# Bayesian Ridge Regression Based Model to Predict Fault Location in HVdc
Network

Timothy Flavin2, Thomas Steiner2, Bhaskar Mitra31 and Vidhyashree Nagaraju2

2Tandy School of Computer Science, The University of Tulsa, Tulsa, OK 74104
USA  3Pacific Northwest National Laboratory, Richland, WA 99354 USA
1Corresponding author: Bhaskar Mitra (email<EMAIL_ADDRESS>
###### Abstract

This paper discusses a method for accurately estimating the fault location in
multi-terminal High Voltage direct current (HVdc) transmission network using
single ended current and voltage measurements. The post-fault voltage and
current signatures are a function of multiple factors and thus accurately
locating faults on a multi-terminal network is challenging. We discuss a novel
data-driven Bayes Regression based method for accurately predicting fault
locations. The sensitivity of the proposed algorithm to measurement noise,
fault location, resistance and current limiting inductance are performed on a
radial three-terminal MTdc network. The test system is designed in Power
System Computer Aided Design (PSCAD)/Electromagnetic Transients including dc
(EMTdc).

###### Index Terms:

Bayes Ridge Regression, Fault location, HVdc, Modular multi-level converter,
MTdc network

##  I Introduction

High Voltage Direct Current (HVdc) transmission corridors are widely used for
long distance transmission. It has emerged as a leading contender compared to
High Voltage Alternating Current (HVac) for bulk power off-shore and on-shore
transmission. They are considered as a viable form of energy transportation
for renewable energy transportation and interconnection of multiple grids of
different frequencies. The Voltage Source Converter (VSC) design allows for
the implementation of interconnected multi-terminal dc (MTdc) networks with
bi-directional power flow. The Modular Multilevel Converter (MMC) has emerged
as a popular choice for VSC-HVdc networks due to salient features of better
scalability and higher operational efficiency [1]. Accurate fault location for
HVdc transmission networks are essential to reduce the downtown and improve
reliability of the system. Accurate location of faults on the network is thus
essential for repair and restoration process.

Several fault location techniques in HVdc systems namely (1) spectral
analysis; (2) travelling wave; (3) impedance method and (4) machine learning
approaches have been discussed previously. Methods incorporating S-transform
to to locate faults have been discussed in [2], although the results of the
proposed method have not been verified with measurements noise. Travelling
based methods are employed on a wide scale for accurate fault location using
(1) single ended and (2) double ended measurements [3].

Double ended measurements provide higher accuracy of fault locations using
synchronized current and voltage measurements. This method involves setting up
and maintenance of robust communication networks and expensive time
synchronized devices [4]. On the contrary single ended measurements are more
convenient. They are cheap but tend to provide inaccurate results as devices
do not have the capability to detect the reflected peak [5]. Its performance
is driven by fault resistance and other network parameters. The reflected
surge waves are weaker thus making their detection difficult. Inability to
capture the reflected surge wave peaks would provide provide inaccurate fault
locations.

Some other methods of fault location has been proposed using digital signal
processing, they require devices with high sampling frequency to achieve the
desired results. Methods involving time-frequency analysis of the fault
transients have been performed using wavelet transform [6], implantation of
such algorithms would require double ended synchronized measurements. Passive
methods of fault locations have been suggested in [7] and [8], although their
efficacy for a real-field example are yet to be demonstrated.

In this paper, we propose a single ended fault location technique using
Bayesian ridge regression coupled with systematic application of various data
pre-processing methods. The single-ended post-fault voltage and current
measurements are used as input to the Bayesian ridge regression model. To
validate the selection of BRR and sequence of pre-processing methods,
combination of seven machine learning methods, five feature extraction, and
two dimensionality reductions were applied with range of values for
configurable parameters. Results suggested that Bayesian ridge model
demonstrated a minimum of 50-70% better predictive capability, while
maintaining third ranking based on information theoretic measures.

The remainder of the paper is organised as follows: Section II describes the
test system and generation of data. Section III describes the Bayesian
Regression methodology and other preprocessing steps. Section IV discusses the
parameters required to assess the model performance. Section V discusses the
results and Section VI concludes the paper with major findings.

##  II Test System and Data Generation

###  II-A Test System Design

The modeling of the three terminal MMC in this paper is based on the design
suggested in [9]. The MMC model consists of 400 half bridge sub-modules (HBSM)
per arm. Hybrid discretization and relaxation algorithms described in [9] are
used to define the numerical stiffness in the differential algebraic
equations. Details about the system parameters are provided in Table I.

Table I: System Parameters | Parameters | Value
---|---|---
ac side | Voltage (L-L RMS) | 333 kV
Length of transmission line 1 & 3 | 100 km
Length of transmission line 2 | 150 km
System Frequency | 60 Hz
Transmission line resistance | 0.03206 $\Omega$/km
dc side | Voltage (L-L) | 640 kV
Length of transmission line | 1000 km
Transmission line resistance | 0.03206 $\Omega$/km
MMC capacity | 1 GW

The model of a radial three-terminal MTdc symmetric monopole is shown in Fig.
1. The system is equipped with hybrid dcCB at the MMC terminals. The dc
transmission lines are designed as frequency dependent models having $6$
conductors with a vertical spacing of $5m$ and horizontal spacing of $10m$
between the conductors.

Figure 1: Radial multi-terminal dc network

###  II-B Data Description

As shown in Fig. 1, a 640 kV dc, radial MTdc transmission network was utilized
for the single-ended fault location. The parameters for the test system are
listed in Table I. Fault events were simulated on various sections of the dc
transmission network using PSCAD/EMTdc. Faults were simulated at a separation
of 25km across the network. Bus faults are typically rare and were not
considered for this study. Scenarios were generated by varying the current
limiting inductance (1mH - 200mH), fault resistance (0.01$\Omega$ \-
200$\Omega$), and change of loads in the transmission system (non-fault
events). The current and voltage at the terminals were sampled at $10kHz$.

Fig. 2 shows the raw current and voltage data generated for this study. Dark
color in the color bar indicates fault simulations that were closer to the
beginning of the network and lighter colored lines represent simulations that
were closer to the end of the network.

(a)

(b)

Figure 2: Single ended measurements

##  III Methodology

This section presents the proposed Bayesian Ridge Regression based model to
predict fault locations with minimum error threshold within MTdc networks.

Five simple pre-processing techniques are considered to process the MTdc data
to predict location of faults based on Bayesian ridge regression. The data is
split into k-folds to test the robustness of the proposed approach and an
average of error was recorded at the end of the process.

Individual stages of processing are described below in the order in which they
are applied:

### Step-1: Downsampling (DS)

Unbalanced datasets tend to have more observations for the response variable
in one category than others and this often produces undesirable results in
predicting or classifying unseen observations. To address this, the data is
downsampled by a integer factor, where one-out-of-$n$ observations is
selected.

### Step-2: Principal Component Analysis (PCA)

Principal component analysis reduces the dimensionality of the data by
extracting prominent features [10]. PCA is defined as the “linear projection
that minimizes the average projection cost, which is the mean squared distance
between the data points and their projections” [11]. Mathematically, minimize

$J=\frac{1}{N}\sum_{i=1}^{N}||x_{n}-\tilde{x}_{n}||^{2}$ (1)

where $\tilde{x}_{n}$ is the average of the data set. The main drawback of PCA
is that the produced features no longer hold real world meaning because they
are now linear combinations of other features.

### Step-3: Low Pass Filter (LPF)

The first two pre-processing steps allow only high frequency signal
components, which could lead to data misinterpretation and is commonly known
as aliasing. To address this, a finite impulse response (FIR) filter is
applied as an anti-aliasing filter. It is a low pass filter, which computes
every $n^{th}$ output through a decimating FIR filter computed for the
$m^{th}$ sample as

$y[m]=\sum_{i=0}^{k-1}x[mN-i]\bullet h[i]$ (2)

where $h[i]$ is the $i^{th}$ response of length $k$.

### Step-4: Fast Fourier Transform (FFT)

The frequency domain components of a time-series can be extracted using
Discrete Fourier Transform (DFT) [12]. FFT is useful to perform the DFT of a
sequence. FFT performs the computation of the DFT matrix as a product of
sparse factors. The DFT for such a sequence can be given as (3),

$X[k]=\sum_{n=0}^{N-1}x[n]e^{-j2\pi kn/N}$ (3)

where _N_ is the length of the signal. The sampling frequency of the signal is
$10kHz$, the maximum represented frequencies are half of the sampling
frequency. We try to capture all the representative frequencies in that range
that direct the accurate detection of a fault location.

### Step-5: Data Normalization

As a last stage in pre-processing, the data is normalized to standardize it
throughout the data. Normalization process is intended to eliminate any
unstructured and redundant data to ensure logical data storage, thus improving
the quality of the data.

Given data $X$, normalization utilizes $l^{2}$-norm approach as

$\mathbf{X}=\frac{x-\mu}{|\mathbf{X}|}$ (4)

where $\mu$ is the mean, $\mathbf{X}$ is the $l^{2}$-norm, and
$|\mathbf{X}|=\sqrt{\left(\sum_{i=1}^{n}|x_{k}|^{2}\right)}$.

###  III-A Bayesian Ridge Regression (BRR)

Bayesian ridge estimates a probabilistic model of the regression problem that
includes regularization parameters.

Given observations $(x_{i},y_{i})$, $i=1,\dots,m$ to be considered as training
sequence and $x_{j}$ of length $k$ as a test fold split during the k-fold
splitting at the beginning, the objective is to find $y_{k}$. In practical
applications, given $(x_{i},y_{i})$, $i=1,\dots,n-1$ as training sequence, the
objective is to predict $y_{n}$ for $x_{n}$ with smallest error.

The model assumes that the distribution of objects and labels, $(x_{i},y_{i})$
respectively are generated by the rule [13]

$y_{i}=w\times x_{i}+\beta$ (5)

where $w$ is a random vector distribution as the Gaussian distribution
parameterized by mean and covariance matrix as $N(0,(\sigma^{2}/a)I)$ and
$\beta$ is distributed with $N(0,\sigma^{2})$.

The conditional distribution for the prediction label $y_{n}$ for the test
object $x_{n}$ is

$N(\widehat{y_{n}},(1+g_{n})\sigma^{2})$

.

Here $g_{n}:=x_{n}^{^{\prime}}(X^{\prime}X+aI)^{-1}x_{n}$, where $X$ is the
design matrix for the training sequence.

Now, the Bayesian prediction interval is

$(B_{L},B^{U}):=\left(\widehat{y}_{n}\sqrt{(}1+g_{n})\sigma_{\epsilon/2}\widehat{y}_{n}+\sqrt{(}1+g_{n})\sigma_{\epsilon/2}\right)$
(6)

##  IV Goodness-Of-Fit Measures

This section lists measures used to assess model performance.

###  IV-A Mean Absolute Percentage Error (MAPE)

Mean absolute percentage error calculates the error as a percentage of how far
off the predicted value is from the actual value

$MAPE=\frac{1}{n}\times\sum_{i=1}^{n}\bigg{\lvert}\frac{\widehat{x}_{i}-x_{i}}{x_{i}}\bigg{\rvert}\times
100$ (7)

where $\widehat{x}_{i}$ is the predicted value and $x_{i}$ is the actual
value.

###  IV-B Mean Absolute Error (MAE)

The performance is measured as the mean absolute (MAE) value of the prediction
errors in kilometers from the target location, defined as follows,

$MAE=\frac{\sum_{i=1}^{n}\left|\widehat{x}_{i}-x_{i}\right|}{n}$ (8)

where $\widehat{x_{i}}$ is the predicted fault location, $x_{i}$ is the actual
location and n is the number of data points.

###  IV-C Predictive Ratio Risk (PRR)

The predictive ratio risk of a model is

$PRR=\sum_{i=k+1}^{n}\left(\frac{\widehat{x}_{i}-x_{i}}{\widehat{x}_{i}}\right)^{2}$
(9)

where the term in the denominator penalizes underestimation of the number of
defects more heavily than overestimation.

###  IV-D Predictive Power (PP)

The predictive power of a model is

$PP=\sum_{i=k+1}^{n}\left(\frac{\widehat{x}_{i}-x_{i}}{x_{i}}\right)^{2}$ (10)

where the term in the denominator penalizes overestimation of the number of
defects.

##  V Results and Discussion

This section demonstrates the proposed model through examples when applied to
data presented in Section II-B. The first example compares exhaustive
combination of data pre-processing and machine learning models with the
proposed approach and the second example summarizes the performance of the
Bayesian ridge regression based model through comparison of goodness-of-fit
measures including MAPE, MAE, PRR, and PP.

###  V-A Experimental Setup and Model Selection Validation

To validate the selection of methods used in the proposed model, an exhaustive
combination of various pre-processing methods and regression methods was
applied to the data considered in Section II-B. A combination of five feature
extraction methods, two dimensionality reduction techniques, and seven machine
learning models were applied in total.

For feature extraction, low pass filter, fast Fourier transform, sklearn’s
Standard Scalar with mean and unit variance set to 0 and 1 respectively,
$l^{2}$-norm based normalization method as described in Section III, and a
square root transformation are considered. For dimensionality reduction, down
sampling and principal component analysis are considered. To validate the
proposed pre-processing model’s efficiency in Section III, combinations of the
seven pre-processing methods with various configurable parameter values were
applied. For the techniques that do not have parameters, the list only
contained two elements, True and False, corresponding to whether or not to
apply the given technique. In order to adhere to laws about loss of
information such as the Nyquist-Shannon sampling theorem [14], preprocessing
techniques were applied in the same order every time. The parameters tested
are listed below in the order that they were applied in each run of the
algorithm.

  * •

LPF: `[500, 300, 250, 200, 150, 100, 50]`

  * •

Down sampling Factor: `[1,3,5,10,100]`

  * •

Fourier Transform: `True, False`

  * •

Data Normalization: `True, False`

  * •

Principal Components: `[no PCA,4,8,12,16,28]`

  * •

Square Root: `True, False`

  * •

Standard Scalar: `True, False`

Applying both standard scalar and data normalization was avoided since this
would result in two levels of normalization. In addition, if the specified
low-pass filter is greater than the Nyquist frequency after being downsampled,
the filter would be useless because the downsampled data would not be sampled
at a high enough frequency to record signals up to the filter frequency.

For each combination of preprocessing strategies, the transformed input data
was passed to each of the seven machine learning regressors [10, 15]
considered including Bayesian Ridge (BayR), Support Vector (SVR), K-Nearest
Neighbors (KNNR), Decision Tree (DTree), Extreme Gradient Boosting (XGB),
Gradient Boosting (GB), and Multi-Layer Perceptron (MLP). This process was
done for both the voltage and current data sets.

Models could be sensitive to traditional train-test split due to smaller
sample size. To account for this this sensitivity, models were validated using
k-fold cross validation with four folds. The error of each model was stored as
the average of the MAE on the test sets of each of the four folds.

Fig. 3 shows mean average error after testing every valid combination of
preprocessing strategies and machine learning models on the input data. The
results were sorted to find the models and pre-processing combinations which
produced the lowest errors. Bayesian ridge regression produces the lowest
errors consistently.

Figure 3: Pre-processing pipeline MAEs grouped by model

###  V-B Goodness-of-fit measure comparison

Table II indicates decision tree as a better model based on information
theoretic measures such as MAPE and MAE, while predictive measures recommend
Bayesian Ridge regression as a better predictive model. BRR presents a
comparatively smaller information theoretic measures with PRR and PP with at
least 50% and 70% better predictive performance compared to all other models.

Table II: Comparison of goodness-of-fit measures of various models Model | MAPE | MAE | PRR | PP
---|---|---|---|---
BayR | 21.77 | 80.68 | 0.4748 | 0.8405
SVR | 106.1 | 219.9 | 2.607 | 63.98
KNNR | 44.80 | 111.37 | 1.135 | 1.209
DTree | 11.62 | 34.03 | 1.361 | 4.981
XGB | 15.84 | 38.11 | 1.15 | 10.08
GB | 34.16 | 76.62 | 0.946 | 5.335
MLP | 55.20 | 117.5 | 1.502 | 14.93

Fig. 4 shows predictions of BRR compared to the actual fault locations for the
current dataset with DS = 3, PCA = 12, LPF = 150, followed by FT and data
normalization. Blue line indicates the ideal relationship between actual and
estimated values.

Figure 4: Bayesian Ridge regression: Predictions vs Actual

Some other models, such as the Decision tree, scored nearly perfectly on the
training data while not generalizing at all on the test data supporting
results presented in Table II. This failure to generalize may also be seen in
some of the more competitive models, such as the MLP. For these reasons, BRR
is the suggested as the most desirable of the models explored here.

##  VI Conclusion

In this paper, we proposed a novel single ended fault location technique using
Bayes Regression. Post fault current and voltage measurements are recorded
that are utilized to develop a probabilistic model to accurately predict fault
locations in HVdc networks. We pre-process the recorded measurements using (1)
Downsampling, this helps to reduce the unbalance in the raw data that could
produce biased results; (2) PCA, helps to extract the prominent fault features
from the data; (3) LPF, this helps to reduce errors caused by aliasing; (4)
FFT, helps to extract the prominent fault frequencies essential for estimating
fault location and (5) Normalization, eliminates any unstructured or redundant
data. The pre-processed data is utilized to develop a probabilistic regression
model using Bayesian ridge. The performance of the prediction model is
evaluated using several statistical measures discussed in Section IV. The
obtained results demonstrates that BRR develops an effective probabilistic
model of the data that provides a better predictive performance compared to
other methods for accurately locating faults in multi-terminal networks.

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