# A Predictive Online Transient Stability Assessment with Hierarchical
Generative Adversarial Networks

Rui Ma, , Sara Eftekharnejad, , Chen Zhong, , Mustafa Cenk Gursoy

###### Abstract

Online transient stability assessment (TSA) is essential for secure and stable
power system operations. The growing number of Phasor Measurement Units (PMUs)
brings about massive sources of data that can enhance online TSA. However,
conventional data-driven methods require large amounts of transient data to
correctly assess the transient stability state of a system. In this paper, a
new data-driven TSA approach is developed for TSA with fewer data compared to
the conventional methods. The data reduction is enabled by learning the
dynamic behaviors of the historical transient data using generative and
adversarial networks (GAN). This knowledge is used online to predict the
voltage time series data after a transient event. A classifier embedded in the
generative network deploys the predicted post-contingency data to determine
the stability of the system following a fault. The developed GAN-based TSA
approach preserves the spatial and temporal correlations that exist in
multivariate PMU time series data. Hence, in comparison with the state-of-the-
art TSA methods, it achieves a higher assessment accuracy using only one
sample of the measured data and a shorter response time. Case studies
conducted on the IEEE 118-bus system demonstrate the superior performance of
the GAN-based method compared to the conventional data-driven techniques

###### Index Terms:

Classification, generative adversarial networks, phasor measurement unit,
transient stability assessment.

##  I Introduction

Transient stability is defined as the ability of all power grid synchronous
machines to preserve synchronism after severe disturbances [1]. The ever-
increasing integration of inverter-based resources and the operation of power
grids under more stressed conditions, has renewed the calls for fast online
transient stability assessment (TSA) tools. An appropriate TSA will grant the
system operators adequate time to take proper mitigative actions, such as
intentional islanding or controlled load shedding.

The conventional TSA methodologies belong to one of three categories: 1) time-
domain simulations, 2) direct methods, and 3) trajectory-based approaches.
Time-domain methods solve a set of high-dimensional and nonlinear differential
algebraic equations to assess the transient stability [2, 3]. These methods
assume complete knowledge of the system model parameters and the operating
conditions. However, their high computational complexity makes them
impractical for near real-time applications. To enable faster transient
stability assessment, direct methods, e.g., Lyapunov method [4, 5, 6] and
extended equal area criterion [7, 8], have been developed. The direct methods
simplify TSA by energy functions that evaluate the dynamic performance of the
system in a shorter time [9, 10]. Yet, because of the need to simplify the
dynamic models, these methods do not scale well to large power systems. The
trajectory-based methods, such as Lyapunov exponents [11] and apparent
impedance [12] methods, are relatively fast. However, at least a few cycles of
post-fault transient data are needed for a precise stability assessment.

With the increased availability of high-resolution synchrophasor data, new TSA
approaches have emerged where physical system constraints and system data
inform each other for a more accurate TSA. Hence, without constructing a
complete dynamic grid model, the dynamic stability status of power grids can
be determined. Machine learning methods such as decision trees [13, 14, 12],
core vector machine [15], support vector machine (SVM) [16], and extreme
learning machine [17, 18] have previously been deployed for TSA using system
measurements, formulating TSA as a two-class (stable and unstable)
classification problem. These data-driven methods require a conversion from
raw Phasor Measurement Unit (PMU) data to selected features, which may result
in the loss of critical dynamic information and adversely affect
classification accuracy. Deep learning methods, such as convolutional neural
network (CNN) [19, 20], long short-term memory (LSTM) [21, 22], gated
recurrent unit (GRU) [23], and the stacked denoising autoencoder (SDAE) [24]
have been shown to address the problem of information loss by directly
utilizing raw PMU data. However, to guarantee the assessment accuracy, longer
durations of post-contingency time series data are needed, which will result
in TSA delays. Longer delays are not desired for online TSA since the window
of opportunity for taking an effective corrective action may be missed.

To address the aforementioned gaps in data-driven TSA methods, a novel data-
driven method is developed in this paper to predict the post-contingency PMU
measurements that are used for classification. By predicting the data, shorter
durations of post-contingency measurements are needed, while ensuring a high
TSA accuracy. The prediction capability is enabled by extending Generative
Adversarial Networks (GAN), a model-free machine learning technique for
generating datasets that closely mimic real datasets. Here, GAN enables
learning the model behavior (system transient response in this case) to
generate fictitious datasets.

A GAN model consists of two neural networks: generative and adversarial
networks. The generative network converts the input data drawn from a Gaussian
distribution to synthetic data. The adversarial network classifies the
generated synthetic data as either real or synthetic. The objective of the
adversarial network is to maximize the difference between the real and
synthetic data, while the generative model aims to minimize this difference
[25]. GAN has been deployed in various domains for generating synthetic
datasets including images [26] and music [27]. In the context of power
systems, GAN has been used to generate missing PMU data [28] or address the
lack of realistic PMU time series data [29, 30, 31]. Different from these
applications that aimed at enhancing the historical data, in this paper, the
GAN model is extended for predicting the future PMU data. The prediction
capability pursued by this work significantly differs from the previous
applications. Hence, a different approach to adopting GAN is required to
generate realistic future data and predict the system transient stability
based on those data. This approach should: a) learn the spatial and temporal
features of the transient data; b) predict the post-fault PMU data; and c)
assess the transient stability of the system. The developed GAN-based TSA
approach addresses these requirements for a fast and accurate stability
assessment.

The novelty of the developed TSA approach stems from the ability to rapidly
and accurately determine transient stability, while only utilizing one sample
of the measured post-contingency PMU data. This is achieved by developing a
hierarchical structure, shown in Fig. 1, for the generative adversarial
networks. Several GAN models are stacked together to construct HGAN. The input
to the hierarchical generative adversarial network (HGAN) is the measured
post-contingency PMU data. The lowest level GAN utilizes the measured PMU data
to predict the measurements for the next sampling time. The subsequent
predictions are used by higher level GANs to further predict the PMU time
series data. With the proposed structure, HGAN requires only a single sample
of the measured data to predict the post-contingency time series transient
data. In addition to the predicted sequence of the transient data, a binary
classifier is embedded in the generative network of each sub-GAN. Each sub-GAN
performs TSA individually, based on the measured one-sample PMU data as well
as the predicted subsequent time series data from the lower-level sub-GANs.
The final TSA is achieved by combining the assessment results from all sub-
GANs, with a goal to reduce the prediction error.

The main contributions of this work are as follows:

  1. 1.

A hierarchical GAN-based TSA is developed to accurately predict the transient
status after a disturbance. Only a single sample of post-contingency PMU
voltage data is needed. Hence, TSA can be performed in a relatively short
period of time;

  2. 2.

The generative and adversarial networks are designed in a manner to retain the
spatial and temporal features of the multivariate PMU time series data and
improve TSA accuracy;

  3. 3.

Case studies are conducted to demonstrate the ability of the developed model
in predicting a sequence of the transient data. In comparison to the
conventional machine learning-based TSA methods that deploy decision trees,
SVM, LSTM, or GRU, the developed model achieves a higher TSA accuracy with
only one sample of PMU measurements.

Figure 1: The structure of the developed HGAN-based TSA

The rest of this paper is organized as follows: Section II briefly describes
the concept of transient stability and formulates the TSA problem. Section III
introduces the conventional GAN model and illustrates each element of the
developed HGAN model. The developed HGAN model is tested on the IEEE 118-bus
system in Section IV. The conclusions are provided in Section V.

##  II Transient Stability Assessment

During a power system disturbance, such as a transmission line fault, large
swings appear in generator rotor angles, bus voltages, or line currents [24].
To ensure a stable grid, all generators should maintain synchronism after the
clearance of the faults. If the synchronism is lost, prompt mitigations such
as controlled islanding need to be taken to prevent further failures. This
remarks the need for an online TSA approach that can promptly and precisely
identify the transient status at the earliest stages of a fault clearance.

Fig. 2 depicts the post-fault dynamics of the generator rotor angles and bus
voltages in the IEEE 118-bus system. A three-phase short circuit is simulated
on line 102 at time instant 0.167s, which is cleared after 5 cycles, and
Generator 1 is the reference machine. As seen, large excursions occur in the
trajectories of the generator rotor angles and bus voltage magnitudes when the
described fault occurs. To classify the system transient stability status, a
commonly used stability index based on rotor angles is used [32]:

$\eta=\frac{360^{\circ}-\Delta\sigma_{max}}{360^{\circ}+\Delta\sigma_{max}}$
(1)

where $\Delta\sigma_{max}$ is the absolute value of the maximum post-fault
angle difference between any two generators. The stability is defined as,

$\delta=\left\\{\begin{matrix}\text{stable;}&\eta>0\\\
\text{unstable;}&\eta\leq 0.\end{matrix}\right.$ (2)

In the transient event depicted in Fig. 2, $\eta=41.3$. Therefore, the IEEE
118-bus system is considered stable after clearing the three-phase fault on
line 102. However, using (1) for determining the transient status requires the
estimation of $\Delta\sigma_{max}$ over a relatively long time (a few
seconds), which is not desired for online TSA. Hence, the objective of the
developed TSA approach, illustrated in Fig. 3(a), is to acquire the transient
status at the earliest stages of a contingency. As illustrated in Fig. 3(b),
unlike the other machine learning-based TSA methodologies that require a large
volume of post-contingency PMU data, the developed method only requires the
first sample of the PMU data after the clearance of the faults. Thus, more
time can be granted for finding the optimal corrective actions. In this work,
the PMU voltage data are selected for TSA for two main reasons: (a) the
voltage phasors are readily available from the PMU measurements; (b) previous
TSA studies have shown that bus voltage data can be utilized to accurately
predict the transient stability [21, 24, 33]. It is also shown in Section IV-B
that only utilizing PMU voltage magnitude data is sufficient to achieve a high
assessment accuracy. However, other PMU measurements that contain the system
response, such as bus voltage angles and line current phasors, can also be
deployed by the developed HGAN-based TSA method. Additionally, to mimic real
power grids, only some buses are equipped with PMUs (e.g., 20 buses in the
IEEE 118-bus system). The PMUs are added in a manner to ensure full system
observability.

Figure 2: The trajectories of the generator rotor angles and voltage
magnitudes in the IEEE 118-bus system. A three-phase short circuit is
simulated on line 102 at 0.167s and is cleared after 5 cycles (0.083 s).

(a) The developed HGAN-based TSA method

(b) The conventional machine learning-based TSA methods

Figure 3: The input and output of the developed HGAN model and the
conventional machine learning-based TSA methods. $x(t)$ is the first sampled
post-contingency PMU data. The number of PMU samples is $N$.

##  III Predictive Online Transient Stability Assessment

A new approach is developed for near real-time transient system assessment,
which is fast, accurate, and requires significantly less PMU data than the
conventional techniques. The developed approach is twofold: 1) predicting the
post-contingency PMU voltage time series data, and 2) assessing the transient
stability using the predicted data. In this section, the fundamentals of the
GAN are introduced first, followed by a detailed description of each element
of the HGAN-based TSA method.

###  III-A The Generative Adversarial Network

GAN is an unsupervised learning approach that has been used in many
applications for generating synthetic images, music, time series data, among
others [34, 35, 26, 27, 36]. GAN learns the distribution of a real dataset and
generates synthetic data that are close to the real data [25]. In the context
of power systems, GAN-based methods have been developed to recover missing PMU
data [28] and generate synthetic PMU datasets [29, 30, 31].

GAN consists of two deep neural networks: generator $G$ and discriminator $D$.
The generator generates synthetic data while retaining the statistical
features of the real data. The discriminator differentiates the real data from
the fictitious data generated by the generator. The two networks $G$ and $D$
are trained iteratively until a Nash equilibrium is achieved, which indicates
that the generated synthetic data and real data cannot be distinguished [28].

For a random Gaussian noise space $\mathbf{Z}$, the generator maps the input
noise $z$ drawn from $\mathbf{Z}$ to a synthetic dataset $\hat{x}$ as,

$\mathbf{G}(z|\theta_{g}):z\rightarrow\hat{x}$ (3)

where $\theta_{g}$ is the neural network parameters of $G$. The objective of
the generator is to map a random noise to synthetic data, such that the
distribution of the synthetic data $P(\hat{x})$ is close to the real data
$P(x)$. The discriminator $D$ estimates the probability,
$D(\cdot|\theta_{d})$, that the input data is real rather than synthetic,
where $\theta_{d}$ the is the neural network parameters of $D$. The
discriminator aims to maximize the difference between the real and synthetic
data so that they could be distinguished,

$\begin{split}\underset{D}{\text{Max}}\leavevmode\nobreak\
V(G,D)=&\mathbb{E}_{x\sim P(x)}[\log D(x)]\\\ &+\mathbb{E}_{z\sim
P(z)}[\log(1-D(\hat{x})]\end{split}$ (4)

where $V(\cdot)$ is the loss function. The first term $\mathbb{E}_{x\sim
P(x)}[\log D(x)]$ is the probability that the discriminator classifies the
real data as real. The second term is the probability that the synthetic data
$\hat{x}$ generated from $G$ is classified as fake by $D$. The objective of
$G$ is to minimize the difference between the real and synthetic data,

$\underset{G}{\text{Min}}\leavevmode\nobreak\ V(G,D)=\mathbb{E}_{z\sim
P(z)}[\log(1-D(\hat{x}))].$ (5)

As the generator and discriminator are trained together, the objective of GAN
is to address a minimax problem:

$\begin{split}\underset{G}{\text{Min}}\underset{D}{\text{Max}}\leavevmode\nobreak\
V(G,D)=&\mathbb{E}_{x\sim P(x)}[\log D(x)]\\\ &+\mathbb{E}_{z\sim
P(z)}[\log(1-D(\hat{x}))].\end{split}$ (6)

The GAN training algorithm is described in Algorithm 1. In each training
episode, by leveraging the stochastic gradient descent algorithm, the
generator and discriminator are updated, such that the synthetic data are
closer to the real data.

1 Initialize the neural network parameters of the generator and discriminator,
i.e., $\theta_{g}$ and $\theta_{d}$ respectively;

2 for  _$ep=1,\cdots,ep_{max}$_ do

3 Obtain a minibatch of the random noise $z$ from $P(z)$, and another
minibatch of the real data $x$ from $P(x)$;

4 Generate the synthetic data $\hat{x}=G(z|\theta_{g})$;

5 Feed the generated synthetic data $\hat{x}$ and the real dataset $x$ into
the discriminator $D$, and obtain the estimated probability $D(x|\theta_{d})$
and $D(\hat{x}|\theta_{d})$;

6 Update $G$ and $D$ by descending their gradients:
$\begin{split}&\triangledown_{\theta_{g}}(\mathbb{E}_{z\sim
P(z)}[log(1-D(\hat{x}))])\\\ &\triangledown_{\theta_{d}}(\mathbb{E}_{x\sim
P(x)}[logD(x)]+\mathbb{E}_{z\sim P(z)}[log(1-D(\hat{x})])\end{split}$

7 end for

Algorithm 1 The GAN training steps

###  III-B Gated Recurrent Units for Feature Extraction from PMU Time Series
Data

When predicting the future dynamic response in a power system based on data,
it is critical to preserve the spatial and temporal features of the time
series data to ensure accuracy. Gated Recurrent Units (GRUs), a variant of
RNN, enable learning the spatial and temporal features of time series data
[37]. Thus, the developed TSA approach leverages GRU to construct the
generator and the discriminator.

Equipped with directed cycles in network connections, RNN retains information
from past data, so that it can be combined with the present data to more
accurately predict the future sequence of data. However, RNN suffers from the
vanishing or exploding gradient problem during training [38]. If the RNN is
not designed properly, the gradient may be exponentially updated towards zero,
referred to as a vanishing gradient, or exponentially diverge to infinity,
referred to as an exploding gradient. Either of these problems challenges the
proper RNN training. To resolve this problem, GRU is equipped with a specific
mechanism to determine the volume of past data, to be added to the output. By
doing so, the problem of vanishing or exploding gradient is eliminated. As
illustrated in Fig. 4, GRU consists of three elements: the reset gate, the
update gate, and the current memory unit [37]. The update gate determines the
volume of past data that are used for predicting the future data sequence,
while the reset gate determines the volume of the stored past information to
be forgotten. The current memory unit uses the output from the reset gate to
store the relevant information from the past data. The output of GRU is the
sum of the output from the three gates. The three gates and the final output
can be described as,

$\displaystyle r_{t}=\sigma(W^{r}x_{t}+U^{t}h_{t-1})$ (.1) $\displaystyle
z_{t}=\sigma(W^{z}x_{t}+U^{z}h_{t-1})$ (.2) $\displaystyle
h_{t}^{{}^{\prime}}=\tanh(Wx_{t}+r_{t}\odot Uh_{t-1})$ (.3) $\displaystyle
h_{t}=z_{t}\odot h_{t-1}+(1-z_{t})\odot h_{t}^{{}^{\prime}}.$ (.4)

In (7), $W$ and $U$ are the weights of the GRU network, and $\odot$ is the
Hadamard (element-wise) product.

Figure 4: The structure of GRU

###  III-C The Developed HGAN-based TSA Method

The developed HGAN-based TSA aims to predict the post-fault transients and
assess the system’s transient stability. The developed method also improves
the GAN structure by learning the spatial and temporal features of the
multivariate PMU time series data. As deploying one type of PMU data is
sufficient to assess the transient stability, the PMU voltage magnitude data
are utilized for TSA in this paper. However, other PMU measurements may also
be used by the HGAN-based TSA model. The measured post-fault PMU voltage
magnitudes are the inputs to the GAN model, i.e., $x(t)$. The data for the
next time instant, i.e., $\hat{x}(t+1)$ are predicted by the aforementioned
GAN structure. However, the measured data $x(t)$ and the predicted data
$\hat{x}(t+1)$ may not have sufficient information to determine the transient
status. To address this deficiency, it is proposed in this paper to stack
multiple GANs to predict a longer sequence of future data to better reveal the
transient status of the system. As a result, a higher transient assessment
accuracy can be achieved.

The structure of the developed HGAN-based TSA is shown in Fig. 1. Assuming
that HGAN has $N$ stacked GANs, the measured PMU post-fault voltage data is
fed into the lowest level GAN, i.e., GAN 1 to predict the data for the next
time instant $\hat{x}(t+1)$. The predicted data is combined with the measured
data $x(t)$ and fed into GAN 2 for subsequent predictions. Therefore, the
HGAN-based TSA with $N$ GANs can generate $N+1$ voltage time series data,
$\\{x(t),\hat{x}(t+1),\cdots,\hat{x}(t+N)\\}$.

A binary classifier is embedded in each generator of the stacked GANs for TSA
classification. In each GAN $k$, the embedded classifier utilizes the input
sequence data $\\{x(t),\hat{x}(t+1),\cdots,\hat{x}(t+k-1)\\}$ to determine the
TSA classification, i.e., stable or unstable. The classification results from
all sub-GANs are then combined by using an average-based ensemble strategy to
determine the final transient stability assessment. The generator has three
objectives: 1) learning the distribution of the real measurements; 2)
predicting the next sequence of the time series data; 3) assessing the
transient stability status. Based on these objectives, the generator loss
function in GAN $k$ is formulated as,

$\begin{split}\mathbf{L}_{G_{k}}=&\log(1-D(\hat{x}(t+k)))+(\hat{x}(t+k)-x(t+k))^{2}\\\
&+y\log(p(y))+(1-y)\log(1-p(y))\end{split}$ (8)

where $y$ is the binary label of the transient data. Here, a zero label is
assigned to unstable transients, and one to stable transients. The probability
that the data is labeled as stable is defined by $p(y)$. In formulating the
loss function in (8), the square of the error, $(\hat{x}(t+k)-x(t+k))^{2}$, is
used to minimize the error of the predicted data $\hat{x}(t+k)$. The cross
entropy loss $y\log(p(y))+(1-y)\log(1-p(y))$ is included so that the generator
correctly classifies the system stability status. In addition, the GRU
structure is used to construct generative and adversarial neural networks to
better learn the spatial and temporal features of the time series data.

The detailed structure of GAN $k\leavevmode\nobreak\ (k\in\\{0,1,\cdots,N\\})$
is illustrated in Fig. 5. The input to GAN $k$ is the measured and predicted
PMU voltage data
$X=\\{x(t),\hat{x}(t+1),\cdots,\hat{x}(t+k-1)\\},\leavevmode\nobreak\
X\in\mathbf{R}^{k\times N_{PMU}}$, where $N_{PMU}$ is the total number of PMUs
deployed in the system. In the generator $G$, the input $X$ is first fed into
the GRU cell to learn the spatial and temporal features of the time series
data. To improve the learning efficiency of the generator, the fully connected
layer is added after the GRU cell. As the generator has two different outputs,
i.e., the predicted label and subsequent data, two parallel fully connected
layers are used to perform these two tasks separately. Furthermore, to
generate the predicted label $y$ and the predicted data $\hat{x}(t+k)$, the
softMax and Sigmoid activation functions are deployed to interpret the fully
connected layer output. The discriminator $D$ also uses GRU to construct the
neural networks. The output of the GRU in the discriminator is sent to the
fully connected layer, followed by a linear activation function that converts
the output to the estimated probability of the input data
$D(\cdot|\theta_{d})$. The estimated probability is then backpropagated to the
generator to update the parameters of the generative network.

Figure 5: The structure of GAN $k$ in the HGAN-based TSA

The prediction accuracy of the upper level GAN depends largely on the accuracy
of the lower GANs. A small prediction error from a lower level GAN will
propagate to the higher levels and eventually lead to a large prediction
error. Hence, in this study, the GANs are trained sequentially. For instance,
when training GAN $k$, the network parameters of the well-trained lower GANs
$i,\leavevmode\nobreak\ (i\in\\{0,\cdots,k-1\\})$, are kept fixed. Otherwise,
repeatedly updating the well-trained networks of the lower GANs when training
the upper GANs will cause an overfitting problem, and thus reduce the
prediction accuracy. The training process of the HGAN-based TSA is given in
Algorithm 2. For each GAN model, deploying the stochastic gradient descent
algorithm, the generator and the discriminator are updated until the
convergence of the cross-entropy loss, or until the training episode reaches
its maximal $ep_{max}$, whichever occurs sooner.

1 Normalize the data such that the maximum normalized voltage is one;

2 for  _$i=1,\cdots,N$_ do

3 Initialize the neural network parameters of the generator and discriminator,
i.e., $\theta_{g}$ and $\theta_{d}$ in GAN $i$, respectively ;

4 for  _$ep=1,\cdots,ep_{max}$_ do

5 Obtain a minibatch of the measured post-fault PMU voltage data $x(t)$ and
$x(t+i)$;

6 Using the well-trained lower GANs $k,\leavevmode\nobreak\
k\in\\{1,\cdots,i-1\\}$ to obtain the sequence data
$\\{x(t),\hat{x}(t+1),\cdots,\hat{x}(t+i-1)\\}$;

7 Feed the obtained sequence data to the generator $G$, generate the data
label $y$, and predict data $\hat{x}(t+i)$ for the next sampling time;

8 Input the predicted data $\hat{x}(t+i)$ and real data $x(t+i)$ to the
discriminator $D$, and obtain the estimated probability
$D(\hat{x}(t+i)|\theta_{d})$ and $D(x(t+i)|\theta_{d})$;

9 Update the generator and discriminator with gradient descent algorithm;

10 end for

11 Save the network parameters $\theta_{g}$ and $\theta_{d}$ for GAN $i$;

12

13 end for

Algorithm 2 The training algorithm for the developed HGAN-based TSA

The trained HGAN-based TSA can be applied online for stability assessment.
Once the first post-fault PMU voltage data, i.e., $v(0)$, is measured, the
HGAN-based model with $N$ sub-GANs generates the subsequent voltage data
$\\{v(0),\hat{v}(1),\cdots,\hat{v}(N)\\}$. The TSA results from these $N$ sub-
GANs will be combined by using an average-based ensemble technique for the
final TSA classification. Here, it is assumed that the final TSA result is
stable if more than half of sub-GANs vote for stable; otherwise, the system is
unstable. The use of the ensemble method reduces the variance of the
prediction errors from all sub-GANs and results in a more accurate and
reliable TSA [39].

##  IV Case Studies

To evaluate the performance of the HGAN-based TSA under various system
operating conditions, the IEEE 118-bus system with 118 buses, 170 lines, and 9
transformers is studied. To demonstrate the effectiveness of the HGAN-based
TSA method, four different baseline methods, namely, decision trees, SVM,
LSTM, and GRU are used for comparison purposes. Furthermore, as the TSA
accuracy of the HGAN-based method relies on PMU measurements, sensitivity
studies are performed to investigate the impact of noise, number, and location
of PMUs on TSA accuracy. The simulations are conducted on an i7 computer with
a 3.2GHz CPU and 64GB RAM.

###  IV-A PMU Data Generation

The post-fault PMU voltage time series for testing and training are generated
with simulations. Various operating conditions (90%, 100%, and 110% of the
nominal loading condition) are simulated under different system typologies,
i.e., normal and one line out of service (due to maintenance). The
contingencies considered in this study are three-phase faults on buses and
transmission lines (located at 30%, 50%, and 80% of the lines). The faults are
cleared after 5 cycles, and the system topology is kept unchanged after
clearing the fault. To minimize the adverse impacts of an imbalanced dataset
on the accuracy of the HGAN-based TSA, 5,000 stable and 5,000 unstable
transients are generated. Furthermore, 80% of the data are used for training,
and the remaining 20% are utilized for testing. For creating stability labels,
the transient stability of each event is determined using the index outlined
in (1). To generate the PMU time series data, the sampling frequency of the
simulation is set to 120 frames per second. Additionally, to mimic real power
grids where not every bus is equipped with PMUs, 20 PMUs are deployed in the
IEEE 118-bus system to ensure full observability [40]. The locations of the
PMUs are given in Table I. The training and testing data can be found at the
IEEE DataPort (https://dx.doi.org/10.21227/6f5v-q924).

Table I: PMU placement in the IEEE 118-bus system

Test System | PMU placement
---|---
IEEE 118-bus system |  | 5, 12, 15, 17, 32, 37, 49, 56,
---
59, 67, 69, 70, 71, 77, 80, 85,
92, 96, 100, 105

###  IV-B Performance Analysis of the HGAN-based TSA

—Performance Analysis: Upon obtaining the training and testing datasets that
contain transient events, each level GAN of the developed TSA model is
trained, following the training steps outlined in Algorithm 2. The training
parameters are provided in Table II. It is found that the HGAN-based model
with three GANs is sufficient for an accurate stability assessment. Although
more GANs generate longer duration of voltage time series data, and thus
better represent a transient pattern, a small error in lower level GANs could
propagate through the stacked GANs, and eventually lead to a large cumulative
error. The learning rates of the generator and discriminator are set to 0.001
and 0.0001, respectively. A larger learning rate can increase the learning
speed, while the results may converge to a suboptimal solution. Additionally,
to address the overfitting problem during the training of a GAN, the neural
network parameters of the lower level GANs are not updated.

Table II: Parameters of the HGAN-based TSA model

Parameters | Value
---|---
Number of GANs | 3
Number of episode | 20000
Learning rate of generator / discriminator | 1e-3 / 1e-4
Number of GRU layers / hidden layers | 2 / 30
Batch size | 128

To demonstrate the learning ability of the HGAN-based model, the cross-entropy
loss and the squared error of each sub-GAN in (8) are depicted in Fig. 6. To
check for overfitting, the testing data is used to test the model in each
training step. Note that the testing data are not used to train the HGAN-based
model. As seen from Fig. 6, for all three GANs, the cross-entropy loss and the
squared error dramatically decrease at the beginning of the training phase and
converge after that. For instance, the cross-entropy loss for training data in
GAN 1 drops from 0.7 to 0.33, and the squared error of the training data
decreases from 0.1965 to 0.0002. It is also found that with the predicted data
from lower GANs, the cross-entropy loss of GAN 2 and 3 drops faster than the
loss in GAN 1\. The faster drop indicates that combining the predicted and the
measured data leads to more distinguishable transient features for the GAN
model, and thus the learning speed is increased. The cross-entropy loss and
the squared error reduction indicate that the developed model is effectively
trained to improve the classification and prediction accuracy. Comparisons of
the training and testing loss and error demonstrate that no overfitting
occurs. This is concluded by the fact that the cross-entropy loss of the
testing data does not rise as the loss of the training data decreases. In
other words, both these values maintain a consistent difference.

Figure 6: The cross-entropy loss and the squared error for each of the three
GANs in the HGAN-based model.

Upon training the developed HGAN model, the testing data, i.e., 1000 stable
and 1000 unstable events, are used to evaluate the classification accuracy of
each GAN. The classification results are presented in Table III. With the
single measured PMU voltage data, the classification accuracy of the lowest
level GAN 1 is 95.35%. However, combining the real measured data and the
predicted data from the lower level GANs yields a classification accuracy of
99.95% for both GAN 2 and GAN 3\. Since a single sampled data carries limited
transient information, the classification accuracy of GAN 1 is only 95.3%.
With more predicted data that hold the spatial and temporal features of a
transient event, the hidden oscillation patterns of the event are more
apparent to the classifiers and result in higher classification accuracy. The
confusion matrix of each GAN is depicted in Fig. 7. For GAN 1, it can be
observed that all unstable transient data are correctly identified, while 8.6%
of the stable events are incorrectly classified as unstable. Benefiting from
the predicted PMU voltage data from GAN 1, in GAN 2, the percentage of
misclassified stable events drops to 0.1%. Combining the classification
results from these three GANs yields an accuracy of 99.95% for TSA. The
computation time for TSA using the developed model with three stacked GANs is
0.00298s, which is a reasonable time for near real-time analysis.

Table III: TSA classification accuracy of each GAN in the HGAN-based TSA

| GAN 1 | GAN 2 | GAN 3
---|---|---|---
Accuracy | 95.35 % | 99.95 % | 99.95 %

Figure 7: The confusion matrix of each GAN in the HGAN-based TSA model

—Comparison with Baseline Methods: To further demonstrate the effectiveness of
the developed data-driven TSA approach, its performance is compared with the
conventional data-driven baseline methods that deploy other machine learning
techniques, namely, decision trees, SVM, LSTM, and GRU. The inputs to the
baseline methods are the measured post-fault PMU voltage data. To achieve the
highest classification accuracy, the LSTM hyperparameters, namely, the
learning rate, training iteration, number of layers, hidden layers, and batch
size are set to 0.0005, 40000, 2, and 40, and 128 respectively. Similarly,
with various trials on GRU, the optimal learning rate, training iteration,
number of layers, and hidden layers are found to be 0.0005, 40000, 2, and 50,
respectively. The comparisons between the baseline methods and the HGAN-based
TSA are presented in Table IV and Fig. 8. Compared to SVM, LSTM, and GRU, the
decision trees and the HGAN-based TSA achieve higher classification accuracy,
i.e., 99.5% and 99.95%, respectively with only one sample of PMU measurements.
On the other hand, to reach a similar TSA classification accuracy, the HGAN-
based method requires less response time, i.e., 1.359 cycles. Although the
SVM, GRU, and LSTM based-TSA methods can achieve the similar classification
accuracy with around 2 cycles, the reduction of 0.65 cycles achieved by the
developed HGAN-based method is critical for online TSA. The transient
instability issues propagate rapidly in the grids. Hence, a faster TSA allows
more time for mitigative actions to prevent further severe consequences. The
response time is the combination of the waiting time for sufficient PMU
samples and the computation time of the TSA method. Although the decision
tree-based TSA achieves a relatively high classification accuracy with one
sample data, decision trees do not have the ability to predict the transients,
while the developed HGAN-based model predicts the transients. This predictive
capability makes the developed method suitable for other applications, e.g.,
cascading failure analysis and power system planning, that would need
transient analysis. Hence, the HGAN-based model outperforms the other four
baseline methods from both the accuracy and applicability perspectives.

Table IV: TSA classification accuracy of the baseline methods and the HGAN-
based TSA with only one sample of PMU measurements (20 PMUs are used)

Method | TSA Classification Accuracy
---|---
Decision tree | 99.5%
SVM | 95.8%
LSTM | 94.75%
GRU | 94.8%
HGAN | 99.95%

Figure 8: Comparison of response time between the baseline methods and HGAN-
based TSA method. In total, 20 PMUs are used in the IEEE 118-bus system. To
achieve the same accuracy as the HGAN-based method, other methods would use
more samples of PMU measurements and thus require longer response time.

###  IV-C Impact of PMU Measurement Variations

The PMU measurements used in this study are obtained from simulations and are
thus noise free. However, in practice, PMU measurements are contaminated by
noise. Hence, the sensitivities of the HGAN-based TSA method to noisy PMU
measurements are analyzed. Following the study in [41], and [42], white
Gaussian noise is added to the simulated PMU data. Four different signal-to-
noise ratios (SNR), namely, 50dB, 60dB, 70dB, and 80dB, are considered. The
classification accuracies of the HGAN-based method under different noise
conditions are presented in Fig. 9. The classification results in Fig. 9 show
that the noise mainly affects the performance of GAN 1. Larger noise will lead
to a lower classification accuracy in GAN 1. However, it is observed that the
classification accuracy of GAN 2 and GAN 3 are above 99%, and are not impacted
by the addition of noise. Another interesting observation is that among the
five different noise conditions, GAN 1 achieves the highest classification
accuracy when a noise of 80dB SNR is added to the measurements. This accuracy
is even higher than using clean PMU data. This observation can be explained by
the known fact that adding a small noise to the input data leads to a better
performance of the GAN models, as they learn more general features of the
transient data for a better classification performance [43].

Figure 9: The TSA classification accuracy of each sub-GAN under different
noise conditions. The testing data are used for evaluation.

In the existing power systems, PMUs are not deployed on every bus. Hence,
depending on where the PMUs are located, the performance of the HGAN-based TSA
could vary. To evaluate the performance of the developed approach under
varying PMU placement scenarios, five scenarios of different PMU placements
are investigated. The PMU locations for these five scenarios are listed in
Table V. The classification accuracy for each GAN under these five scenarios
is depicted in Table VI. Results in Table VI show that the location of PMUs
significantly affects the classification performance for GAN 1, where the
accuracy ranges from 94.55% to 97.9%. Benefiting from the predicted data, the
impact of the location is decreased in GAN 2 and GAN 3, and the classification
accuracy is improved. It can also be observed that the classification accuracy
slightly drops in GAN 3, due to the accumulation of small prediction errors in
lower GANs.

Table V: PMU location scenarios for investigating the impact of PMU placement
on the developed TSA approach

Scenario | Location
---|---
1 | 5, 12, 15, 17, 37, 49, 59, 69, 71, 77, 80, 85, 92, 96, 100, 105
2 | 5, 12, 15, 17, 37, 56, 67, 69, 70, 71, 77, 80, 92, 96, 100, 105
3 | 5, 12, 15, 17, 37, 49, 67, 69, 70, 71, 77, 80, 92, 96, 100, 105
4 | 5, 12, 15, 17, 32, 37, 49, 56, 59, 71, 77, 80, 85, 92, 96, 100
5 | 5, 12, 15, 17, 32, 37, 49, 56, 59, 69, 71, 77, 80, 92, 96, 100

Table VI: The TSA classification accuracy of each GAN with different PMU
locations. The testing data are used for evaluation. A total number of 16 PMUs
are used

PMU Location Scenario | TSA Classification Accuracy (%)
---|---
GAN 1 | GAN 2 | GAN 3
1 | 94.55 | 98.55 | 97.90
2 | 97.15 | 99.40 | 98.40
3 | 96.65 | 99.65 | 99.15
4 | 97.90 | 98.55 | 97.55
5 | 95.40 | 98.25 | 97.20

Table VII: TSA classification accuracy of each sub-GAN with different number
of PMUs

Number of PMUs | GAN 1 | GAN 2 | GAN 3
---|---|---|---
6 | 86.90% | 84.65% | 88.20%
8 | 91.20% | 91.15% | 77.95%
10 | 94.20% | 94.55% | 95.20%
12 | 95.75% | 99.60% | 99.65%
14 | 96.05% | 99.93% | 99.92%
16 | 96.65% | 99.65% | 99.15%
18 | 96.25% | 99.40% | 99.91%
20 | 95.35% | 99.95% | 99.95%

The impact of the number of PMUs on the HGAN-based TSA is also investigated.
The classification results for different numbers of PMUs are given in Table
VII. It can be observed that increasing the number of PMUs improves the TSA
accuracy for GAN 1. When the number of PMUs is larger than 12, the
classification accuracy of GAN 2 and 3 are around 99%. However, with less than
10 PMUs the TSA classification accuracy of GAN 2 and GAN 3 drops, indicating
that the predicted data from the lower GANs are not close to the real ones.
The prediction error propagates to higher GANs and leads to a large cumulative
error, which drastically degrades the classification accuracy. For example,
with 8 PMUs, the classification accuracy is 91.15% in GAN 2 and drops to
77.95% in GAN 3 due to the propagation of the error. Hence, it can be
concluded that more PMU data improve the classification accuracy. However,
when the number of PMUs is larger than a threshold, e.g., 12 in the IEEE
118-bus system, the classification accuracy remains fixed in higher level
GANs.

###  IV-D Discussions

Here, the potential applications of the developed HGAN-based TSA approach, its
limitations, and possible solutions are discussed. The developed HGAN-based
TSA utilizes the predicted transient data to determine the transient stability
status of a system. With the specific design of the generator and the GRU
network, the predicted voltage time series data retain the spatial and
temporal features of the real data. Examples of the predicted post-fault
voltage time series for both stable and unstable transients are given in Fig.
10. The relative rotor angles of the stable and unstable transient events are
shown in Fig. 10(a) and Fig. 10(b), respectively. The HGAN-based TSA has five
GANs, and thus can predict the voltage of the subsequent five time steps
$\\{\hat{v}(2),\cdots,\hat{v}(6)\\}$, using the measured post-fault voltage
data $v(1)$. It can be observed in Figs. 10(c) and 10(e) that the time series
patterns of the two types of real events are different. Compared with the real
transient events, only using the measured PMU voltage data from the first time
instant, i.e., $v(1)$, the HGAN-based model predicts the oscillation features
of these two events, and for the most part, generates similar time series data
as the real measurements. With the prediction capability, the developed HGAN-
based approach is not solely limited to TSA, but can also be deployed in other
applications (e.g., cascading failure analysis) that can benefit from the
ability to predict how transient events progress.

Although the developed TSA approach is promising for transient analysis, its
performance relies on the training data, like any other data-driven model.
Here, as the transient data are generated from simulations, the training
dataset is ensured to be balanced, i.e, the number of stable and unstable
transients is close. However, the real-world transient training datasets are
imbalanced, i.e., the ratio between the unstable and stable events is less
than 5%. This imbalance occurs since the unstable transient events are high-
impact but low-frequency events. An imbalanced training dataset may adversely
affect the classification accuracy. To address the problem of imbalanced data,
two strategies, i.e., data resampling and redesigning the classifier
algorithm, are suggested to improve the HGAN-based TSA method. The data
resampling methods, such as oversampling and undersampling methods, can
rebalance the ratio between the two types of events [44]. In addition to
resampling the training data, the classifier algorithm can be adjusted, such
that the learning process is more sensitive to the correct identification of
the unstable events. For instance, in the cost-sensitive method introduced in
[45], the misclassification cost is modified such that a higher cost is
assigned to the wrong prediction of the unstable events.

(a) Relative rotor angles of a real stable transient

(b) Relative rotor angles of a real unstable transient

(c) Real stable transient

(d) Predicted stable transient

(e) Real unstable transient

(f) Predicted unstable transient

Figure 10: Comparison between the real and predicted post-fault stable and
unstable transients. The developed TSA approach has 5 GANs and the number of
PMUs is 20. The PMU voltage measurements are normalized using a Min-Max
scaling method.

##  V Conclusion

This study developed a novel GAN-based TSA approach to promptly and accurately
assess the post-fault transient status of a power system using PMU-provided
voltage measurements. Unlike the conventional GAN model, the generator in this
study is redesigned, such that the modified GAN model predicts the sequence
data and learns the distribution of the real data. Benefiting from the
specific hierarchical structure of multiple GANs, the developed HGAN-based
model maintains the spatial and temporal features of the multivariate PMU time
series data. Therefore, with only one sample of PMU measurement, the developed
TSA approach improves the stability assessment accuracy. Case studies
conducted on the IEEE 118-bus system demonstrate that, with the help of the
predicted data, the TSA accuracy can reach 99.95%. Compared with the other
four machine-learning-based methods, i.e., decision trees, SVM, GRU, and LSTM,
the HGAN-based TSA achieves a higher classification accuracy and a shorter
response time. Additionally, to demonstrate the robustness of the HGAN-based
TSA, the impacts of measurement noise, location, and the number of available
measurements are investigated. It is observed that the prediction capability
of the HGAN-based TSA approach makes it robust to these variations and helps
improve the classification accuracy. Following proper offline training, the
developed TSA can be applied in near real-time to assess the post-fault
stability of a power system. Additionally, as the HGAN model enables learning
the spatial and temporal features of the system transients, the predicted
sequence data can be used for various applications that benefit from the
ability to predict the transient response of a system.

In the future, the classifier embedded in the generator can be redesigned to
be robust to imbalanced training datasets. The encoder and decoder can also be
added to the generator to further improve the predicted unstable transient
data.

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