9.1 Introduction

The considerable environmental benefits of integrating renewable generation into the grid have encouraged several governmental policies around to world. Since the non-synchronous generation (wind, wave and solar power) is required to behave like conventional synchronous generation units, (full-scale power supply) the high voltage power electronics converters have become attractive to be selected for the integration into either the transmission or distribution power levels [1].

A wider use of non-synchronous generation relies on the use of Full Rated Converters (FRCs). This is due to them enabling multiple control features which include controlling active and reactive power [2], assuring voltage ride through capability [3] in order to deal with variable speed wind turbines [4], and adding Maximum Power Point Tracking (MPPT) algorithms [5]. However, the generation is completely decoupled from the system, and is consequently unable to contribute dynamically to the system [6, 7]. Thus, if the synthetic inertia control option is not added, the decoupling displaces synchronous machines [8].

The reduction of the inertia in the power grid has provoked a global concern by the system operators, power planners and researchers. Several reports have shown the experience with large inclusion of non-synchronous generation and the dynamical challenges during the past years [9].

Some of the main challenges which have been reported include the following:

  • The first experienced challenge is the decrease in the system inertial response which affects the dynamical response [10,11,12].

  • The second challenge due to non-synchronous integration is the impact on small signal stability. Several studies have shown that the large-scale inclusion of wind power degrades the damping of the electromechanical modes. This displace them to a different stability operation region, thereby affecting the response of the system under small and large disturbances [13, 14].

A study conducted in [15] analyses the impact of the effect of high penetration of photovoltaic (PV) on small signal stability. Due to the reduced system inertia the study result showed a reduction in the damping torques of the system. A transient and small signal stability analyses with a gradual inclusion of PV are developed in [16], the eigenvalue results show that a displacement of conventional units have a great impact on the oscillatory modes.

  • The third challenge is the coherency of power systems and how it can be affected by the use of high power electronic converters. One relevant study shows that the large-scale inclusion of wind power changes the coherency of the synchronous generators coherent groups [17], however additional research is required in understanding how the non-synchronous generation affects the coherency in power systems.

The phenomenon of coherency in large interconnected power systems is presented when some generators swing together after a disturbance [18, 19]. To identify coherent groups, different methods have been studied during the last years in literature. Time and frequency-domain methods have been used to analyse the coherency of generators [20, 21]. Partitioning-based slow coherency methods have been presented in [22, 23].

Support vector clustering is applied for coherency studies in [24], computing compact and separated clusters based on the measurements obtained.

Some approaches applying machine learning and computational intelligence techniques for coherent identification have also been investigated in [25, 26]. In [27], neural networks, as patterns recognition and classificators sets are used. In a more recent contribution [28], where neural networks are also used, a fast method for security assessment is proposed. Fuzzy clustering methods have also been applied in [29] and [30] with auto-configuring training, and c-means clustering methods, respectively.

Another related method is presented in [31], which applies maximum spanning tree partitioning, to obtain the strongest connections in the network after a disturbance.

One method is developed in [32], where the flocking agent-interaction method is applied for different scenarios including rapid clustering identification.

Digital signal processing techniques have been applied in this topic. One important contribution is presented in [33], where the wavelet transform is used to obtain the phase relations according to the signals, and is used to determine the common frequencies of the generators involved. Another important approach is given in [34], where the Hilbert-Huang transform is tracking the generator coherency instantaneously.

Coherency-based graph theory has been studied in [35] and [36], where the topological network structure is analysed. In this method the generators are clustered according to the sub-networks and cut-sets obtained.

In [17], the coherency including wind farms is studied. The determination of the coherent groups is done by the rotor angle response observation after a fault.

The Koopman operator has proven to be a suitable method for coherency identification in power systems [37]. Koopman Mode Analysis (KMA) provides a graphical tool based on linear transformations on Hilbert spaces to analyse (non-linear) Hamiltonian systems. This linear, infinite-dimensional operator is defined for any non-linear dynamical system [38, 39]. One important characteristic of the Koopman operator is its ability to capture the full pattern information of large complex dynamical systems like power grids. Also, in [40], an islanding method is proposed and it is shown that the Koopman operator can determine the static connectivity of a system in a similar way as graph theory does.

This document follows the theory presented in [41], and also applies the technique developed in [37] and [42]. Otherwise, the same mentioned authors, in a most recent contribution of KMA presented in [43], the authors show the application in the 2006 European Grid-wide disturbance [44]. It is demonstrated the versatility of the KMA by decomposing the power exchanges between the operative areas in order to not only observe the coupled swing dynamics, but use the KMA to diagnose the instabilities using the data obtained from a real past case [43].

The main contributions of this document are the follow ones. The first one, is the coherent groups identification including non-synchronous generations in power systems which has been barely studied in literature before, and second one, to address the notion of how the reduction of inertia by the large penetration of power converters change affects the coherency in power systems from the data. Note the Koopman Analysis requires only the measurements of the system, not the model, which makes a robust and practical graphical tool. Moreover, the analysis is illustrated in a test system under the consideration of different penetration level scenarios and fault locations illustrating gradually the coherent groups conformation.

This document is an extended version of [2, 70] and is structured as follows. In Section 9.2, the problem formulation of coherency identification in non-synchronous integration is presented. In Section 9.3, the preliminaries of the theory of the Koopman operator, the definition of KM and the coherency definition are reviewed. In Section 9.4, the coherency in the Nordic test system is studied under different faults. Section 9.5 presents the case studies regarding the gradual increasing of non-synchronous generation integration on the test system. Finally, the conclusions and future work of this research are given.

9.2 Problem Formulation

The identification of coherent groups is of importance for the development, deployment and implementation of control schemes to improve the system transmission capability [45]. Having identified coherent groups, it is possible to classify those generators that are oscillating together, in order to understand the dynamic behaviour of the system under disturbances, and design appropriate controllers to protect it against them [46].

Coherency identification can be used in Wide Area Monitoring Protection and Control (WAMPAC) systems [47] in different ways. For instance, controlled islanding uses the coherency identification to know how to split the network into different sub-groups, and to avoid a blackout of the complete system [48, 49].

A dynamic preventive observation of the coherent groups can confine cascading faults within smaller self-sustainable islands making the grid more robust under disturbances [50].

Self-healing schemes can also be improved through the observation of the coherent groups to protect, apply control actions and restore the system after a fault [51].

The identification of coherent groups can also be helpful to locate, design or tune Power System Stabilizers (PSS) [52, 53]. Furthermore, the design of Flexible AC Transmission Systems (FACTS) controllers can also take advantage of the coherent groups identification as shown in [54], where the feedback measurements are selected based on the dominant machines of the identified groups.

Moreover, cyber-security schemes can take advantage of the coherent identification as a tool for monitoring the dynamic changes instantaneously and upgrade the control decisions faster [55, 56]. Furthermore, it is possible to propose dynamic security assessments to make the response of the protection systems more efficient and adaptable, and also to conceive advanced alert systems in order to avoid large area disturbances [57, 58].

Coherency identification methods can be classified into two basic groups [59]. The first one is based on linearised models of power systems [60, 61]. These methods might not be completely suitable since they do not capture the post-fault dynamics. The second group can be called measurement-based methods, because they rely on the signals obtained from the system, either off-line or on-line methods, to bring a more precise dynamic observation of the system [62].

9.2.1 Koopman Self-clustering Optimization

One of the biggest concerns in Wide Area Measurement Systems (WAMS) is the effective computing techniques and measurement technologies which allow to visualize the dynamics of the systems and use the data for short/long term power system planning [63, 64]. Koopman Mode Analysis (KMA), in its current stage of development, is capable of receiving the signals from simulation or measured data, providing the clusters of the coherent groups. Certainly, the method has the potential to be a real-time coherency method; but further research is needed. Otherwise, the computational effort of the analysis is based only in the mathematical (numerical) calculation which can be carried out by any of the processors developed in the current technological age, or it can be easily embedded in a Hardware in-the-Loop (HiL) system or even a Floating Point Gate Array (FPGA). However, these two latest aspects are not the purpose of this document.

The visualization of the method also plays an important role, e.g. from the Transmission System Operator (TSO) point of view. The display of the results should be easy to understand by the operator, who is responsible for planning the interconnection of HVDC lines or large non-synchronous generation, and determine decisions on-time, propose control strategies to prevent undesirable events, etc. The results, and the final results display obtained by the use of the KMA (phase vs. amplitude diagrams), can bring a friendly interpretation of the dynamics variations in the power systems supervised.

One of the main robust characteristic of KMA is that can handle several data (e.g from data-receivers or predicted models) providing an assessment for short/long-term futuristic power planing and update the control systems based on the analysis given [43].

Partitioning power networks (islanding) for protecting the grid of blackouts and large collapses using KMA has an important advantage, the model-free and the relying on dynamics data in the network [40].

Coherency identification using KMA belongs to the measurement-based group since it only requires sampled data [65]. Apart of the mentioned advantages, KMA has some important characteristics which makes it a powerful tool. The first, and most important one is that it deals with non-linear dynamics. Secondly, it does not need any training process, such as neural networks or support vector machines, because this method lies on the spectrum calculation of the Koopman Operator.

The aim of this paper is to show that the KMA can be used for identification of coherent swings and generators in systems under the effect of the decoupling of full converters. In this paper, non-synchronous generation is modelled by aggregated full converter units with the same power rating as that of the replaced synchronous generation.

9.3 Koopman Operator Preliminaries

9.3.1 Koopman Operator, Eigenvalue and Mode

Consider the following dynamics described by a discrete-time non-linear system evolving on a smooth manifold [40, 66].

$$\begin{aligned} x_{k+1}=F(x_{k}) \end{aligned}$$
(9.1)

where, \(x_{k} \in M\) is the state variable belonging to state space M, and \(F : M\rightarrow M\) is a non-linear, vector-valued function. The Koopman Operator is a linear operator \(\mathscr {U}\) that acts on scalar-values functions on M in the following manner: for \(g : M\rightarrow \mathbb {R}\), \(\mathscr {U}\) maps g into a new function \(\mathscr {U}g\) given by

$$ \mathscr {U}g(x)=g(F(x)) $$

Although the dynamical system is non-linear and evolves on a finite-dimensional space, the Koopman operator is linear, but infinite-dimensional. The Koopman eigenfunctions \(\varphi _{j}:\) \(M\rightarrow \mathbb {\mathbb {C}}\) and the Koopman eigenvalues (KEs) \(\lambda _{j}\in \mathbb {C}\) associated to \(\mathscr {U}\), are defined as

$$ \mathscr {U}\varphi _{j}(x)=\lambda _{j}\varphi _{j}(x),\qquad j=1,2,\ldots $$

Consider \(g:M\rightarrow \mathbb {R}^{p}\) a vector-valued observable. If each \(g_{i}\) of the components in \(\mathbf {{{\varvec{g}}}}\) lies within the span of eigenfunction \(\varphi _{j}\), then the time-evolution of observable \({{\varvec{g}}}\left( {{\varvec{x}}}_{k}\right) \) from \({{\varvec{g}}}\left( {{\varvec{x}}}_{0}\right) \) is expanded as follows:

$$\begin{aligned} {{\varvec{g}}}\left( {{\varvec{x}}}_{k}\right) =\sum _{j=1}^{\infty }U^{k}\varphi _{j}\left( {{\varvec{x}}}_{0}\right) {{\varvec{v}}}_{j}=\sum _{j=1}^{\infty }\lambda _{j}^{k}\varphi _{j}\left( {{\varvec{x}}}_{0}\right) {{\varvec{v}}}_{j} \end{aligned}$$
(9.2)

where, \({{\varvec{v}}}_{j}\)is the vector-valued coefficient of the decomposition and is called the j-th Koopman Mode (KM) [67]. This decomposition is based on the properties of the point spectrum of U, and the analysis based on (9.2) is called Koopman Mode Analysis (KMA). The KMA enables the extraction of single-frequency modes from data on fully non-linear dynamics from the temporal behaviour of the corresponding KM. The phase of \(\lambda _{j}\) determines its frequency, and the magnitude determines the growth rate.

The computation of Koopman eigenvalues and KMs is effectuated by a modified version of Arnoldi algorithm described in [68], where it is shown that the Ritz value \(\tilde{\lambda }_{j}\) and vector \(\tilde{\textit{v}}_{j}\) approximate the Koopman eigenvalue \(\lambda _{j}\) and factor \(\varphi _{j}\left( {{\varvec{x}}}_{0}\right) {{\varvec{v}}}_{j}\) in the expansion in terms of a finite truncation.

The input of the algorithm is the \(N+1\) sampled data \(\left\{ {{\varvec{g}}}\left( {{\varvec{x}}}_{0}\right) ,{{\varvec{g}}}\left( {{\varvec{x}}}_{1}\right) ,\ldots ,{{\varvec{g}}}\left( {{\varvec{x}}}_{N}\right) \right\} \). The outputs are N pairs of Koopman eigenvalues and KMs. The finite sum expansion is expressed by:

$$\begin{aligned} {\left\{ \begin{array}{ll} {{\varvec{g}}}\left( {{\varvec{x}}}_{k}\right) = &{} \sum _{j=1}^{N}\tilde{\lambda }_{j}^{k}\tilde{{{\varvec{v}}}_{j}}\qquad k=0,\ldots ,N-1\\ {{\varvec{g}}}\left( {{\varvec{x}}}_{N}\right) = &{} \sum _{j=1}^{N}\tilde{\lambda }_{j}^{N}\tilde{{{\varvec{v}}}_{j}}+{{\varvec{r}}} \end{array}\right. } \end{aligned}$$
(9.3)

where, \({{\varvec{r}}}\) is a residue with the approximation error.

9.3.2 Coherency in the Koopman Mode

By denoting \(\tilde{v}_{ji}\) as the i-th element of \({{{\varvec{v}}}_{{\varvec{j}}}}\), a coherent group of KMs is identified based on the amplitude coefficient \(A_{ji}:=\mid \tilde{v}_{ji}\mid \) and the initial phase \(\alpha _{ji}:=arg(\tilde{v}_{ji})\) for each mode j and observable i (e.g. rotor angle \(\delta _{i}\) and voltage angle \(\theta _{i}\)).

Coherency for KMs is defined in [51] as follows. For given finite N modes \(\left\{ \tilde{v}_{1},\ldots ,\tilde{v}_{N}\right\} \) and fixed constants \(\left( \varepsilon _{1},\varepsilon _{2}\right) \), two observables \(\left\{ g_{k},g_{v}\right\} \) are called \(\left( \varepsilon _{1},\varepsilon _{2}\right) \)-coherent with respect to mode j if

$$ {\left\{ \begin{array}{ll} \mid A{}_{j,k}-A{}_{j,v}\mid &{}<\varepsilon _{1}\\ \mid \alpha {}_{j,k}-\alpha {}_{j,v}\mid &{} <\varepsilon _{2} \end{array}\right. } $$

In this case, \({{\varvec{g}}}\left( {{\varvec{x}}}_{k}\right) \) contains swing dynamics of synchronous machines in a power system, so it is possible to group the oscillatory components with similar amplitude \(A_{ji}\) and initial phase \(\alpha _{ji}\) of machines to state them as coherent. This is illustrated in the Figure 9.1.

Fig. 9.1
figure 1

Groups of (\(\varepsilon _{1}\), \(\varepsilon _{2}\))-coherent observables illustrated in a phase versus amplitude plot

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9.3.3 Illustrative Example

Let us consider a simple example of KMA. The signals shown in Figure 9.2 represent the measurements obtained from an oscillatory signal. Notice these signals are not coupled.

$$\begin{aligned} \begin{aligned} g_{1} =&0.5sin\left( 2\pi f_{1}t \right) \\ g_{2} =&0.8sin\left( 2\pi f_{2}t-\pi /2 \right) \\ g_{3} =&sin\left( 2\pi f_{3}t \right) \end{aligned} \end{aligned}$$
(9.4)

Here \(f_{1}\), \(f_{2}\), and \(f_{3}\), are chosen as 6, 2.6, 0.7 Hz respectively. These signals are depicted in Figure 9.2 over a time period of 4 s.

Fig. 9.2
figure 2

Three sine waves of different frequency corresponding to (9.4). The 2.6 Hz sine wave is displaced by a \(\pi /2\) rads phase shift

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Now, KMA is applied with a sampling frequency of \(f_{s}=60\) Hz to the data. \(N+1=240\) samples are acquired giving \(N=239\) modes. The modes are now listed based on the Growth Rate (GR) which is related to the damping in case of sampled dynamics. The five modes with the largest GRs are listed in Table 9.1.

Table 9.1 Dominant Koopman modes obtained for the data shown in Figure 9.2
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In this manner, the sampled data has been decomposed into a set of Ritz values \(\tilde{\lambda }\) and vectors \({{\varvec{v}}}\). The dominant frequencies are identified by applying KMA to the sampled data. Modal dynamics for Modes 1-3 are depicted in Figure 9.3. It is seen that for each mode, essentially only one observable (out of three) contributes (the one corresponding to the sine wave of the same frequency as the mode). The sum over all modes according to (9.2)–(9.3) reconstructs the sampled data.

Fig. 9.3
figure 3

The modal “dynamics” of the three dominant modes. Each of the modes has three contributions corresponding to the three measured observables (the sine waves \(g_{1}\), \(g_{2}\), \(g_{3}\))

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9.4 Application to the Nordic System

The single-line diagram of the Nordic test system is shown in Figure 9.4. This system contains 32 high voltage buses, 20 synchronous generators with different types of generation (circled in the figure), in four geographical identified area. The North and External area are hydro-dominated while the south and central areas have a mixture of nuclear, thermal and coal power plants. Central area has the highest level consumption whereas the North area has the lowest level. The transmission system is designed for 400 kV (19 buses) with some regional systems at 220 kV (2 buses) and 130 kV (11 buses). The details of the system, such as unit rating, line data, dynamic data, and loading conditions, are given in [69]. Power System Stabilizers (PSS) have been located in the following synchronous machines: 1042, 1043, 4011, 4042, 4047, 4051, 4062, 4063.

Fig. 9.4
figure 4

Nordic 32 test system

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Table 9.2 List of selected faults—Nordic 32-bus test system
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9.4.1 Numerical Simulation

The setting of numerical simulation is as follows. The constants and power loads are the same as in [69]. All numerical simulations discussed in this paper are performed using the software DigSilent Power Factory®. Then, some disturbances (three-phase short circuit faults) in the system are supposed in order to trigger the swing curves signals to analyse. The location of the faults are based on the previous study of this system presented in [70], however the Critical Clearing Time (CCT) differs due to the different dynamic settings in the system (Table 9.2).

9.4.2 Koopman Modes and Eigenvalues Analysis

KM have been computed based on the measurements obtained, which are the rotor angle \(\delta _{i}\) of the synchronous generators. For computation, the observable \({{\varvec{g}}}\left( {{\varvec{\delta }}}\right) \) is chosen, where \({{\varvec{\delta }}}=\left[ \delta _{1},\ldots ,\delta _{7},\delta _{9},\ldots ,\delta _{20}\right] ^{T}\). The symbol T indicates transpose in vectors. It is used on the simulation outputs obtained from the software, where the uniform sampling period \(T_{s}=1/(f_{s}),\) with \(f_{s}=100\) Hz and the number of samples \(N=1001\). An appropriate number of samples should be selected to capture a large quantity enough of snapshots.

Consider a fault located in the bus number \(B_{4047}\) located in the Central area. Figure 9.5 shows the time response of the \(\Delta \delta _{i}\). Note that \(G_{8}\) (\(G_{4011}\)) is the reference machine.

Fig. 9.5
figure 5

Rotor angle time response after the fault at bus \(B_{4047}\)

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The coherent generators extraction are obtained from the KMs decomposition. In this document, the mode of interest is the one with the largest norm. The frequency related to this mode is 0.6Hz. Table 9.3 shows the numerical computation of KEs and KMs, Mode 1 to Mode 10.

Table 9.3 Result of Koopman modes after fault in B4047
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The distribution of \(A_{8,i}\) versus \(\alpha _{8,i}\) is plotted in Figure 9.5. The circle points for generators show the different cluster groups obtained. For this mode, the phase \(\alpha _{8,i}\) clusters two main generator coherent groups \(CG_{1}\) (1012, 1013, 1014, 4012, 4071, 4072) and \(CG_{2}\) (\(G_{1043}\), \(G_{4041}\), \(G_{4042}\), \(G_{4047}\), \(G_{4051}\), \(G_{1042}\), \(G_{4062}\), \(G_{4063}\), \(G_{2032}\), \(G_{4021}\), \(G_{4031}\), \(G_{1021}\) and \(G_{1022}\)) and different subgroups. Figure 9.6 show the time response of these sub-groups respectively. Observing \(CG_{1}\) and \(CG_{2}\) groups and the time response is possible to match which groups have positive and negative rotor angle, or in other words, the accelerating and de-accelerating groups respectively.

Fig. 9.6
figure 6

Coherent sub-groups identified after the fault at bus \(B_{4047}\)

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Following the same process, it is possible to identify the coherent groups/subgroups with the different faults stated above. Table 9.4 shows the corresponding KMs and their frequencies according to the rest of the faults. Figure 9.7a–d show the distribution of the largest Koopman Modes.

Same groups (\(CG_{1}\)and \(CG_{2}\)) and subgroups are obtained, nevertheless some slightly differences can be observed. For the fault located in bus \(B_{4063}\), it can be seen in Figure 9.7c that \(G_{4062}\) and \(G_{4063}\) are more related to the central group of generators than other cases. Sub-group \(S_{7}\) can be joined with sub-group \(S_{6}\). The other generators remained to the same sub-groups.

Table 9.4 Largest Koopman modes for the different faults
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Fig. 9.7
figure 7

Distribution of initial phases and amplitude coefficients for different faults

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9.5 Test Cases: Gradual Increasing of Non-synchronous Generation

The impact of the integration of non-synchronous generation on the test system coherency is analysed by replacing some of the synchronous generators with back-to-back FRC with the same active and reactive power outputs in order to guarantee the same initial conditions. Note that the power outputs are fixed through the simulation. Two scenarios are tested: first one, replaces synchronous generators in the Central-South area while the second one replaces in the North-External area.

9.5.1 Scenario Central-South

Case 1 (\(C_{1})\) to Case 5 (\(C_{5}\)) represent the replacement of synchronous generation by the integration of non-synchronous generation based FRC gradually, in order to analyse different levels of power penetration. For example, \(C_{1}\) considers the replacement of one generation only, and \(C_{2}\) considers the replacement of two generators including the one in \(C_{1}\), and so on. It is assumed that the dispersed generation is connected to one established substation. These five scenarios are summarised in Table 9.5:

Table 9.5 Generator replacement for each case
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Initially, KM are evaluated for fault at \(B_{4047}\) for cases presented above. Table 9.6 shows the KM variation according to the replacement by non-synchronous in the system. It can be seen that the largest mode frequency stays in the same range. The same behaviour is obtained in the rest of the cases; however, the frequency tables variation are not presented here.

Table 9.6 Koopman largest mode variation (fault 4047)
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KM for \(C_{2}\) is plotted in Figure 9.8. The replacement by non-synchronous generation in the first two cases does not affect the coherent groups/sub-groups analysed before. Figure 9.8 shows the KM for \(C_{3}\) and the significant changes for cases \(C_{4}\) and \(C_{5}\) with the circle points variation of the generators concerned. After the third case, with the replacement of generator \(G_{4062}\), generator \(G_{4063}\) (both in the South area) become to swing and be more coherent with generators in the Central area. Generators in the North and External area keep swinging together.

Fig. 9.8
figure 8

Coherency identification including non-synchronous generation (after fault at bus \(B_{4047}\))

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For the fault in bus \(B_{4031}\), first two cases of non-synchronous generation replacement do not show change in the coherent sub-groups as the former analysis. From the third case, significant changes can be observed in the KM sub-groups. The sub-groups based on the amplitude coefficient of the respective mode show that the Central area have become more separated. Specially in case 4, where generator \(G_{1042}\) moves closer to \(G_{4063}\) (Figure 9.9).

Fig. 9.9
figure 9

Coherency identification including non-synchronous generation (after fault at bus \(B_{4031}\))

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Generator \(G_{1042}\) is becoming more separate in \(C_{4}\) for post-faults in buses \(B_{4063}\), \(B_{4012}\) and \(B_{4021}\)as it can be seen in Figures 9.10 and 9.11 (Figure 9.12).

Fig. 9.10
figure 10

Coherency identification including non-synchronous generation (after fault at bus \(B_{4063}\))

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Fig. 9.11
figure 11

Coherency identification including non-synchronous generation (after fault at bus \(B_{1021}\))

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Fig. 9.12
figure 12

Coherency identification including non-synchronous generation (after fault at bus \(B_{1012}\))

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9.5.2 Scenario North-External Area

This scenario considers the replacement of some generators in the North and External area. The location of these new cases in the system are presented in Table 9.7. It is applied the same gradual replacement from the previous scenario.

Table 9.7 Generator replacement for each case (North-Ext)
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The replacement by non-synchronous generation in the first two cases remains the sub-groups previously identified for the same faults located. After the replacement of the third generator, for the fault located in bus \(B_{4031}\), makes the generator \(G_{1021}\) be more coherent with the sub-group of generators \(G_{4012}\) and \(G_{1014}\), whereas for the faults located in buses \(B_{1021}\) and \(B_{1012}\) the generator \(G_{1021}\) be more coherent with the generators \(G_{2032}\), \(G_{4031}\) and \(G_{4021}\). This is shown in Figures 9.13, 9.14 and 9.15 (Table 9.8).

Fig. 9.13
figure 13

Coherency identification including non-synchronous generation (after fault at bus \(B_{4031}\))

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Fig. 9.14
figure 14

Coherency identification including non-synchronous generation (after fault at bus \(B_{1021}\))

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Fig. 9.15
figure 15

Coherency identification including non-synchronous generation (after fault at bus \(B_{1012}\))

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Table 9.8 Koopman largest mode variation (fault 4031)
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9.6 Conclusions and Future Work

The document presents an original contribution to the application of the KMA for the coherency pattern identification of power systems system with a gradual large inclusion of non-synchronous generation.

The application of KMA to the studied power system (Nordic 32 system) showed the variation of the coherent groups through the gradual inclusion of power converters. The results show that if a minor replacement of synchronous machines the coherency is not affected, but after the replacement of the 13% of the power in the system, some significant changes in the coherent groups can be identified.

The coherent groups in one operative area can be altered by the increasing inclusion of non-synchronous generation becoming either groups apart or be more coherent with other groups. A general comment can be made in relation to the fault location since the coherency phenomena is related to it. After studying several faults incorporating the gradual increasing of full converters. The effect of a fault close to a specific generator in the form of acceleration is propagated, and reflected around the closer generators in the form of synchronizing power. Therefore, with the replacement of synchronous machines (inertia reduction), the coherent groups separate different, as the amplitude coefficient separation in the KM showed.

The applied coherency identification method provides a direct calculation and a powerful graphical visualisation tool for observing the coherent groups/subgroups according to the measurements obtained while the dynamics are changing.

This tool has the potential to be integrated in real-time simulator systems and be improved by a sliding window, which means can be applied in situational awareness alert system or control loops design involved in FRC.

KMA application is concise with the slow coherency theory, however with the reduction of inertia in the system, the fault location change the coherent groups in the system.

Future work requires the addition of PSS to non-synchronous generation and other control loops, e.g. synthetic inertia in order to diminish the oscillations.