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Signal processing techniques for sensing based generator coherency analysis
Electrical Power and Energy Systems 104 (2019) 215–221
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Signal processing techniques for sensing based generator coherency analysis Gianluca Bruno, Enrico Maria Carlini, Cosimo Pisani
⁎
T
TERNA, Italian System Operator, Via Palmiano 101, 00138 Rome, Italy
A R T I C LE I N FO
A B S T R A C T
Keywords: Coherency Power system Clustering
In this paper generator coherency analysis of power system is investigated via some signal processing techniques. Sensor data analysis here designed is based on the fusion of advanced signal processing techniques for sensingbased coherency identification, including k-means and fuzzy k-means clustering, agglomerative hierarchical cluster tree, and Independent Component Analysis (ICA). Detailed results are presented and discussed in order to prove the effectiveness of the techniques and carry out a comparative assessment.
1. Introduction The large scale deployment of advanced sensor networks for acquiring and processing synchronized and spatially-distributed measurements is considered one of the most important prerequisite for improving the reliability and the security of large-scale interconnected power systems. This has stimulated the conceptualization of Wide Area Measurement Systems (WAMS), which process phasor data acquired from key buses by Global Positioning System (GPS)-synchronized sensing devices, called Phasor Measurement Units (PMUs) [1]. The large stream of time-synchronized phasor measurements acquired by the PMUs, if properly processed and integrated with other traditional sensors (such as remote terminal units, digital fault recorders, etc.), can be adopted to enhance the power systems observability. This improves the “situational awareness” of Transmission System Operators (TSOs) by enabling advanced proactive functions, such as system integrity protection schemes, adaptive protection, and dynamic on-line security analysis (DSA) [2]. These applications could play a strategic role in modern electrical transmission networks, which are frequently pushed to operate very close to their stability limits. The identification of the coherent group of generators, here referred as Generation Coherence Analysis – GCA, represents strategic and valuable information [3]. In this context, the term coherent means that, after the disturbance onset, the generators exhibit similar rotor angle swing curves, which are so close to each other that they can be assumed to oscillate together [4]. One fundamental remark in coherency analysis is that the formation of coherent groups depends on both the nature and location of the disturbance [5]. Several methods have been proposed in literature for GCA, which can be classified in two main categories: model-based methods and measurement based methods. Model-based methods mainly rely on the ⁎
availability of a power system dynamic model, which is typically linearized around the current operating point. Although the adoption of these methods has been widely explored in the power system literature, their real-time deployment on large and interconnected power systems is highly challenging due to their huge computation demand. Moreover, they need detailed information on the modeling parameters of each power system component, which is not readily available, or affected by large uncertainties. To overcome these limitations, the adoption of measurement-based coherency identification methods has been proposed in the literature [6]. These approaches try to extract actionable knowledge from the grid sensors data streaming, such as generator rotor angle and speed, bus voltage magnitude and phase. The modern literature on measurement-based GCA is vast and [7–11] outlines the main contributions, the open problems, and the research challenges characterizing this emerging research domain. The analysis of these papers reveals that, although several signal processing based techniques have been proposed for GCA, an experimental assessment of their performances on a real and complex operation scenario is still at its infancy [12]. Armed with such a vision, in this paper generator coherency analysis of power system is investigated via some signal processing techniques. Sensor data analysis here designed is based on the fusion of advanced signal processing techniques for sensing-based coherency identification, including k-means and fuzzy k-means clustering, agglomerative hierarchical cluster tree, and Independent Component Analysis (ICA). Detailed results are presented and discussed in order to prove the effectiveness of the techniques and carry out a comparative assessment. The remainder of the paper is organized as follows. Section 2 collected the main literature contributions regarding GCA. In Section 3 the
Corresponding author. E-mail address: [email protected] (C. Pisani).
https://doi.org/10.1016/j.ijepes.2018.06.020 Received 9 February 2018; Received in revised form 7 May 2018; Accepted 5 June 2018 Available online 12 July 2018 0142-0615/ © 2018 Elsevier Ltd. All rights reserved.
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theoretical foundations of the adopted sensing-based coherency identification algorithms are analyzed. In Sections 4 the main results are presented and discussed. The conclusive remarks and the future developments are summarized in Section 5.
computational complexity. This important feature makes the proposed solution particularly suitable for an on-line application.
2. Related works
The measurement based coherency identification algorithms here analyzed are presented in the following with related mathematical framework.
3. Measurement-based coherency identification algorithms
GCA is one of the most fundamental tool for implementing dynamic equivalencing of power systems [13]. Thus, it has attracted large research efforts aimed at defining effective methodologies for identifying coherent areas in interconnected power systems. In particular, GCA in coherency based DE techniques has been traditionally addressed by deploying linearized power system models. These solution techniques are very straightforward but, as outlined in [6], they could be not suitable for GCA of large-scale power systems in the presence of critical disturbances, due to their inability to describe complex non-linear system dynamics. This has limited their deployment in power system control centers, and stimulated research for alternative solutions based on more advanced modeling techniques. In particular, the solution method proposed in [14] combines the Taylor-series expansion of the generator rotor angles at three different transient phases, with a measure of the electrical coupling among the generators, obtained by defining a distance measure based on the system admittance matrix. According to this approach, the generators’ coherency is identified by analyzing the epsilon decompositions of the power flow equations Jacobian matrix. This makes the proposed approach mathematically straightforward, and its algorithmic complexity linear. On the other hand it requires the precise knowledge of detailed parameters of the power systems under analysis, which are very difficult to estimate, and are affected by large uncertainties [15]. To overcome this limitation, more advanced solution methodologies, based on signal processing techniques, have been proposed in the literature. In particular, in [16] a line vulnerability index, which is obtained by processing the post-fault line transient potential energy and the bus voltage magnitudes, is adopted to classify coherent areas in large-scale power transmission systems. The same problem has been solved in other papers by means of spectral analysis techniques, which include: Fast Fourier Transform (FFT) of the generator rotor angles expanded via Taylor series [17], Fourier analysis of generator speed measurements [18], Hilbert Huang Transform of the phase differences among inter-area oscillations and swing curves [19], normalized spectral clustering algorithm and ICA of the generator speed and bus angle signals [20,21], wavelet phase difference analysis of low frequency electromechanical oscillations [22] and multiflock-based analysis of the generator frequencies and phases [23]. Recently, attention has been given to pattern recognition techniques based on neural networks [24] and clustering techniques some of which are presented in Section 3. Although the performance of these methods have been successfully validated on some experimental test-beds, their deployment on realistic transmission systems is still in its infancy, and needs to be researched [25]. In this context, several open problems need to be fixed including the improvement of the computing efficiency, the enhancement of the algorithm scalability, and the complexities in managing large data-sets. These issues are particularly relevant in the context of the ENTSO-E Continental synchronous area, where the complexities deriving by the interconnection of the national power systems, the need for accurate monitoring of the strategic energy corridors. In trying to address these issues, generator coherency analysis of power system is investigated via some signal processing techniques. The main idea is to extract actionable intelligence from measured data-sets by properly combining multiple signal processing techniques, such as kmeans clustering techniques, agglomerative hierarchical cluster tree and ICA. The outcomes of this measurement-based coherency identification paradigm is expected to match with a model-based coherency identification algorithm (e.g. directional cosine [4]), but with a lower
3.1. K-means K-means is the most simple and popular unsupervised learning algorithm able to deal with clustering problems. The aim of the method is to classify an ensemble of measurements, as the case of the generator rotor measurements (i.e. rotor angle or speed) in a certain number of clusters, k, fixed a priori. Each cluster is defined by a centroid properly placed in the clustering space: different centroid locations imply different results. One approach to initialize the algorithm is to position each centroid as much as possible far away from each other. Then, each point belonging to a given measurements ensemble has to be associated to the nearest centroid. At this point the new k centroids can be updated as barycenters of the clusters resulting from the previous step and a new binding has to be performed among the points of the measurements ensemble and the nearest new centroids. The procedure is iterated until no more change of centroids position is obtained. In mathematical terms, given an ensemble of observations x = [x1, x2, …, xn]T, i.e. generator rotor measurements, where each of them is a d-dimensional real vector, k-means clustering collects the n observations into k (≤n) sets, S = {S1, S2, …, Sk} by minimizing the following objective function: k
F=
∑ ∑
‖x−μj ‖2
(1)
i = 1 x ∈ Si
hence: k
argmin = S
∑ ∑
‖x−μj ‖2
i = 1 x ∈ Si
(2)
whereas ‖x − μj‖ is a properly chosen measure of distance (e.g. Euclidean or L1 distance etc.), among n data measurements x and the cluster center μj. Each individual xp in x must be assigned to only one cluster. Standard algorithm for solution of (2) is proposed by Lloyd in [26] and consists in two steps. Starting from an initial guess about k means, the algorithm assigns each observation to the cluster whose mean, mk, yields the least within-cluster sum of squares:
Si(t ) = {x p : ‖x p−mi(t ) ‖2 ⩽ ‖x p−mj(t ) ‖2 ∀ j, 1 ⩽ j ⩽ k }
(3)
Each xp is hence assigned to only one element of S(t) while the new means, to be the centroids of the observations in the new clusters, are updated as follows:
mi(t + 1) =
1 |Si(t ) |
∑ xj ∈ Si(t )
xj (4)
The main drawbacks of the method are that (i) there is no guarantee of convergence to a global minimum of the (1) and (ii) the results are significantly sensitive to the initial randomly selected cluster centers. By running the algorithm multiple times and by recourse to the physical knowledge about the investigated power system both effects can be mitigated. 3.2. Fuzzy k-means The highlighted mathematical framework suggests how k-means could be a good potential candidate for extension to work with fuzzy theory. In particular, while in k-means algorithm each observation, xp, cannot be assigned to more than one cluster, in fuzzy k-means this can 216
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3.4. Independent component analysis
be done by defining proper membership levels indicating the strength of the affiliation between xp and a given cluster. Fuzzy k-means algorithm [27] aims to partition x according to the same philosophy of the standard k-means algorithm. The main difference relies in the objective function to minimize which is: k
G=
Independent Component Analysis is an advanced computational technique able to decompose a multivariate signal into statistically independent and non-Gaussian elemental components (i.e. independent components (ICs)) without any prior information about both the sources and the mixing parameters that have produced the original signal. Spectral ICA [21], demonstrated to be superior to time-domain ICA since invariant to the time delays and phase lags, is considered in our investigations. The former, by directly operating on the normalized power spectra of the measurements is able to provide single-peak and narrow-band ICs which correspond to a frequency of the estimated oscillatory source. Hence, by denoting with X the power spectra of the measurements obtained for instance from the application of discrete Fourier transform, ICA assumes that each individual spectrum is a linear mixture of hidden and independent processes, specifically the ICs. ICs are arranged in the following transformation as row vectors of S:
n
∑∑ i=1
wijm ‖x j−μi ‖2
j=1
(5)
hence: k
argmin = S
n
∑∑ i=1
wijm ‖x j−μi ‖2
j=1
(6)
Apart from the summation indexes exchange performed for the sake of clarity with respect to the k-means formulation, the membership values term wij appears in the objective function that is expressed as:
wij =
1 k
∑ t=1
(
‖xi − cj ‖ ‖xi − ct ‖
)
X = AS
2 m−1
where A is called mixing matrix that has to be estimated together with S from the knowledge of the normalized spectra X. This is accomplished by firstly making a preliminary sphering or prewhitening of the measurements in X through the linear transformation in the sequel:
(7)
whereas m, which assumes real values not less than one, a.k.a. fuzzifier, determines the level of cluster fuzziness. A large m value results in smaller memberships wij and fuzzier clusters. In the particular case of an unitary m value, wij converge to 0 or 1 which means a crisp partitioning. When no information about the clustering problem is available, m is typically set to 2. Therefore fuzzy k-means algorithm returns both the ensemble of the cluster centers μ = M = {M1, M2, …, Mk} and the membership function matrix W, a.k.a. partition matrix. The elements wij ∈ W can vary in the range [0,1] and measure the grade of membership of each observation in each cluster. The extreme values indicate no membership and full membership respectively: grade of membership is related inversely to the distance of the observation from cluster center. Although the different fuzzifications [27] permits exploring alternative clustering solutions, fuzzy k-means is very similar to the standard k-means algorithm. Actually it suffers from the same drawbacks pointed out previously.
V = MX = MAS = BS
IC i = s i = (b i)TV
X = TS R + R
(14)
where T is the reduced mixing matrix, SR contains the three dominant ICs and R is the residual of the process.
max{d (a, b): a ∈ Ab ∈ B}
(8)
min{d (a, b): a ∈ Ab ∈ B}
(9)
∑ ∑
(13)
Scaling, sign modifications and order sorting operations are needed to obtain physically meaningful results about the really dominant ICs [21]. In particular for the sake of visualization in a three-dimensional space, Eq. (11) can be reformulated as:
4. Simulation results
Hierarchical clustering methods aim at yielding a hierarchy of clusters typically expressed via a dendrogram for the sake of explicit visualization. Through an agglomerative or a divisive strategy, all the observations x are recursively merged moving up the hierarchy or splitted moving down the hierarchy [28]. As in the previous algorithms, in hierarchical clustering, a proper metric (a measure of distance between pairs of observations) has to be chosen. Furthermore, once the metric is established, a linkage criterion establishing the clusters dissimilarity as a function of the pairwise distances of observations in the sets must be defined. Through the linkage criterion, the distance between ensembles of observations are determined as a function of the pairwise distances between same observations. The following three examples show, specifically, maximum, minimum and average linkage clustering:
a∈A b∈B
(12)
Since the components of V are mutually uncorrelated, the problem of finding an arbitrary full-rank matrix A can be reduced to the one of finding an orthogonal matrix B, whose columns are the sought ICs, by maximizing the kurtosis of:
3.3. Hierarchical clustering
1 |A||B|
(11)
4.1. IEEE 68 bus To validate the performance of the proposed framework, the GCA of the IEEE 68 bus 16 machine is here considered. This benchmark represents a reduced order model of the New England and New York interconnected system [4]. Five geographical areas interconnected by 86 transmission lines can be identified. The total active and reactive power required by the system total load is PL = 18.233, 9 MW, QL = 2.188, 4 MVAr. All the loads are modeled as conforming load. The generated power is PG = 18.408, 2 MW. All the generators of the test system (G1–G16) are represented by a sub-transient model. All the specific information related to the benchmark components data, such as transmission lines or generators data can be found in [4]. A dynamic simulation has been performed to acquire the time behavior of rotor measurements during a network perturbation [30]. More specifically, a three-phase fault has been applied on the transmission line connecting the buses 4–5 at time instant t = 0.1 s which is subsequently cleared by the line protections in accordance with the ordinary fault clearing and reclosing times. The rotor measurements are depicted in Fig. 1a and b along an observation window of 5 s. Sampling frequency fs is aligned with the one of the Italian WAMS, fs = 50 Hz. Model based approach like cosine directional confirms the presence of five main clusters: C1 = {G1, G2, G3, G4, G5, G6, G7, G8, G9}, C2 = {G10, G11, G12, G13},
d (a, b) (10)
where A and B are two ensembles of measurements and d the chosen metric. Several other linkage criteria have been proposed, among the best known and employed one finds the Ward’s criterion [29]. 217
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a)
45 40
Euclidean distance [ rad ]
Generator rotor angle [ rad ]
2.5
2
1.5
1
0.5
35 30 25 20 15 10 5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0
5
Time [ s ]
6
7
4
5
8
9
1 10 11 12 13 14 16 15
Fig. 2. Benchmark outcomes.
b)
Table 1 K-Means Clustering Results.
1.01
Generator rotor speed [ p.u. ]
2
Generator number [ # ]
1.015
1.005
1
0.995
0.99
3
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Cluster number
Generator number
C1 C2 C3 C4 C5
{ G2, G3, G4, G5, G6, G7, G9}, G1 G8 G10 { G11, G12, G13, G14, G15, G16}
aligned with the ones of the reference system. The three dominant ICs are considered for the sake of visualization, IC1, IC2 and IC3. ICs have been properly adjusted in their peaks and ordered/scaled as hinted in the Section 3 to be physically meaningful. Their dominance has been established by calculating the ICs’ energy via Hilbert operator [3]. Fig. 4 depicts the results of the rows of T in the ICs space: the triplets closest to each other represent coherent generators belonging to the same cluster. As can be appreciated, the results are aligned with the ones of the benchmark. From these outcomes obtained, ICA needs to be considered, together with hierarchical clustering, in a real deployment of a framework that could process spatially measured synchrophasors acquired on large interconnected systems.
5
Time [ s ]
Fig. 1. Generator rotor measurements.
C3 = {G14}, C4 = {G15} and C5 = {G16} [4]. As can be noticed, generators G14, G15 and G16 are very large equivalents and each of them forms a single generator group. The remaining two clusters, C1 and C2, are representative of the New England and New York area respectively. Generator rotor angles are fed to all the considered measurement based identification algorithms. Agglomerative hierarchical clustering is firstly considered to show that, by fixing a certain Euclidean distance among the rotor angle time series, the benchmark outcomes are confirmed. Ward’s linkage criterion is chosen for the purpose. Actually as it can be noted, a proper cut of the dendrogram in Fig. 2 permits to obtain the aforementioned clusters arrangement. Nonetheless the partition solution provided by this clustering algorithm is implicitly not unique as the definition of coherency in [8]. On the contrary the application of k-means and fuzzy k-means algorithms with fixed cluster number (i.e. five) provides results consistent from themselves but not aligned with the ones of the benchmark. Table 1 summarizes the results of k-means algorithm. The results of fuzzy k-means algorithm, which are equivalent to the ones of k-means algorithm in Tab. I as expected, are here shown in pictorial manner through the bar diagram of the partition matrix W in Fig. 3. In this application W is a c-by-g matrix with c the cluster number and g the machine number. Therefore, for each machine one has a score column vector 1-by-c whose elements sum is unitary. The highest value of the latter permits to associate the considered machine in the related cluster. On the contrary, ICA application on the single sided power spectra of the mean centered rotor angle time trends over a range of frequency up to the Nyquist frequency provides aggregation results exactly
4.2. Dynamic Study Model The Dynamic Study Model aims at reproducing the global dynamic performance of synchronously interconnected power system of Continental Europe with reference to systems inertia, frequency containment reserve and dominant inter-area oscillation modes [31]. The goal of DSM is to model the fully synchronously interconnected power system of Continental Europe with the purpose to set up a robust, clear and facilely dynamic model to reproduce the same results in different simulation tools. The model is specifically made up to represent local phenomena (such as voltage transients, system protection of lines, generators), it is appropriate for describing mean frequency transients (system inertia, frequency containment reserve) and dominant inter-area oscillation modes. The load flow model consists in the electrical topology of the synchronously interconnected power system of Continental Europe and in the steady-state information for all production sites regarding a peak demand case in 2020. The modeling of system dynamics is carried out through standard dynamic models for synchronous machines and control devices both. To 218
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C
C
3
C
5
2
C
1
0.9
0.8
Membership grade
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
12
14
16
Generator number [ # ] Fig. 3. Fuzzy k-means clustering results.
Fuzzy k-means provides similar results to the previous approach, while on the contrary simple k-means clustering provides a different but at the same time not-consistent partitioning of generators: it seems for instance that Switzerland that is very often a neutral point from oscillatory point of view instead oscillate with Turkey in far east (see Table 2). Even in this case spectral ICA is able to provide correct indications about generators coherency providing consistent clusters. In particular as it can be seen from Fig. 6 four main clusters can be identified: Cluster 1 - Turkey and Greece, Cluster 2 Spain and Portugal, Cluster 3 Denmark and Switzerland, Cluster 4 – Italy. Cluster 1 and 2 are normally involved in East-West ENTSO-E CE interarea modes while Cluster 3 and 4 in North-South verifying the modal properties of this large interconnected system. 5. Conclusion and future works Fig. 4. ICA clustering results.
In this paper some propaedeutic analyses are carried out for a potential future development of a computational framework for GCA aimed at processing the data acquired by the synchronized, and spatially distributed sensors of the major European WAMS is conceptualized. The proposed solution aims at extracting actionable intelligence from measured data-sets by applying multiple signal processing techniques and is very useful in case of network split. Detailed results demonstrated the ability of the proposed framework in performing GCA in simulated environment. This important feature represents a strategic tool in improving the system resilience to dynamic perturbations, which represents one of the most critical issues to address in modern power systems. On the basis of these results, the future directions of this research will be oriented in enhancing the GCA computational framework with novel methodologies for adaptive dynamic equivalence. Adaptive is referred to the fact that, as highlighted in the introduction section, In particular, the final goal is to process the coherency data in order to identify a reduced order dynamic model for the entire ENTSO-E continental area. This equivalent model could support the implementation of fast contingency analyses, e.g. transient stability studies, or the deployment of special protection schemes supporting islanding control. To this aim a more pervasive penetration of sensors, and the design of
ensure a common approach in different simulation tools, synchronous machines are modeled with a governor (GOV), an automatic voltage regulator (AVR) and a power system stabilizer (PSS). As previously done on IEEE test system a network perturbation is applied, specifically a loss of a huge power plant in Spain, and hence rotor angles from the major generators in the following control areas are sampled: Spain (ES), Portugal (PT), Turkey (TR), Denmark (DK), Italy (IT), Greece (GR) and Switzerland (SW). This reduced set of generators is designed since sufficient enough for the purposes of this analysis: actually they are located at the borders (or the center in case of CH) of the system so they are particularly sensible to the inter-area oscillations which very often drive the coherency of the interconnected system. Rotor angles time series are hence provided to the measurementbased coherency identification algorithms of Section 3. Agglomerative hierarchical clustering is firstly considered to show in Fig. 5 that, however it is fixed a certain Euclidean distance among the rotor angle time series there are some violation of the ordinary modal properties of the system: for instance it seems that Greece and Spain generators oscillate together while they normally oscillate against. 219
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Fig. 5. Agglomerative clustering outcomes.
Table 2 K-Means Clustering Results.
[3]
Cluster number
Generator number
C1 C2 C3 C4
{ CH, DE, IT} PT ES { CH, TR}
[4] [5] [6]
[7] [8] [9]
[10] [11]
[12] [13] [14] [15] [16]
[17]
Fig. 6. ICA outcomes.
[18]
new interoperable frameworks for real-time data sharing among the national TSOs are currently under development in Continental Europe.
[19] [20]
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[21]
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Signal processing techniques for sensing based generator coherency analysis Gianluca Bruno, Enrico Maria Carlini, Cosimo Pisani
⁎
T
TERNA, Italian System Operator, Via Palmiano 101, 00138 Rome, Italy
A R T I C LE I N FO
A B S T R A C T
Keywords: Coherency Power system Clustering
In this paper generator coherency analysis of power system is investigated via some signal processing techniques. Sensor data analysis here designed is based on the fusion of advanced signal processing techniques for sensingbased coherency identification, including k-means and fuzzy k-means clustering, agglomerative hierarchical cluster tree, and Independent Component Analysis (ICA). Detailed results are presented and discussed in order to prove the effectiveness of the techniques and carry out a comparative assessment.
1. Introduction The large scale deployment of advanced sensor networks for acquiring and processing synchronized and spatially-distributed measurements is considered one of the most important prerequisite for improving the reliability and the security of large-scale interconnected power systems. This has stimulated the conceptualization of Wide Area Measurement Systems (WAMS), which process phasor data acquired from key buses by Global Positioning System (GPS)-synchronized sensing devices, called Phasor Measurement Units (PMUs) [1]. The large stream of time-synchronized phasor measurements acquired by the PMUs, if properly processed and integrated with other traditional sensors (such as remote terminal units, digital fault recorders, etc.), can be adopted to enhance the power systems observability. This improves the “situational awareness” of Transmission System Operators (TSOs) by enabling advanced proactive functions, such as system integrity protection schemes, adaptive protection, and dynamic on-line security analysis (DSA) [2]. These applications could play a strategic role in modern electrical transmission networks, which are frequently pushed to operate very close to their stability limits. The identification of the coherent group of generators, here referred as Generation Coherence Analysis – GCA, represents strategic and valuable information [3]. In this context, the term coherent means that, after the disturbance onset, the generators exhibit similar rotor angle swing curves, which are so close to each other that they can be assumed to oscillate together [4]. One fundamental remark in coherency analysis is that the formation of coherent groups depends on both the nature and location of the disturbance [5]. Several methods have been proposed in literature for GCA, which can be classified in two main categories: model-based methods and measurement based methods. Model-based methods mainly rely on the ⁎
availability of a power system dynamic model, which is typically linearized around the current operating point. Although the adoption of these methods has been widely explored in the power system literature, their real-time deployment on large and interconnected power systems is highly challenging due to their huge computation demand. Moreover, they need detailed information on the modeling parameters of each power system component, which is not readily available, or affected by large uncertainties. To overcome these limitations, the adoption of measurement-based coherency identification methods has been proposed in the literature [6]. These approaches try to extract actionable knowledge from the grid sensors data streaming, such as generator rotor angle and speed, bus voltage magnitude and phase. The modern literature on measurement-based GCA is vast and [7–11] outlines the main contributions, the open problems, and the research challenges characterizing this emerging research domain. The analysis of these papers reveals that, although several signal processing based techniques have been proposed for GCA, an experimental assessment of their performances on a real and complex operation scenario is still at its infancy [12]. Armed with such a vision, in this paper generator coherency analysis of power system is investigated via some signal processing techniques. Sensor data analysis here designed is based on the fusion of advanced signal processing techniques for sensing-based coherency identification, including k-means and fuzzy k-means clustering, agglomerative hierarchical cluster tree, and Independent Component Analysis (ICA). Detailed results are presented and discussed in order to prove the effectiveness of the techniques and carry out a comparative assessment. The remainder of the paper is organized as follows. Section 2 collected the main literature contributions regarding GCA. In Section 3 the
Corresponding author. E-mail address: [email protected] (C. Pisani).
https://doi.org/10.1016/j.ijepes.2018.06.020 Received 9 February 2018; Received in revised form 7 May 2018; Accepted 5 June 2018 Available online 12 July 2018 0142-0615/ © 2018 Elsevier Ltd. All rights reserved.
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theoretical foundations of the adopted sensing-based coherency identification algorithms are analyzed. In Sections 4 the main results are presented and discussed. The conclusive remarks and the future developments are summarized in Section 5.
computational complexity. This important feature makes the proposed solution particularly suitable for an on-line application.
2. Related works
The measurement based coherency identification algorithms here analyzed are presented in the following with related mathematical framework.
3. Measurement-based coherency identification algorithms
GCA is one of the most fundamental tool for implementing dynamic equivalencing of power systems [13]. Thus, it has attracted large research efforts aimed at defining effective methodologies for identifying coherent areas in interconnected power systems. In particular, GCA in coherency based DE techniques has been traditionally addressed by deploying linearized power system models. These solution techniques are very straightforward but, as outlined in [6], they could be not suitable for GCA of large-scale power systems in the presence of critical disturbances, due to their inability to describe complex non-linear system dynamics. This has limited their deployment in power system control centers, and stimulated research for alternative solutions based on more advanced modeling techniques. In particular, the solution method proposed in [14] combines the Taylor-series expansion of the generator rotor angles at three different transient phases, with a measure of the electrical coupling among the generators, obtained by defining a distance measure based on the system admittance matrix. According to this approach, the generators’ coherency is identified by analyzing the epsilon decompositions of the power flow equations Jacobian matrix. This makes the proposed approach mathematically straightforward, and its algorithmic complexity linear. On the other hand it requires the precise knowledge of detailed parameters of the power systems under analysis, which are very difficult to estimate, and are affected by large uncertainties [15]. To overcome this limitation, more advanced solution methodologies, based on signal processing techniques, have been proposed in the literature. In particular, in [16] a line vulnerability index, which is obtained by processing the post-fault line transient potential energy and the bus voltage magnitudes, is adopted to classify coherent areas in large-scale power transmission systems. The same problem has been solved in other papers by means of spectral analysis techniques, which include: Fast Fourier Transform (FFT) of the generator rotor angles expanded via Taylor series [17], Fourier analysis of generator speed measurements [18], Hilbert Huang Transform of the phase differences among inter-area oscillations and swing curves [19], normalized spectral clustering algorithm and ICA of the generator speed and bus angle signals [20,21], wavelet phase difference analysis of low frequency electromechanical oscillations [22] and multiflock-based analysis of the generator frequencies and phases [23]. Recently, attention has been given to pattern recognition techniques based on neural networks [24] and clustering techniques some of which are presented in Section 3. Although the performance of these methods have been successfully validated on some experimental test-beds, their deployment on realistic transmission systems is still in its infancy, and needs to be researched [25]. In this context, several open problems need to be fixed including the improvement of the computing efficiency, the enhancement of the algorithm scalability, and the complexities in managing large data-sets. These issues are particularly relevant in the context of the ENTSO-E Continental synchronous area, where the complexities deriving by the interconnection of the national power systems, the need for accurate monitoring of the strategic energy corridors. In trying to address these issues, generator coherency analysis of power system is investigated via some signal processing techniques. The main idea is to extract actionable intelligence from measured data-sets by properly combining multiple signal processing techniques, such as kmeans clustering techniques, agglomerative hierarchical cluster tree and ICA. The outcomes of this measurement-based coherency identification paradigm is expected to match with a model-based coherency identification algorithm (e.g. directional cosine [4]), but with a lower
3.1. K-means K-means is the most simple and popular unsupervised learning algorithm able to deal with clustering problems. The aim of the method is to classify an ensemble of measurements, as the case of the generator rotor measurements (i.e. rotor angle or speed) in a certain number of clusters, k, fixed a priori. Each cluster is defined by a centroid properly placed in the clustering space: different centroid locations imply different results. One approach to initialize the algorithm is to position each centroid as much as possible far away from each other. Then, each point belonging to a given measurements ensemble has to be associated to the nearest centroid. At this point the new k centroids can be updated as barycenters of the clusters resulting from the previous step and a new binding has to be performed among the points of the measurements ensemble and the nearest new centroids. The procedure is iterated until no more change of centroids position is obtained. In mathematical terms, given an ensemble of observations x = [x1, x2, …, xn]T, i.e. generator rotor measurements, where each of them is a d-dimensional real vector, k-means clustering collects the n observations into k (≤n) sets, S = {S1, S2, …, Sk} by minimizing the following objective function: k
F=
∑ ∑
‖x−μj ‖2
(1)
i = 1 x ∈ Si
hence: k
argmin = S
∑ ∑
‖x−μj ‖2
i = 1 x ∈ Si
(2)
whereas ‖x − μj‖ is a properly chosen measure of distance (e.g. Euclidean or L1 distance etc.), among n data measurements x and the cluster center μj. Each individual xp in x must be assigned to only one cluster. Standard algorithm for solution of (2) is proposed by Lloyd in [26] and consists in two steps. Starting from an initial guess about k means, the algorithm assigns each observation to the cluster whose mean, mk, yields the least within-cluster sum of squares:
Si(t ) = {x p : ‖x p−mi(t ) ‖2 ⩽ ‖x p−mj(t ) ‖2 ∀ j, 1 ⩽ j ⩽ k }
(3)
Each xp is hence assigned to only one element of S(t) while the new means, to be the centroids of the observations in the new clusters, are updated as follows:
mi(t + 1) =
1 |Si(t ) |
∑ xj ∈ Si(t )
xj (4)
The main drawbacks of the method are that (i) there is no guarantee of convergence to a global minimum of the (1) and (ii) the results are significantly sensitive to the initial randomly selected cluster centers. By running the algorithm multiple times and by recourse to the physical knowledge about the investigated power system both effects can be mitigated. 3.2. Fuzzy k-means The highlighted mathematical framework suggests how k-means could be a good potential candidate for extension to work with fuzzy theory. In particular, while in k-means algorithm each observation, xp, cannot be assigned to more than one cluster, in fuzzy k-means this can 216
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3.4. Independent component analysis
be done by defining proper membership levels indicating the strength of the affiliation between xp and a given cluster. Fuzzy k-means algorithm [27] aims to partition x according to the same philosophy of the standard k-means algorithm. The main difference relies in the objective function to minimize which is: k
G=
Independent Component Analysis is an advanced computational technique able to decompose a multivariate signal into statistically independent and non-Gaussian elemental components (i.e. independent components (ICs)) without any prior information about both the sources and the mixing parameters that have produced the original signal. Spectral ICA [21], demonstrated to be superior to time-domain ICA since invariant to the time delays and phase lags, is considered in our investigations. The former, by directly operating on the normalized power spectra of the measurements is able to provide single-peak and narrow-band ICs which correspond to a frequency of the estimated oscillatory source. Hence, by denoting with X the power spectra of the measurements obtained for instance from the application of discrete Fourier transform, ICA assumes that each individual spectrum is a linear mixture of hidden and independent processes, specifically the ICs. ICs are arranged in the following transformation as row vectors of S:
n
∑∑ i=1
wijm ‖x j−μi ‖2
j=1
(5)
hence: k
argmin = S
n
∑∑ i=1
wijm ‖x j−μi ‖2
j=1
(6)
Apart from the summation indexes exchange performed for the sake of clarity with respect to the k-means formulation, the membership values term wij appears in the objective function that is expressed as:
wij =
1 k
∑ t=1
(
‖xi − cj ‖ ‖xi − ct ‖
)
X = AS
2 m−1
where A is called mixing matrix that has to be estimated together with S from the knowledge of the normalized spectra X. This is accomplished by firstly making a preliminary sphering or prewhitening of the measurements in X through the linear transformation in the sequel:
(7)
whereas m, which assumes real values not less than one, a.k.a. fuzzifier, determines the level of cluster fuzziness. A large m value results in smaller memberships wij and fuzzier clusters. In the particular case of an unitary m value, wij converge to 0 or 1 which means a crisp partitioning. When no information about the clustering problem is available, m is typically set to 2. Therefore fuzzy k-means algorithm returns both the ensemble of the cluster centers μ = M = {M1, M2, …, Mk} and the membership function matrix W, a.k.a. partition matrix. The elements wij ∈ W can vary in the range [0,1] and measure the grade of membership of each observation in each cluster. The extreme values indicate no membership and full membership respectively: grade of membership is related inversely to the distance of the observation from cluster center. Although the different fuzzifications [27] permits exploring alternative clustering solutions, fuzzy k-means is very similar to the standard k-means algorithm. Actually it suffers from the same drawbacks pointed out previously.
V = MX = MAS = BS
IC i = s i = (b i)TV
X = TS R + R
(14)
where T is the reduced mixing matrix, SR contains the three dominant ICs and R is the residual of the process.
max{d (a, b): a ∈ Ab ∈ B}
(8)
min{d (a, b): a ∈ Ab ∈ B}
(9)
∑ ∑
(13)
Scaling, sign modifications and order sorting operations are needed to obtain physically meaningful results about the really dominant ICs [21]. In particular for the sake of visualization in a three-dimensional space, Eq. (11) can be reformulated as:
4. Simulation results
Hierarchical clustering methods aim at yielding a hierarchy of clusters typically expressed via a dendrogram for the sake of explicit visualization. Through an agglomerative or a divisive strategy, all the observations x are recursively merged moving up the hierarchy or splitted moving down the hierarchy [28]. As in the previous algorithms, in hierarchical clustering, a proper metric (a measure of distance between pairs of observations) has to be chosen. Furthermore, once the metric is established, a linkage criterion establishing the clusters dissimilarity as a function of the pairwise distances of observations in the sets must be defined. Through the linkage criterion, the distance between ensembles of observations are determined as a function of the pairwise distances between same observations. The following three examples show, specifically, maximum, minimum and average linkage clustering:
a∈A b∈B
(12)
Since the components of V are mutually uncorrelated, the problem of finding an arbitrary full-rank matrix A can be reduced to the one of finding an orthogonal matrix B, whose columns are the sought ICs, by maximizing the kurtosis of:
3.3. Hierarchical clustering
1 |A||B|
(11)
4.1. IEEE 68 bus To validate the performance of the proposed framework, the GCA of the IEEE 68 bus 16 machine is here considered. This benchmark represents a reduced order model of the New England and New York interconnected system [4]. Five geographical areas interconnected by 86 transmission lines can be identified. The total active and reactive power required by the system total load is PL = 18.233, 9 MW, QL = 2.188, 4 MVAr. All the loads are modeled as conforming load. The generated power is PG = 18.408, 2 MW. All the generators of the test system (G1–G16) are represented by a sub-transient model. All the specific information related to the benchmark components data, such as transmission lines or generators data can be found in [4]. A dynamic simulation has been performed to acquire the time behavior of rotor measurements during a network perturbation [30]. More specifically, a three-phase fault has been applied on the transmission line connecting the buses 4–5 at time instant t = 0.1 s which is subsequently cleared by the line protections in accordance with the ordinary fault clearing and reclosing times. The rotor measurements are depicted in Fig. 1a and b along an observation window of 5 s. Sampling frequency fs is aligned with the one of the Italian WAMS, fs = 50 Hz. Model based approach like cosine directional confirms the presence of five main clusters: C1 = {G1, G2, G3, G4, G5, G6, G7, G8, G9}, C2 = {G10, G11, G12, G13},
d (a, b) (10)
where A and B are two ensembles of measurements and d the chosen metric. Several other linkage criteria have been proposed, among the best known and employed one finds the Ward’s criterion [29]. 217
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a)
45 40
Euclidean distance [ rad ]
Generator rotor angle [ rad ]
2.5
2
1.5
1
0.5
35 30 25 20 15 10 5
0
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2.5
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3.5
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Time [ s ]
6
7
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9
1 10 11 12 13 14 16 15
Fig. 2. Benchmark outcomes.
b)
Table 1 K-Means Clustering Results.
1.01
Generator rotor speed [ p.u. ]
2
Generator number [ # ]
1.015
1.005
1
0.995
0.99
3
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Cluster number
Generator number
C1 C2 C3 C4 C5
{ G2, G3, G4, G5, G6, G7, G9}, G1 G8 G10 { G11, G12, G13, G14, G15, G16}
aligned with the ones of the reference system. The three dominant ICs are considered for the sake of visualization, IC1, IC2 and IC3. ICs have been properly adjusted in their peaks and ordered/scaled as hinted in the Section 3 to be physically meaningful. Their dominance has been established by calculating the ICs’ energy via Hilbert operator [3]. Fig. 4 depicts the results of the rows of T in the ICs space: the triplets closest to each other represent coherent generators belonging to the same cluster. As can be appreciated, the results are aligned with the ones of the benchmark. From these outcomes obtained, ICA needs to be considered, together with hierarchical clustering, in a real deployment of a framework that could process spatially measured synchrophasors acquired on large interconnected systems.
5
Time [ s ]
Fig. 1. Generator rotor measurements.
C3 = {G14}, C4 = {G15} and C5 = {G16} [4]. As can be noticed, generators G14, G15 and G16 are very large equivalents and each of them forms a single generator group. The remaining two clusters, C1 and C2, are representative of the New England and New York area respectively. Generator rotor angles are fed to all the considered measurement based identification algorithms. Agglomerative hierarchical clustering is firstly considered to show that, by fixing a certain Euclidean distance among the rotor angle time series, the benchmark outcomes are confirmed. Ward’s linkage criterion is chosen for the purpose. Actually as it can be noted, a proper cut of the dendrogram in Fig. 2 permits to obtain the aforementioned clusters arrangement. Nonetheless the partition solution provided by this clustering algorithm is implicitly not unique as the definition of coherency in [8]. On the contrary the application of k-means and fuzzy k-means algorithms with fixed cluster number (i.e. five) provides results consistent from themselves but not aligned with the ones of the benchmark. Table 1 summarizes the results of k-means algorithm. The results of fuzzy k-means algorithm, which are equivalent to the ones of k-means algorithm in Tab. I as expected, are here shown in pictorial manner through the bar diagram of the partition matrix W in Fig. 3. In this application W is a c-by-g matrix with c the cluster number and g the machine number. Therefore, for each machine one has a score column vector 1-by-c whose elements sum is unitary. The highest value of the latter permits to associate the considered machine in the related cluster. On the contrary, ICA application on the single sided power spectra of the mean centered rotor angle time trends over a range of frequency up to the Nyquist frequency provides aggregation results exactly
4.2. Dynamic Study Model The Dynamic Study Model aims at reproducing the global dynamic performance of synchronously interconnected power system of Continental Europe with reference to systems inertia, frequency containment reserve and dominant inter-area oscillation modes [31]. The goal of DSM is to model the fully synchronously interconnected power system of Continental Europe with the purpose to set up a robust, clear and facilely dynamic model to reproduce the same results in different simulation tools. The model is specifically made up to represent local phenomena (such as voltage transients, system protection of lines, generators), it is appropriate for describing mean frequency transients (system inertia, frequency containment reserve) and dominant inter-area oscillation modes. The load flow model consists in the electrical topology of the synchronously interconnected power system of Continental Europe and in the steady-state information for all production sites regarding a peak demand case in 2020. The modeling of system dynamics is carried out through standard dynamic models for synchronous machines and control devices both. To 218
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C
C
3
C
5
2
C
1
0.9
0.8
Membership grade
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
12
14
16
Generator number [ # ] Fig. 3. Fuzzy k-means clustering results.
Fuzzy k-means provides similar results to the previous approach, while on the contrary simple k-means clustering provides a different but at the same time not-consistent partitioning of generators: it seems for instance that Switzerland that is very often a neutral point from oscillatory point of view instead oscillate with Turkey in far east (see Table 2). Even in this case spectral ICA is able to provide correct indications about generators coherency providing consistent clusters. In particular as it can be seen from Fig. 6 four main clusters can be identified: Cluster 1 - Turkey and Greece, Cluster 2 Spain and Portugal, Cluster 3 Denmark and Switzerland, Cluster 4 – Italy. Cluster 1 and 2 are normally involved in East-West ENTSO-E CE interarea modes while Cluster 3 and 4 in North-South verifying the modal properties of this large interconnected system. 5. Conclusion and future works Fig. 4. ICA clustering results.
In this paper some propaedeutic analyses are carried out for a potential future development of a computational framework for GCA aimed at processing the data acquired by the synchronized, and spatially distributed sensors of the major European WAMS is conceptualized. The proposed solution aims at extracting actionable intelligence from measured data-sets by applying multiple signal processing techniques and is very useful in case of network split. Detailed results demonstrated the ability of the proposed framework in performing GCA in simulated environment. This important feature represents a strategic tool in improving the system resilience to dynamic perturbations, which represents one of the most critical issues to address in modern power systems. On the basis of these results, the future directions of this research will be oriented in enhancing the GCA computational framework with novel methodologies for adaptive dynamic equivalence. Adaptive is referred to the fact that, as highlighted in the introduction section, In particular, the final goal is to process the coherency data in order to identify a reduced order dynamic model for the entire ENTSO-E continental area. This equivalent model could support the implementation of fast contingency analyses, e.g. transient stability studies, or the deployment of special protection schemes supporting islanding control. To this aim a more pervasive penetration of sensors, and the design of
ensure a common approach in different simulation tools, synchronous machines are modeled with a governor (GOV), an automatic voltage regulator (AVR) and a power system stabilizer (PSS). As previously done on IEEE test system a network perturbation is applied, specifically a loss of a huge power plant in Spain, and hence rotor angles from the major generators in the following control areas are sampled: Spain (ES), Portugal (PT), Turkey (TR), Denmark (DK), Italy (IT), Greece (GR) and Switzerland (SW). This reduced set of generators is designed since sufficient enough for the purposes of this analysis: actually they are located at the borders (or the center in case of CH) of the system so they are particularly sensible to the inter-area oscillations which very often drive the coherency of the interconnected system. Rotor angles time series are hence provided to the measurementbased coherency identification algorithms of Section 3. Agglomerative hierarchical clustering is firstly considered to show in Fig. 5 that, however it is fixed a certain Euclidean distance among the rotor angle time series there are some violation of the ordinary modal properties of the system: for instance it seems that Greece and Spain generators oscillate together while they normally oscillate against. 219
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Fig. 5. Agglomerative clustering outcomes.
Table 2 K-Means Clustering Results.
[3]
Cluster number
Generator number
C1 C2 C3 C4
{ CH, DE, IT} PT ES { CH, TR}
[4] [5] [6]
[7] [8] [9]
[10] [11]
[12] [13] [14] [15] [16]
[17]
Fig. 6. ICA outcomes.
[18]
new interoperable frameworks for real-time data sharing among the national TSOs are currently under development in Continental Europe.
[19] [20]
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