while below L3 is determined stable and the stability assessment of the samples fall into these two regions is ensured correct. For the cases in GR, calculate the instability probability.
3.2.1 CSVM
3.2.2 ASVM
It is worth noting that (12) is exactly the same as (10), but the relative size of \({\varvec{\uplambda}}\) and \({\varvec{\upbeta}}\) are different: for ASVM, the stable samples corresponding to a larger Lagrangian multiplier, whereas CSVM is the opposite.
3.3 Stability assessment rules based on novel SVM
3.3.1 Basic process
The assessment rules are constructed by using CSVM and ASVM to train two boundaries, and the rules naturally divide the input space into three non-overlapping regions: stable region, unstable region and GR.
Stability assessment rules
Type | Result of CSVM | Result of ASVM | Assessment result | Output \(y_{pg}\) |
---|---|---|---|---|
1 | Stable | Stable | Stable | 1 |
2 | Unstable | Stable | Cannot judge | 0 |
3 | Stable | Unstable | Cannot judge | 0 |
4 | Unstable | Unstable | Unstable | −1 |
As shown in Table 1, if the system is judged stable, the output \(y_{pg} = 1\); if the system is judged unstable, the output \(y_{pg} = - 1\); and if the results of CSVM and ASVM are different, the output \(y_{pg} = 0\), which means the samples fall into GR. Combine (10) and (12) and then the mathematical description of stability assessment rule are following (according to the value of the system \({\varvec{X}}_{pg}\) in the input space):
1) When \(f_{CSVM} \left( {{\varvec{X}}_{pg} } \right) > 0\) and \(f_{ASVM} \left( {{\varvec{X}}_{pg} } \right) > 0\), the post-fault system is stable.
2) When \(f_{CSVM} \left( {{\varvec{X}}_{pg} } \right) < 0\) and \(f_{ASVM} \left( {{\varvec{X}}_{pg} } \right) < 0\), the post-fault system is unstable.
3) For other situations, the post-fault system falls in GR, and other methods are needed to make further judgement.
3.3.2 Evaluation index
3.3.3 Instability probability assessment
The larger the I_{DD} value indicates that the distance from the sample to the unstable region is greater than the distance from the sample to the stable region, so that the I_{DD} value can characterize the relative positional relationship between the sample in GR and other regions. The instability probability curve can be fitted according to the proportion of unstable samples under different I_{DD} values. In the real-time application, if the result of the assessment is in GR, calculate the I_{DD} value of the sample and the instability probability is evaluated by using the instability probability curve.
The process of generating the instability probability curve is: ① generate a large number of samples in GR; ② calculate the I_{DD}; ③ calculate proportion of unstable samples under different I_{DD}; ④ obtain instability probability curve by Platt correction.
This optimization problem can be solved by likelihood methods [20].
4 Case study
4.1 IEEE 39-bus system
This section validates the stability assessment and demonstrates the TSA process based on the IEEE 39-bus system. There are 10 generators, 19 loads and 46 lines in this system. Fix the pre-fault operation mode, and make transient stability simulation on the basic of that. In order to avoid islanding, 35 of the 46 lines are involved in the fault scan, and the location of the fault is at both ends of the line. Therefore, there are 70 kinds of faults, the number of which is from 1 to 70. The fault is set as follow: three-phase short circuit occurs at the end of the line, and after a period of time the fault line is removed. Record the active power, reactive power of the line, phase angle of bus voltage and generator information at the moment of fault clearing time as alternative input features. A total of 2500 samples are generated, and 2000 of which are randomly selected as the training samples, which are used to train the model, while the other 500 are the test samples, using for testing the model.
1) Feature and parameter selection
Alternative feature types and their number
Alternative feature type | Alternative feature number |
---|---|
Amplitude of bus voltage | 39 |
Phase angle of bus voltage | 39 |
Active power of line | 46 |
Reactive power of line | 46 |
Power angle of generator | 10 |
Rotor speed of generator | 10 |
Active power output of generator | 10 |
Reactive power output of generator | 10 |
Based on these 15 features, use 5-fold cross validation method to test the 400 groups of C and γ, and balance the classification effect and the overfitting problem, and finally determine the parameters C = 1.2, γ = 0.5.
2) Result of the proposed strategy
Training and testing results of TSA considering conservativeness
Stage | N _{1} | N _{2} | N _{3} | Evaluation indicator | ||
---|---|---|---|---|---|---|
I_{PG} (%) | I_{PFD} (%) | I_{PFA} (%) | ||||
Training | 0 | 0 | 81 | 4.05 | – | – |
Testing | 0 | 0 | 24 | 4.80 | 0 | 0 |
The instability probability curve in Fig. 5 is Sigmoid curve. It indicates that in this case the I_{DD} value varies from − 0.1 to 0.1. With the increase of I_{DD}, the probability of unstable samples decreases, which conforms with its physical meaning: I_{DD} value reflects the distance from the sample point to GR, and the larger it is, the sample point is closer to the boundary of CSVM and the stable region. When I_{DD} value is less than − 0.1, the instability probability is close to 100%; when I_{DD} value is bigger than 0.1, the instability probability is close to 0; and when I_{DD} value equals 0, the instability probability is about 50%. When in real-time application, samples that fall into GR can be used to calculate the instability probability based on their I_{DD} values, which can help operators make decision as a reference.
3) Result of traditional SVM
Training and test results of stability assessment rules based on traditional SVM in IEEE 39-bus system
Stage | Stable result of classification | Unstable result of classification | Evaluation indicator | ||||
---|---|---|---|---|---|---|---|
Stable samples | Unstable samples | Stable samples | Unstable samples | A_{c} (%) | N _{1} | N _{2} | |
Training | 1267 | 7 | 37 | 689 | 97.8 | 7 | 37 |
Test | 309 | 0 | 16 | 175 | 96.8 | 0 | 16 |
The training and test accuracy can reach 97%, which means for the most cases, the method of TSA under the concept of stability region can correctly judge the post-fault system stability. Further analysis of the misclassified samples: during the training stage, the stability assessment rules judge 7 unstable samples as stable, 37 stable samples as unstable, that is, 7 false dismissals and 37 false alarms. In the test stage, the stability rule judges16 stable samples as unstable, which means 16 false alarms. Due to the influence of sample quality, dimension of input features, simulation model error and other factors, the two sorts of samples are not completely divisible. Therefore, the accuracy of evaluation cannot reach 100%, and false dismissals and false alarms can not be avoided. Although this stability assessment rule shows high accuracy in training and test, it cannot be applied in real time application.
4.2 Case study based on provincial power system
This case is based on Hubei power grid of summer operation mode. The training and test samples are generated by scanning the 426 faults. The faults are set as three-phase short-circuit faults on 500 kV or 220 kV lines, and the fault line is being cut off after a period of time. For the fault of each location, select a certain range of fault clearing time to simulate. Record power angle of generator, rotor speed, active power and reactive power of line as alternative features, and 3100 in total. Finally 12780 samples are generated, 10176 of which are selected as training samples randomly, while the left 2604 as test samples.
Set the stop criterion of feature selection as \(- \Delta I_{PG} < 0.1\%\), and 20 features are selected, which are also the active power and reactive power of line. It is worth noticing that, compared with IEEE 39-bus system, the amount of input features needed by Hubei power grid is not directly proportional to the increase in scale. This is due to the fact that the crucial factors that affect the stability of the grid no not increase proportionally with the increase of scale.
Training and test results of stability assessment rules based on CSVM and ASVM
Stage | N _{1} | N _{2} | N _{3} | Evaluation indicator | ||
---|---|---|---|---|---|---|
I_{PG} (%) | I_{PFD} (%) | I_{PFA} (%) | ||||
Training | 0 | 0 | 892 | 8.77 | – | – |
Testing | 0 | 0 | 234 | 8.99 | 0 | 0 |
Training and test result of stability assessment rules based on traditional SVM in HBPS
Stage | N _{1} | N _{2} | A_{c} (%) |
---|---|---|---|
Training | 134 | 53 | 98.16 |
Testing | 27 | 33 | 97.70 |
Similar to small system, there are no false dismissal and false alarm in the training and test results of stability assessment, considering the conservativeness, while the GR is about 9%. Although the traditional SVM-based stability assessment rules can reach 98% of the classification accuracy, false dismissal exits. Therefore, the comparison between Table 4 and Table 5 shows the proposed method is more in line with the actual operational requirements of the power system. Compared with the small system case, the factors affecting the stability of the large system are more scattered. The number of input features of the stability assessment rule needs to be increased to provide enough information. As the input features increase, the GR will gradually decrease. The requirements for the size of GR should be determined according to the requirements of assessment conservativeness in actual operation.
5 Conclusion
This paper proposes a strategy using CSVM and ASVM for real-time TSA. In this strategy, GR is defined to solve the problem that traditional analysis methods cannot avoid false alarms and missed alarms, and it ensures the assessment accuracy outside GR. Also, in order to adapt to the TSA rule, new indicators are proposed, especially the quantitative indicators of GR. In addition, the indicator I_{DD} is proposed to describe the stability probability of samples in GR, and instability probability curve is fitted to be taken as reference for the operators. Although it is very difficult to realize online real-time TSA, this paper has made useful explorations. There is still a long distance from real engineering practice, but the works of this paper show great application prospect.
Notes
Acknowledgements
This work was supported by Science and Technology Project of State Grid Corporation of China, National Natural Science Foundation of China (No. 51777104) and China State Key Laboratory of Power System (No. SKLD16Z08).
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