# A Multi-scale Time-series Dataset with Benchmark for Machine Learning in
Decarbonized Energy Grids

Xiangtian Zheng  Texas A&M University, Department of Electrical and Computer
Engineering, College Station, 77840, USA  Nan Xu  University of Southern
California, Computer Science Department, Los Angeles, 90007, USA  Loc Trinh
University of Southern California, Computer Science Department, Los Angeles,
90007, USA  Dongqi Wu  Texas A&M University, Department of Electrical and
Computer Engineering, College Station, 77840, USA  Tong Huang  Texas A&M
University, Department of Electrical and Computer Engineering, College
Station, 77840, USA  S. Sivaranjani  Texas A&M University, Department of
Electrical and Computer Engineering, College Station, 77840, USA  Yan Liu
University of Southern California, Computer Science Department, Los Angeles,
90007, USA  corresponding author(s): Le Xie<EMAIL_ADDRESS>and Yan Liu
<EMAIL_ADDRESS> Le Xie  Texas A&M University, Department of Electrical
and Computer Engineering, College Station, 77840, USA  Texas A&M Energy
Institute, College Station, 77840, USA  corresponding author(s): Le Xie
<EMAIL_ADDRESS>and Yan Liu<EMAIL_ADDRESS>
###### Abstract

The electric grid is a key enabling infrastructure for the ambitious
transition towards carbon neutrality as we grapple with climate change. With
deepening penetration of renewable resources, the reliable operation of the
electric grid becomes increasingly challenging. In this paper, we present
PSML, a first-of-its-kind open-access multi-scale time-series dataset, to aid
in the development of data-driven machine learning (ML)-based approaches
towards reliable operation of future electric grids. The dataset is
synthesized from a joint transmission and distribution electric grid to
capture the increasingly important interactions and uncertainties of the grid
dynamics, containing power, voltage and current measurements over multiple
spatio-temporal scales. Using PSML, we provide state-of-the-art ML benchmarks
on three challenging use cases of critical importance to achieve: (i) early
detection, accurate classification and localization of dynamic disturbances;
(ii) robust hierarchical forecasting of load and renewable energy; and (iii)
realistic synthetic generation of physical-law-constrained measurements. We
envision that this dataset will provide use-inspired ML research in safety-
critical systems, while simultaneously enabling ML researchers to contribute
towards decarbonization of energy sectors.

## Background & Summary

The electric grid is one of the largest sources of carbon emissions, and is
expected to play a key role in tackling climate change [1]. The electricity
sector around the world is undergoing a major transition towards carbon
neutrality with deepening penetration of renewable energy resources and
vehicle electrification. The variability of renewable energy resources along
with growing electricity demand and system vulnerability under extreme weather
events pose pressing technological challenges during this transition [2].
Conventional physics-based modeling, optimization and control tools are
becoming inadequate in these evolving systems due to the high degree of
uncertainty and variability in power generation, consumption, and
environmental factors such as climate change.

During this period of energy sector transition, there are enormous
opportunities for artificial intelligence (AI) and machine learning (ML)-based
methods [3] to improve grid operations ranging from more accurate forecasting
of renewables and load [4, 5, 6], to planning [7, 8, 9], real-time monitoring
[10, 11, 12], control [13, 14] and protection [15]. Conversely, power systems
are highly nonlinear dynamical systems with interesting physical phenomena
over various time scales; indeed, we believe that the breadth of problems
available in this domain can stimulate the development of new algorithms,
tools, and techniques in ML.

In order to foster advances that are mutually beneficial to both the ML and
power system communities, it is necessary to develop well-documented and
calibrated open-source datasets and use cases that are relevant to real-world
power engineering problems, while simultaneously being accessible and usable
to ML researchers with limited backgrounds in power/energy systems. There have
been attempts at developing ML benchmarks for various power system tasks such
as renewable [16, 17] and load forecasting [18, 19, 20], and fault and anomaly
detection [21, 22, 23, 24]. Other researchers have attempted to accelerate
algorithm development by providing online simulation platforms for specific
tasks, such as the L2RPN competition [25, 26] and the oscillation source
location contest [27].

However, the development of a cross time-and-spatial scale, open-source
dataset from the power engineering domain that can be utilized by the broader
ML community is still at a nascent stage, with several gaps in existing
sources. Firstly, most benchmarks for ML in power systems employ datasets that
are either scattered across multiple independent system operators, as in the
case of load and renewable data, or not publicly available, as in the case of
dynamic data. Secondly, the identification of relevant problems, dataset
generation, and implementation of dedicated ML-based algorithms, all require
deep knowledge of the power engineering domain and diverse power system
simulation tools. This lack of coherent comprehensive datasets along with
well-defined tasks is the key barrier for ML communities to contribute to
power system problems. Finally, there is a lack of consistent domain-relevant
assessment metrics against which different ML algorithms can be compared.

In this paper, we bridge this gap by creating a comprehensive open-source
dataset along with associated use cases and benchmarks that are relevant to
the power system research community. The dataset contains minute-level real-
world load, weather and renewable time series data over 3 years from 66 areas
across the U.S., minute-level synchrophasor measurements of 1 year in 3
scenarios, and millisecond-level synchrophasor measurements in more than 1000
disturbance cases. This is synthesized from a joint transmission (bulk) and
distribution (retail) electric grid that contains a rich and diverse set of
energy resources and dynamic events. This dataset is self-contained and
coherent across transmission and distribution-level dynamics at multiple time-
scales, making it ideally suited for developing multi-scale ML-based
algorithms. We then benchmark the performance of both traditional algorithms
from the power engineering domain and common ML-based algorithms for three
tasks of critical importance to the power engineering community, namely
forecasting, monitoring, and simulating. Specifically, we consider the
following use cases: (i) early detection, accurate classification and
localization of dynamic disturbance events; (ii) robust hierarchical
forecasting of load and renewable energy; and (iii) realistic synthetic
generation of physical-law-constrained measurement time series.

In summary, we develop one of the first comprehensive open-source datasets
with associated use cases and benchmarks from the power systems domain that
can be leveraged by ML researchers interested in advancing the state-of-the-
art in time series forecasting, classification, and generation, while
contributing towards future zero-carbon energy systems. The full dataset and
benchmark codes can be downloaded from GitHub [28] and Zenodo [29].

## Methods

In this section, we describe the method used to create the multi-scale time-
series dataset along with the benchmark models, which includes co-simulation
model development, data generation methods, and benchmark algorithms for key
tasks. A brief workflow overview is shown in Fig. 1, including (i) load and
renewable data collection and generation, (ii) power, voltage and current data
generation, and (iii) machine learning benchmarks for key tasks. The process
starts with collecting the real-world weather and load time series data. We
then generate the solar and wind generation profiles based on the physical
renewable generation models given the corresponding weather data. To obtain
the multi-scale measurement data, we conduct steady-state power flow
simulation under different system conditions of load and renewable generation
and perform transient dynamic simulation under various random disturbances by
a novel joint transmission and distribution (T+D) grid simulation platform. We
then benchmark the performance of traditional algorithms from the power
engineering domain and common ML-based algorithms for selected key tasks of
interests. Details about the methods for each step are elaborated in the rest
of this section. Details about the source code and method implementations are
explained in the Code availability section.

Remark: The voltage and current measurement time series are simulation data.
The reason that we include simulation data in the PSML dataset consists of
three aspects. First, real operational data of the power grid are typically
confidential and most are forbidden to be publicly shared, due to policies
such as critical energy/electric infrastructure information (CEII). Second,
some high-impact events that are challenging for analysis, such as forced
oscillations under resonance conditions, are rarely observed in real-world
operational data. Considering such a small amount of challenging events is
insufficient for the training, testing, and validation of ML algorithms.
Third, while real measurement data can reflect the impact of a small amount of
renewable energy in today’s power systems, it cannot capture the dynamics of
the future grid with deep renewable penetration.

### Co-simulation Model Development

We create a joint T+D grid simulation platform that consists of one PSS/E
23-bus transmission system [30] and two IEEE 13-bus distribution systems [31]
as shown in Fig. 2-a. The model of the bulk transmission grid is implemented
by PSS/E, modified from the original PSS/E 23 bus test system that has 6
thermal generators and 7 load buses. We replace one thermal generator with a
wind turbine model to represent renewable-rich scenarios. We also connect two
load buses to the distribution grid distribution systems while keeping the
rest of the load buses connected to lumped load. To better comply with the
ratings and size of the benchmark distribution systems, all load capacities
are reduced by roughly 40% of their original values. The model of the
distribution grids is the IEEE 13-bus feeder [31] implemented by OpenDSS,
which are connected to the corresponding load buses in the transmission system
model. In each distribution grid, we respectively attach solar photovoltaic
(PV) and power inverter models to load buses, which represents aggregated
residential roof-top solar generation.

As OpenDSS has no template of solar generation dynamic models, we build an
inverter-interfaced solar PV dynamic model in Python scripts that can be used
as a customized dynamic model compatible with OpenDSS. Following the
literature [32], the model consists of several typical controller components
as shown in Fig. 2-b and the dynamics are summarized as follows. The dynamics
of the power calculator follows Eq. 1 and 2, where $i_{\text{o}}$ and
$v_{\text{o}}$ are instantaneous current and voltage at the terminal of the
output filter, $P$ and $Q$ are the active and reactive output power,
$\omega_{\text{c}}$ is the upper frequency of the low-pass output filter, and
$i_{\text{od}}$, $i_{\text{oq}}$, $v_{\text{od}}$, and $v_{\text{oq}}$ are
respectively the direct and quadrature components of $i_{o}$ and $v_{o}$. The
dynamics of the frequency droop controller follows Eq. 3-5, where $\omega_{0}$
is the nominal frequency, $P^{*}$, $Q^{*}$ and $E^{*}$ are the dispatched set
points of the active power, reactive power and potential, $v_{\text{d}}^{*}$
and $v_{\text{q}}^{*}$ is the set point of the subsequent voltage controller,
and $M_{\text{f}}$, $D_{\text{f}}$, $M_{\text{v}}$ and $D_{\text{v}}$ are the
predefined parameters of the controller. Specifically, Eq. 3 and 4 describe
the dynamics mimicing traditional synchronous generators while Eq. 5 describe
the way how the droop controller adapts the set point of the subsequent
voltage controller. The dynamics of the voltage and current controller is
elaborated by Eq. 6-11, where $\xi_{\text{d}}$ and $\xi_{\text{q}}$ are the
state variables of the current controller, $i_{\text{ld}}^{*}$ and
$i_{\text{lq}}^{*}$ are the set points of the current controller,
$v_{\text{id}}^{*}$ and $v_{\text{iq}}^{*}$ are the set points of the
inverter, $K_{\text{iv}}$, $F$ and $K_{\text{pv}}$ are the predefined
parameters of the voltage controller, $K_{\text{ic}}$ and $K_{\text{pc}}$ are
the predefined parameters of the current controller, and $C_{\text{f}}$ and
$L_{\text{f}}$ are the capacitance and inductance of the output filter. The
dynamics of the output filter can be described by Eq. 12-15, where
$r_{\text{f}}$, $C_{\text{f}}$ and $L_{\text{f}}$ are the resistance,
capacitance and inductance of the output filter.

$\displaystyle\dot{P}=-\omega_{\text{c}}P+\omega_{\text{c}}(v_{\text{od}}i_{\text{od}}+v_{\text{oq}}i_{\text{oq}})$
(1)
$\displaystyle\dot{Q}=-\omega_{\text{c}}P+\omega_{\text{c}}(v_{\text{od}}i_{\text{od}}+v_{\text{oq}}i_{\text{oq}})$
(2) $\displaystyle\dot{\delta}=\omega-\omega_{0}$ (3) $\displaystyle
M_{\text{f}}\dot{\omega}=-D_{\text{f}}(\omega-\omega_{0})+P^{*}-P$ (4)
$\displaystyle
M_{\text{v}}\dot{v}_{\text{od}}^{*}=D_{\text{v}}(Q^{*}-Q)-(v_{\text{od}}^{*}-E^{*}),\quad
v_{\text{oq}}^{*}=0$ (5)
$\displaystyle\dot{\xi}_{\text{d}}=v_{\text{od}}^{*}-v_{\text{od}},\quad\dot{\xi}_{\text{q}}=v_{\text{oq}}^{*}-v_{\text{oq}}$
(6)
$\displaystyle\dot{\psi}_{\text{d}}=i_{\text{ld}}^{*}-i_{\text{ld}},\quad\dot{\psi}_{\text{q}}=i_{\text{lq}}^{*}-i_{\text{lq}}$
(7) $\displaystyle
i_{\text{ld}}^{*}=K_{\text{iv}}\xi_{\text{d}}+Fi_{\text{od}}+K_{\text{pv}}(v_{\text{od}}^{*}-v_{\text{od}})-\omega_{0}C_{\text{f}}fv_{\text{oq}}$
(8) $\displaystyle
i_{\text{lq}}^{*}=K_{\text{iv}}\xi_{\text{q}}+Fi_{\text{oq}}+K_{\text{pv}}(v_{\text{oq}}^{*}-v_{\text{oq}})-\omega_{0}C_{\text{f}}fv_{\text{od}}$
(9) $\displaystyle
v_{\text{id}}^{*}=K_{\text{ic}}\psi_{\text{d}}+K_{\text{pc}}(i_{\text{ld}}^{*}-i_{\text{ld}})-\omega_{0}L_{\text{f}}i_{\text{lq}}$
(10) $\displaystyle
v_{\text{iq}}^{*}=K_{\text{ic}}\psi_{\text{q}}+K_{\text{pc}}(i_{\text{lq}}^{*}-i_{\text{lq}})-\omega_{0}L_{\text{f}}i_{\text{ld}}$
(11) $\displaystyle
L_{\text{f}}\dot{i}_{\text{ld}}=-r_{\text{f}}i_{\text{ld}}+\omega
L_{\text{f}}i_{\text{lq}}+v_{\text{id}}-v_{\text{od}}$ (12) $\displaystyle
L_{\text{f}}\dot{i}_{\text{lq}}=-r_{\text{f}}i_{\text{lq}}+\omega
L_{\text{f}}i_{\text{ld}}+v_{\text{iq}}-v_{\text{oq}}$ (13) $\displaystyle
C_{\text{f}}\dot{v}_{\text{od}}=\omega
C_{\text{f}}v_{\text{oq}}+i_{\text{ld}}-i_{\text{od}}$ (14) $\displaystyle
C_{\text{f}}\dot{v}_{\text{oq}}=\omega
C_{\text{f}}v_{\text{od}}+i_{\text{lq}}-i_{\text{oq}}$ (15)

To implement the T+D co-simulation, we further create a Python control process
to facilitate workflow coordination and data exchange between PSS/E, OpenDSS,
PV inverter models and input data files. Specifically, the python control
process is able to (i) start and pause the simulation in the PSS/E and OpenDSS
every step, (ii) read and store the updated variables of the transmission and
distribution grid models after each step, and (iii) overwrite the values of
certain variables at the beginning of each step. In such way, we can exchange
the voltage and power data between the transmission and distribution systems
step by step that enables the T+D co-simulation. Please refer to the Code
availability section for the details about the source code and method
implementations.

### Data Generation

This subsection introduces the methods how we collect the real-world data and
generate the measurement data via co-simulation. The source code of the data
collection and simulation implementation can be found in the Github repository
as elaborated in the Code availability section.

#### Load and Renewable Time Series Data Generation

We collect real-world load time series and synthesize active power time series
of renewable generation along with real-world weather data. For collecting the
load power time series, we aggregate hourly real-world load data of
representative 66 load zones ranging from 2018 to 2020, obtained from major
power markets in the U.S. that regulate about 70% of U.S. electricity sales
[33]. To incorporate renewable power and weather time series in PSML, we
collect real weather data of 5-minute resolution of each load zone from 2018
to 2020 from National Solar Radiation Database (NSRDB) [34]. The selected
weather station of each load zone locates around major cities within range. We
calculate the renewable generation power based on the collected weather data
of each load zone. The active power output of residential solar photovoltaic
(PV) is estimated by the System Advisor Model (SAM) [35], based on the solar
radiation-related data. The active power output of wind turbines is estimated
by the location-dependent wind turbine power curves [36] based on the
collected wind-related data. Finally, we aggregate time-stamped load,
renewable and weather data of 66 load zones by interpolation in the PSML
dataset. For the convenience of subsequent simulation, the load data are
further normalized by their 3-year average value, while the renewable data are
normalized by the nominal power values of the physical renewable models.

#### Minute-level Voltage, Current and Power Time Series

The steady-state simulation produces minute-level data of the power
transmission and distribution system using the load and renewable profiles.
These profiles specifies the net real and reactive power consumption across
the system except the buses where dispatch-able thermal generators are
located. To keep the system frequency stable, the total real power generation
from the thermal generators must be exactly the same as the total net real
power consumption. Hence, the real power set-point of all generators are
determined by their capacity limits and load level. A power-flow solution is
performed to obtain the voltage of nodes and current of branches by solving a
set of circuit theory derived algebraic equations (power flow equations) of
the network model. The solution of the algebraic equations is determined by
loads, renewable generation and thermal generation.

In our data generation procedure, we implement an iterative approach to create
an uniform power-flow solution across the transmission system and multiple
distribution systems. Each distribution system is represented as an equivalent
load bus whose load value equals the sum of all nodes in the distribution
system. We start with flat voltage level of 1.0 per-unit across the entire
transmission system, change the loads and generator set-points according to
the time-series data and solve standard power flow to obtain the voltage at
the buses that has distribution system models. The voltage solutions are
passed to each distribution system model as the voltage of the equivalent
infinite source representing the transmission system. The distribution systems
are then solved using the ’actual’ voltage to obtain updated values of the
real and reactive power consumption, as the load power consumption are related
to the system voltage. The total powers of distribution systems are then
passed back to the transmission system model and are used to update the power
flow solution. This process is repeated until the voltage difference between
two iterations converges to be less than a very small tolerance factor. The
detailed simulation procedure is listed in Algorithm 1 and the source code of
our implementation using PSS/E and OpenDSS can be found in our GitHub
repository.

Read transmission system case file in PSS/E with default parameters

Select hour from wind profile and set wind generators outputs

Select hour from load profile and scale load capacities

Initialize voltages all non-generator buses $V^{0}$ to 1 per-unit (flat start)

Initialize iteration counter $n=1$

while $||V^{n}-V^{n-1}||_{2}>\epsilon$ do

for $d$ in list of distribution circuits do

Read distribution system case files in OpenDSS with default parameters

Select hour from load and PV profiles and set net bus loads

Set source voltage to node voltage $V^{n-1}_{d}$ in transmission where the
distribution circuit is located

Solve power flow and calculate total P and Q from all 3 phases

In transmission, adjust $P_{d}$ and $Q_{d}$ for the bus $d$ that has
distribution circuit

end for

Solve power flow using the updated P/Q values from distribution systems

Obtain all node voltages $V^{n}$ from results

end while

Simulation has converged, extract data to output file

Algorithm 1 Iterative Steady-State Power Flow Method for T+D Joint Simulation

#### Millisecond-level Voltage, Current and Power Time Series

In power systems there are many disturbances with different extent of types
and severity, common ones include faults (short circuit between conductors or
with the ground), unexpected equipment tripping and forced oscillation. These
disturbances must be detected and handled as quickly as possible, or they may
cause cascading failures and the damage can be out of control. Specifically,
disturbances must be cleared within the Critical Clearing Time (CCT), usually
at the order of 100 ms, after which the larger system will become unstable.
Before the occurrence of a disturbance, the power system, at a larger time-
scale, is assumed to be operating around a stable equilibrium point that is
determined by the steady-state power flow solution related to generation and
demand profiles. The initial conditions of dynamic components, including
generators, wind turbines, solar PV systems and their many control devices,
are determined by continuing random variations of the generation and demand
capacity. During disturbances, the dynamics of the grid can be described by
differential algebraic equations (DAEs). A transient simulation is essentially
the process of solving those DAEs and obtaining timeseries of voltages,
currents frequencies and other state variables. Fig. 1 encapsulates the
simulation mechanism, and Fig. 3 illustrates several typical events of
interested in co-simulation.

The approach to perform T+D co-simulation for transient events use a similar
iterative algorithm to ensure that the data obtained from different systems
and simulators are closely correlated with each other. The iteration process
of exchanging voltage and power data between the bulk transmission system
model and distribution system models is repeated for every time step in
transient simulation. Beside the power system circuit and component models in
PSS/E and OpenDSS, interver-interfaced solar PV dynamic models are implemented
in Python and run separately. The simulation model developed for generating
synthetic time series in PSML possesses the following features to obtain high-
fidelity data: (i) The dynamic model of both the transmission and distribution
systems are benchmark systems which are extensively used in power system
research; (ii) we model the impact of deep penetration of renewables by
incorporating detailed models of renewable generation and representative load
and weather patterns in the U.S.; and (iii) the interaction between the
transmission and the distribution systems in the fast time scale is modeled in
the transient simulation, which is not captured in existing publicly available
synthetic datasets, e.g., the oscillation dataset [27].

Select hour from wind, solar and load profiles

Solve steady-state power flow (Algorithm 1) as starting point

Create a disturbance with random on-time $T_{on}$, clear time $T_{c}$ and
parameters

Create model objects for all PV generators and initialize with solar profile

Initialize transient simulation process in PSS/E using steady-state solution

Initialize time counter $t=0$

while $t<$ max step $T$ do

Add/clear disturbance if $t==T_{on}$ or $T_{c}$

for $d$ in list of distribution circuits do

Read distribution system case files in OpenDSS with default parameters

Select hour from load and PV profiles and set net bus loads

Set source voltage to node voltage $V^{t}_{d}$ in transmission where the
distribution circuit is located

Solve PV dynamic models and get net P/Q from PV generators

Solve power flow and calculate total P and Q from all 3 phases

In transmission, adjust $P_{d}^{t}$ and $Q_{d}^{t}$ for the bus $d$ that has
distribution circuit

end for

Advance OpenDSS transmission system transient simulation by one step

end while

Collect timeseries measurements and store to output file

Algorithm 2 Transient Event Simulation Method for T+D Joint Simulation

### Machine Learning Benchmarks

In this subsection, we select three types of grid-domain use cases for ML
approaches. The use cases are (i) event detection, classification and
localization, (ii) forecasting of renewable generation and load; and (iii)
synthetic synchrophasor data generation. The reason that these use cases are
selected is that they essentially can be formulated as classical ML problems
which have been extensively studied during the past half-century. As a result,
methods developed in the ML communities have great potential to provide
solutions to these power grid use cases. Compared with conventional approaches
heavily relying on grid physical models and network topology (line
connectivity), such as the line outage detection algorithm [37], and the
energy approach to forced oscillation localization [38], one attractive
advantage of the ML-based approaches is that they do not require availability
of information on grid physical model and topology. In addition, the sparsity,
size, and scale of these time-series measurements provide a unique playground
for the advancement of new ML methods. In what follows, we introduce the goal
of each use case and present the benchmark of the performance of popular
learning methods in terms of solving these three types of power system
problems. We refer readers to our Github repository for more details on data
structure and instructions on use cases.

#### Event Detection, Classification and Localization

Renewable energy resources, such as wind/solar farms, are not as dispatchable
as conventional fossil fuel generators due to their stochastic nature. As a
result, those renewables introduce uncertain disturbances which may compromise
the safe operation of the grid. Therefore, it is imperative for Independent
System Operators (ISOs) to accurately recognize disturbances and perform
corrective measures timely so as to ensure the safety of the grid. The health
of the power grid is monitored by sensors such as synchrophasors/phasor
measurement units (PMUs). These sensors stream time-stamped measurements to
ISOs. Based on these streaming measurements, ISOs may have the following three
questions: (i) When is an event happening; (ii) What type of event is
happening? and (iii) Where is the source that caused the event? Answering
these questions is critical to maintaining reliable operation of a power grid
integrated with rich renewable energy resources.

The streaming measurements can be denoted by $X\in\mathbb{R}^{T\times M}$,
where $T$ is the number of time stamps by now and $M$ is the number of
measurements. _Event detection_ aims to answer the first ISO question by
recognizing the disturbance occurrence once it takes place, hence a model
$\mathcal{H}$ is learned to be able to identify the disturbance occurrence
given sequence $X$, i.e., $\mathcal{H}:X\rightarrow\\{0,1\\}$. Suppose the
event takes place at time $\tau$: when $T<\tau$, the model is expected to be
quiet without any alarms ($0$ predicted); when $T\geq\tau$, the model should
alarm as soon as possible ($1$ predicted). _Event classification_ answers the
second ISO question based on streaming sensor measurements. Given the
measurement $X$, the objective of this task is to learn a model $\mathcal{F}$
that can classify the underlying event type $y$, i.e.,
$\mathcal{F}:X\rightarrow y$. In PSML, $y$ is a subset of disturbances
$\mathcal{C}$ where $\mathcal{C}:=$ $\\{$branch fault, branch tripping, bus
fault, bus tripping, generator tripping, forced oscillation$\\}$. _Event
localization_ focuses on locating events (for branch fault, branch tripping,
bus fault, bus tripping, generator tripping) or the root cause of events (for
forced oscillations) by observing measurements. We are aimed at learning a
model $\mathcal{G}$ that can map measurement $X$ to the bus(es) $z$ nearest to
the events detected or the root cause of the events, i.e.,
$\mathcal{G}:X\rightarrow z$, where $z$ is a subset of buses $\mathcal{Z}$ in
the entire system. It is worth noting that compared with the size of the whole
grid, the sensor coverage might be sparse in practice, rendering the tasks of
event detection, classification and localization more challenging.

We select the following representative benchmark algorithms for this task. We
implement _InceptionTime_ , _MC-DCNN_ and _ResNet_ (with the help of sktime-dl
[39] package), and _MLSTM-FCN_[40] in Tensorflow. We implemented all other
methods by ourselves in Pytorch except _TapNet_ [41] and _MiniRocket_ [42].
For deep learning approaches, we use grid search to select general
hyperparameters such as layer size, number of layers, normalization approach,
etc.

  * •

Power domain: Event localization is implemented by calculating the event
signature of each PMU, where the event signature is estimated by several
statistical parameters including Shannon entropy, standard deviation, range,
mean difference and crest factor as introduced in the literature [43]. The PMU
with the most dominant event signature indicates the location of the event.

  * •

Traditional machine learning methods: with generally good performance across
different time series datasets, 1-nearest neighbor (1-NN) related approaches
have been widely employed as standard benchmarks in the their corresponding
benchmarks, e.g., UCR [44] and UEA [45]. We consider three measures for sample
distance computation: Euclidean and dynamic time warping with each feature
dimension treated independently (_DTW-i_) or dependently (_DTW-d_). We also
adopt _MiniRocket_ , where kernel transformations are firstly applied to time
series followed by simple linear classifiers for time series classification
[42].

  * •

Convolutional Neural Networks: benefiting from the deep convolutional neural
networks (CNNs) and residual connections, vanilla and different variants of
CNNs are consider to perform classification tasks: (i) InceptionTime [46]: an
ensemble of deep CNNs inspired by Inception-v4 [47] architecture; (ii) MLSTM-
FCN [40]: concatenation of LSTM of CNN for feature representation learning
with an additional squeeze-and-excitation block for further performance
improvement; (iii) ResNet [48]: adaptation of residual network from images
[49] to multivariate time series; (iv) MC-DCNN [50]: multi-channels deep CNNs
where different temporal patterns are transformed firstly and then learned
through separate convolutional layers; and (v) TapNet [41]: an attentional
prototype network was incorporated into the convolutional layers to learn
latent features for time series classification.

  * •

Other deep learning methods: we also analyze the performance of general deep
neural networks (i.e., fully-connected neural network) and specific time
series deep models (i.e., RNN and its two variants [51, 52], transformer [53])
is evaluated.

For the training process, we randomly select 439 time-series from the
millisecond transient PMU data as training samples and the remaining 110 time-
series for testing. Each sequence has a metadata associated with the event
type similar to the classification use case, i.e. branch_fault, branch_trip,
bus_fault, bus_trip, gen_trip. Each time-series has a sequence length of 960
observations, representing 4 seconds in the system recorded at 240 Hz. There
are 91 dimensions for each time-series, including voltage, current and power
measurements across the transmission system. In the test process, with class
imbalance in consideration, we adopt balanced accuracy to obtain performance
of different classification methods for event classification and localization:
balanced_acc = (sensitivity \+ specificity) / 2. Since both too early (false
positive where no event happens yet) or too late (false negative where
damage/cost accumulates along time without alarm or action) event detection
are undesired, we leverage the macro-averaged mean absolute error, which
considers the divergence between actual and predicted labels for ordinal
regression on imbalanced datasets [54].

#### Load and Renewable Energy Forecasting

The ultimate goal of the power grid is to balance generation and load. Today,
this is mostly achieved by load forecasting and generation scheduling based on
the forecast before real-time operation. In real-time operation, the
relatively small mismatch between scheduled generation and actual load is
compensated by dispatchable generation units that can respond quickly, i.e.,
the spinning reserve. The spinning reserve may rely on fossil fuel and incur
high operational costs. However, with increasing renewable integration, this
operational paradigm is not feasible without accurate renewable and load
forecasting. Renewable generations, e.g., wind/solar farms, have their maximum
output apparent power determined by weather, which cannot be actively
increased (but can be curtailed), due to their stochastic and volatile nature.
A poor forecast of renewable generation therefore leads to a large amount of
expensive, fossil fuel-based spinning reserve being committed. Compounding the
challenge, loads will become less predictable in the future grid, due
stochastic loads like electric vehicles, and small-probability yet high-impact
events, e.g., COVID-19 pandemic (Fig. 4) and the Texas winter storm in 2021.
Therefore, accurate forecasting of renewable generation and load is critical
to support reliable operation of the future grid.

We focus on the following two important subtasks: (i) Point Forecast (PF):
given a time sequence in the past $k$ time steps, the current time $t$ and the
forecasting horizon $\tau$, we have the past targets $y_{t-k:t}$, the past
observations $x_{t-k:t}$, as well as the past, current and future known
variables $u_{t-k:t+\tau}$ (e.g., date, holiday). We aim to predict the
targets $\tau$ time steps ahead:
$\hat{y}_{t+\tau}=r(y_{t-k:t},x_{t-k:t},u_{t-k:t+\tau})$. (ii) Prediction
Interval (PI): Uncertainty quantification can provide more reference
information for decision-making and has received growing research and
industrial interests these years [55, 56]. For high-quality uncertainty
quantification, we would also like to obtain the prediction interval,
$[\hat{y}^{L},\hat{y}^{H}]$, to cover the ground truth $y$ with at least the
expected tolerance probability, i.e., $p=0.95$ in our load and renewable
energy forecasting task.

We select the following representative benchmark algorithms for this task. We
implement the time-series models with statsmodels [57], the traditional
machine learning models with sklearn [58], _N-BEATS_ [59], _WaveNet_ [60],
_TCN_ [61], _LSTNet_ [62], _DeepAR_ [63], _Informer_ [64] and _Neural ODE_
[65] with codes published officially (or unofficially), and all other deep
learning approaches by ourselves in Pytorch. For deep learning approaches, we
use grid search to select general hyperparameters such as layer size, number
of layers, normalization approach, etc.

  * •

Time-series models: besides the _naive_ method takes the current value
directly as the prediction, we also consider autoregressive integrated moving
average (ARIMA) and exponential smoothing (ETS).

  * •

Traditional machine learning methods: we select the top four widely used
machine learning methods in load and renewable energy forecasting literature
[66]: support vector regression (SVR), random forest (RF), gradient boosted
decision trees (GBDT), and linear regression (LR).

  * •

Multilayer perceptron: besides fully-connected neural newotks (FNN) and
extreme learning machines (ELM), we also list performance of N-BEATS, a deep
stacked neural architecture based on backward and forward residual links,
which outperformed the winner of M4 competition [67].

  * •

Convolutional Neural Networks: we study performance of vanilla CNN, WaveNet
[60] composed of dilated causal convolutions for audio generation, Temporal
convolutional neural networks (TCN [68]) with additional residual blocks.

  * •

Recurrent Neural Networks: both basic recurrent neural networks (vanilla RNN,
LSTM [51] and GRU [52]) and advanced variants are studied: LSTNet [62] with
patterns extracted from convolutional layers and fed to recurrent neural
networks, DeepAR [63] with output from recurrent neural networks as likelihood
parameters for probabilistic forecasting.

  * •

Transformer-based: we list performance from vanilla transformer [53] and its
variant, informer [69], with self-attention distilling and generative style
decoder for long sequence forecasting.

  * •

Neural ODE: motivated by the Euler discretization of continuous
transformations in residual networks and recurrent neural network decoders,
Neural ODEs parameterize the derivative of hidden state using a neural network
and compute the network output with a differential equation solver [65]

Given load and renewable energy data recorded from 66 locations in minute-
level, we split the sequence from each location according to years firstly and
have three cases: (i) use data from Jan to Nov in 2018 for training and Dec in
2018 for testing; (ii) use data from Jan, 2018 to Nov, 2019 for training and
Dec in 2019 for testing; (iii) use data from Jan, 2019 to Nov, 2020 for
training and Dec in 2020 for testing. Noted that we adopt the rolling strategy
during testing, so that testing data before current time step is observable
for model forecasting. In the test process, we adopt three commonly leveraged
metrics in load and renewable energy forecasting literature [66]: root mean
square error (RMSE), mean absolute error (MAE) and mean absolute percentage
error (MAPE). Following practice in the M4 competition [56], the performance
of generated point intervals is evaluated using the mean interval score (MIS)
[70]:

$\text{MIS}=\frac{1}{N}\sum_{i=1}^{N}(\hat{y}^{u}_{i}-\hat{y}^{l}_{i})+\frac{2}{a}(\hat{y}^{l}_{i}-y_{i})\mathcal{I}(y_{i}<\hat{y}^{l}_{i})+\frac{2}{a}(y_{i}-\hat{y}^{u}_{i})\mathcal{I}(y_{i}>\hat{y}^{u}_{i})$
(16)

where $N$ is the number of instances for each prediction horizon,
$\mathcal{I}$ is the indicator function with value $1$ when the inequality
holds and $0$ otherwise, and $a=0.05$ for $95\%$ prediction intervals
generation.

#### Synthetic Time-series Generation

A major hurdle in applying deep learning models to power system problems is
usually the lack of sufficient and high-quality datasets for training, as it
is well-known that more eventful data usually lead to better classification
performance [71, 72, 73]. The accessibility of real-world power grid PMU
measurement data is limited due to the regulation CEII [74] for national
security and sensitivity concerns. While researchers recently have contributed
to the creation of large-scale synthetic simulation models [75] for analysis
[76, 77], there are always gaps between simulation models and real-world
systems and the unique values of real-world PMU time series data cannot be
exploited for research purposes. It is therefore critical to investigating
methods for synthesizing power system datasets that follow the same properties
of the real system data while complying with physical laws for the network and
its underlying dynamic behaviors.

This task involves multi-channel time series generation, for which the
training data are disturbance-induced dynamic voltage, current and power
measurements across power grids. The expected outputs are dynamic voltage,
current and power measurements that preserve certain dynamic patterns and
physical laws. The key challenges that distinguish this task from normal image
generation are: (i) multi-channel time series are governed by unknown
algebraic and differential equations derived from physical laws, and (ii)
dynamic time series incorporate discrete disturbance events. Our evaluations
are carried out over simulated voltage, current and power data from PSML.

We select the following representative benchmark algorithms for this task. All
models are trained with a fixed hidden dimensionality of 256, a fixed number
of two or three layers for recurrent networks, and a tuned dropout ratio $\in$
{0.0, 0.5}.

  * •

NaiveWGAN: we show the performance of a naive GAN architecture (MLP generator
and discriminator) with the Wasserstein loss [78].

  * •

RCGAN: a conditional recurrent GAN architecture [79] is tested that leverage
recurrent generator and discriminator and conditioned on auxiliary
information.

  * •

COT-GAN: we test a recurrent GAN trained with a Causal Optimal Transport (COT)
loss suitable for learning time dependent data distributions [80].

  * •

TimeGAN: we list the performance of a recurrent GAN architecture that combines
unsupervised GAN learning with a supervised teacher-forcing component in the
loss function [81].

  * •

DoppelGANger: we test a state-of-the-art GAN architecture [82] that leverages
two generators and discriminators to first generate auxiliary metadata before
generating the time-series.

For training and testing the models, we set the first 400 millisecond
transient time-series as training samples and the next 150 time-series for
testing, where each time-series has a sequence length of 960 observations,
representing 4 seconds in the system recorded at 240 Hz, and has 91
dimensions. As the task is to synthesize multiple realistic-looking PMU
streams that respect the physical constraints from real PMU streams, we define
the following metrics to assess the quality of generated data: (i) Fidelity:
samples should be indistinguishable from the real data. We train a post-hoc
time-series classification model (by optimizing a 2-layer LSTM) to distinguish
between sequences from the original and generated datasets and report the
error, and (ii) Diversity: samples should be distributed to cover the real
data. We apply PCA analyses on both the original and synthetic datasets
(flattening the temporal dimension) and visualize how closely the
distributions are in 2D space.

## Data Records

The dataset is hosted on Zenodo [29]. The folders in the dataset are organized
by data type, including minute-level load and renewable data, minute-level PMU
measurements data, and millisecond-level PMU measurements data.

In the folder of the minute-level load and renewable data, we store the data
in the CSV files by zonal location.

  * •

File ISO_zone_#.csv: for example, _CAISO_zone_1.csv_ is a CSV file containing
munute-level laod, renewable and weather data from 2018 to 2020 in the zone 1
of CAISO.

    * –

Field _time_ : time of minute resolution.

    * –

Field _load_power_ : time of minute resolution.

    * –

Field _wind_power_ : time of minute resolution.

    * –

Field _solar_power_ : time of minute resolution.

    * –

Field _DHI_ : diffuse horizontal irradiance.

    * –

Field _DNI_ : direct normal irradiance.

    * –

Field _GHI_ : global horizontal irradiance.

    * –

Field _Dew Point_ : dew point in degree Celsius.

    * –

Field _Solar Zeinth Angle_ : angle in degree between the sun’s rays and the
vertical direction.

    * –

Field _Wind Speed_ : wind speed in meter per second.

    * –

Field _Relative Humidity_ : relative humidity in percentage.

    * –

Field _Temperature_ : temperature in degree Celsius.

In the folder of the minute-level PMU measurements data, we store the data in
the subfolders by scenario setting. Each subfolders contains a TXT file and a
CSV file that respectively store the metadata and simulation measurements
data.

  * •

Folder case #: for example, _case 0_ is a folder corresponding to the scenario
setting #0.

    * –

File pf_input_#.txt: the TXT file contains the metadata of the selected load
and renewable data for the simulation.

    * –

File pf_result_#.csv: the CSV file contains the voltage at buses and power on
branches in the transmission system via T+D simulation.

      * *

Field _time_ : time of minute resolution.

      * *

Field _Vm_#_ : voltage magnitude in per unit at the bus #.

      * *

Field _Va_#_ : voltage angle in rad at the bus #.

      * *

Field _P_#_#_#_ : for example, _P_3_4_1_ means the active power in the #1
branch from the bus 3 to 4.

      * *

Field _Q_#_#_#_ : for example, _Q_5_20_1_ means the reactive power in the #1
branch from the bus 5 to 20.

In the folder of the millisecond-level PMU measurements data, we seperate the
data into two folders by oscillation type, namely forced oscillation and
natural oscillation. In each folder, the data are organized in the following
way.

  * •

Folder row_#: for example, _row_0_ is a folder corresponding to the
disturbance scenario #0.

    * –

File dist.csv: the CSV file contains the three-phased voltage at nodes in the
distribution system.

      * *

Field _Time(s)_ : time of millisecond resolution.

      * *

Field _#.#.#_ : for example, _3005.633.1_ means the per-unit voltage magnitude
of the phase A at the bus 633 of the distribution grid, the one connecting to
the bus 3005 in the transmission system..

    * –

File trans.csv: the CSV file contains the voltage at buses and power on
branches in the transmission system.

      * *

Field _Time(s)_ : time of millisecond resolution.

      * *

Field _VOLT #_ : voltage magnitude in per unit at the bus #.

      * *

Field _POWR # TO # CKT #_ : for example, _POWR 151 TO 152 CKT ’1 ’_ means the
active power transferring in the #1 branch from the bus 151 to 152.

      * *

Field _VARS # TO # CKT #_ : for example, _VARS 151 TO 152 CKT ’1 ’_ means the
reactive power transferring in the #1 branch from the bus 151 to 152.

Besides, the algorithm codes are available on Github [28]. In the folder
_Code_ , we organize the codes for algorithm reproduction as follows. Please
refer to the Usage Notes for the details of installation, package usage and
navigation.

  * •

File _dataloader.py_ : Pytorch data loaders with both data processing and
splitting included.

  * •

File _evaluator.py_ : evaluators to support fair comparison among different
approaches.

  * •

Folder _BenchmarkModel_ : .

    * –

Folder _EventClassification_ : the folder contains all files for the event
classification task.

      * *

Folder _configs_ : the folder contains all trained model configurations (YAML
files).

        * ·

File _#.yaml_ : for example, _RNN.yaml_ stores the model parameters of the RNN
model.

      * *

Folder _models_ : the folder contains all model codes (PY files).

        * ·

File _#.py_ : for example, _RNN.py_ implements the RNN model.

      * *

File _evaluating.py_ : codes to evaluate the trained benchmark models for
event classification.

      * *

File _processing.py_ : codes to process the millisecond-level measurements
data.

      * *

File _requirements.txt_ : the required Python packages for model use.

    * –

Folder _LoadForecasting_ : the folder contains all files for the forecasting
task.

      * *

Folder _configs_ : the folder contains all trained model configurations (YAML
files).

        * ·

File _#.yaml_ : for example, _FNN.yaml_ stores the model parameters of the FNN
model.

      * *

Folder _models_ : the folder contains all model codes (PY files).

        * ·

File _#.py_ : for example, _FNN.py_ implements the FNN model.

      * *

File _evaluating.py_ : the codes to evaluate the trained banchmark models for
forecasting.

      * *

File _processing.py_ : the codes to process the minute-level load and
renewable data.

      * *

File _requirements.txt_ : the required Python packages for model use.

    * –

Folder _SyntheticDataGeneration_ : the folder contains all files for the
synthetic data generation task.

      * *

Folder _##_ : for example, _RGAN_ contains all files for the RGAN model.

        * ·

File _requirements.txt_ : the required Python packages for model use.

        * ·

Other files: other files contained in the folder depends on the model, of
which the usage are instructed in the README file.

      * *

File _evaluating.py_ : the codes to evaluate the trained banchmark models for
forecasting.

      * *

File _processing.py_ : the codes to process the minute-level load and
renewable data.

## Technical Validation

### Dataset Explorations

#### Load and Renewable Time Series

The zonal load and weather data are collected from multiple real-world
sources, of which the fidelity is guaranteed by manual data quality control.
The associated physical models for renewable estimation are commonly used in
the power system. We illustrate daily load and renewable power profiles in
Fig. 4, which shows seasonal disparity and strong variation of renewable
energy. We can also observe a significant load reduction during the COVID-19
pandemic, which is valuable for investigating the impacts of unprecedented
events on the energy sector.

#### Minute-level Voltage, Current and Power Time Series.

To demonstrate the data fidelity, we illustrate the power spectral density
analysis that quantifies the periodicity on one selected 1-year-long minute-
level synthetic PMU measurement data. Specifically, we calculate the power
spectral density of the voltage angle at each transmission system bus. Fig. 5
shows the average power spectral density, where the highest power density
appear at the period of 12 hours, 24 hours and 1 week. It is consistent with
our prior observation in other real-world PMU dataset that the major periodic
patterns are derived from the loads that mainly possess periods of 12 hours,
24 hours and 1 week.

#### Millisecond-level Voltage, Current and Power Time Series.

Fig. 6 visualizes some typical events obtained from the co-simulation,
compared with the profiles by traditional simulation. It shows that the co-
simulation method reveals more details. We further perform modal analysis on
the generated power time series to demonstrate the fidelity. We observe in
Fig. 7 that the generated power time series data possess only few dominant
modes of high energy, which match our prior knowledge.

### Machine Learning Benchmark Evaluation

#### Event Detection, Classification and Localization

As listed in Table 1, we evaluate methods from four main categories: (i) PMU
score from the power community, (ii) standard classification benchmarks based
on 1-nearest neighbor (1-NN), (iii) convolutional neural networks and (iv)
other popular deep learning methods. For both event classification and
detection, we observe in general that approaches composed of convolutional
neural networks perform much better than 1-NN based standard benchmarks and
other deep learning approaches. However, to localize the event across the
grid, the majority of deep learning approaches fail to achieve competitive
performance as 1-NN approaches: only _MC-DCNN_ reaches above $0.40$ balanced
accuracy besides 1-NN Euclidean, 1-NN DTW-i and 1-NN DTW-d. In analyzing PMU
measurements from the transmission system, we attribute the success of
convolutional neural networks to their explicit spatial correlation modeling,
where the voltage and current evolve along time according to both the external
oscillation events and the inherent network connectivity.

Based on the above observations, we expect more accurate classifications can
be obtained by proposing more powerful deep learning methods from but not
limited to the following directions. We hope that these interesting directions
will motivate the ML community in addressing the challenging problems of event
detection, classification and localization for dynamical systems that exhibit
tight multi-scale spatio-temporal coupling.

  * •

Graph neural networks with both spatial and temporal dependencies: event
localization is a great challenge when only temporal dependencies are modeled
in deep learning approaches. Given bus locations and their connectivity, graph
neural networks may be promising in modeling spatial dependencies and locating
the actual event bus.

  * •

Incorporating contrastive learning into representation learning: by comparing
two instances ($x_{i}$ and $x_{j}$) rather than learning the mapping from $X$
to $y$, 1-NN based approaches outperform deep learning approaches in event
localization. Recently, representation learning based on comparing three
instances (that is, one anchor, positive, and negative sample in triplet loss)
or two instances (e.g., two similar samples by transformation or same
annotated label in Siamese approaches) shows effectiveness in capturing
underlying patterns and further benefits downstream tasks in domains like
computer vision [83], reinforcement learning [84], etc.

#### Load and Renewable Energy Forecasting

We list the performance of different forecasting methods on 1-hour ahead load
forecasting in years from 2018 to 2020 in Table 2, while we list the
performance on all the forecasting tasks on our Github. For short-term load
forecasting, exponential smoothing outperforms the other benchmarks both in
point forecast and prediction interval, while deep learning approaches fail to
achieve competitive performance on par with time series models and traditional
machine learning methods. Similar observations can be discovered in both
short- and long-term forecasting of wind and solar. As visualized in Fig. 8,
we can observe strong periodicity in both the observational features (such as
wind speed, relative humidity, and temperature) as well as target features
(such as solar power, wind power, and load) from the year 2018 to year 2020.
However, deep learning approaches fail to capture such significant trends for
accurate future forecasting due to their limited memorizing capabilities. For
extremely long time series, it’s difficult to efficiently extract and leverage
useful past time steps without tedious feature engineering [85]. Taking our
forecasting task as an example: simply enlarging the scope of considered
historical data to cover information from the previous $1440$-th, $10,080$-th,
$43,200$-th time step in our minute-level data for the reference of the same
time in the last day, same weekday in the previous week, and same day in the
previous month, is both time-consuming during processing crowded useless
information and will deteriorate forecasting performance in the end.

Accordingly, we suggest the following directions for further exploration. We
believe that this task can motivate the development of novel deep learning
based forecasting algorithms that take into account long-term memory and
exploit spatio-temporal patterns in the data.

  * •

Memory network to remember and utilize past history efficiently: approaches
such as memory networks [86] could be potential solutions to identify key
information from long histories for real-time forecasting.

  * •

Cross-learning: using information from multiple series to predict individual
ones has shown promising results in top approaches of past Kaggle competitions
[87]. Taking into account load and renewable energy time series from other
locations, such as nearby locations in the same time zone or with similar
social and economic patterns, could potentially enhance forecasting accuracy.

#### Synthetic Time-series Generation

We observe that current SOTA time-series generation methods cannot properly
capture the necessary characteristics from PMU data to generate realistic
time-series. This is reflected in the fidelity metrics in Table 3, where a
post-hoc 2-layer LSTM can easily separate real vs. generated samples. Time-GAN
[81] and COT-GAN [80] are current SOTA methods published in NeurIPS 19 and 20
respectively and they both achieve relatively low auto-correlation and cross-
correlation error compared to real data (Fig. 9-a). In addition, we apply PCA
analyses on both the original and synthetic datasets (flattening the temporal
dimension) and visualize how closely the distributions are in 2D space (Fig.
9-c). We observe that overall, the methods also fail to cover the underlying
data distribution. DoppelGANger [82] is the only method that can model
metadata, i.e., fault type in this case, and can generate sensible results.
However, it still struggles to learn the distribution of the metadata (Fig.
9-b). One of the main challenges from this time-series dataset is its size
(dimensionality and sequence lengths). Compared to the datasets from their
original papers, fast-sampled PMU data from PSML nearly double or triple the
number of observations at 960 observations with 91 channels for measurements
including voltage magnitude, voltage phase angle, current magnitude, current
phase angle, real power, reactive power, and frequency. Current generation
approaches leveraging recurrent networks do a poor job of modelling the long-
term temporal correlations seen in the data. For long time series, RNNs take
many passes to generate the entire samples, which causes them to forget long
temporal correlation. Besides, one particular challenge in the power grid data
is that each dimension of the time series cannot be handled separately, since
the whole system is governed by the Kirchhoff’s voltage and current laws at
each snapshot. This provides an interesting direction for future generation
work to address not only the scalability problem but also the constrained
generation problem.

## Usage Notes

The dataset and codes are licensed under the CC BY 4.0, meaning everyone can
use it only for non-commercial research purpose. We recommend users to follow
the guidance on Github [28], including the details of installation, package
usage, dataset navigation, and code navigation.

## Code availability

A step-by-step guidance and the source-code for dataset generation and machine
learning benchmarks can be found on GitHub [28]. Specifically, we provide
ready-to-use Pytorch data loaders with both data processing and splitting
included, and also share the code of evaluators to support fair comparison
among different ML-based algorithms, of which the dependencies and usage are
also descibed on Github [28](https://github.com/tamu-engineering-
research/Open-source-power-dataset).

## References

  * [1] Climate Central.  Emissions Sources (2020) (Available: https://www.climatecentral.org/gallery/graphics/emissions-sources-2020 [Online], 2020).
  * [2] Xie, L., Singh, C., Mitter, S. K., Dahleh, M. A. & Oren, S. S.  Toward carbon-neutral electricity and mobility: Is the grid infrastructure ready?  _Joule_ (2021).
  * [3] Rolnick, D. _et al._ Tackling climate change with machine learning.  _arXiv preprint arXiv:1906.05433_ (2019).
  * [4] Toubeau, J.-F., Bottieau, J., Vallée, F. & De Grève, Z.  Deep learning-based multivariate probabilistic forecasting for short-term scheduling in power markets.  _IEEE Transactions on Power Systems_ 34, 1203–1215, 10.1109/TPWRS.2018.2870041 (2019).
  * [5] Yang, C., Thatte, A. A. & Xie, L.  Multitime-scale data-driven spatio-temporal forecast of photovoltaic generation.  _IEEE Transactions on Sustainable Energy_ 6, 104–112, 10.1109/TSTE.2014.2359974 (2015).
  * [6] Wang, X., Wang, Y., Shi, D., Wang, J. & Wang, Z.  Two-stage wecc composite load modeling: A double deep q-learning networks approach.  _IEEE Transactions on Smart Grid_ 11, 4331–4344, 10.1109/TSG.2020.2988171 (2020).
  * [7] Yan, Z. & Xu, Y.  Real-time optimal power flow: A <italic>lagrangian</italic> based deep reinforcement learning approach.  _IEEE Transactions on Power Systems_ 35, 3270–3273, 10.1109/TPWRS.2020.2987292 (2020).
  * [8] Ning, C. & You, F.  Deep learning based distributionally robust joint chance constrained economic dispatch under wind power uncertainty.  _IEEE Transactions on Power Systems_ 1–1, 10.1109/TPWRS.2021.3096144 (2021).
  * [9] Li, M., Wei, W., Chen, Y., Ge, M.-F. & Catalao, J. P. S.  Learning the optimal strategy of power system operation with varying renewable generations.  _IEEE Transactions on Sustainable Energy_ 1–1, 10.1109/TSTE.2021.3088951 (2021).
  * [10] Xie, L., Chen, Y. & Kumar, P. R.  Dimensionality reduction of synchrophasor data for early event detection: Linearized analysis.  _IEEE Transactions on Power Systems_ 29, 2784–2794, 10.1109/TPWRS.2014.2316476 (2014).
  * [11] Huang, T., Freris, N. M., Kumar, P. R. & Xie, L.  A synchrophasor data-driven method for forced oscillation localization under resonance conditions.  _IEEE Transactions on Power Systems_ 35, 3927–3939, 10.1109/TPWRS.2020.2982267 (2020).
  * [12] Huang, T., Gao, S., Long, X. & Xie, L.  A neural lyapunov approach to transient stability assessment in interconnected microgrids.  In _Proceedings of the 54th Hawaii International Conference on System Sciences_ , 3330 (2021).
  * [13] El Helou, R., Kalathil, D. & Xie, L.  Fully decentralized reinforcement learning-based control of photovoltaics in distribution grids for joint provision of real and reactive power.  _IEEE Open Access Journal of Power and Energy_ 8, 175–185, 10.1109/OAJPE.2021.3077218 (2021).
  * [14] Xu, H., Domínguez-García, A. D. & Sauer, P. W.  Optimal tap setting of voltage regulation transformers using batch reinforcement learning.  _IEEE Transactions on Power Systems_ 35, 1990–2001, 10.1109/TPWRS.2019.2948132 (2020).
  * [15] Wu, D., Zheng, X., Kalathil, D. & Xie, L.  Nested reinforcement learning based control for protective relays in power distribution systems.  In _2019 IEEE 58th Conference on Decision and Control (CDC)_ , 1925–1930, 10.1109/CDC40024.2019.9029268 (2019).
  * [16] Voyant, C. _et al._ Machine learning methods for solar radiation forecasting: A review.  _Renewable Energy_ 105, 569–582 (2017).
  * [17] Dolara, A., Grimaccia, F., Leva, S., Mussetta, M. & Ogliari, E.  Comparison of training approaches for photovoltaic forecasts by means of machine learning.  _Applied Sciences_ 8, 228 (2018).
  * [18] Yildiz, B., Bilbao, J. I. & Sproul, A. B.  A review and analysis of regression and machine learning models on commercial building electricity load forecasting.  _Renewable and Sustainable Energy Reviews_ 73, 1104–1122 (2017).
  * [19] Feng, C. & Zhang, J.  Assessment of aggregation strategies for machine-learning based short-term load forecasting.  _Electric Power Systems Research_ 184, 106304 (2020).
  * [20] Almalaq, A. & Edwards, G.  A review of deep learning methods applied on load forecasting.  In _2017 16th IEEE international conference on machine learning and applications (ICMLA)_ , 511–516 (IEEE, 2017).
  * [21] Zainab, A., Refaat, S. S., Syed, D., Ghrayeb, A. & Abu-Rub, H.  Faulted line identification and localization in power system using machine learning techniques.  In _2019 IEEE International Conference on Big Data (Big Data)_ , 2975–2981 (IEEE, 2019).
  * [22] Cui, M., Wang, J. & Yue, M.  Machine learning-based anomaly detection for load forecasting under cyberattacks.  _IEEE Transactions on Smart Grid_ 10, 5724–5734 (2019).
  * [23] Hink, R. C. B. _et al._ Machine learning for power system disturbance and cyber-attack discrimination.  In _2014 7th International symposium on resilient control systems (ISRCS)_ , 1–8 (IEEE, 2014).
  * [24] Mohammadpourfard, M., Weng, Y. & Tajdinian, M.  Benchmark of machine learning algorithms on capturing future distribution network anomalies.  _IET Generation, Transmission & Distribution_ 13, 1441–1455 (2019).
  * [25] Marot, A. _et al._ Learning to run a power network challenge for training topology controllers.  _Electric Power Systems Research_ 189, 106635 (2020).
  * [26] Marot, A. _et al._ Learning to run a power network challenge: a retrospective analysis.  _arXiv preprint arXiv:2103.03104_ (2021).
  * [27] IEEE PES, NASPI.  2021 IEEE-NASPI oscillation source location contest (Available: http://web.eecs.utk.edu/~kaisun/Oscillation/2021Contest/ [Online], 2021).
  * [28] Zheng, X. _et al._ PSML: A Multi-scale Time-series Dataset for Machine Learning in Decarbonized Energy Grids (Code).  _Zenodo_ https://doi.org/10.5281/zenodo.5663995 (2021).
  * [29] Zheng, X. _et al._ PSML: A Multi-scale Time-series Dataset for Machine Learning in Decarbonized Energy Grids (Dataset).  _Zenodo_ https://doi.org/10.5281/zenodo.5130612 (2021).
  * [30] SIEMENS.  PSS/E (Available: https://new.siemens.com/global/en/products/energy/energy-automation-and-smart-grid/pss-software/pss-e.html [Online], 2021).
  * [31] Schneider et.al., K. P.  Analytic considerations and design basis for the ieee distribution test feeders.  _IEEE Transactions on Power Systems_ 33, 3181–3188, 10.1109/TPWRS.2017.2760011 (2018).
  * [32] Huang, T., Gao, S. & Xie, L.  A neural lyapunov approach to transient stability assessment of power electronics-interfaced networked microgrids.  _IEEE Transactions on Smart Grid_ (2021).
  * [33] Ruan, G. _et al._ A cross-domain approach to analyzing the short-run impact of covid-19 on the us electricity sector.  _Joule_ 4, 2322–2337 (2020).
  * [34] National Renewable Energy Laboratory.  NSRDB: National Solar Radiation Database (Available: https://nsrdb.nrel.gov/ [Online], 2021).
  * [35] Blair, N. _et al._ System advisor model, sam 2014.1. 14: General description.  Tech. Rep., National Renewable Energy Lab.(NREL), Golden, CO (United States) (2014).
  * [36] U.S. Energy Information Administration.  Form EIA-860 with previous formdata (Available: https://www.eia.gov/electricity/data/eia860/ [Online], 2021).
  * [37] Tate, J. E. & Overbye, T. J.  Line outage detection using phasor angle measurements.  _IEEE Transactions on Power Systems_ 23, 1644–1652, 10.1109/TPWRS.2008.2004826 (2008).
  * [38] Maslennikov, S., Wang, B. & Litvinov, E.  Dissipating energy flow method for locating the source of sustained oscillations.  _International Journal of Electrical Power & Energy Systems_ 88, 55–62 (2017).
  * [39] Large, J., Bostrom, A., Fawaz, H. I. & Löning, M.  sktime-dl (Available: https://github.com/sktime/sktime-dl.git [Online], 2019).
  * [40] Karim, F., Majumdar, S., Darabi, H. & Harford, S.  Multivariate lstm-fcns for time series classification.  _Neural Networks_ 116, 237–245 (2019).
  * [41] Zhang, X., Gao, Y., Lin, J. & Lu, C.-T.  Tapnet: Multivariate time series classification with attentional prototypical network.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , vol. 34, 6845–6852 (2020).
  * [42] Dempster, A., Schmidt, D. F. & Webb, G. I.  Minirocket: A very fast (almost) deterministic transform for time series classification.  In _Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining_, 248–257 (2021).
  * [43] Pandey, S., Srivastava, A. K. & Amidan, B. G.  A real time event detection, classification and localization using synchrophasor data.  _IEEE Transactions on Power Systems_ 35, 4421–4431 (2020).
  * [44] Dau, H. A. _et al._ The ucr time series archive.  _IEEE/CAA Journal of Automatica Sinica_ 6, 1293–1305 (2019).
  * [45] Bagnall, A. _et al._ The uea multivariate time series classification archive, 2018.  _arXiv preprint arXiv:1811.00075_ (2018).
  * [46] Fawaz, H. I. _et al._ Inceptiontime: Finding alexnet for time series classification.  _Data Mining and Knowledge Discovery_ 34, 1936–1962 (2020).
  * [47] Szegedy, C., Ioffe, S., Vanhoucke, V. & Alemi, A. A.  Inception-v4, inception-resnet and the impact of residual connections on learning.  In _Thirty-first AAAI conference on artificial intelligence_ (2017).
  * [48] Wang, Z., Yan, W. & Oates, T.  Time series classification from scratch with deep neural networks: A strong baseline.  In _2017 International joint conference on neural networks (IJCNN)_ , 1578–1585 (IEEE, 2017).
  * [49] He, K., Zhang, X., Ren, S. & Sun, J.  Deep residual learning for image recognition.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 770–778 (2016).
  * [50] Zheng, Y., Liu, Q., Chen, E., Ge, Y. & Zhao, J. L.  Time series classification using multi-channels deep convolutional neural networks.  In _International conference on web-age information management_ , 298–310 (Springer, 2014).
  * [51] Hochreiter, S. & Schmidhuber, J.  Long short-term memory.  _Neural computation_ 9, 1735–1780 (1997).
  * [52] Chung, J., Gulcehre, C., Cho, K. & Bengio, Y.  Empirical evaluation of gated recurrent neural networks on sequence modeling.  _arXiv preprint arXiv:1412.3555_ (2014).
  * [53] Vaswani, A. _et al._ Attention is all you need.  In _Advances in neural information processing systems_ , 5998–6008 (2017).
  * [54] Baccianella, S., Esuli, A. & Sebastiani, F.  Evaluation measures for ordinal regression.  In _2009 Ninth international conference on intelligent systems design and applications_ , 283–287 (IEEE, 2009).
  * [55] Khosravi, A., Nahavandi, S., Creighton, D. & Atiya, A. F.  Comprehensive review of neural network-based prediction intervals and new advances.  _IEEE Transactions on neural networks_ 22, 1341–1356 (2011).
  * [56] Makridakis, S., Spiliotis, E. & Assimakopoulos, V.  The m4 competition: 100,000 time series and 61 forecasting methods.  _International Journal of Forecasting_ 36, 54–74 (2020).
  * [57] Seabold, S. & Perktold, J.  statsmodels: Econometric and statistical modeling with python.  In _9th Python in Science Conference_ (2010).
  * [58] Buitinck, L. _et al._ API design for machine learning software: experiences from the scikit-learn project.  In _ECML PKDD Workshop: Languages for Data Mining and Machine Learning_ , 108–122 (2013).
  * [59] Oreshkin, B. N., Carpov, D., Chapados, N. & Bengio, Y.  N-BEATS: Neural basis expansion analysis for interpretable time series forecasting.  In _International Conference on Learning Representations_ (2020).
  * [60] Oord, A. v. d. _et al._ Wavenet: A generative model for raw audio.  _arXiv preprint arXiv:1609.03499_ (2016).
  * [61] Bai, S., Kolter, J. Z. & Koltun, V.  An empirical evaluation of generic convolutional and recurrent networks for sequence modeling.  _arXiv:1803.01271_ (2018).
  * [62] Lai, G., Chang, W.-C., Yang, Y. & Liu, H.  Modeling long-and short-term temporal patterns with deep neural networks.  In _The 41st International ACM SIGIR Conference on Research & Development in Information Retrieval_, 95–104 (2018).
  * [63] Salinas, D., Flunkert, V., Gasthaus, J. & Januschowski, T.  Deepar: Probabilistic forecasting with autoregressive recurrent networks.  _International Journal of Forecasting_ 36, 1181–1191 (2020).
  * [64] Zhou, H. _et al._ Informer: Beyond efficient transformer for long sequence time-series forecasting.  In _The Thirty-Fifth AAAI Conference on Artificial Intelligence, AAAI 2021, Virtual Conference_ , vol. 35, 11106–11115 (AAAI Press, 2021).
  * [65] Chen, R. T., Rubanova, Y., Bettencourt, J. & Duvenaud, D.  Neural ordinary differential equations.  _arXiv preprint arXiv:1806.07366_ (2018).
  * [66] Ahmad, T., Zhang, H. & Yan, B.  A review on renewable energy and electricity requirement forecasting models for smart grid and buildings.  _Sustainable Cities and Society_ 55, 102052 (2020).
  * [67] Oreshkin, B. N., Carpov, D., Chapados, N. & Bengio, Y.  N-beats: Neural basis expansion analysis for interpretable time series forecasting.  _arXiv preprint arXiv:1905.10437_ (2019).
  * [68] Lea, C., Flynn, M. D., Vidal, R., Reiter, A. & Hager, G. D.  Temporal convolutional networks for action segmentation and detection.  In _proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , 156–165 (2017).
  * [69] Zhou, H. _et al._ Informer: Beyond efficient transformer for long sequence time-series forecasting.  In _Proceedings of AAAI_ (2021).
  * [70] Gneiting, T. & Raftery, A. E.  Strictly proper scoring rules, prediction, and estimation.  _Journal of the American statistical Association_ 102, 359–378 (2007).
  * [71] Gaing, Z.-L.  Wavelet-based neural network for power disturbance recognition and classification.  _IEEE transactions on power delivery_ 19, 1560–1568 (2004).
  * [72] Okumus, H. & Nuroglu, F. M.  Power system event classification based on machine learning.  In _2018 3rd International Conference on Computer Science and Engineering (UBMK)_ , 402–405 (IEEE, 2018).
  * [73] Zheng, X., Wang, B., Kalathil, D. & Xie, L.  Generative adversarial networks-based synthetic pmu data creation for improved event classification.  _IEEE Open Access Journal of Power and Energy_ 8, 68–76, 10.1109/OAJPE.2021.3061648 (2021).
  * [74] Federal Energy Regulatory Commission.  Critical Energy/Electric Infrastructure Information (Available: https://www.ferc.gov/ceii [Online], 2021).
  * [75] Birchfield, A. B., Xu, T., Gegner, K. M., Shetye, K. S. & Overbye, T. J.  Grid structural characteristics as validation criteria for synthetic networks.  _IEEE Transactions on power systems_ 32, 3258–3265 (2016).
  * [76] Xu, Y. _et al._ US test system with high spatial and temporal resolution for renewable integration studies.  In _2020 IEEE Power & Energy Society General Meeting (PESGM)_, 1–5 (IEEE, 2020).
  * [77] Wu, D. _et al._ An open-source extendable model and corrective measure assessment of the 2021 texas power outage.  _Advances in Applied Energy_ 4, 100056, https://doi.org/10.1016/j.adapen.2021.100056 (2021).
  * [78] Arjovsky, M., Chintala, S. & Bottou, L.  Wasserstein generative adversarial networks.  In _International conference on machine learning_ , 214–223 (PMLR, 2017).
  * [79] Esteban, C., Hyland, S. L. & Rätsch, G.  Real-valued (medical) time series generation with recurrent conditional gans.  _arXiv preprint arXiv:1706.02633_ (2017).
  * [80] Xu, T., Wenliang, L. K., Munn, M. & Acciaio, B.  Cot-gan: Generating sequential data via causal optimal transport.  _arXiv preprint arXiv:2006.08571_ (2020).
  * [81] Yoon, J., Jarrett, D. & van der Schaar, M.  Time-series generative adversarial networks.  In _Proceedings of the 33rd International Conference on Neural Information Processing Systems_ , 5508–5518 (2019).
  * [82] Lin, Z., Jain, A., Wang, C., Fanti, G. & Sekar, V.  Using gans for sharing networked time series data: Challenges, initial promise, and open questions.  In _Proceedings of the ACM Internet Measurement Conference_ , 464–483 (2020).
  * [83] Taigman, Y., Yang, M., Ranzato, M. & Wolf, L.  Deepface: Closing the gap to human-level performance in face verification.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 1701–1708 (2014).
  * [84] Grill, J.-B. _et al._ Bootstrap your own latent: A new approach to self-supervised learning.  _arXiv preprint arXiv:2006.07733_ (2020).
  * [85] Wang, B. _et al._ Deep uncertainty quantification: A machine learning approach for weather forecasting.  In _Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_, 2087–2095 (2019).
  * [86] Weston, J., Chopra, S. & Bordes, A.  Memory networks.  _arXiv preprint arXiv:1410.3916_ (2014).
  * [87] Bojer, C. S. & Meldgaard, J. P.  Kaggle forecasting competitions: An overlooked learning opportunity.  _International Journal of Forecasting_ 37, 587–603 (2021).
  * [88] Goodfellow, I. _et al._ Generative adversarial nets.  _Advances in neural information processing systems_ 27 (2014).

## Author contributions statement

Conceptualization, L.X. and Y.L.; Data collection, X.Z.; Co-simulation
platform, D.W., T.H. and X.Z.; Benchmark, N.X. and L.T.; Writing - original
draft, X.Z., S.S., N.X., L.T. and D.W.; Writing - review and editing, all
authors; Visualization, X.Z., N.X. and L.T.; Project administration, L.X. and
Y.L.

## Competing interests

The authors declare no competing interests.

## Figures & Tables

Figure 1: Conceptual block diagram of the data flow of the transmission +
distribution co-simulation platform used to create PSML. While the simulation
is a closely integrated process that combines all types of input data, results
with different time-scales are generated at different simulation stages.
Figure 2: Diagram of the join transmission and distribution (T+D) simulation
model. a. One PSS/E 23-bus system and two IEEE 13-bus systems implement the
co-simulation by exchanging the real-time voltage and power information, which
are respectively simulated by the PSS/E and OpenDSS. b. Details of the
inverter-based solar PV dynamic model in the distribution system. Figure 3:
Visualization of different types of disturbances in the PSS/E 23-bus
transmission system that induce the transient millisecond-level voltage,
current and power measurements. Figure 4: Illustration of daily load, solar
and wind power profiles with 1-minute resolution sampled in Houston, capturing
seasonal disparity, strong variation of renewables, and unprecedented load
drop during pandemic. The solid lines represent the average and shaded areas
represent standard deviation. Figure 5: Power spectral density of minute-level
PMU measurements of 1 year. It demonstrates the data fidelity by the high
power density at the periods of 12 hours, 24 hours and 1 week, which is
consistent with our prior observations in other real-world PMU dataset. Figure
6: Illustration of voltage magnitude time series of millisecond resolution in
cases of different types of disturbances, where the black solid profiles are
critical features captured by the novel T+D co-simulation, some of which are
missed by the red dashed lines obtained conventional transmission system
simulation alone. Figure 7: Modal analysis of millisecond-level PMU
measurements, showing limited number of dominant modes of high energy. Figure
8: Visualization of feature trends in three years from one sampled location of
the forecasting task. Figure 9: Performance of generative methods. a.
Autocorrelation on PMU datasets for all generative methods. b. Distribution of
generated faults for DoppelGANger. c. PCA coverage evaluation on millisecond-
level eventful PMU datasets.

[capbesideposition=left,top]table[]

Table 1: Classification performance on simulation measurements for event
detection, classification and localization. We present $avg\pm stdev$ values
for experiments with $10$ random seeds.

Categories Methods Classification $\uparrow$ Localization $\uparrow$ Detection
$\downarrow$ (Balanced Acc) (Balanced Acc) (Macro MAE) Power Domain PMU score
[43] — 0.266 — Traditional Machine Learning Methods 1-NN Euclidean [45, 44]
0.537 0.402 36.465 1-NN DTW-i [45, 44] 0.610 0.463 53.928 1-NN DTW-d [45, 44]
0.598 0.474 53.709 MiniRocket [42] 0.690 $\pm$ 0.022 0.208 $\pm$ 0.226 53.908
$\pm$ 3.358 Convolutional Neural Networks Vanilla CNN 0.564 $\pm$ 0.058 0.168
$\pm$ 0.053 40.458 $\pm$ 12.686 InceptionTime [46] 0.715 $\pm$ 0.040 0.243
$\pm$ 0.047 43.743 $\pm$ 10.605 MLSTM-FCN [40] 0.742 $\pm$ 0.029 0.285 $\pm$
0.023 31.873 $\pm$ 5.400 ResNet [48] 0.725 $\pm$ 0.049 0.232 $\pm$ 0.044
38.578 $\pm$ 9.569 MC-DCNN [50] 0.726 $\pm$ 0.019 0.437 $\pm$ 0.030 38.107
$\pm$ 5.675 TapNet [41] 0.653 $\pm$ 0.018 0.397 $\pm$ 0.065 58.251 $\pm$ 1.974
Other Deep Learning Methods Fully-connected Neural Network 0.583 $\pm$ 0.042
0.245 $\pm$ 0.035 54.131 $\pm$ 9.964 Vanilla RNN 0.504 $\pm$ 0.045 0.224 $\pm$
0.037 57.184 $\pm$ 4.285 LSTM [51] 0.544 $\pm$ 0.049 0.248 $\pm$ 0.043 56.434
$\pm$ 2.851 GRU [52] 0.653 $\pm$ 0.029 0.332 $\pm$ 0.062 55.550 $\pm$ 2.090
Vanilla Transformer [53] 0.612 $\pm$ 0.041 0.340 $\pm$ 0.090 46.824 $\pm$
0.866

[capbesideposition=left,top]table[]

Table 2: Performance on 1-hour-ahead load point forecast and interval
prediction.

Categories Methods Point Forecast 95% Pred. Interval RMSE $\downarrow$ MAE
$\downarrow$ MAPE $\downarrow$ MIS $\downarrow$ 2018 2019 2020 2018 2019 2020
2018 2019 2020 2018 2019 2020 Time-series Models Naive 0.041 0.039 0.032 0.029
0.029 0.024 0.032 0.032 0.027 0.203 0.175 0.190 ARIMA 0.031 0.027 0.024 0.022
0.020 0.017 0.023 0.022 0.019 0.113 0.095 0.083 ETS 0.029 0.026 0.022 0.021
0.019 0.017 0.022 0.021 0.018 0.107 0.088 0.076 Traditional Machine Learning
Methods SVR 0.057 0.058 0.044 0.046 0.048 0.037 0.053 0.057 0.043 0.181 0.182
0.185 RF 0.033 0.032 0.028 0.026 0.026 0.022 0.029 0.028 0.024 0.099 0.093
0.095 GBDT 0.033 0.032 0.024 0.026 0.026 0.020 0.029 0.029 0.023 0.095 0.087
0.091 LR 0.026 0.027 0.030 0.020 0.022 0.024 0.021 0.024 0.026 1.141 0.871
0.100 Multilayer Perceptron ELM 0.229 0.145 0.184 0.196 0.119 0.146 0.233
0.140 0.170 0.531 0.427 0.573 FNN 0.088 0.091 0.114 0.073 0.076 0.085 0.083
0.089 0.100 0.240 0.248 0.257 N-BEATS [67] 0.090 0.091 0.084 0.067 0.072 0.068
0.072 0.082 0.077 0.209 0.207 0.236 Convolutional Neural Networks Vanilla CNN
0.103 0.065 0.089 0.085 0.049 0.058 0.099 0.053 0.067 0.263 0.243 0.243
WaveNet [60] 0.159 0.109 0.123 0.132 0.087 0.103 0.156 0.093 0.120 0.438 0.454
0.659 TCN [68] 1.336 0.966 1.133 1.330 0.960 1.130 1.478 1.100 1.277 2.777
1.996 2.359 Recurrent Neural Networks Vanilla RNN 0.154 0.094 0.084 0.122
0.075 0.068 0.137 0.088 0.080 0.330 0.273 0.230 LSTM [51] 0.070 0.136 0.167
0.055 0.114 0.101 0.061 0.136 0.120 0.244 0.286 0.302 GRU [52] 0.087 0.132
0.092 0.071 0.108 0.074 0.081 0.127 0.086 0.216 0.286 0.201 LSTNet [62] 0.065
0.056 0.0826 0.050 0.046 0.067 0.055 0.053 0.075 0.188 0.328 0.227 DeepAR [63]
0.068 0.110 0.093 0.055 0.086 0.078 0.059 0.101 0.090 1.752 1.515 1.531
Transformer -based Transformer [53] 0.108 0.121 0.132 0.086 0.095 0.101 0.098
0.114 0.119 0.265 0.280 0.298 Informer [69] 0.129 0.110 0.073 0.102 0.086
0.059 0.118 0.101 0.067 0.252 0.215 0.214 - Neural ODE [65] 0.117 0.265 0.151
0.098 0.249 0.128 0.102 0.270 0.137 0.737 0.709 0.801

[capbesideposition=left,top]table[]

Table 3: Run-time and fidelity metrics on millisecond-level eventful PMU
datasets. Auto- and cross-correlation are calculated as the sum of the
absolute difference between the correlation coefficients computed from real
and generated samples.

Method Autocorrelation $\downarrow$ Cross- Discriminative Hours $\downarrow$
correlation $\downarrow$ Score $\downarrow$ NaiveGAN [88] 663.13 7175.78 0.492
$\pm$ 0.022 5 RCGAN [79] 667.89 7607.29 0.499 $\pm$ 0.002 27 COT-GAN [80]
112.42 2532.69 0.435 $\pm$ 0.016 6 TimeGAN [81] 72.56 1361.45 0.481 $\pm$
0.008 52 DoppelGANger [82] 86.76 3994.45 0.447 $\pm$ 0.004 22