Practical Implementation and Operational Experience of Dynamic Mode Decomposition in Wide-area Monitoring Systems of Italian Power System

Andrea Vicario; Alberto Berizzi; Giorgio Maria Giannuzzi; Cosimo Pisani

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Practical Implementation and Operational Experience of Dynamic Mode Decomposition in Wide-area Monitoring Systems of Italian Power System PDF

- ORCID：
Andrea Vicario
✉
- ORCID：
Alberto Berizzi
✉
- ORCID：
Giorgio Maria Giannuzzi
✉
- ORCID：
Cosimo Pisani
✉

the Department of Energy at Politecnico di Milano, Milan, Italy； Terna S.p.A., Rome, Italy

Updated：2023-05-23

DOI：10.35833/MPCE.2021.000509

Abstract

This study presents the assumptions and strategies for the practical implementation of the dynamic mode decomposition approach in the wide-area monitoring system of the Italian transmission system operator, Terna. The procedure setup aims to detect poorly damped interarea oscillations of power systems. Dynamic mode decomposition is a data-driven technique that has gained increasing attention in different fields; the proposed implementation can both characterize the oscillatory modes and identify the most influenced areas. This study presents the results of its practical implementation and operational experience in power system monitoring. It focuses on the main characteristics and solutions identified to reliably monitor the interarea electromechanical modes of the interconnected European power system. Moreover, conditions to issue an appropriate alarm in case of critical operating conditions are described. The effectiveness of the proposed approach is validated by its application in three case studies: a critical oscillatory event and a short-circuit event that occurred in the Italian power system in the previous years, and a 15-min time interval of normal grid operation recorded in March 2021.

Keywords

Power system control; power system dynamics; wide-area monitoring system (WAMS); dynamic mode decomposition

I. Introduction

THE growth of non-programmable renewable energy sources (RESs) (likely to accelerate in the next few years owing to the de-carbonizing policies adopted in Europe and worldwide) has enabled transmission system operators (TSOs) to identify the most suitable functions to guarantee the security of power systems. Accordingly, the significant deployment of phasor measurement units (PMUs) for wide-area monitoring systems (WAMSs) may be of paramount importance, provided suitable methods and algorithms to elaborate the massive stream of data that are available to both extract significant information in real time and provide alarms in the case of critical operating conditions.

Presently, interarea oscillations are critical because they are likely to move a large amount of power across interconnections unless they are immediately damped. Consequently, these oscillations can trigger cascading events, potentially leading to large blackouts.

Moreover, the occurrence of poorly damped interarea oscillations is increasing (for example, in the European power system [

1], [2]). Hence, to preserve system stability, it is necessary to ① promptly identify them in both perturbed and normal operating conditions, ② characterize them in real time, and ③ determine and establish suitable countermeasures.

The characterization of the electromechanical modes aims to estimate their magnitude, frequency, and damping. Many algorithms are available in the technical literature to simultaneously perform these three goals, starting from PMU data with different properties in terms of accuracy and robustness. For instance, they can be based on the Hilbert transform [

3]-[5], which enables the estimation of the damping of the oscillatory modes with high accuracy, particle swarm optimization [6], characterized by a good accuracy of the estimated parameters, or the wavelet-based method [7], which decomposes signals into functions of both time and frequency domains. However, a complete characterization of the oscillatory modes, identifying the areas involved and the phase displacement of oscillations (mode shapes) that cannot be performed using the aforementioned methodologies. Other methods have also been investigated, for example, those based on principal component analysis [8], which can provide a robust and accurate evaluation of magnitude, frequency, and damping of oscillatory modes, giving at the same time information on the areas more affected by each mode. However, this methodology cannot completely characterize the mode shapes associated in terms of phase displacement.

Dynamic mode decomposition (DMD) has recently gained the interest of researchers in the field of power system, both for post-disturbance and ambient data analysis, owing to its accuracy, robustness, and information content, resulting from optimization over a time window. In addition, the computational complexity of DMD is limited, which enables the real-time exploitation of its properties. DMD provides modal decomposition, where each mode comprises spatially correlated structures that have the same linear behavior over time. First applied to fluid dynamics in [

9] and deeply studied in [10], DMD has been proposed in many alternative algorithms, mainly developed to overcome some of its intrinsic weaknesses, particularly in the case of noisy datasets.

References [

11] and [12] proposed the first version of the DMD to monitor electromechanical modes; the “exact DMD formulation”, presented in [10], has been adopted for modal estimation of power systems in [13]. Optimized DMD [14] or robust DMD, based on robust principal component analysis [15], is the alternative method of avoiding corruption and noise in datasets. Block-enhanced DMD [16], which is based on the Hankel matrix, improves the ability to capture mode information from ambient data. This approach (also called data stacking) was proposed in [17] and combined with an optimal hard threshold to select the best model order to deal with noise. Nonlinear observables [18] can extend the DMD to better capture the system dynamics. Randomized DMD combined with data stacking [19] can increase the computing efficiency without losing accuracy. In [20], the output-only observer/Kalman filter identification was used to process the ambient data, followed by the DMD to characterize the frequency and damping.

Finally, DMD was applied to identify synchronous machine coherency under post-fault conditions [

21]. In [22], the rotor angle and acceleration of synchronous machines during a fault were predicted using DMD. In [22], extended-DMD was used for dynamic state estimation in real time, whereas DMD was used for inertia estimation in [23]. In [24], the output results of the DMD were used as a reference to validate a machine learning approach.

Based on a review of the technical literature, research on DMD theory has been developed and has been applied to both synthetic and real data (e.g., [

11], [13], [16], [18], [19] based on the analysis of real ring-down events and ambient data). Particularly, DMD proved its effectiveness in the analysis of data recorded during oscillatory events.

However, its ability during fast transient such as short circuits or normal load variation and the possibility of issuing alarms in case of critical operating conditions is yet to be reported. Moreover, practical DMD implementation in a TSO control room is still undocumented. This study aims to fill this gap by providing the operational experience of using DMD in a TSO control center where the data stream from WAMS is processed. The DMD output was used to monitor and control the Italian power system in its interconnected operation with the European power system.

The main contributions of this study are as follows.

1) Application of DMD to detect and characterize oscillatory interarea modes (frequency, amplitude, damping, and mode shapes), which must be reliable and robust under both transient (under different types of perturbations) and normal operating conditions.

2) Determination of conditions to issue alarms for the control room to trigger possible countermeasures.

3) Analysis of DMD properties in the presence of ambient signals characterized by factors such as noise, changes of operating conditions, and topology.

The aforementioned contributions were identified after a one-year test campaign conducted on the Italian power system. The results presented in this study show very good robustness under all different power system conditions and very good accuracy compared with other DMD approaches tested. In addition, the identification of mode shapes was accurate and in agreement with the operational experience of engineers in control rooms and ENTSO-E studies.

The remainder of this paper is organized as follows. The DMD theory is presented in Section II, whereas Section III focuses on its practical implementation in Terna control center, with the underlying criteria to issue alarms and trigger control actions. Section IV presents some selected results focusing on the operating conditions and perturbations that can be critical to accurately identifying modal power system properties. Two events occurred in the Italian power system in the last few years, and a 15-min time interval of normal grid operation recorded in March 2021 was analyzed. The results show very good robustness and accuracy of the adopted technique and implementation. Finally, conclusions are drawn in Section V.

II. DMD Theory

This section presents the theory of the DMD implemented for the security monitoring module and the identification of the dominant modes and their features. The DMD utilizes singular value decomposition (SVD) to obtain dimensionality reduction in high-dimensional systems [

25]. The DMD adopted uses the exact DMD [10], [13] combined with the block-enhanced formulation, as proposed by [16], [17] for power systems.

A. Exact DMD Architecture

This subsection briefly discusses the theory of exact DMD [

10], [13]. Data are collected from a generic nonlinear system with unknown dynamics. The data from the measurements are used to approximate the dynamics with locally linear systems:

\frac{d x}{d t} = A x

(1)

where $x$ is a vector representing the states of the dynamic system at a generic time t; and A is the constant matrix describing the dynamic system. Discretizing (1) with sampling time $Δ t$ , we can obtain:

x_{k + 1} = B x_{k}

(2)

B = e^{A Δ t}

(3)

Matrices $A$ and $B$ have the same eigenvectors $ϕ_{j}$ and their eigenvalues are such that $λ_{k} = e^{ω_{k} Δ t}$ , where $λ_{k}$ is the k^th eigenvalue of $B$ and $ω_{k}$ is the k^th eigenvalue of $A$ .

The DMD performs a low-rank projection of $B$ , indicated by $\tilde{B},$ which optimally fits the measured data by minimizing the error $ε$ :

ε = {‖x_{k + 1} - \tilde{B} x_{k}‖}_{2}

(4)

This approximation holds only over the sampling window where $\tilde{B}$ is built. To compute $\tilde{B}$ and minimize $ε$ across all snapshots $k = 1,2, \dots, m$ , the n measurements for each of the $m$ snapshots can be arranged into two data matrices $X$ and $X'$ and the exact DMD can be carried out. The measurements are stored in a matrix $X$ organized as follows: data related to the same snapshot are stored in the same column and data from the same PMU are stored in the same row. $X'$ has the same structure; however, data are time-shifted by $Δ t$ .

X = \overset{m - 1 s n a p s h o t s}{\vec{[\begin{matrix} | & | & \dots & | \\ x_{1} & x_{2} & \dots & x_{m - 1} \\ | & | & \dots & | \end{matrix}]}}

(5)

(6)

Thus, (2) can be rewritten as:

X' = B X

(7)

Hence, matrix $B$ is written as:

B = X' X^{†}

(8)

where † represents the Moore-Penrose pseudoinverse [

26]. Thus, rank truncation can be performed to consider only a limited number r of dominant modes; the exact DMD approach computes a rank-reduced representation in terms of a proper orthogonal decomposition (POD) projected matrix

\tilde{B}

. Hence, the data matrix can be approximated using its SVD:

X \approx U_{r} Σ_{r} V_{r}^{*}

(9)

where $U_{r}$ , $Σ_{r}$ , and $V_{r}$ have the dimensions of $n \times r$ , $r \times r$ , and $(m - 1) \times r$ , respectively; and $*$ denotes the conjugate transpose. Thus, $B$ can be efficiently projected onto the POD modes, and the upper triangular matrix $\tilde{B}$ is obtained as:

\tilde{B} = U_{r}^{*} (X' V_{r} Σ_{r}^{- 1} U_{r}^{*}) U_{r} = U_{r}^{*} X' V_{r} Σ_{r}^{- 1}

(10)

$\tilde{B}$ defines the low-dimensional linear model of a dynamic system. To identify the mode properties, frequencies, damping, and mode shapes, the eigenvalue problem for $\tilde{B}$ is solved as:

\tilde{B} W_{r} = W_{r} Λ_{r}

(11)

where the columns of matrix $W_{r}$ ( $r \times r$ ) are the eigenvectors; and $Λ_{r}$ ( $r \times r$ ) is a diagonal matrix containing the eigenvalues $λ_{r}$ of $\tilde{B}$ . Finally, $B$ can be reconstructed from $W_{r}$ and $Λ_{r}$ ; a subset of the eigenvalues of $B$ is provided by $Λ_{r}$ , whereas a subset of its eigenvectors $ϕ_{j}$ is provided by the columns of $Φ_{r}$ ( $n \times r$ ) (the “exact DMD modes”):

Φ_{r} = X' V_{r} Σ_{r}^{- 1} W_{r}

(12)

Finally, the approximated solution for all future time is:

x (t) \approx \sum_{j = 1}^{r} ϕ_{j} e^{ω_{j} t} b_{r j} = Φ_{r} e^{Ω_{r} t} b_{r}

(13)

where $ω_{j} = l n (λ_{r j}) / Δ t$ is the j^th eigenvalue of $A$ in the continuous-time domain; $b_{r}$ is a vector, whose elements $b_{r j}$ are the initial amplitudes of each DMD mode; and $Ω_{r}$ is a diagonal matrix containing the continuous eigenvalues $ω_{j}$ . This can be computed from the first snapshot $x_{1}$ at $t = 0$ in (13).

x_{1} \approx Φ_{r} b_{r}

(14)

Because $Φ_{r}$ is generally a non-square matrix, $b_{r}$ is computed by finding the best-fit solution using the least-squares method:

b_{r} = Φ_{r}^{†} x_{1}

(15)

The frequency $f$ and damping ratio $ξ$ of the identified modes are computed from the continuous-time eigenvalues ( $ω_{j} = α \pm i β$ ):

f = \frac{β}{2 π}

(16)

ξ = \frac{- α}{\sqrt[]{α^{2} + β^{2}}}

(17)

Finally, the mode shapes of the processed measurements are obtained from the columns of $Φ_{r}$ in (12).

B. Block-enhanced Formulation Considering Time-delay Coordinates for DMD

This study adopts a block-enhanced formulation using an augmented set of coordinates built by considering the time-delay coordinates. Time-delay coordinates [

25] are used to reconstruct the dynamics of systems that do not have sufficient measurements, even allowing the estimation of modal parameters from a single PMU. They can be obtained by stacking s times the vector of measurements

x

, building the

(n \times s) \times (m - s + 1)

Hankel matrix

H

H = [\begin{matrix} \begin{array}{l} [\begin{matrix} | \\ x_{1} \\ | \end{matrix}] \\ [\begin{matrix} | \\ x_{2} \\ | \end{matrix}] \\ ⋮ \\ [\begin{matrix} | \\ x_{s} \\ | \end{matrix}] \end{array} & \begin{matrix} \begin{array}{l} [\begin{matrix} | \\ x_{2} \\ | \end{matrix}] \\ [\begin{matrix} | \\ x_{3} \\ | \end{matrix}] \\ ⋮ \\ [\begin{matrix} | \\ x_{s + 1} \\ | \end{matrix}] \end{array} & \begin{matrix} \begin{matrix} \dots & [\begin{matrix} | \\ x_{m - s + 1} \\ | \end{matrix}] \\ \dots & [\begin{matrix} | \\ x_{m - s + 2} \\ | \end{matrix}] \\ ⋮ \\ \dots & [\begin{matrix} | \\ x_{m} \\ | \end{matrix}] \end{matrix} \end{matrix} \end{matrix} \end{matrix}]

(18)

Thus, the block-enhanced formulation involves applying the exact DMD to matrices $X$ and $X^{'}$ derived from $H :$

X = [\begin{matrix} | & | & \dots & | \\ H_{1} & H_{2} & \dots & H_{m - s} \\ | & | & \dots & | \end{matrix}]

(19)

X^{'} = [\begin{matrix} | & | & \dots & | \\ H_{2} & H_{3} & \dots & H_{m - s + 1} \\ | & | & \dots & | \end{matrix}]

(20)

If the measurement set is low-dimensional, the rank of $H$ may be increased using more time-delay coordinates. The number of time-delay coordinates s may be increased until the system reaches full rank [

10].

C. Mode Ranking

The DMD can extract the most important spatio-temporal patterns and eigenvalues from a dataset; however, the mode ranking resulting from the DMD is not necessarily in agreement with the energy content. Therefore, a criterion should be determined to select the dominant modes (accordingly, [

27], [28] first identify dominant modes from data-driven estimation methods). In [10], the initial mode amplitude evaluation is proposed according to (15); however, this solution only considers the initial conditions, not the mode evolution over time. Different solutions can be found in previous studies to associate each mode with a pseudo-energy. For example, in [11], the ranking was based on a combination of the right eigenvectors, the right singular vector, and its singular values. The method based on [29] is instead adopted in [13] to rank the modes and identify the dominant modes. For each mode

j,

the associated energy

E_{j}

is computed as:

E_{j} = |b_{r j} e^{ω_{j} T}|

(21)

where $T$ is the time window considered. Here, the ranking is not based on the behavior of a single point in time but takes into account the whole time window; modes are finally ranked according to their energy and those presenting the largest values are assumed to be dominant. As confirmed by simulations and tests, the DMD can generate spurious modes with damping and frequency similar to actual modes [

13] for different system orders and different sizes of the time window considered. This is the only drawback observed during all the tests performed on the DMD implemented.

Finally, another criterion to identify dominant modes in real time is based on [

30], which utilizes the Riccati equation, searching for the optimal mode amplitude

b_{r j}

that minimizes the

l_{2}

error between

X

and the approximation in (22) (more details are provided in [30]).

X \approx [\begin{matrix} | & | & | \\ ϕ_{r 1} & ϕ_{r 2} & \dots \\ | & | & | \end{matrix}] [\begin{matrix} b_{r 1} & 0 & \dots & 0 \\ 0 & b_{r 2} & \dots & 0 \\ ⋮ & ⋮ & ⋮ \\ 0 & 0 & \dots & b_{r j} \end{matrix}] [\begin{matrix} \begin{matrix} 1 & λ_{r 1} & \dots \\ 1 & λ_{r 2} & \dots \\ ⋮ & ⋮ \\ 1 & λ_{r j} & \dots \end{matrix} & \begin{array}{l} λ_{r 1}^{m - 1} \\ λ_{r 2}^{m - 1} \\ ⋮ \\ λ_{r j}^{m - 1} \end{array} \end{matrix}]

(22)

where $ϕ_{r i}$ is the i^th column of matrix $ϕ_{r}$ .

The computed optimal $b_{r j}$ is then associated with each identified mode. This index is selected for the final implementation because the tests reported in Section IV indicate that the first two approaches may result in false alarms.

III. Practical Implementation of DMD Approach

This section describes the practical implementation of the DMD approach in Terna control center. The practical choice of the most suitable type of DMD, filtering of input data, selection of the most suitable index to issue alarms, and information visualization to control room operators are described.

As the DMD approach implemented is tailored for the Italian power system within continental Europe, the main interest of the Italian TSO is to detect the interarea modes involving the Italian power system. The modes with the highest participation factors of Italian power plants are the principal focus, especially those characterized by a frequency ranging from 0.24 to 0.35 Hz, as involved in real events [

2] and previous studies [13]. This mode, also known as the South-North (S-N) European mode, shows that the Italian power system is in phase opposition to the rest of the northern European power system. Because the largest oscillatory events are better observable in southern Italy [2], [13], most of the processed PMUs are placed in this portion of the system.

The DMD implemented in Terna control center uses the exact DMD with the block-enhanced formulation and elaborates the data stream (sampled at 50 Hz, i.e., a sample every 20 ms) from the n PMUs available from the WAMS system (generally, 20 PMUs are processed). The input data stream is a 20 s sliding window; this time-window length has been selected as a good compromise between the stability of the results over time and the quick response of the monitoring tool. It can be changed by the user if necessary.

Data streams from the field are properly processed to address issues such as noise or missing elements that could influence the real data flow: ① data detrending; ② data packet reconstruction, i.e., fill missing values; ③ data filtering via a Hilbert bandpass filter to maintain only typical frequencies of the main European interarea modes (0.10-0.50 Hz), discarding slower (control) modes and faster (local) modes [

31].

All available voltages, frequencies, magnitudes, angles, and real and reactive power flows (measured by PMUs) can be used as inputs to the DMD. However, rank truncation typically detects and discards redundant information. Busbar frequencies alone can be used for this application and provide good observability of the interarea mode (thus saving computation time, according to the real-time requirements of the tool). Moreover, as power systems generally present a low-rank pattern [

13], based on experience, a rank truncation to the first eight singular values is sufficiently accurate to observe the main interarea modes of the interconnected European power system.

Among the outcomes of the DMD, a very important properity is the knowledge of mode shapes. It is a key factor compared with other approaches, as it enables the understanding of the coherency of the system in real time. This knowledge is important, particularly in the case of re-dispatching actions, power reduction, or even disconnection of the generator(s). Generators with the highest participation factor have the highest damping effect on ongoing oscillation. The main goal of the proposed application is to issue an alarm for control room operators in the case of critical interarea oscillations. Hence, the observed modes should be ranked and a suitable index should be selected to be compared with a threshold to generate an alarm or apply an automatic control action. The monitoring of individual mode amplitudes and the relevant damping alone might lead to false alarms; adopted as indices, the initial amplitude from (15) shows unsatisfactory performance, whereas the choice of (21) shows unstable behavior, as demonstrated in Section IV.

Based on the aforementioned findings, the monitoring system triggers an alarm if all the following three conditions are fulfilled for each 20 s sliding window processed.

1) The DMD detects the presence of at least one mode with a frequency within the aforementioned range (0.24-0.35 Hz). Notably, the control room displays continuously show modal frequency, amplitude, and associated mode shapes.

2) The optimal amplitude $b_{r j}$ for the 0.24/0.35 Hz mode computed according to (22) is higher than the threshold set according to experience. This threshold is currently set to be 0.10 p.u., a compromise between the need to capture all critical events and, simultaneously avoiding false alarms. Many tests have been conducted on real power systems to fix this threshold. It has been selected after long-term validation, considering the different operating conditions of the Italian power system (low/high demand), different grid topologies, generation patterns (traditional v.s. renewable), and side effects such as PMU noise (estimated with a signal-to-noise ratio of 45 dB according to [

32]), number of PMU processed, packet losses, and outliers in the main data stream.

3) The peak-to-peak frequency deviation is higher than 60 mHz for at least 3 cycles, proving an ongoing event.

If all these conditions are fulfilled, the alarm is set to be ON, and the operator is aware that a critical interarea oscillation is in place. Manual control actions will be performed (reduction of the real power flows in critical sections or tie lines, generator re-dispatching/disconnection, load reduction, and network topology changes, based on the system condition and operator knowledge). Automatic control actions based on wide-area power oscillation damping control activation [

33], [34] (control of the active power generated) will also be implemented soon in Terna control center.

Figure 1 shows the overall flowchart of the proposed procedure.

Fig. 1 Overall of flowchart of proposed procedure.

IV. Tests and Results

In this section, the main results obtained from the practical DMD implementation are presented and discussed, with reference to three different operating conditions, to prove both its reliability and robustness to operating conditions: a significant oscillatory event, a short-circuit event, often erroneously understood by monitoring tools as an oscillation, and a 15-min time interval of normal grid operation. Particularly, the results of the exact DMD and block-enhanced DMD are compared with three different ranking approaches [

10], [13], [30] in terms of detection speed.

Frequency measurements are known to be influenced by significant noise. In the preliminary tests, frequency, voltage, and power measurements are employed to feed the DMD, as heterogeneous measurements can be used for mode identification. An increasing number of signals might improve the ability to detect a critical oscillation if the added signals exhibit high observability for this mode. Meanwhile, highly correlated signals do not always improve the quality of identification but increase computing time [

27]. Presently, the procedure is based on frequency measurements only, suitably filtered, also because the alarm is set to be ON based on a peak-to-peak frequency deviation. Moreover, not all Italian critical tie lines are currently covered by the PMU to provide real and reactive power measurements and to feed the approach.

A. Oscillatory Event on December 3, 2017

The oscillatory event that occurred on December 3, 2017 was first extensively discussed and presented in [

2] and [13] and was characterized by an undamped frequency oscillation up to approximately 300 mHz in southern Italy. It began at 01:09 and reached its maximum deviation at 01:15. Figure 2 shows the frequencies of the oscillatory event recorded at different locations of the Italian power system.

Fig. 2 Frequencies of oscillatory event on December 3, 2017.

These oscillatory events can lead to emergencies. Therefore, the monitoring system should correctly classify the event, provide proper interpretations to operators, and suggest the most suitable corrective actions.

The frequency, damping, and amplitude of the first three modes identified by the block-enhanced DMD along with trigger status are shown in Fig. 3. Mode 2 presents a frequency close to 0.29-0.30 Hz and damping less than 5% all the time, as shown in Fig. 4; its amplitude, i.e., the optimal $b_{r j}$ computed according to (22), shows an increasing pattern from 00:10 until 00:14. The corresponding mode shapes during the maximum frequency deviation are shown in Fig. 5. With regard to the correctness of the approach, the estimated mode shapes resulting from the DMD were compared in [

13] with the traditional modal analysis results, thereby successfully validating the approach. Moreover, regarding the European power system, Fig. 5 is consistent with the studies conducted by the System Protection and Dynamics Group of ENTSO-E [2].

Fig. 3 Frequency, damping, and amplitude of first three modes identified by block-enhanced DMD along with trigger status.

Fig. 4 Frequency and damping of mode 2.

Fig. 5 Mode shapes during the maximum frequency deviation.

This event is used as a benchmark and to tune the implemented DMD; if in 2017 the DMD had been online in Terna control center, it would have issued an alarm (the trigger signal in Fig. 3) at 00:11, a few seconds after the beginning of the oscillatory event in a pronounced manner. This trigger would have been possible because all the tripping conditions described in Section III are verified. A mode with the frequency characteristics of the S-N European mode is identified, the optimal $b_{r j}$ exceeds the threshold of 0.10, and PMU frequency measurements deviate by more than 60 mHz from 50 Hz for three cycles. Hence, the DMD would have enabled prompt control by control room operators.

The other two modes identified by the block-enhanced DMD, i.e., one around 0.40-0.45 Hz (mode 1) and one around 0.20 Hz (mode 3), both show negative or zero damping values. Nevertheless, their amplitudes are not sufficient to trigger an alarm. These modes suggest the presence of unstable behavior, as their damping fluctuates significantly, even showing high negative values. However, Fig. 6 confirms that these modes are not physical because the relative frequencies do not appear in the signal spectrum; they are spurious modes caused by the DMD approximating the nonlinear system dynamics in a least-squares manner over a certain time span [

13]. The proposed analysis of the optimal

b_{r j}

can filter these modes, and it is also useful in practical implementations. Further, damping alone cannot be used as a reliable index for grid monitoring (associated amplitude, mode shapes, and frequency must also be considered).

Fig. 6 Power spectrum of oscillatory event on December 3, 2017.

Other DMD-based approaches were tested on the same event shown in Fig. 2. During this assessment, attention was paid to the trigger signal to evaluate the ability of the monitoring tool to capture the alarm conditions in a timely manner.

Figure 7 shows the result of applying the exact DMD approach (not block-enhanced) using (22) as the amplitude to trigger the alarm. Two modes (1 and 2) are identified in the range of 0.24-0.35 Hz; a third mode close to 0.10 Hz sometimes appears. Furthermore, the amplitudes, i.e., the optimal $b_{r j}$ , of modes 2 and 3 present spikes that render the identification of spurious modes possibly generated by the DMD approximation difficult. Finally, the trigger signal presents discontinuities, thus putting the operator in trouble about the decision to issue a control action.

Fig. 7 Frequency, damping, and amplitude identified by exact DMD adopting optimal $b_{r j}$ along with trigger status.

Figure 8 shows the results obtained by the procedure applying the block-enhanced DMD adopting $E_{j}$ of (21) to trigger the alarm [

13]. Even if the trigger signal is correctly set to be 1 at the beginning of the oscillatory event at 00:11, Terna discards this approach for the mode amplitude evaluation, as the threshold to be compared with

E_{j}

over time cannot be easily defined.

E_{j}

is indeed dependent on the value of T considered and the number of PMUs processed; that is, it depends on the structure of the monitoring system (energy in Fig. 8 ranging from 0 to 10). Hence, fixing a threshold to trigger the alarm on

E_{j}

would be difficult, whereas the use of (22) is proven to be more efficient.

Fig. 8 Results obtained by procedure applying block-enhanced DMD adopting $E_{j}$ of (21) to trigger alarm.

Figure 9 shows the results obtained by applying the block-enhanced DMD combined with the initial amplitude (15) proposed by [

10], where the trigger signal presents discontinuities, particularly at 00:13, as the mode amplitude (red line) is reduced below the threshold, which causes Terna to also discard this approach and adopt the block-enhanced DMD, together with the optimal amplitude from (22).

Fig. 9 Results obtained by applying block-enhanced DMD combined with initial amplitude proposed by [

10].

B. Short-circuit Event in February 2019

The second event analyzed in this study differs significantly from the first event and is a short-circuit event that occurred in the high-voltage network of the Italian power system in February 2019, as shown in Fig. 10.

Fig. 10 PMU data acquired in February 2019.

This event is studied to verify the selectivity of the oscillation detection, as the monitoring tool aims to issue alarms only in the case of interarea oscillation. Therefore, it is crucial to validate that the monitoring tool can distinguish interarea oscillations from other types of oscillations. The power spectrum of the short-circuit event in February 2019 is shown in Fig. 11, where a small peak can be observed at approximately 0.25 Hz.

Fig. 11 Power spectrum of short-circuit event in February 2019.

The results obtained by the proposed approach are shown in Fig. 12. The monitoring system identifies three modes ranging from 0.20 to 0.40 Hz, characterized by low or even negative damping. However, despite their negative damping, the system correctly does not issue any alarm (as the S-N mode amplitude does not exceed the predefined threshold of 0.1), thus providing the correct interpretation of the phenomenon monitored.

Fig. 12 Results obtained by proposed approach along with trigger status.

C. Normal Grid Operation in March 2021

The third case considers all possible issues that might occur in a power system in the interval of 15 min of normal grid operation in March 2021, as shown in Fig. 13.

Fig. 13 PMU data acquired in March 2021.

The goal is to assess the ability of the adopted implementation to correctly identify the modes in place, although their amplitudes are not sufficiently high to make the situation critical. It can be observed that the frequency deviates around the nominal value by a maximum of 40 mHz. This deviation does not represent a critical condition; therefore, it is needless issuing an alarm to the operator. However, a poorly damped mode (blue oscillation) is present, which should be tracked by the monitoring tool. The power spectrum of the time window considered is shown in Fig. 14, which shows a peak at 0.29 Hz.

Fig. 14 Power spectrum of time window considered.

The proposed approach accurately identifies a mode at approximately 0.30 Hz (mode 2 in Fig. 15), whose energy content (measured by the value of the optimal $b_{r j}$ from (22)) oscillates close to the threshold of 0.10. Even if its associated amplitude sometimes exceeds the defined threshold, no alarm signal is issued because the peak-to-peak frequency deviation is less than 60 mHz. Other modes are identified at approximately 0.40 and 0.20 Hz; however, their energy is very low and hence they are filtered out.

Fig. 15 Frequency, damping, and amplitude identified by proposed approach along with trigger status.

D. Range of Effectiveness of Proposed Approach

Concerning the assessment of the suitability of using a fixed threshold, Figs. 16 and 17 present the probability density function of the modal amplitude and frequency estimated in 2021 (S-N European mode). The average modal amplitude is well below the threshold of 0.10 p.u., whereas the frequency is centered at approximately 0.30 Hz. The two figures prove the quality of the selected approach and settings against the variability of the operating conditions such as the topology and generation pattern.

Fig. 16 Probability density function of modal amplitude estimated in 2021.

Fig. 17 Probability density function of modal frequency estimated in 2021.

Table I lists the average computation time of 20 s sliding window (computed considering data detrending, data packet reconstruction, data filtering, and mode identification) of the events shown in this study and two additional events (events 1 and 2). Computations have been performed on an Intel^® Core^TM i7-9750H CPU @ 2.60 GHz laptop. The average values are always less than 100 ms, making the proposed approach suitable for real-time monitoring using a reasonable number of inputs.

Table I Average Computation Time of 20 s Sliding Window

Event	Number of processed PMUs	Time length of data analyzed (min)	Average computational time of 20 s sliding window (s)
Oscillatory event of 2017	12	25	0.06
Short-circuit event of 2019	17	17	0.07
Normal grid operation of 2021	19	15	0.07
Event 1	22	10	0.07
Event 2	33	16	0.10

V. Conclusion

This study presents the practical implementation of the block-enhanced DMD approach adopted by Terna in its control room for the real-time monitoring of frequency oscillations in the Italian power system and the approach along with the criteria set to generate a reliable alarm signal in the case of critical interarea oscillations in a real power system. Such criteria are the outcome of a massive testing campaign conducted in 2020 and 2021. Its adoption in the control room makes it possible to alert operators in the case of sustained interarea oscillations. The approach utilizes the most interesting properties of the DMD approach such as decomposition over time and space, optimization over a time window, accuracy, robustness, information content, limited computation complexity, and the possibility of exploiting its properties in real time. Different variants are tested and the best approach is set and tuned. The performance is validated using the presented results and analyses. The adopted indices, threshold set, and overall monitoring system are demonstrated to be very reliable and robust under different operating conditions, e.g., low/high load, low/high RES penetration, and network topology changes.

References

ENTSO-E. (2017, Jul.). Analysis of CE inter-area oscillation of 1st December 2016. [Online]. Available: https://eepublicdownloads.entsoe.eu [Baidu Scholar]

ENTSO-E. (2018, Mar.). Oscillation event 03.12.2017 system protection and dynamics WG. [Online]. Available: https://eepublicdownloads.entsoe.eu [Baidu Scholar]

E. M. Carlini, G. M. Giannuzzi, R. Zaottini et al., “Parameter identification of interarea oscillations in electrical power systems via an improved Hilbert transform method,” in Proceedings of 2020 55th International Universities Power Engineering Conference (UPEC), Turin, Italy, Sept. 2020, pp. 1-8. [Baidu Scholar]

D. Lauria and C. Pisani, “On Hilbert transform methods for low frequency oscillations detection,” IET Generation, Transmission & Distribution, vol. 8, no. 6, pp. 1061-1074, Jun. 2014. [Baidu Scholar]

D. Lauria and C. Pisani, “Improved non-linear least squares method for estimating the damping levels of electromechanical oscillations,” IET Generation, Transmission & Distribution, vol. 9, no. 1, pp. 1-11, Jan. 2015. [Baidu Scholar]

F. Bonavolonta, L. P. D. Noia, A. Liccardo et al., “A PSO-MMA method for the parameters estimation of interarea oscillations in electrical grids,” IEEE Transactions on Instrumentation and Measurement, vol. 69, no. 11, pp. 8853-8865, Jun. 2020. [Baidu Scholar]

R. Vaz, G. R. Moraes, E. H. Z. Arruda et al., “Event detection and classification through wavelet-based method in low voltage wide-area monitoring systems,” International Journal of Electrical Power & Energy Systems, vol. 130, p. 106919, Sept. 2021. [Baidu Scholar]

A. Bosisio, A. Berizzi, G. R. Moraes et al., “Combined use of PCA and Prony analysis for electromechanical oscillation identification,” in Proceeding of 7th International Conference on Clean Electrical Power: Renewable Energy Resource Impact, Otranto, Italy, Jul. 2019, pp. 62-70. [Baidu Scholar]

P. J. Schmid, “Dynamic mode decomposition of numerical and experimental data,” Journal of Fluid Mechanics, vol. 656, pp. 5-28, Jul. 2010. [Baidu Scholar]

J. N. Kutz, S. L. Brunton, B. W. Brunton et al., Dynamic Mode Decomposition. Philadelphia: SIAM, 2016. [Baidu Scholar]

E. Barocio, B. C. Pal, N. F. Thornhill et al., “A dynamic mode decomposition framework for global power system oscillation analysis,” IEEE Transactions on Power Systems, vol. 30, no. 6, pp. 2902-2912, Dec. 2015. [Baidu Scholar]

S. Mohapatra and T. J. Overbye, “Fast modal identification, monitoring, and visualization for large-scale power systems using dynamic mode decomposition,” in Proceeding of 19th Power Systems Computation Conference, Genoa, Italy, Aug. 2016, pp. 1-8. [Baidu Scholar]

A. Berizzi, A. Bosisio, R. Simone et al., “Real-time identification of electromechanical oscillations through dynamic mode decomposition,” IET Generation, Transmission & Distribution, vol. 14, no. 19, pp. 3992-3999, Jul. 2020. [Baidu Scholar]

L. Wang, G. Cai, Z. Chen et al., “Dominant inter-area oscillation mode identification using local measurement and modal energy for large-scale power systems with high grid-tied VSCs penetration,” International Journal of Electrical Power & Energy Systems, vol. 117, p. 105697, May 2020. [Baidu Scholar]

I. Scherl, B. Strom, J. K. Shang et al., “Robust principal component analysis for modal decomposition of corrupt fluid flows,” Physical Review Fluids, vol. 5, no. 5, p. 054401, May 2020. [Baidu Scholar]

D. Yang, H. Gao, G. Cai et al., “Synchronized ambient data-based extraction of interarea modes using Hankel block-enhanced DMD,” International Journal of Electrical Power & Energy Systems, vol. 128, p. 106687, Jun. 2021. [Baidu Scholar]

M. Zuhaib and M. Rihan, “Identification of low-frequency oscillation modes using PMU based data-driven dynamic mode decomposition algorithm,” IEEE Access, vol. 9, pp. 49434-49447, Mar. 2021. [Baidu Scholar]

N. Mohan, K. P. Soman, and S. S. Kumar, “Wide-area monitoring of power system using dynamic mode decomposition on nonlinear observables,” in Proceedings of 15th IEEE India Council International Conference (INDICON), Coimbatore, India, Dec. 2018, pp. 1-5. [Baidu Scholar]

A. Alassaf and L. Fan, “Randomized dynamic mode decomposition for oscillation modal analysis,” IEEE Transactions on Power Systems, vol. 36, no. 2, pp. 1399-1408, Jul. 2021. [Baidu Scholar]

S. Zhou, G. Cai, D. Yang et al., “Ambient data-driven oscillation modes extraction using output-only observer/Kalman filter identification and dynamic mode decomposition,” in Proceeding of 2021 IEEE 4th International Electrical and Energy Conference (CIEEC), Wuhan, China, May 2021, pp. 1-6. [Baidu Scholar]

A. Saija, K. Sonam, F. Kazi et al., “Coherency identification in multimachine power systems using dynamic mode decomposition,” in Proceeding of European Control Conference (ECC), St. Petersburg, Russia, Jul. 2020, pp. 1342-1347. [Baidu Scholar]

M. Netto, V. Krishnan, L. Mili et al., “Real-time modal analysis of electric power grids-the need for dynamic state estimation,” in Proceedings of International Conference on Probabilistic Methods Applied to Power Systems (PMAPS), Liege, Belgium, Sept. 2020, pp. 1-7. [Baidu Scholar]

D. Yang, B. Wang, G. Cai et al., “Data-driven estimation of inertia for multiarea interconnected power systems using dynamic mode decomposition,” IEEE Transactions on Industrial Informatics, vol. 17, no. 4, pp. 2686-2695, Apr. 2021. [Baidu Scholar]

C. Olivieri, F. de Paulis, A. Orlandi et al., “Estimation of modal parameters for inter-area oscillations analysis by a machine learning approach with offline training,” Energies, vol. 13, no. 23, p. 6410, Dec. 2020. [Baidu Scholar]

S. L. Brunton and J. N. Kutz, Data-driven Science and Engineering. Cambridge: Cambridge University Press, 2019. [Baidu Scholar]

J. C. A. Barata and M. S. Hussein, “The Moore-Penrose pseudoinverse: a tutorial review of the theory,” Brazilian Journal of Physics, vol. 42, no. 1-2. pp. 146-165, Apr. 2012. [Baidu Scholar]

R. B. Leandro, A. S. e Silva, I. C. Decker et al., “Identification of the oscillation modes of a large power system using ambient data,” Journal of Control, Automation and Electrical Systems, vol. 26, no. 4, pp. 441-453, Apr. 2015. [Baidu Scholar]

D. J. Trudnowski, J. W. Pierre, N. Zhou et al., “Performance of three mode-meter block-processing algorithms for automated dynamic stability assessment,” IEEE Transactions on Power Systems, vol. 23, no. 2, pp. 680-690, May 2008. [Baidu Scholar]

J. Kou and W. Zhang, “An improved criterion to select dominant modes from dynamic mode decomposition,” European Journal of Mechanics–B/Fluids, vol. 62, pp. 109-129, Mar.-Apr. 2017. [Baidu Scholar]

M. R. Jovanović, P. J. Schmid, and J. W. Nichols, “Sparsity-promoting dynamic mode decomposition,” Physics of Fluids, vol. 26, no. 2, pp. 1-22, Feb. 2014. [Baidu Scholar]

P. Kundur, N. J. Balu, and M. G. Lauby, Power System Stability and Control. New York: McGraw-Hill, 1994. [Baidu Scholar]

M. Brown, M. Biswal, S. Brahma et al., “Characterizing and quantifying noise in PMU data,” in Proceeding of 2006 IEEE PES General Meeting, Boston, USA, Jul. 2016, pp. 1-7. [Baidu Scholar]

C. Zhang, Y. Zhao, L. Zhu et al., “Implementation and hardware-in-the-loop testing of a wide-area damping controller based on measurement-driven models,” in Proceeding of 2021 IEEE PES General Meeting, Washington DC, USA, Jul. 2021, pp. 1-5. [Baidu Scholar]

L. Zhu, W. Yu, Z. Jiang et al., “A comprehensive method to mitigate forced oscillations in large interconnected power grids,” IEEE Access, vol. 9, pp. 22503-22515, Feb. 2021. [Baidu Scholar]

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