17.1 Introduction

In order to provide an economical and efficient power supply, bulk power exchange between remote locations is essential. Thus, power transmission at extra-high voltage (EHV) levels is a common practice nowadays. However, as with any part of the power system, the elements that constitute the EHV transmission system are exposed to a number of factors that can cause critical faults. In fact, abnormalities introduced during fault events in EHV networks can not only produce catastrophic damage in the equipment but also lead the entire power system into unstable operation. Additionally, in spite of continuously growing energy demand, factors like de-regulated market, economics, rights-of-way, and environmental restrictions have limited the building of new transmission lines. As a result, existing transmission lines are forced to operate at high loadings, close to their operating limits. Consequently, it is imperative to develop protection devices as well as protection schemes with high-speed performance, which decrease the effect of faults on power system operation and also become an economic way to increase power transfer in a network without investment in new transmission lines.

Over the last three decades, a lot of research has been conducted in the development of ultra-high-speed protection algorithms. Particularly in the late 1970s and '80s, a number of researchers presented significant advances in this area. However, the available technology at that time was not adequate for the high computation requirements of such protection schemes, and therefore they could not take advantage of sophisticated techniques for transient analysis.

Despite the important technological advances in microprocessor based relays, the basic principle of transmission line protection has not changed for more than half a century [1]. This principle is based on extracting the power frequency components of current and voltage signals by means of so-called phasor estimation techniques [2]. These techniques use analog and digital filtering to attenuate the non-power frequency signals caused by fault generated transients and extract only information concerning the power frequency component. Thus, the success of phasor estimation techniques depends on how well the non-power frequency components are filtered out. However, in certain applications, particularly in EHV and ultra-high voltage (UHV) transmission lines, fault signals are prone to be highly contaminated with non-power frequency transients. Thus, the accuracy of phasor estimation is reduced [1, 3]. In fact, these algorithms suffer not only from accuracy problems but also from unwanted responses, noise, disturbance rejections, and reliability issues [3]. Additionally, the filtering process involves an inherent delay that limits the operation time to at least the length of the time window of the filter (normally one cycle of the fundamental frequency for phasor estimation techniques).

A solution to improving the operation time of transmission line protection is reducing the relay data window. However, for data windows shorter than one cycle, information on the steady-state power frequency component decreases whereas the transient content becomes more important. These shortcomings suggest that, for developing faster protection schemes, the analysis must be done over the entire frequency spectrum contained in electrical signals [4]. Therefore, identification of complex waveforms is required instead of the evaluation of periodic characteristics [1]. In this field, the term Transient Based Protection appears, which is based on analysis of the entire frequency spectrum contained in transient current and voltage signals appearing after a fault. This principle proposes that high frequency components contained in transient signals provide valuable information for achieving fast and accurate protection [4, 5].

The development of Electromagnetic Transients Programs (EMTP) and real-time digital simulators [6] established the fundamental basis for modeling and analysis of power system transients. In this field, more accurate models have been introduced over the years to represent the complexities of power systems. In fact, nowadays one can affirm with certainty that simulations using EMTP provide a faithful replica of real power systems. Therefore, these simulations allow gathering a (theoretically) infinite number of observations from a modeled power system. In this context, data mining and signal processing tools play an important role, since they allow relevant information to be obtained from simulation-generated data in order to achieve a meaningful understanding of the real phenomena.

It is well known that faults appear in a stochastic manner. Hence, for a power system exposed to a number of external factors, it is almost impossible to predict when, where, and how a fault will arise. However, despite the innumerable variables that influence the features of a fault, all the faults that can take place in a transmission line have the same mean where they appear and propagate, and then they share a common characteristic. Therefore, the patterns that govern fault generated transients help us to develop protection algorithms, based on the knowledge acquired from data mining and signal processing of a representative sample of fault generated transients.

In this context, the data mining technique known as Empirical Orthogonal Functions (a.k.a. Principal Components Analysis) emerges as a powerful tool for power system applications [7–11]. This technique is used to find a small set of orthogonal functions that gives a good interpretation concerning fault generated transients. Results show that fault signals decomposed in terms of these orthogonal basis functions exhibit well-defined patterns, which can be used for recognizing the main features of fault events such as inception angle, fault type, and fault location. Signals of less than a quarter of cycle can be used, hence high-speed protection algorithms can be designed utilizing these orthogonal basis functions.

The organization of this chapter is as follows: Section 17.2 introduces the theoretical background of Empirical Orthogonal Decomposition. Next, applications of Empirical Orthogonal Functions for transmission line protection are provided in Section 17.3. A case study and evaluation of the protection scheme is presented in 17.4. Finally, conclusions are drawn in Section 17.5.

17.2 Empirical Orthogonal Functions

The term Empirical Orthogonal Functions (EOFs) was first mentioned by Edward Lorenz in 1956 as mechanism to find a simplified but optimal representation of climate data with time and space dependence [12]. Subsequently, this technique has been widely employed in oceanography and meteorology for data compression and pattern extraction [13, 14]. Contrary to other feature extraction techniques that employ exhaustive supervised training, EOFs are easy to implement and do not require adjustment of control parameters. They depend solely on the nature of the input data.

In scientific literature, the EOF technique is also referred to as Principal Components Analysis (PCA) [15–19]. Actually both techniques, EOFs and PCA, are the same. However, for signal processing, the term EOF is more appropriate than PCA, because the term EOF and the theory behind it were founded on the analysis of time series structures. EOFs allows decomposition of a signal in terms of orthogonal basis functions that give an optimal representation of the data. Therefore, it leads to a better understanding of the signal structure.

17.2.1 Formulation

Let the vector be a time series representing a p-length data window captured from the ith discrete signal of a set of n measured signals. Then this set of signals can be represented by the n×p data matrix X:

17.1

where n is the number of observed signals and p is the number of samples per signal.

For signal analysis it will be further assumed that each discrete signal (time series) is an observation, and each sample is a feature. The element x_i,j is a generic element of X, which represents the value of the feature j in the observation i.

EOFs are based on the assumption that any observation x_i(t) can be expanded in terms of orthogonal basis functions so that:

17.2

where f₁(t), f₂(t),…, f_p(t) are the orthogonal basis functions, which are called the EOFs, and q_ij is the decomposition coefficient of x_i(t) on f_j(t). The number of EOFs is equal to the number of features (p), thus (17.2) expands x_i(t) without loss of information [20]. However, p could be a large number; then for dimensionality reduction, it is desirable to select a small number of EOFs. Therefore, the series is truncated as follows:

17.3

Here, is the error associated with the truncation, and r is the number of selected EOFs, . In matrix form, (17.3) gives:

17.4

where contains the coefficients q_ij for i=1,2,…n, and j=1,2,…r; is the matrix containing in its columns the r-EOFs [ f₁(t), f₂(t), …, f_r(t)]; and contains in its rows the decomposition errors [ϵ₁(t), ϵ₂(t), …, ϵ_n(t)]′. The detailed expression for (17.4) is then:

17.5

where the decomposition errors are formed by the truncated orthogonal basis functions :

17.6

From the above equation, it can be demonstrated that (n×r matrix of zeros), because of the fact that for (definition of orthogonality). Additionally, without loss of generality it can be assumed that any orthogonal function f_j(t) has norm 1, which means that (r×r identity matrix). Therefore, the decomposition coefficients are obtained by:

17.7

That is:

17.8

It is desirable to find a set of r-EOFs that retain as much as possible of the information of X. That is, E in (17.4) has to be minimized. The solution to the problem states that EOFs are obtained by:

17.9

Each eigenvector f_j(t) has an associated eigenvalue λ_j, which represents the amount of information that f_j(t) retains from X after the decomposition. Thus, those eigenvectors with higher associated eigenvalue acquire more information from input data. The term explained variability (EV) is used as a measure of the amount of information retained by each EOF.

17.10

The order of the EOFs is defined in accordance to their explained variability. That is, the first of the EOFs is the one having the greatest associated eigenvalue.

17.3 Applications of EOFs for Transmission Line Protection

Based on the characteristics and patterns of fault generated transients, this section introduces the proposed protection algorithms. Figure 17.1 presents the general protection scheme: the first stage consists of the formulation of the EOFs, which is the basis for pattern extraction of fault generated transients affecting the protected transmission line. This process is done offline and requires modeling of the protected transmission line and associated power system. Fault simulation is then performed to obtain a fault database, which is then used for attaining the EOF and the associated scores X(q) that represent the fault database in the EOF domain.

The online algorithm begins with signal acquisition that consists of measuring of analog signals from the transducer and transformation to digital signals through an A/D converter. An algorithm is then implemented for event detection: once a potential fault is detected, the wavefront of this event is extracted from the signal; this wavefront must have the same sampling frequency and number of samples as the stated EOF. The detected event defined by the discrete signal x(t) is then decomposed with the selected EOF, which gives a set of values that define the event in the EOF domain x(q). This signal is then compared with precomputed faults that serve as reference for implementation of protection functions. Finally, the protection scheme has to decide which action should be executed.

17.3.1 Fault Direction

When the power system is not radial, faults can be fed by both line ends. Then defining the fault direction is indispensable for adequate fault location and guaranteeing selectivity in power system protection. For this protection function, decomposition using the two first EOFs is used; that is, fault current and voltage signals are represented by:

17.11

17.12

The directional function requires comparison of and , then f₁(t) and f₂(t) obtained for forward fault currents are used. This is not a general rule—EOFs obtained from voltage signals can also be used and the results are similar.

The general scheme for directional evaluation is presented in Figure 17.2. After a fault is detected, the relay extracts the superimposed quantities of current and voltage signals sampled with the same data window and sampling frequency used for obtaining f₁(t) and f₂(t). Then, both discrete signals are decomposed with f₁(t) and f₂(t), which results in:

17.13

17.14

The angle between current and voltage vectors in the EOF space is given by:

17.15

Where and are the norm of and , respectively, and is the dot product. Expressed in terms of the decomposition coefficients, (17.15) gives:

17.16

A recognized feature of forward faults is that they cause superimposed voltages and currents with different sign [3]. Thus, the rule to determine the fault direction is: ϕ close to 180° denotes fault in the forward direction and ϕ close to 0° denotes a reverse (backward) fault (see Figure 17.3). Hence, a forward fault has to fulfill:

17.17

Where δ gives a confidence interval for the possible variations of ϕ, and must be set in accordance to precomputed training faults.

17.3.2 Fault Classification

In order to minimize the impact on power quality and continuity after a contingency, it is required to disconnect as few devices as possible from the power network. Single phase to ground faults have a higher probability of occurrence than multiphase faults. Thus, single pole tripping (SPT) schemes are implemented for improving selectivity in transmission system protection. SPT enhances the stability, power transfer capabilities, reliability, and availability of a transmission system during and after a single phase to ground fault [21, 22]. Furthermore, accurate SPT accompanied with automatic reclosing allows greater improvement of power system stability [23]. Therefore, in order to enable SPT and auto reclosing schemes, a method for classifying faults is essential.

17.3.2.1 Required EOF

For fault direction, data analysis of fault data is focused only on the first two EOFs, which normally contain more than 90% of data variability. However, information provided by f₁(t) and f₂(t) is not sufficient for fault classification, since faults of different types can have similar values in this space. Hence, in order to achieve an effective fault classification, those EOFs with smaller variances should be also considered, since they provide significant information of fault characteristics.

17.3.2.2 Fault Type Surfaces

A training database in the EOF space X(q) contains the coefficients of training faults decomposed in terms of the r selected EOFs. X(q) includes observations of the six possible fault types (ABC, AB, ABG, AC, ACG, and AG), which describes a perfectly defined r-dimension pattern, where fault location and fault inception angle patterns are relevant.

During the fault simulation stage, only a representative sample of possible faults is generated for training data. With the purpose of covering the entire fault type surface, formed in the r-dimensional space given by the selected EOF, the reference data is enlarged with estimated observations created by interpolation in accordance with the fault location and inception angle patterns.

17.3.2.3 Defining the Fault Type

Figure 17.4 presents the scheme for fault classification. After an event is detected, the relay extracts the superimposed current quantity sampled with the same data window and sampling frequency of the EOF. The discrete signal obtained is decomposed with the selected EOF, and the resultant vector is:

17.18

The fault type is identified using k-nearest neighbor classification (kNN). Here, the evaluated signal is assigned to the class of its nearest (most similar) sample in the database Y(q). Different metrics can be used to find the k-nearest neighbor points, for instance: city block, Euclidean, or cosine distance. Cosine distance (17.19) is recommended since it evaluates the angle between the two vectors, and it is independent from vectors magnitude. This metric improves classification for ground contact faults, because fault resistance influences the current magnitude but not its angle.

17.19

where is a reference fault current that belongs to the reference database Y(q), and r is the number of selected EOFs. Additionally, when the evaluated event is a fault occurring in the protected transmission line, the minimum distance between and Y(q) is small; then, a maximum distance d_ref can be established as a first criterion to classify the event as an internal fault (inside the protected transmission line), non-fault event, or external fault.

In order to improve the security of this classification method, an additional criterion is included for choosing the more severe fault type in the case that the k-nearest neighbor classification faces a conflict—for instance, two surfaces with similar nearest neighbors to the evaluated fault.

17.4 Study Case

17.4.2 The Power System and Transmission Line

The Western System Coordinating Council (WSCC) 3-Machine 9-Bus test system [28] (see Figure.17.5) is evaluated in ATP/EMTP. The longest transmission line (line B6-B9, 180 km) is studied. Line B6-B9 and its adjacent lines B4-B6 (96 km) and B8-B9 (108 km) are modeled using the frequency dependent model of J. Martí [26], which provides an accurate and reliable time domain model for electromagnetic transients simulations. See model data in Appendix A. Fault signals are registered in relay R₆ located at bus B6.

17.4.4 Training Data Matrix

The training data matrix is composed by faults described in Table 17.2. From a data mining perspective, every discrete current or voltage fault signal x_i(t) is an observation, and the samples of each signal are their features. This set of signals is represented by the n×p data matrix X shown in (17.1).

17.4.4.1 Data Window

Defining the time window length is a crucial aspect in the design of a protection algorithm. The longer the time window, the more information it provides, whereas a shorter time window yields high-speed decision making. In the present analysis, time windows in the order of 600 µs are used, which is long enough to observe the evident patterns of fault generated transients and avoids reflected waves from the remote bus affecting the wave shape of signals measured by relay R₆ (see Figure. 17.6). A methodology such as that proposed in [29] can be used to detect the initial wavefront.

17.4.4.2 Sampling Frequency

In a transmission line, frequencies in the order of some hundreds of kHz are expected in fault generated transients. By virtue of Nyquist's theorem, the sampling frequency should be twice that frequency if the entire frequency spectrum is taken into account, which means an additional cost in the sampling process during A/D conversion. However, the sampling frequency should be distinguished depending on the protection function to be implemented, for instance a directional function is associated with the two firsts EOFs only, which are related to low frequency patterns and therefore they are scarcely affected by high frequency components [14, 16]. Hence, not too high a sampling frequency is required for pattern based directional protection. On the other hand, fault classification and location can require the identification of high frequency patterns, thus a high sampling frequency is necessary in this function. In the following sections, various sampling frequencies are studied in order to show the effect of this variable on the signal patterns.

17.4.5 Signal Conditioning

17.4.5.1 Superimposed Component

The pre-fault component is removed from fault signals, by means of (17.20), in order to obtain the superimposed component introduced by the fault:

17.20

where:

1. is the generated deviations from the steady-state signals;
2. x(t)is the discrete signal obtained from the model;
3. x_S(t) is the pre-fault signal.

A common practice to derive the incremental magnitudes is using the signals measured exactly one cycle before the actual measurement. This can be achieved, for instance, by using a delta filter [30]. Another practice is using high-pass filters to suppress the steady-state components so that only the high frequency components will be present in the relaying signals [3].

The data matrix of fault signals (voltages or currents) is formed with the superimposed components :

17.21

17.4.5.2 Centering the Variables

An essential condition for applying EOFs is removing the mean of each variable from the data set in order to center the data on the origin. This process does not change the EOFs, because they are obtained from the variance–covariance matrix, which is computed from the centered data. However, for a new signal that is going to be compared with predefined faults, it must be centered using the mean vector of the training data matrix.

17.4.5.3 Scaling

In some cases, such as for implementing the pattern based directional function, data can be scaled to set data in a common scale independent from measurement units. In this case, the coefficients of the training matrix are scaled between -1 and 1 by dividing by the maximum absolute coefficient; this last operation does not affect the data shape.

For protection functions such as fault classification and location, data scaling is not recommended since the fault magnitude contributes to adequate fault classification.

17.4.6 Energy Patterns

An alternative for categorizing a transient event is by evaluating the energy associated to voltage and current waveforms. In signal processing, the energy E_i of a p-length discrete signal is computed by:

17.22

Where is the time interval between each sample of the discrete signal. The mean power P_i and the RMS value in the interval are given by:

17.23

17.24

The RMS value as a measure of energy was evaluated for the group of fault current signals generated with the conditions detailed in Table 17.2. Results highlight that faults of the same type follow a defined pattern governed by a fault inception angle pattern and a fault location pattern (see Figures 17.7 and 17.8).

Combination of both characteristics (location and inception angle) for single phase to ground faults (AG) with is presented in Figure 17.9. Here is can be seen that faults of the same type, in this case single phase to ground faults, have a defined place in the demarked surface.

The data window plays an important role in the patterns defined by the RMS values of fault currents. Longer data windows have more information and hence allow major differentiation between fault signals. For example, see in Figure 17.10 how the location of a fault can be distinguished using longer data windows. Nonetheless, longer signals (in terms of time) contain transient components altered at remote nodes, and therefore they are affected by the configuration of those nodes (see Figure 17.11). Although this aspect is not a major limitation, it has to be considered in the design of the protection algorithm.

In general, the energy associated with faulted signals is higher than the energy of non-fault signals. However, some faults can generate signals with relatively low associated energy: such is the case of high impedance faults (HIF) or faults that could occur with small inception voltage. On the other hand, some non-fault transients can produce signals with high energy: for instance, transmission line or capacitor bank switching. A mechanism to distinguish faults arising on the protected device and external faults is also necessary since both can produce high energy signals. Here, EOF decomposition plays an important role because EOFs can be used to decompose energy patterns into an n-dimension space that retains the dominant patterns of data processed.

17.4.7 EOF Analysis

17.4.7.1 Computing the EOFs

The training data matrix is composed of the faults detailed in Table 17.2. That is, the data matrix contains 6 fault types, 12 fault inception angles, and 19 fault locations. That gives a total of 1368 (6×12×19) faults. Each current signal is a 600 µs discrete-time series. The number of samples in each signal will depend on the sampling frequency. Different sampling frequencies ( f_s) are studied in order to show the incidence of this variable in the EOF. The left singular vectors and their related eigenvalues of the data matrix are computed. Explained variability EV of the first eight EOF is summarized in Table 17.4.

fs		EV(%)a
		EOF1	EOF2	EOF3	EOF4	EOF5	EOF6	EOF7	EOF8
1 MHz	1 µs	79.08	12.90	3.96	2.28	0.53	0.45	0.22	0.14
200 kHz	5 µs	79.70	12.50	4.04	2.17	0.62	0.38	0.19	0.11
100 kHz	10 µs	82.80	11.25	3.63	1.58	0.31	0.15	0.09	0.05
50 kHz	20 µs	88.86	8.84	1.71	0.26	0.12	0.06	0.04	0.02
20 kHz	50 µs	94.37	4.83	0.58	0.10	0.05	0.03	0.01	0.01

^a Variability percentage from original data retained by each EOF.

From the results presented in Table 17.4, we can draw out some important features of the EOFs:

1. a. The two first EOFs (EOF₁ and EOF₂) represent the dominant pattern since they attain more than 90% of EV in all the cases. This is in fact one of the advantages of EOF decomposition: that most of the variation presented in the original signals is retained in the first EOF.
2. b. The EV of EOF₁ is greater for lower sampling frequencies. This means that the main EOF (EOF₁) is associated with a low frequency pattern.

From the last remark, we can state a relationship between the EOF order and the frequency. From experiences with different transmission lines it has been observed that the first EOF associated with larger eigenvalues is related to low frequencies patterns. That is, the higher the order of the EOF, the higher its frequency (see Figure 17.12). This statement is also mentioned in [14] where Hannachi et al. maintain that “high/low power are associated respectively with low/high frequency variability. Hence low frequency and large scale patterns tend to capture most of the variance observed in the system.” The aforementioned property makes the first EOFs (principal components) immune to noise as is also stated in [16].

Figure 17.13 shows the waveform of EOF₁ and EOF₂ for different sampling frequencies. As can be seen, they are barely affected by the sampling frequency. This occurs because these EOFs represent low frequency patterns and therefore they are not altered by the high frequencies of fault generated transients.

17.4.7.2 Fault Patterns Using EOF

Consider . For this sampling frequency, it can be observed in Table 17.4 that the first three EOFs have more than 99% of the overall explained variability. That is, the entire group of analyzed faults can be decomposed in terms of only three EOFs with less than 1% loss of information:

17.25

Expression (17.8) shows that coefficients q_ij are computed by correlating the ith signal with the associated jth EOF. The coefficients q_ij also presents fault location and fault inception angle patterns similar to energy patterns (see Figures 17.17 and 17.18), presented in Section 14.5. However, the orthogonal decomposition allows representing these patterns in an optimal multidimensional space. For instance, in Figure 17.14, faults generated along the line with the same fault inception angle are projected into a 2-dimensional space conformed by the first two EOF. In this projection, different fault types are well-differentiated.

Results show that even though the fact that faults may appear under (theoretically) infinite stochastic conditions (i.e., fault location, inception angle, fault resistance, and number of involved phases), the waveforms of faulted signals follow a well-defined deterministic behavior. Therefore, it is not necessary to consider the infinite possible fault conditions in order to attain generalized EOFs. In fact, just a small representative sample of different fault conditions is required to achieve a good representation of the total population.

17.4.8 Evaluation of the Protection Scheme

17.4.8.1 Fault Direction

Phase A to ground (AG) fault currents projected in the EOF plane are presented in Figure 17.15. Here, twelve clusters, corresponding to the twelve θ₀ used in the training data, are defined. Each point in Figure 17.15a represents a fault with specific θ₀ and fault location. The corresponding forward fault voltages are presented in Figure 17.15b: it can be observed that forward voltage signals in the EOF plane appear around 180° away from the respective current signal.

On the other hand, backward faults cause voltages and currents with similar angle in the EOF plane denoted by f₁(t) and f₂(t). Figure 17.16 presents the current and voltage signals generated by backward single phase to ground faults (AG).

17.4.9 Fault Classification

In order to obtain reference surfaces for fault classification, a cubic spline data interpolation is employed. Fault location patterns are interpolated from 0 to 180 km with a 2 km step; Angle patterns are interpolated from 0 to 355° with 5° step. Examples of fault location and inception angle interpolation are presented in Figures 17.17 and 17.18. After interpolation, original data grows from 1368 observations to a database with 39 312 observations per fault type, which are stored in matrix Y(q). Figure 17.19 shows the three-dimensional surface of ABG and AG faults for inception angles from 0° to 355°, plotted with interpolated data.

17.4.9.1 Classification

The faults detailed in Table 17.5 were evaluated with the proposed classification scheme. At first, the training stage considered different classification parameters in order to find the best kNN classifier. For the present case study, the optimal classification parameters are as follows. Sampling frequency: 200 kHz; selected EOF: from 1 to 12; distance metric: cosine; number of nearest neighbors: 1. Results are presented in Table 17.6: it can be seen that accuracy for AG faults is 100%, which means that no multiphase fault was classified as a single phase to ground fault AG, which is important for security. Nevertheless, accuracy for multiphase faults is reduced; most confusion appears between faults that involve the same phases, that is, AB with ABG or AC with ACG. Almost all the AG faults that where classified as two phase faults correspond to faults with fault resistance . Recall that the reference database only considered AG for . Hence, classification accuracy could be improved by incorporating one or two more AG surfaces for different fault resistances.

FaultType	Inception Angle (deg)	Rf(Ω)	Fault Locationfrom B6 (km)
ABC	0, 30,…, 330	–	5, 15, 25, …175
ABG	0, 30,…, 330	5 15	5, 15, 25, …175 1, 5, 10, …180
AB	0, 30,…, 330	–	5, 15, 25, …175
ACG	0, 30,…, 330	5 15	5, 15, 25, …175 1, 5, 10, …180
AC	0, 30,…, 330	–	5, 15, 25, …175
AG	0, 30,…, 330	40 0, 20, 100, 300	5, 15, 25, …175 1, 5, 10, …180

	ABC	ABG	AB	ACG	AC	AG	Accuracy
ABC	213	0	0	4	0	0	98.16%
ABG	0	632	8	0	0	4	98.14%
AB	2	28	208	2	0	3	85.60%
ACG	1	0	0	638	6	16	96.52%
AC	0	0	0	16	210	26	83.33%
AG	0	0	0	0	0	1943	100.00%
Sensitivity	98.61%	95.76%	96.30%	96.67%	97.22%	97.54%	97.07%

17.5 Conclusions

Empirical orthogonal functions appear to be a promising alternative for signal processing applied to fault diagnosis. Results show that in spite of the fact that faults may appear under infinite stochastic conditions such as fault location, inception angle, fault resistance, pre-fault current, and number of involved phases, the waveforms of faulted signals follow a well-defined deterministic pattern. The proposed methodology can obtain precise information on a fault by evaluating a very short signal (600 µs for the presented case study), which is really valuable for implementation in high-speed protection schemes.

The proposed protection scheme uses kNN for fault classification and location, which is a very simple classification method. In fact, we highlighted that classification performance can be improved. Readers are encouraged to evaluate more sophisticated classification methods such as support vector machines and other kernel based classifiers.

Appendix 17.A

Study Cases: WECC 9-bus, ATPDraw Models and Parameters

Figure 17.20 shows the WECC 9-bus power system. Lines B4-B5, B5-B7, and B7-B8 are implemented with the 3-phase distribution parameter transposed line Clarke model; transmission lines B4-B6, B6-B9, and B8-B9 are modeled in a more detailed fashion. These lines have three transposition points and eight line segments for simulating faults along the line—see Figure 17.21. Every line segment uses the JMarti model with the data presented in Table 17A.1. Particular data for each line are presented in Tables 17A.2, 17A.3 and 17A.4.

The resultant impedances and admittances for lines B4-B6, B6-B9, and B8-B9 are presented in Table 17A.5.

References

1. 1 Q. H. Wu, Z. Lu, and T. Ji, Protective Relaying of Power Systems Using Mathematical Morphology, 1st ed. Liverpool: Springer, 2009.
2. 2 M. S. Sachdev et al., “Understanding Microprocessor-Based Technology Applied to Relaying,” Power System Relaying Committee, Atlanta, 2009.
3. 3 A. T. Johns and S. K. Salman, Digital Protection for Power Systems, 1st ed. London: Peter Peregrinus, 1997.
4. 4 Z. Q. Bo, F. Jiang, Z. Chen, X. Z. Dong, G. Weller, and M. A. Redfern, “Transient Based Protection for Power Transmission Systems,” in IEEE PES Winter Meeting, Singapore, 2000, vol. 3, pp. 1832–1837.
5. 5 Z. Q. Bo, G. Weller, F. T. Dai, and Q. X. Yang, “Transient Based Protection for Transmission Lines,” in Proceedings of International Conference on Power System Technology (POWERCON), 1998, vol. 2, pp. 1067–1071.
6. 6 N. Watson and J. Arrillaga, Power Systems Electromagnetic Transients Simulation, 1st ed. London: Institution of Engineering and Technology, 2002.
7. 7 A. R. Messina and V. Vittal, “Extraction of Dynamic Patterns From Wide-Area Measurements Using Empirical Orthogonal Functions,” IEEE Trans. Power Syst., vol. 22, no. 2, pp. 682–692, May 2007.
8. 8 P. Esquivel and A. R. Messina, “Complex Empirical Orthogonal Function analysis of wide-area system dynamics,” in 2008 IEEE Power and Energy Society General Meeting - Conversion and Delivery of Electrical Energy in the 21st Century, 2008, pp. 1–7.
9. 9 R. P. Aguilar, F. E. Perez, E. A. Orduna, and N. J. Castrillon, “Empirical orthogonal decomposition applied to transmission line protection,” in 2011 IEEE PES Conference on Innovative Smart Grid Technologies (ISGT Latin America), 2011, pp. 1–7.
10. 10 R. Aguilar, F. Perez, E. Orduna, and C. Rehtanz, “The Directional Feature of Current Transients, Application in High-Speed Transmission-Line Protection,” IEEE Trans. Power Deliv., vol. 28, no. 2, pp. 1175–1182, 2013.
11. 11 J. C. Cepeda and D. G. Colomé, “Benefits of empirical orthogonal functions in pattern recognition applied to vulnerability assessment,” in Transmission Distribution Conference and Exposition - Latin America (PES T D-LA), 2014 IEEE PES, 2014, pp. 1–6.
12. 12 E. N. Lorenz, “Empirical Orthogonal Functions and Statistical Weather Prediction,” Dept. of Meteorology, MIT, Massachusetts, Scientific Report 1, 1956.
13. 13 I. T. Jolliffe, Principal Component Analysis, 2nd ed. New York: Springer-Verlag, 2002.
14. 14 A. Hannachi, I. T. Jolliffe, and D. B. Stephenson, “Empirical Orthogonal Functions and Related Techniques in Atmospheric Science: a Review,” Int. J. Climatol., vol. 27, no. 9, pp. 1119–1152, Jul. 2007.
15. 15 E. Vazquez, I. I. Mijares, O. L. Chacon, and A. Conde, “Transformer Differential Protection Using Principal Component Analysis,” IEEE Trans. Power Deliv., vol. 23, no. 1, pp. 67–72, Jan. 2008.
16. 16 P. Jafarian and M. Sanaye-Pasand, “A Traveling-Wave-Based Protection Technique Using Wavelet/PCA Analysis,” IEEE Trans. Power Deliv., vol. 25, no. 2, pp. 588–599, Apr. 2010.
17. 17 R. Aguilar, F. Pérez, and E. Orduña, “High-Speed Transmission Line Protection Using Principal Component Analysis, a Deterministic Algorithm,” IET Gener. Transm. Distrib., vol. 5, no. 7, pp. 712–719, Jul. 2011.
18. 18 J. A. Morales and E. Orduna, “Patterns Extraction for Lightning Transmission Lines Protection Based on Principal Component Analysis,” IEEE Lat. Am. Trans., vol. 11, no. 1, pp. 518–524, Feb. 2013.
19. 19 Y. Wang, J. Zhou, Z. Li, Z. Dong, and Y. Xu, “Discriminant-Analysis-Based Single-Phase Earth Fault Protection Using Improved PCA in Distribution Systems,” IEEE Trans. Power Deliv., vol. 30, no. 4, pp. 1974–1982, Aug. 2015.
20. 20 D. Randall, “Empirical Orthogonal Functions,” Department of Atmospheric Science, Colorado State University, Colorado, USA, Jan. 2003.
21. 21 F. Calero and D. Hou, “Practical considerations for single-pole-trip line-protection schemes,” in 58th Annual Conference for Protective Relay Engineers, 2005., 2005, pp. 69–85.
22. 22 D. W. P. Thomas, M. S. Jones, and C. Christopoulos, “Phase Selection Based on Superimposed Components,” Transm. Distrib. IEE Proc. - Gener., vol. 143, no. 3, pp. 295–299, May 1996.
23. 23 W. M. Al-hassawi, N. H. Abbasi, and M. M. Mansour, “A neural-network-based approach for fault classification and faulted phase selection,” in Canadian Conference on Electrical and Computer Engineering, 1996, 1996, vol. 1, pp. 384–387 vol. 1.
24. 24 H. W. Dommel, “Digital Computer Solution of Electromagnetic Transients in Single-and Multiphase Networks,” IEEE Trans. Power Appar. Syst., vol. PAS-88, no. 4, pp. 388–399, Apr. 1969.
25. 25 A. Ametani, N. Nagaoka, Y. Baba, and T. Ohno, Power System Transients: Theory and Applications. CRC Press, 2013.
26. 26 J. R. Marti, “Accuarte Modelling of Frequency-Dependent Transmission Lines in Electromagnetic Transient Simulations,” IEEE Trans Power App Syst, vol. PAS-101, no. 1, pp. 147–157, Jan. 1982.
27. 27 G. D. Guidi and F. E. Perez, “MATLAB program for systematic simulation over a transmission line in alternative transients,” presented at the International Conference on Power Systems Transients, Vancouver, Canada, 2013.
28. 28 P. M. Anderson and A. A. Fouad, Power System Control and Stability, 2nd ed. John Wiley & Sons, 2002.
29. 29 R. Aguilar, F. Perez, and E. Orduna, “Ultra-sensitive disturbances detection on transmission lines using principal component analysis,” in 24th IEEE Canadian Conference on Electrical and Computer Engineering (CCECE), Ontario, Canada, 2011, pp. 93–98.
30. 30 G. Benmouyal and J. Roberts, “Superimposed quantities: Their true nature and application in relays,” in 30th Annual Western Protective Relay Conference, Spokane, 2003.