# Graph Neural Network and Koopman Models for Learning Networked Dynamics: A
Comparative Study on Power Grid Transients Prediction ††thanks: The Pacific
Northwest National Laboratory (PNNL) is operated by the Battelle Memorial
Institute for the U.S. Department of Energy under Contract No. DE-
AC05-76RL01830 . The authors gratefully acknowledge the funding support from
the Data Model Convergence (DMC) Initiative at PNNL.

Sai Pushpak Nandanoori1, Sheng Guan2, Soumya Kundu1, Seemita Pal1, Khushbu
Agarwal1,
Yinghui Wu2, Sutanay Choudhury1
1Pacific Northwest National Laboratory
Richland, WA 99354, USA
(emails: {saipushpak.n, soumya.kundu, seemita.pal,
khushbu.agarwal<EMAIL_ADDRESS>
2Case Western Reserve University
Cleveland, OH 44106, USA
(emails: {sxg967<EMAIL_ADDRESS>
###### Abstract

Continuous monitoring of the spatio-temporal dynamic behavior of critical
infrastructure networks, such as the power systems, is a challenging but
important task. In particular, accurate and timely prediction of the (electro-
mechanical) transient dynamic trajectories of the power grid is necessary for
early detection of any instability and prevention of catastrophic failures.
Existing approaches for prediction of dynamic trajectories either rely on the
availability of accurate physical models of the system, use computationally
expensive time-domain simulations, or are applicable only at local prediction
problems (e.g., a single generator). In this paper, we report the application
of two broad classes of data-driven learning models – along with their
algorithmic implementation and performance evaluation – in predicting
transient trajectories in power networks using only streaming measurements and
the network topology as input. One class of models is based on the Koopman
operator theory which allows for capturing the nonlinear dynamic behavior via
an infinite-dimensional linear operator. The other class of models is based on
the graph convolutional neural networks which are adept at capturing the
inherent spatio-temporal correlations within the power network. Transient
dynamic datasets for training and testing the models are synthesized by
simulating a wide variety of load change events in the IEEE 68-bus system,
categorized by the load change magnitudes, as well as by the degree of
connectivity and the distance to nearest generator nodes. The results confirm
that the proposed predictive models can successfully predict the post-
disturbance transient evolution of the system with a high level of accuracy.

Notice of Publication: This work is currently under review in a journal.

##  1 Introduction

Critical infrastructure networks, such as traffic networks and power systems,
are often modeled as networked dynamical systems. Accurate learning of the
spatio-temporal dependencies of the node-level attributes, which constitute
the state of the dynamical network, is essential to monitoring and control of
such systems. Power systems, for example, are increasingly operated closer to
their stability limits (e.g., high loading) [1], which, when coupled with the
fast timescales associated with the electro-mechanical transients, makes
timely and accurate prediction of trajectories absolutely critical to energy
security and resilience. Enabling accurate predictions of the transient
dynamic states help detect the possible state violations into the future and
thereby help predict possible line or link failures which when allowed to
persist may lead to cascading failures. Moreover, emerging data-driven control
techniques in power systems, such as deep reinforcement learning [2, 3], rely
on accurate time-series prediction of trajectories for the learning and
exploration tasks. Even though modern power network comprises of various
sensors, such as the phasor measurement units (PMUs), continually relaying
measurements to the control center, robust prediction of transient dynamic
trajectories remains challenging due to its enormity, complexity and
nonlinearity.

State-of-the-art. Data-driven learning of predictive models for nonlinear
dynamical systems has received some attention in recent times [4, 5, 6]. The
work [4] presents a review of the different neural network-based learning
models for nonlinear dynamical systems, including different variants of
recurrent networks, e.g., long-short-short-term memory (LSTM) models [5, 6].
In the context of power systems dynamics, transient stability assessment and
prediction has been an active area of research over the years, as summarized
in a recent review [1]. It was noted that time-domain simulation-based
approaches often yield the most accurate results, but are burdened by extreme
computational costs. On the other hand, machine learning (ML) methods were
found to be used most frequently, even though their applications were largely
limited to the transient stability classification problem (yes/no question),
as opposed to time-series prediction. In an early ML application for time-
series prediction of transients, [7] uses recurrent radial basis functions and
artificial neural networks for generators’ angles and angular velocities
prediction, albeit in small networks. LSTM has been used in [8] for time-
series prediction of post-contingency trajectories, but only for a single
generator. In [9], a transient stability prediction method based on deep
belief network and LSTM network is developed to predict generator rotor angle
trajectories in the IEEE 39-bus network. However, this predictive model relies
on an extensive list of input features (including detailed generator model
parameters) which are often unavailable, unreliable, or expensive to identify
[10]. In contrast, the methods proposed in this work rely only on streaming
PMU measurements (that includes frequency and voltage magnitudes).

We highlight here the difference between the task of multi-timestep prediction
of transient trajectories, as covered in this work, and the problem of dynamic
state estimation which has received some attention in the power systems
community [11]. In dynamic state estimation, the states of the system at the
current time-point are estimated using available measurements and some
knowledge of the system model. In contrast, this article is concerned with the
time-series prediction of the transient trajectories over a look-ahead period
(of a few seconds) using available measurements, without any prior knowledge
of the system model.

Enabling Models. As explained above, the existing ML methods for transient
prediction either place strong assumptions on the availability of extensive
system features (e.g., detailed generator models), or limit themselves to only
local prediction problems (e.g.transient trajectories of a single generator),
and hence are not applicable to system-wide transient prediction problems
under limited information (e.g.access to noisy measurements data only). In
this article, we focus on data-driven learning models that rely only on
streaming measurements from the sensors (e.g., PMUs) across the network and,
perhaps, some limited information such as the topology of the network and the
locations of the sensors. Specifically, we choose two classes of learning
models – one based on the Koopman operator theory and the other based on the
graph convolutional neural networks – which we demonstrate to have the
expressive capability to encode the nonlinearity in power system transient
dynamics, and capture the dynamic behavior across the network, without
requiring detailed physical models of the system. The following are the two
major enabling techniques considered in this article for robust transient
prediction:

  1. •

predictive models that incorporate Koopman operators to learn the complex
nonlinear dynamic behavior of power system transients in the form of infinite-
dimensional linear maps in some lifted space of observables; and

  2. •

graph convolutional neural networks that model power systems as dynamic
graphs, and exploit their spatial-temporal-topological correlation for
accurate prediction.

The choice of these two methods allows us to perform a comparison between
predictive models that attempt to learn the underlying nonlinear dynamics
governing the evolution of system trajectories, and those that learn the
inherent spatio-temporal correlations in a dynamical network. In what follows,
we provide brief summaries of the applications of the above two predictive
models in related problems.

Koopman Operator Theory. The Koopman operator theory (KOT) introduced in [12],
represents the nonlinear dynamical system as a globally linear system on a
space of (observable) functions. The evolution of these observables is given
by the Koopman operator, a linear, infinite dimensional operator. Finding such
an infinite-dimensional operator for generic nonlinear systems is
computationally intractable. Nevertheless, several efficient approaches have
been developed to construct a finite-dimensional approximation of the Koopman
operator, including dynamic mode decomposition (DMD) [13, 14], extended DMD
[15], robust DMD [16], and deep DMD (deepDMD) [17, 18, 19, 20]. Koopman
operator-based techniques have found many applications in analysis of
nonlinear systems, especially in fluid mechanics [13], synthetic biology [20,
21, 22] and energy systems [23, 24, 25, 26, 27, 20, 28]. Majority of the data-
driven KOT applications in power systems have focused on nonlinear modal
analysis (via the use of Koopman modes), e.g.coherency detection [24, 25],
attack identification [26] or anomaly diagnosis [27], with a few also
considering time-series prediction [27, 20, 29]. The work in [27] used DMD for
transient prediction in power systems, assuming simpler swing-dynamic models
of the generators and access to full-state measurements, both of which do not
generalize to real-world systems. The work in [20], on the other hand,
combined KOT with deep learning tools to develop a deepDMD model for power
systems transient prediction and demonstrated its advantages over EDMD, albeit
on a test system using simpler swing-dynamic models of the generators and
full-state measurements. In a more recent work on analytical construction of
KOT-based models [29], authors only consider a single-machine infinite-bus
example and do not address the feasibility of the approach (e.g.in terms of
accuracy and scalability) to realistic multi-machine networks. Related works
have also explored the use of KOT-based predictive models for feedback control
design [30].

Graph Neural Networks. An alternative option to tackle the non-linearity and
spatial-temporal dynamics of power systems is to exploit the emerging graph
neural networks (GNNs) [31, 32]. Generally speaking, the GNNs are a class of
deep models that represent a graphical network as node embeddings and learn
the best representation (by optimizing learning objective function) that can
be used for downstream tasks. In contrast to the conventional predictive
models, such as recurrent neural networks (RNNs) and convolutional neural
networks (CNNs)[33], the GNNs are able to jointly characterize node and
topological features and thus have been adopted for network learning. CNNs are
known to be limited to the regular Euclidean data computation, where the
localized convolutional filters and pooling operators are difficult to be
defined in the non-Euclidean power-grid data. RNNs on the other hand, are not
optimized to process graphs, as they require to stack feature of nodes in a
specific order, whereas there exists no natural order in power grid data. In
[34], a novel GNN model named the Spatio-Temporal Graph Convolutional Network
(STGCN) was developed for mid/long-term time-series forecasting of traffic
networks. Specifically, instead of applying regular convolutional and
recurrent units, the STGCN deployed a fully convolutional structure on the
time axis to extract spatio-temporal features from graph-structured traffic
data, resulting in faster training with fewer parameters. Application of GNN
to power systems thus far, however, has been sparse. In a recent work [35],
though, the authors used a GNN model for short-term (hours to days) load
forecasting in residential buildings, and demonstrated its capability to
capture the latent spatial dependencies of different houses within some
geographic area.

Contributions. The main contributions of the paper are that:

  1. 1)

We construct data-driven variants of KOT-based models (namely, deepDMD and
robust DMD) for multi-step prediction of the power system transients, using
only a single time-step measurement. The KOT-based predictive models learn the
inherent nonlinear dynamics purely from available measurements, without any
prior information of the physics-based system models. To the best of authors'
knowledge, this is the first time that the transient predictions on a large
power system (IEEE 68 bus) has been studied where the synthetic data is
obtained using higher-order synchronous generator models (eighth order) and
KOT-based methods have been applied using only critical measurements such as
frequency and voltage. Only the work in [20] among the existing works
discusses the transient predictions however, on a small power network (IEEE 39
bus) with simplified swing equation model for the synchronous generators
(second order).

  2. 2)

In a novel application, we implement, train, and test a deep STGCN model
(coupled with an incremental inference strategy) for multi-step transient
prediction in power systems. With the knowledge of the sensor measurements and
the connectivity between the sensor nodes, the STGCN jointly learns the
underlying temporal and spatial features for predicting the transient
trajectories. It is important to mention that the STGCN model (in the
literature [34]) is originally introduced for one-time step traffic prediction
only.

  3. 3)

The synthetic spatio-temporal time-series measurements which corresponds to
different load disturbances of the IEEE 68 bus system are generated as part of
this work using GridSTAGE [36] and are made accessible to the wider community
[37]. Moreover, the KOT-based methods and STGCN model development codes
developed as part of this research are also made accessible online [38] for
reproducible research.

  4. 4)

Detailed comparative evaluation of the data-driven learning models are
performed to demonstrate the feasibility of their application to power network
transient prediction. Specifically, numerical studies are carried out to
evaluate the robustness to noisy measurements, accuracy across spatio-temporal
scales, sensitivity to hyper-parameters, and computational time costs.

The rest of the paper is structured as follows. Sec. 2 and Sec. 3 provide
brief technical description of the KOT-based and the GNN-based models,
respectively. Sec. 4 describes the simulation environment, the procedures for
generating synthetic datasets, and the performance metrics. The numerical
results from the detailed performance evaluation of the predictive models are
presented in Sec. 5. Finally, we conclude the article in Sec. 6, along with a
discussion of the future directions.

##  2 Koopman Operator-Based Prediction

We start with a revisit of the mathematical background for KOT (see [39, 15,
40] and the references therein). Consider the following discrete-time
nonlinear system

$\displaystyle Z_{t+1}=F(Z_{t}),$ (1)

where $Z_{t}\in\mathcal{M}\subseteq\mathbb{R}^{m}$ are the $m$-dimensional
system states, and $F:\mathcal{M}\to\mathcal{M}$ is a continuously
differentiable system map. Note that, in this work, we consider the map
$F(\cdot)$ to be unknown. We denote by $\mathcal{F}(\mathcal{M})$ the space of
all scalar-valued functions of the system states $\boldsymbol{z}$.

###### Definition 1

(Koopman Operator) The Koopman operator,
$\mathbb{U}:\mathcal{F}(\mathcal{M})\to\mathcal{F}(\mathcal{M})$, associated
with the dynamical system (1) is defined as an infinite-dimensional linear
operator which acts on any scalar-valued observable function
$\phi\in\mathcal{F}(\mathcal{M})$ of the system (1) as follows:

$\displaystyle[\mathbb{U}\phi](\boldsymbol{z})=\phi(F(\boldsymbol{z})).$ (2)

The evolution of observables is given by the Koopman operator, $\mathbb{U}$,
which is linear and the function space, $\mathcal{F}(\mathcal{M})$ is
invariant under the action of the Koopman operator. Moreover as the space of
functions is infinite dimensional, the Koopman operator which maps these
functions is also infinite-dimensional, and this process of mapping is usually
referred to as lifting. Koopman operator is also a positive operator such that
for any $\phi(\boldsymbol{z})\geq 0$, we obtain
$[\mathbb{U}\phi](\boldsymbol{z})\geq 0$ [40].

Remark. In the case where the mathematical model (1) is not known explicitly,
a Koopman operator can be identified from sampled time-series data obtained
from experiments or sensor measurements.

In practice, a finite dimensional approximate of the Koopman operator is
computed. In the scope of this work, Robust DMD (that results in the dimension
of the approximate Koopman operator equal to the number of measurements) and
deepDMD (that results in the Koopman operator whose size depends on the chosen
observable space) are implemented to build predictive models for the power
network from PMU data.

DeepDMD and Robust DMD. We adapt the deepDMD model introduced in [20] and
extend it to the case where partial measurements are available at each bus.
While [20] assumes full-state measurements (i.e., all the dynamic states at
each bus of the system are measured), this present work assumes a more
realistic scenario in which only a subset of states at any bus are measured.
Let us denote by $X_{t}\in\mathbb{R}^{n}$ the $n$-dimensional measurements
(e.g., frequencies and voltages) available at any time-instant $t$ from the
PMUs installed in the power network. Henceforth, we will assume to have access
to only the measurements $X_{t}$, and not the underlying system states (nor
the system model).

We define $\boldsymbol{\psi}:\mathbb{R}^{n}\to\mathbb{R}^{q}$ as a
$q$-dimensional vector-valued observable function of the measurements $X_{t}$
as follows:

$\displaystyle\boldsymbol{\psi}(X_{t}):=\begin{bmatrix}\psi_{1}(X_{t})&\psi_{2}(X_{t})&\cdots&\psi_{q}(X_{t})\end{bmatrix}^{\top}$

where each $\psi_{i}$ ($i=1,2,\dots,q$) is a scalar-valued observable function
of the measurements $X_{t}$ .

Learning Objective. The objective of the deepDMD is to learn such a mapping
$\boldsymbol{\psi}$ and the associated linear operator (the approximate
Koopman) in (2) which governs the evolution of the observable functions on the
lifted observables-space. While the observable function $\boldsymbol{\psi}$
maps the time-series data to the observable space, we also need an inverse
mapping to retrieve the measurements from the observable space and this can be
achieved in two different ways. The first one includes identifying the
function $\boldsymbol{\varphi}\equiv\boldsymbol{\psi}^{-1}$ such that we have
$X_{t}=\boldsymbol{\varphi}(\boldsymbol{\psi}(X_{t}))$ . The second one
involves appending the actual measured states ($X_{t}$) with observables to
obtain measurement-inclusive observables, defined as below:

$\displaystyle\boldsymbol{\Psi}(X_{t})=\begin{bmatrix}X_{t}\\\
\boldsymbol{\psi}(X_{t})\end{bmatrix}.$

Since the measurements are accessible directly, there is no need for an
inverse mapping in the case of measurement-inclusive observable functions. Due
to its simplicity, we adopt this second approach of constructing measurement-
inclusive observables in this paper. Note its similarity to the notion of
state-inclusive observables in [20, 41].

The learning task involves identifying a suitable observable function
$\boldsymbol{\Psi}$, and a corresponding Koopman operator $\mathcal{K}$ such
that $\boldsymbol{\Psi}(X_{t+1})=\mathcal{K}\boldsymbol{\Psi}(X_{t})$, at all
time-points $t$ . Since in deepDMD, this learning task is performed via a
neural network, we denote the observable as $\boldsymbol{\Psi}(X_{t},\Theta)$
(instead of $\boldsymbol{\Psi}(X_{t})$), where $\Theta$ represents the neural
network weights and biases. Consider any sequence of $N+1$ measurement
snapshots $\left[{X}_{0},X_{1},\dots,X_{N-1},X_{N}\right]$. Let us define the
following:

$\displaystyle Y_{p}$
$\displaystyle:=\left[X_{0},\,X_{1},\,\dots,\,X_{N-2},\,X_{N-1}\right],$
$\displaystyle Y_{f}$
$\displaystyle:=\left[X_{1},\,X_{2},\,\dots,\,X_{N-1},\,X_{N}\right].$

where both $Y_{p}$ and $Y_{f}$ are a collection of measurements from $N$ time-
points, and $Y_{f}$ is constructed from $Y_{p}$ by shifting the measurements
by one time-point. Stacking up, the following Koopman operator relationship
$\boldsymbol{\Psi}(Y_{f},\Theta)=\mathcal{K}\boldsymbol{\Psi}(Y_{p},\Theta)$
should hold. Therefore, the learning cost function for the deepDMD is given by

$\displaystyle\min_{\mathcal{K},\Theta}$
$\displaystyle\parallel\boldsymbol{\Psi}(Y_{f},\Theta)-\mathcal{K}\boldsymbol{\Psi}(Y_{p},\Theta)\parallel_{F}^{2}+\lambda_{1}\parallel\mathcal{K}\parallel_{F}^{2}+\lambda_{2}\parallel\Theta\parallel_{1}$
(3)

where $\lambda_{1}>0$ and $\lambda_{2}>0$ indicate the weights for the
regularization costs for the Koopman operator and the neural network weights
and biases respectively. These regularization costs help avoid overfitting of
the learned models and additionally, the regularization cost on the Koopman
operator promotes robustness. We illustrate deepDMD in Fig. 1.

Figure 1: Computational overview for deepDMD.

Robust DMD [16] is seen as a special case of (3) such that
$\boldsymbol{\Psi}(X_{t}):=X_{t}$ , which gives us the following cost
function:

$\displaystyle\min_{\mathcal{K}}$ $\displaystyle\parallel
Y_{f}-\mathcal{K}Y_{p}\parallel_{F}^{2}+\lambda_{1}\parallel\mathcal{K}\parallel_{F}^{2}\,.$
(4)

Note that for a given value of $\lambda_{1}$, one can obtain an analytical
solution to (4) such that

$\displaystyle{\cal K}=Y_{f}\left(Y_{p}+\lambda_{1}I\right)^{\dagger}$ (5)

where $I$ is the identity matrix and $(\cdot)^{\dagger}$ denotes the Moore-
Penrose pseudo inverse of $(\cdot)$.

Multi-Step Prediction. The Koopman operator establishes a linear relationship
between two consecutive measurement vectors in the space of observables.
Repeat application of the Koopman operator on the current measurement
($X_{t}$) therefore allows us to generate a sequence of predicted measurements
over some prediction window $H$ as below:

$\displaystyle\boldsymbol{\Psi}(\widehat{X}_{t+h}|X_{t})=\mathcal{K}^{h}\,\boldsymbol{\Psi}(X_{t})\quad
h=1,2,\dots,H\,.$ (6)

Recall that the predicted measurements $\\{\widehat{X}_{t+h}\\}_{h=1}^{H}$ are
retrieved trivially from the measurement-inclusive observable
$\boldsymbol{\Psi}$ .

##  3 Graph Neural Network-Based Prediction

We next describe an alternative method that can holistically exploit network
topology and spatio-temporal information. Graph neural networks (GNNs) have
been verified to be effective for predictive analysis over network data, by
jointly considering node and topological features. For this work, we adopt the
Spatio-Temporal Graph Convolution Network (STGCN), a form of GNN introduced in
[34] for time-series traffic forecasting, and extend it to support flexible
power system transient problem with variable lengths of prediction window
(e.g.1-10 s). Specifically, we develop an enhanced STGCN with an incremental
inference strategy to make it adaptive to variable testing prediction lengths,
while the model is still trained only once. This helps us reduce the re-
training effort and make STGCN adaptive to time-series prediction of power
system transients.

Notations. We represent the cyber-physical power network as an attributed
graph, such that $G_{t}=(V,E,X_{t})$ represents a snapshot of the network at
time $t$, with $V$ as the (fixed) set of vertices (or, nodes) representing the
buses with sensors (PMU); $E\subseteq V\times V$ as the (fixed) edges between
the sensor nodes; and $X_{t}\in\mathbb{R}^{n}$ as the (time-stamped)
$n$-dimensional node feature vector representing the sensor measurements from
across the network at time $t$ . Moreover, $A$ denotes the fixed adjacency
matrix representing the connections between the sensor nodes.

Given past $M$ measurements, $\\{X_{j}\\}_{j=t-M+1}^{t}$, of the power
network, the problem is to predict the most likely measurements in the next
$H$ time-steps, $\\{X_{i}\\}_{i=t+1}^{t+H}$, as follows:

$\displaystyle\\{\widehat{X}_{i}\\}_{i=t+1}^{t+H}$
$\displaystyle=\operatorname*{argmax}_{\\{X_{i}\\}_{i=t+1}^{t+H}}\log
P\left(\\{X_{i}\\}_{i=t+1}^{t+H}\left|\\{X_{j}\\}_{j=t-M+1}^{t}\right.\right)$

Overview of STGCN. In a nutshell, an STGCN [34] learns the best representation
of a power grid with graph convolutional operators based on spectral graph
convolution. A graph convolution operator “$*_{\mathcal{G}}$” can be defined
as:

$g_{\theta}*_{\mathcal{G}}X_{t}=Ug_{\theta}(\Lambda)U^{\top}X_{t}\,,$ (7)

where $g_{\theta}(\cdot)$ is the graph kernel parameterized by the vector
$\theta\in\mathbb{R}^{K}$ (for some $K$) in the Fourier domain, and $X_{t}$ is
the time-stamped node feature vector (e.g.frequency measurements).
$U\in\mathbb{R}^{n\times n}$ refers to the graph Fourier basis that is the
matrix of eigenvectors of the normalized graph Laplacian $L$ =
$I_{n}-D^{-\frac{1}{2}}AD^{-\frac{1}{2}}\in\mathbb{R}^{n\times n}$, where
$I_{n}$ is the $n$-dimensional identity matrix, and $D\in\mathbb{R}^{n\times
n}$ is the diagonal degree matrix with $D_{ii}=\Sigma_{j}A_{ij}\,\forall i$ as
derived from the topology of the power network. $\Lambda\in\mathbb{R}^{n\times
n}$ is the diagonal matrix of eigenvalues of the normalized graph Laplacian
$L$, and the filter $g_{\theta}(\Lambda)$ is diagonal too. As pointed out in
[42, 34], graph convolution is the multiplication of the graph Fourier
transform $U^{\top}X_{t}$ of a signal $X_{t}$ with a filter of graph kernel
$g_{\theta}$.

An Incremental Inference STGCN Model. Fig. 2 shows the architecture of the
enhanced STGCN model, developed to support the variable prediction lengths in
the power system transient trajectories forecasts. Following the work of [34],
the model consists of spatio-temporal convolutional blocks (ST-blocks), each
built by two gated temporal convolution layers and one spatial graph
convolution layer in a sandwich structure. After stacking several ST-blocks,
an output block integrates the comprehensive features extracted from spatial
and temporal domains and outputs a single-step prediction. Extending the work
of [34], our STGCN shuffles the prediction and feeds the output back into the
model to generate forecasts with varying prediction length $H$. It
concatenates the predictions for the next $H$ time-steps and treats temporal
predictions as an autoregressive task. In the following, we briefly outline
each of the components.

Figure 2: Architecture of our STGCN framework.

Graph Spatial Convolution. Existing ML models for power systems time-series
prediction (e.g., LSTM [9, 8]) do not take into account the underlying spatial
structure of the network. The spatial graph convolution layer within the ST-
block allows us to extract meaningful patterns and representations in the
spatial domain by performing graph convolution directly on graph-structured
sensor measurements (e.g., PMUs data). To resolve the high
$\mathcal{O}\left(n^{2}\right)$ computation cost of the kernel $g_{\theta}$,
we adopt similar approximation strategies as in [34]. In particular, by
approximating
$g_{\theta}(\Lambda)\approx\sum_{k=0}^{K-1}\theta_{k}T_{k}(\widetilde{\Lambda})$
via Chebyshev polynomials $T_{k}(\cdot)$ of orders up to $(K-1)$, where
$\widetilde{\Lambda}=2\Lambda/\lambda_{\max}\\!-\\!I_{n}$ and $\lambda_{\max}$
is the largest eigenvalue of $L$, the computation cost can be brought down to
$\mathcal{O}(K\,|{E}|)$ [43]. Restricting to $K=2$ (first-order polynomials)
and assuming $\lambda_{\max}\approx 2$, $g_{\theta}$ can be further
approximated as follows [44]:

$\displaystyle
g_{\theta}*_{\mathcal{G}}X_{t}\approx\theta(\tilde{D}^{-\frac{1}{2}}\tilde{A}\tilde{D}^{-\frac{1}{2}})X_{t}\,,$
(8)

where $\theta=\theta_{0}\\!=\\!-\theta_{1}$, $\tilde{A}\\!=\\!A+I_{n}$ and
$\tilde{D}_{ii}\\!=\\!\sum_{j}\tilde{A}_{ij}$ , leading to efficient
implementation as the matrix product [44].

Gated Temporal Convolution STGCN [34] applies causal convolution [45] with a
width-$K_{t}$ kernel in time dimension and gated linear unit (GLU) [46] as a
non-recurrent approach for fast training (denoted as Gated CNN in Fig. 2). The
input is an $M\times n\times C_{i}$-dimensional tensor representing a sequence
of $M$ graph embeddings (corresponding to the past snapshots) with $C_{i}$
channels. The output with $C_{o}$ channels is an $(M-K_{t}+1)\times n\times
C_{0}$-dimensional tensor which shrinks the size by $(K_{t}-1)$ in the
temporal dimension.

Output Block and Model Learning [34]. One ST-block shrinks the input feature
tensor by $2(K_{t}-\\!1)$ in the temporal dimension. After stacking two ST-
blocks, STGCN attaches an extra gated temporal convolution layer with a fully-
connected layer as the output block. The final output
$\widehat{X}_{t+1}\in\mathbb{R}^{n}$ is a measurement prediction
(e.g.frequency) for all the sensor nodes in the power network, obtained as a
minimizer to the following $\mathcal{L}_{2}$ loss function during training:

$L(\widehat{X};\Theta)=\sum_{t}\left\|\widehat{X}\left(\\{X_{j}\\}_{j=t-M+1}^{t},\Theta\right)-X_{t+1}\right\|^{2}$
(9)

where $\Theta$ refers to the trainable parameters, and $X_{t+1}$ are the
available ground truth measurements.

Incremental Inference. As a unique feature, our STGCN model supports an
incremental prediction process for real-time power grid prediction. While
existing models are trained for a specific predictive window $H$, we developed
an adaptive STGCN to support arbitrary prediction length $H^{\prime}$. Let us
consider, without losing the generality, that the STGCN model is trained based
on a prediction window length of $H$, while the inference task for a testing
case has a prediction window length of $H^{\prime}>H$ (the case
$H^{\prime}\leq H$ is trivial). We firstly generate the next $H$ predictions
and then shuffle the time-series input with predicted values to construct the
new input. Specifically, we augment the observed measurements with the
predicted ones $\\{\widehat{X}_{i}\\}_{i=t+1}^{t+H}$ and form a new input
$\\{X_{t+H-M+1},\ldots,X_{t},\widehat{X}_{t+1},\ldots,\widehat{X}_{t+H}\\}$ to
generate the prediction snapshot $\widehat{X}_{t+H+1}$, and continue until we
predict measurements up to $\widehat{X}_{t+H\textquoteright}$.

Hyperparameter Selection Following a grid search strategy with 5-fold cross-
validation, we select the optimized length of observation window ($M$) to 200
time-points, i.e.4s window at the PMU sampling rate of 50 per second (see Sec.
4).

##  4 Simulation Setup

The performance of the proposed predictive models are evaluated in a wide
range of scenarios generated using the GridSTAGE simulation framework [36]. We
use the IEEE 68-bus system (illustrated in Fig. 3) to develop and test the
three predictive models: Robust DMD, deepDMD, and STGCN. Data corresponding to
the IEEE 68-bus system is generated over various scenarios where each scenario
corresponds to a unique configuration (e.g., load changes) of the power
network. Different criteria such as the proximity of the loads to the
generators, degree of connectivity of loads, etc, are considered for selecting
locations for load changes for creating scenarios (see Table 1).

Data Generation. GridSTAGE (Spatio-Temporal Adversarial scenario GEneration)
is a recently developed framework for simulation of adversarial scenarios and
generation of multivariate spatio-temporal data in cyber-physical systems
[36], and has been used in [26] to test real-time attack identification
algorithm. GridSTAGE is developed based on Matlab, and leverages Power System
Toolbox (PST) [47] where the evolution of power network is governed by
nonlinear differential equations that includes synchronous machine models,
excitation systems, turbine governors, and automatic generation control.

Detailed instructions on generating data scenarios with different system
topologies, attack characteristics, load characteristics, sensor
configuration, control parameters can be found in [36]. We use GridSTAGE to
generate spatio-temporal time-series data with load changes across the network
for the IEEE $68$ bus system. The PMU data is considered to report high
frequency measurements at 50 snapshots per second. In practice, factors such
as outdated calibration and inherent communication channel noises lead to
inaccuracy in the PMU measurements in the range of 80-90 dB of signal-to-noise
ratio (SNR) [48]. To capture this effect, the synthetic PMU measurements from
GridSTAGE have been corrupted by noise with an SNR value of 85 dB.

Scenarios generation. Latin hypercube sampling (LHS) is applied to generate
multiple scenarios representing different combinations of changes in the
active power load magnitudes across the network. Under every scenario, at each
load bus, the load changes are implemented as a pulse where the pulse width is
chosen to be 0.25 seconds, that is, the load magnitude is either increased or
decreased, held there for 0.25 seconds and reverted. This time instance is
treated as time $t=0$ for collecting the time-series data. All the loads were
changed at the same time and the magnitude of load change is different at each
load bus which is chosen based on the LHS. LHS is a statistical method that
allows us to generate random samples of load changes (deviations from nominal)
from multivariate distributions. We group the load change magnitudes broadly
into three classes: low, medium, and high, to generate a wide range of
scenarios representative of different loading conditions for the final
simulations. The transient simulations data for the power network are
generated and recorded for training and testing from over 330 different
scenarios summarized in Table 1. Out of those, $30$ scenarios are chosen for
training and validation of the individual models (Robust DMD, deepDMD, and
STGCN), while the remaining 300 are used for rigorous testing and performance
evaluation. In the scope of this study, the Koopman and STGCN models are
trained assuming PMUs are located at each bus. However, if PMUs are located
only at a subset of buses in the system, then the models trained on the
available PMUs enables prediction only at those corresponding buses.

Table 1: A summarized list of the different load change scenarios simulated
for generating extensive datasets for training and testing of the three
predictive models: Robust DMD, deepDMD and STGCN. Three SD values are chosen
to represent different magnitudes of load changes: low, medium and high.
Various strategic locations for load changes are selected to generate 330
scenarios out of which 30 scenarios were used for training, validation and the
remaining 300 for testing. To quantify the prediction performance, RMSE and
MAPE errors are chosen and they are computed by taking the average of these
errors over 20 testing scenarios for a given choice of SD value. Color coding:
cells with the lowest error values are shown in green; cells with values
around $50^{th}$ percentile are shown in yellow; while cells with the highest
values are shown in red.

|  |  | Robust DMD | deepDMD | STGCN
---|---|---|---|---|---
| Case
---
Index
| Load Change
---
Location
| Load Change
---
Magnitude
| RMSE [Hz]
---
($\times 10^{-3}$)
| MAPE
---
($\times 10^{-5}$)
| RMSE [Hz]
---
($\times 10^{-3}$)
| MAPE
---
($\times 10^{-5}$)
| RMSE [Hz]
---
($\times 10^{-3}$)
| MAPE
---
($\times 10^{-5}$)
| Loads 1-hop | low | 6.51 | 7.01 | 11.28 | 13.34 | 7.57 | 8.84
1 | away from | medium | 9.4 | 9.79 | 11.2 | 12.78 | 10.47 | 12.16
| generators | high | 10.73 | 11.26 | 18.25 | 21.16 | 14.56 | 16.96
| Loads 2-hops | low | 5.42 | 6.06 | 9.6 | 11.64 | 6.28 | 8.54
2 | away from | medium | 6.28 | 7.37 | 11.35 | 12.79 | 7.56 | 10.03
| generator buses | high | 7.64 | 8.79 | 13.05 | 15.17 | 12.25 | 16.73
| Loads 3-hops | low | 5.12 | 5.8 | 6.38 | 7.92 | 2.82 | 3.43
3 | away from | medium | 5.65 | 6.42 | 8.14 | 9.9 | 3.54 | 4.06
| generator buses | high | 6.04 | 7.04 | 8.73 | 10.63 | 4.45 | 5.21
| Loads with | low | 4.6 | 4.39 | 6.24 | 7.39 | 1.13 | 1.45
4 | high degree of | medium | 5.37 | 5.98 | 7.44 | 9.29 | 2.43 | 2.89
| connectivity | high | 5.51 | 6.13 | 8.28 | 10.43 | 2.51 | 2.87
| Loads with | low | 6.49 | 7.5 | 11.51 | 13.88 | 10.71 | 13.92
5 | low degree of | medium | 9.81 | 11.57 | 16.96 | 18.86 | 16.48 | 21.78
| connectivity | high | 11.48 | 13.76 | 19.09 | 20.67 | 19.38 | 25.85

Figure 3: Single-line diagram of IEEE 68-bus system with generator and load
locations. Tie lines connecting the two areas are highlighted.

Model Implementation and Configuration. All the models are implemented with
TensorFlow and the respective codes are available online [38]. The KOT-based
predictive models (Robust DMD and deepDMD) are trained and tested on the
servers with 2.6 GHz Intel core i7 processor and 16GB of RAM. For deepDMD, we
found neural networks with 4 layers can identify the observables in the latent
space (for the IEEE 68 bus system). Robust DMD and deepDMD are trained with
the regularization weights $\lambda_{1}=0.01$, $\lambda_{2}=0.005$,
respectively. All of these KOT-based models use both frequency and voltage
magnitude states as training data. Therefore, these methods yield frequency as
well as voltage magnitude predictions. On the other hand, the STGCN model is
optimized on Google CoLab service with a machine of 4 CPU cores (Intel Xeon @
2.20 GHz), 25 GB RAM, and an NVIDIA P100 GPU. We use only the frequency state
to train STGCN model for frequency prediction, following the setting in [34]
that assumes the training and testing data refer to the same measurement.

Performance Metrics. For convenience and uniformity across scenarios, we will
denote the current time as $t=0$ and consider the predicted measurements
$\\{\widehat{X}_{t}\\}_{t=1}^{H}$ at the next $H$ time-steps $t=1,2,\dots,H$ .
We use the following error metrics for quantifying and comparing the
prediction accuracy of the proposed models: root mean square error (RMSE) and
mean absolute percentage error (MAPE), as defined below for each sensor
location (bus-$i$):

$\displaystyle\text{RMSE}_{i}$
$\displaystyle:=\sqrt{\frac{1}{H}\sum_{t=1}^{H}(X_{t,i}-\widehat{X}_{t,i})^{2}}$
$\displaystyle\text{MAPE}_{i}$
$\displaystyle:=\frac{1}{H}\sum_{t=1}^{H}\left|\frac{X_{t,i}-\widehat{X}_{t,i}}{X_{t,i}}\right|\,,$

where $X_{t,i}$ and $\widehat{X}_{t,i}$ respectively denote the actual (ground
truth) and predicted values of frequency at location bus-$i$ at time $t$ . As
the objective is to predict the frequencies acrosss the IEEE 68-bus network,
the RMSEi and MAPEi are defined for each bus-$i$. For the purpose of reporting
these errors in Table 1, however, the worst case (or, maximum) values of RMSE
and MAPE across all the buses, i.e.$\max_{i}\text{RMSE}_{i}$ and
$\max_{i}\text{MAPE}_{i}$ , are considered.

##  5 Performance Evaluation

In this section, we present a detailed analysis of the performances of the
different predictive models based on their overall, temporal and spatial
accuracy, sensitivities to model parameters and time costs.

Figure 4: Actual and predicted frequencies corresponding to two randomly
chosen disturbances using Robust DMD, deepDMD and STGCN at generator buses 57,
64, 66 for a period of 10 seconds (500 time-steps as per the given PMU
sampling frequency.) The top plots correspond to disturbance - 1 (load changes
at buses 20, 23, 25, 36, 41, 42) and the bottom plots correspond to
disturbance - 2 (load changes at buses 4, 8, 12, 15, 18, 27, 44, 46, 47, 48,
51). The solid lines show the actual noisy measurements whereas the dotted
lines indicate the predictions.

Illustrative Examples. Before presenting the detailed performance analysis,
let us illustrate the working of the predictive models on a couple of load
change scenarios. Specifically, using measurements corrupted by noise as the
initial observation, we verify the accuracy of the predictive models by
investigating how well they can “recover” the actual (ground truth)
frequencies. Figure 4 illustrates the actual and predicted frequencies at 3
arbitrarily chosen generator buses under two different disturbance scenarios:
1) (top) disturbance - 1 correspond to the load changes at buses 20, 23, 25,
36, 41, 42; and 2) (bottom) disturbance - 2 correspond to the load changes at
buses 4, 8, 12, 15, 18, 27, 44, 46, 47, 48, 51. Notice that during the
transient period the model predictions demonstrate some undershoot –
e.g.Robust DMD, deepDMD for disturbance-2 from 4s and STGCN for disturbance-1
throughout the prediction window – or overshoot behavior – e.g.deepDMD
predictions under disturbance-1 and 2 around 2s.

In the STGCN prediction for disturbance-1, we notice from Fig. 4 that the
initial predicted state of the generators change abruptly. Moreover, this
behavior is not seen in the disturbance-2.

###### Remark 2

Based on our evaluation study on the different scenarios, the abrupt jump in
the predictions (immediately when the prediction is started) is only seen with
the STGCN model when the initial disturbance is relatively high (similar to
the disturbance-1 scenario in Fig. 4).

Overall, all the models successfully “recover” the post-disturbance transient
evolution of the power network frequencies, with very small RMSE ($<21\times
10^{-3}\,$Hz) and MAPE ($<23\times 10^{-5}$) values at all locations at all
time-points.

Overall Accuracy. Next, we report the overall accuracy of these predictive
models under various cases listed in Table 1. Each of the five cases refers to
a selection strategy for the load change locations, while under each case
three different choices have been considered to represent different magnitudes
of load changes: low, medium and high. For each of the 15 combinations of the
load change location and magnitude, a set of 22 scenarios have been generated.
Out of those 330 scenarios, 30 (2 from each of the 15 combinations of load
change location and magnitude) were used for training and validation, while
the rest 300 were used for extensive testing. In general, all models can
maintain good accuracy, e.g.RMSE values within $1.13\times 10^{-3}\,$Hz and
$19.38\times 10^{-3}\,$Hz and MAPE values within $1.45\times 10^{-5}$ and
$25.85\times 10^{-5}$. Among all the cases studied, the best accuracy is
achieved with the STGCN in case 4 with low load magnitude changes (with
$1.13\times 10^{-3}\,$Hz RMSE and $1.45\times 10^{-5}$ MAPE), possibly
attributed to its exploitation of spatio-temporal and topological features
embedded in the power network. On the flip side, though, the performance of
the STGCN was found to be more sensitive to the changes of spatial and network
topological features such as degree of connectivity of the load change
location (e.g.cases 4 and 5), in contrast to Robust DMD which displays a much
smaller variance in RMSE and MAPE across all the scenarios.

Furthermore, we make the following general observations for all the predictive
models:

  1. 1.

when the magnitude of load disturbance is increased, the prediction errors of
all the models are nearly doubled;

  2. 2.

higher predictive errors are observed when the load disturbance is near a
generator bus (i.e.1-hop) rather than farther away from one (e.g.2- or
3-hops); and

  3. 3.

prediction errors are small when the load change is at buses with high degree
of connectivity (4-5 neighbors) as opposed to low degree of connectivity (1-2
neighbors).

The results summarized above (and in Table 1) demonstrate the overall
performance and comparative analysis of the three predictive models in
generating transient trajectory forecasts robustly across various scenarios.
Next we further explore the robustness of the performance on temporal and
spatial scales.

Temporal Accuracy. To evaluate the model performance on the temporal scale, we
take a closer look at how the prediction error evolves as a function of time
over a prediction window of 10s. In order to do, we collect the prediction
errors $(X_{t,i}\\!-\\!\widehat{X}_{t,i})$ across all scenarios, and report
two statistical measures – the mode (i.e.the most frequently occurred value)
and a corresponding 90% confidence interval – of the errors at each time-point
within the prediction window. Figure 5 shows the modes and the corresponding
90% confidence intervals of the prediction error values at each time-point
within the prediction window. Notice that the modes of the prediction error
are nearly zero for all the models uniformly across the prediction window.
Moreover, the confidence interval reveals relatively wider distribution of
error values during the initial transients period, i.e.immediately after a
disturbance, before gradually tapering off. Interestingly, the error
distributions are larger for the deepDMD method, as compared to Robust DMD and
STGCN. Overall, all the predictive models capture the transient evolution of
the system frequencies well.

Figure 5: Prediction error of frequencies with $90\%$ confidence interval
under several testing scenarios using Robust DMD, deepDMD and STGCN at every
time-point.

Spatial Accuracy. We next evaluate the model performance on the spatial scale,
by evaluating the prediction accuracy at each bus on the network. Our goal is
to understand the impact of bus locations in the network on the transient
prediction performance. For all the test scenarios, the prediction errors
($X_{t,i}-\widehat{X}_{t,i}$) are aggregated at each bus as averaged value
over all time-points, as reported in Fig. 6. We observe that the modes of the
prediction errors with respect to each predictive model remain close to zero
at every bus. We notice some non-uniformity in how the 90% confidence interval
is spatially distributed across the network, e.g.the predictive errors display
a relatively wider range of values for load buses 18-to-30 and generator buses
53-to-63. Interestingly, though, the error distribution (i.e.the 90%
confidence intervals) appears to follow the same pattern for each of the
predictive models, indicating that the spatial error distribution perhaps
reveals an inherent characteristic of power system network under study.
Finally, while all the predictive models demonstrate robust performance on the
spatial scale across all scenarios, the error distributions of deepDMD are
generally larger than those of Robust DMD and STGCN.

Figure 6: Prediction error of frequencies with $90\%$ confidence interval
under several testing scenarios using Robust DMD, deepDMD and STGCN; mode
error aggregated over all buses.

Robustness. In summary, the results presented above demonstrate the robustness
of the performance of the predictive models from the following two
perspectives:

  1. 1.

all the predictive models limit the temporal and spatial errors in the range
of $[-0.025,0.04]$ Hz (90% confidence intervals), as per Figs. 5-6, across all
scenarios in Table 1.

  2. 2.

all the test studies for prediction performance were carried out under sensor
measurement noise of 85 dB.

Sensitivity to Model Parameters. The deepDMD model is seen to provide reliable
predictions over a range of choice of hyperparameters, as shown in Fig. 7,
with respect to the number of hidden layers, number of units per layer,
activation functions and the batch size. This demonstrates that the deepDMD is
robust to the choice of hyperparameters.

Figure 7: RMSE for several choice of hyperparameters for case-4 with SD = 13.
Similar pattern is seen over other cases as well.

STGCN-based predictive model requires a history of observations for predicting
future state evolution. Figure 8 shows the effect of observation length (1s,
2s, 3s, 4s) on the accuracy of the predictions using STGCN. Perhaps as
expected, longer observation window leads to better prediction results.
Specifically, as the length of the observation window increases from 1s to 4s,
STGCN records 83.7% improvement in RMSE. Note that a 4s observation window was
used for all STGCN prediction results reported above.

Figure 8: STGCN prediction performance (for a period of 10s) as a function of
observation window after the disturbance.

Time Cost. Given the same training environment, the training of deepDMD takes
on average 10-60 minutes depending on the choice of hyperparameters. Since,
Robust DMD admits an analytical solution as shown in (5), training time is
only around 5s. In CoLab environment, STGCN takes on average 50 minutes to
train for the same amount of training data. All models are quite feasible for
fast prediction. For a prediction window of 10s (i.e.500 data points at 50 PMU
samples per second), both robust DMD and deepDMD has an average inference cost
of 0.1s, while that of STGCN ranges from 1.2s (with 1s observation window) to
2.1s (with 4s observation window). Note the trade-off between the inference
time cost and the prediction RMSE in STGCN performance with varying
observation window length.

Summary. In general, all the predictive models demonstrate robust performance
under noisy measurements with high accuracy (RMSE within $19.38\times 10^{-3}$
Hz and MAPE within $25.85\times 10^{-5}$), on both temporal and spatial
scales, across all of the 300 load change scenarios listed in Table 1,
alongside demonstrating feasibility of practical implementation with fast
inference (0.1-2.1s for a 10s prediction window). In particular, KOT-based
methods (Robust DMD and deepDMD) model the power system evolution as a linear
dynamical system (in lifted space), do not require the knowledge of the
network topology, and deliver very fast inference (0.1s for 10s prediction
window). STGCN, on the other hand, achieves best accuracy by taking advantage
of enriched spatio-temporal features in (and is thus more sensitive to)
available historical data and network topology, and, with the proposed
enhancement on incremental inferencing, easily adapts to variable prediction
window length, much like the KOT-based methods.

##  6 Conclusion and Future Work

Using the example of power systems transient dynamics, we presented a
comparative study of different data-driven models, based on the Koopman
operator theory and the graph convolutional neural network, for robust multi-
timestep prediction. The Koopman-based models (namely, Robust DMD and deepDMD)
require only a single time-point observation of the measurements, but need
more than one type of system measurements (such as frequency and voltage
magnitude). The graph convolutional neural network model (namely, STGCN), on
the other hand, uses an additional information on the network topology and
needs a history of observations (e.g., frequency) to simultaneously exploit
the spatio-temporal and topological features of the system. Using IEEE 68-bus
network as example, synthetic training and testing datasets were created (and
made publicly accessible) by simulating a wide range of load change scenarios,
categorized by the magnitudes of load changes, the degree of connectivity of
the load change location, and the distance of the load change location to
nearest generator nodes. Numerical studies were performed to evaluate and
verify the performance of the models with respect to the robustness to noisy
measurements, accuracy across spatio-temporal scales, sensitivity to hyper-
parameters and computational time costs. This has been a preliminary
investigation of two classes of complementary methods for learning (and
predicting) nonlinear dynamical system behavior, using power systems as an
example. Future studies will involve combining the best of the Koopman-based
and graph convolution neural network models, along with distributed learning
techniques, to enable robust models for large-scale dynamical networks.

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