After a fault has occurred, the
sound pole will be affected by the fault pole due to the presence of
line coupling and a certain fault current component will be induced.
Therefore, it is essential that fault pole selection is carried out by
selecting a suitable setting value.
When a positive fault occurs on the DC line, the distance between the
positive fault waveforms at 4 and 5 is smaller and the negative fault
waveforms are larger, so the ratio of the positive and negative fault
waveforms is smaller and smaller than the setting value; when a negative
fault occurs on the DC line, the distance between the positive fault
waveforms at 4 and 5 is larger and the negative fault waveforms are
smaller, so the ratio of the positive and negative fault waveforms is
larger and larger than the setting value; when a bipolar fault occurs,
the difference between the positive and negative fault waveforms is
smaller and the ratio of the positive and negative fault waveforms is
around the setting value. When a bipolar fault occurs, the difference
between the positive and negative fault waveforms is smaller, and the
ratio of the positive and negative fault waveforms is around the setting
value. Therefore, the ratio of the positive and negative fault waveform
distances is used to construct a pole selection criterion, which can be
used to identify fault and sound poles.
Measurement of current fault component waveform similarity
Considering the following three characteristics of the current fault
component, the similarity of its waveform is measured comprehensively:
The overall distribution characteristic, i.e., the overall similarity of
the fault waveform is reflected by the magnitude of the distance of the
sampling points at the same time of the curve;
(2) Overall dynamic characteristics, i.e., similarity is measured by
comparing the corresponding fault waveforms in the sampling time period;
(3) Local dynamic characteristics, i.e., similarity is measured by
comparing the average rate of change of the fault waveform at the same
sampling interval.
The Euclidean distance is used to
measure the true distance between two points in m-dimensional space. It
can reflect the overall distribution characteristics of waveform
similarity.
The fault current components at protection installation points 4 and 5
are extracted and defined as XP4, XN4, XP5 and XN5 respectively, where
XP4 = (x1, x2, …, xn), XN4 = (y1, y2, …, yn), XP5 = (z1, z2, …,
zn), XN5 = (q1, q2, …, qn). Then the Euclidean distance of the
waveform at the protection installation points on both sides can be
obtained as equations (1) and (2).
\begin{equation}
D_{\text{WP}}\left(X_{P4},X_{P5}\right)=\sqrt{\sum_{i=1}^{n}\left(x_{i}-z_{i}\right)^{2}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)\nonumber \\
\end{equation}\begin{equation}
D_{\text{WN}}\left(X_{N4},X_{N5}\right)=\sqrt{\sum_{i=1}^{n}\left(y_{i}-q_{i}\right)^{2}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)\nonumber \\
\end{equation}Where DWP and DWN represent the waveform Euclidean distance between the
positive and negative poles of the protection device points on both
sides respectively. n represents the number of sampling points, which
can be selected according to actual accuracy requirements.
Unlike Euclidean distance, DTW (Dynamic Time Warping) does not carry out
distance calculation strictly according to the corresponding value of
the same sampling point of the sequence. It adopts the idea of dynamic
programming to adjust different sampling points of sampling sequence,
explore the relationship between corresponding elements, and optimize
the curved path.
The algorithm can well describe the overall dynamic characteristics of
curves and measure the overall shape similarity between sequences. It is
suitable for the situation where the fault waveforms of two converter
stations have good overall similarity, but they are not completely
aligned at the sampling points.
For the sampling sequence XP4 and
XP5, the n × m distance matrix DP is constructed, where the elements are
DP (i, j) in equation (3); for XN4 and XN5 the distance matrix DN is
constructed, where the elements DN (i, j) in equation (4). Equation (3)
represents the Euclidean distance between the sampling sequence XP4 and
the XP5 sequence points xi and zj , and Equation (4) represents the
Euclidean distance between the sampling sequence XN4 and the XN5
sequence points yi and qj.
\begin{equation}
D_{P}\left(i,j\right)=\sqrt{{(x_{i}-z_{j})}^{2}}\ \ \ \ \ \ 1\leq i\leq n,1\leq j\leq m\ \ \ \ \ \ \ \ \ \ \ \ (3)\nonumber \\
\end{equation}\begin{equation}
D_{N}\left(i,j\right)=\sqrt{{(y_{i}-q_{j})}^{2}}\ \ \ \ \ \ 1\leq i\leq n,1\leq j\leq m\ \ \ \ \ \ \ \ \ \ \ \ (4)\nonumber \\
\end{equation}The data set consisting of each
group of adjacent elements in DP (i, j) and DN (i, j) is called the
curved path of the collected waveform and is denoted as L = {l1 , l2 ,
⋯ls , ⋯, lk } and R = {r1 , r2 , ⋯rs , ⋯, rk }. Where the element ls
is the coordinate of the sth point on the path L, i.e., ls = (i, j), and
the element rs is the coordinate of the sth point on the path R, i.e.,
rs = (i, j). At the same time, the paths L and R should satisfy the
following two-part constraints. One is that the selected paths need to
contain all sampling points, i.e., l1 = (1, 1) and ls = (n, m), the
other is that each sampling point needs to match with the adjacent
sampling points, i.e., if ls = (i, j), then ls+1 = (a, b) satisfies 0 ≤
a - i ≤ 1 and 0 ≤ b - j ≤ 1. Therefore, the DTW distance between the
sampling sequences XP4 and XP5 is defined by equation (5), and the DTW
distance between the sampling sequence XN4 and XN5 is defined as
equation (6).
\begin{equation}
D_{\text{Pdtw}}\left(X_{P4},X_{P5}\right)=\operatorname{}{\sum_{s=1}^{k}{D\left(L_{s}\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (5)}}\nonumber \\
\end{equation}\begin{equation}
D_{\text{Ndtw}}\left(X_{N4},X_{N5}\right)=\operatorname{}{\sum_{s=1}^{k}{D\left(R_{s}\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (6)}}\nonumber \\
\end{equation}Where D(Ls ) and D(Rs ) are the cumulative distances of the curved paths
L and R. DPdtw and DNdtw represent the waveform DTW distances between
the positive and negative poles of the protection installation points on
each side, respectively.
4.3 Local dynamic mapping
Because the sampling sequences of XP4, XN4, XP5 and XN5 are discrete,
considering the accuracy of the algorithm, it can be converted into a
morphological sequence of XP4’, XN4’, XP5’ and XN5’ with n-1 length.
Among them, there exists:
XP4=(x1’,x2’,…xn’),XN4=(y1’,y2’,…yn’),XP5=(z1’,z2’,…zn’),XN5=(q1’,q2’,…qn’).
These morphological sequences can better describe the local dynamic
characteristics of current fault component waveforms, reflect its trend
information at each sampling interval. Where, the element xi’ is shown
in (7), yi’, zi’, qi’ and xi’ are the similar expressions.
\begin{equation}
x_{i}^{{}^{\prime}}=\frac{x_{i+1}-x_{i}}{t}\text{\ \ \ \ \ \ \ \ }i=1,2,\cdots,n-1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (7)\nonumber \\
\end{equation}In (7), ∆t is the time interval.
4.4 Entropy Weight Method
Entropy weight method is an objective weight method, which uses entropy
to express the characteristics of the amount of information, that is,
the greater the difference between each evaluation object is, the more
information it contains, the smaller its entropy is. The entropy weight
method only highlights the local difference, that is, the greater the
level difference between the evaluation objects is, the greater the
weight of the index is and the greater the impact on the evaluation
results is.
4.5 Similarity Description of Overall Current Fault Component
Waveform
Based on the analysis of the fault characteristics of the current fault
waveform in Section 3, combining Euclidean distance and DTW distance, a
method of measuring the similarity of current fault waveforms with
overall and local characteristics is proposed. Therefore, the distance
of positive and negative current fault waveforms at the protection
installation points on both sides are shown in equations (8) and (9).
\begin{equation}
D_{\text{allP}}=\alpha D_{\text{WP}}\left(X_{P4},X_{P5}\right)+\beta D_{\text{Pdtw}}\left(X_{P4},X_{P5}\right)+\gamma D_{\text{Pdtw}}\left(X_{P4}^{{}^{\prime}},X_{P5}^{{}^{\prime}}\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (8)\nonumber \\
\end{equation}\begin{equation}
D_{\text{allN}}=\alpha D_{\text{WN}}\left(X_{N4},X_{N5}\right)+\beta D_{\text{Ndtw}}\left(X_{N4},X_{N5}\right)+\gamma D_{\text{Ndtw}}\left(X_{N4}^{{}^{\prime}},X_{N5}^{{}^{\prime}}\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (9)\nonumber \\
\end{equation}Where DallP and DallN represent the distance between the positive and
negative current fault waveforms at the protection installation point on
both sides respectively. The smaller the value is, the higher the
similarity between the two sides of the current fault waveform is. DWP
(XP4 , XN4 ) and DWN (XP4 , XN4 ) reflect the overall distribution
characteristics of the waveform respectively; DPdtw (XP4 , XN4 ) and
DNdtw (XP4 , XN4 ) reflect the overall dynamic characteristics of the
waveform respectively; DPdtw (XP4, , XN4, ) and DNdtw (XP4, , XN4, )
reflect the local dynamic characteristics of the waveform respectively.
α, β and γ are the weights of each measure of distance respectively,
which are taken as 0.4, 0.3 and 0.3 according to the actual sampling
accuracy.
Protection Principal Implementation Process
5.1 Current Fault Waveform Extraction
When a fault occurs, the current will change and the current fault
waveform will show different change features depending on the fault
type. From the current fault characteristics in Section 3.1, the current
fault component can be extracted by equation (10).
\begin{equation}
I_{s}\left(t\right)=I_{S}\left(t\right)-I_{S}\left(t-T\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (10)\nonumber \\
\end{equation}Where: t is the sampling moment; Is (t) is the actual current value at
t; T is the selected sampling moment before the fault occurs, generally
being taken n sampling period before the fault occurs. Considering the
stability of the system operation, it is taken two sampling periods
before the fault occurs. ΔIs (t) is the current fault component
extracted from the protection measurement installation points at t.
5.2.1 Protection start criterion: When a fault occurs, the current
fault component rises or falls and the protection start criterion can be
constructed based on the magnitude of the current change. By extracting
the change of the current fault component at each protection measurement
installation, the protection start criteria are given in equation (11).
\begin{equation}
\left\{\ \begin{matrix}I_{P}>K_{set1}=0.1kI_{n}\\
I_{N}<K_{set2}=-0.1kI_{n}\\
\end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (11)\right.\ \nonumber \\
\end{equation}Where: ΔIP and \(I_{N}\ \)are the
current fault components of the positive and negative currents
respectively; Kset 1 andKset 2 are the threshold values of
the starting current of the positive and negative currents respectively;k is the reliability coefficient, generally ranging from 1.2 to
1.5; In is the rated current. If the start
criteria are satisfied, further identification between fault and
non-fault intervals is carried out.
Using equations (8) and (9) to calculate the combined distance between
the positive and negative current fault waveforms at each protection
measurement installation respectively, denoted as DPX and DNX, the
identification criterion between the fault and non-fault intervals is
determined by equation (12).
\begin{equation}
\left\{\begin{matrix}\left|D_{\text{PX}}\right|<D_{\text{set}1}\\
\left|D_{\text{NX}}\right|<D_{\text{set}1}\\
\end{matrix}\right.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (12)\nonumber \\
\end{equation}Where: X is the variable and takes the value of each protection
measurement installation point labeled as 1-8, that is P1-P8 and N1-N8
points shown in Fig.1, |·| indicates the modal value.
When a fault occurs in different converter station intervals, the
waveform distances DPX and DNX of positive and negative current fault
components in the corresponding fault intervals are smaller than the
preset setting values. However, the opposite is true for the non-failure
interval, the waveform distance values of them are greater than the
preset setting value.
Considering the topology structure and operating conditions of DC power
grid, the setting value Dset1 is taken as 0.8. When the current fault
component and waveform distance between the two measurement points
satisfy both equations (11) and (12), the protection of the
corresponding fault interval is activated and the non-fault interval
protection is blocked. According to equation (11) and equation (12), the
fault interval can be reliably identified.
5.2.2 Internal and External Fault Identification Criteria: After
ensuring that the non-fault line can operate correctly, further internal
and external fault identifications are carried out.When the fault
interval is determined, taking the flat-wave reactor as the boundary
condition, the stations can be
classified
into DC line segment and non-DC line segment. To ensure reliable
operation of the protection on the DC line, faults in the non-DC line
section should be excluded.
From the analysis in Section 3.2.1, it can be seen that when a fault
occurs in the DC line, the fault current waveform at the protection
installation between the converter stations changes, showing positive
correlation, high similarity and small fault distance; when a fault
occurs outside the DC line, the fault current waveform at the protection
installation of the converter stations changes, showing negative
correlation, low similarity and large fault distance;
Extract the minimum distance value of current fault waveform of the DC
line between the two-end converter stations, as shown in equation (13).
\begin{equation}
D_{1}=\min{(\left|D_{\text{allP}}\right|,\left|D_{\text{allN}}\right|)}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (13)\nonumber \\
\end{equation}According to the difference of current fault waveform distance between
internal and external faults, the internal and external faults criterion
can be constructed as equation (14)..
\begin{equation}
D_{1}<D_{\text{set}2}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (14)\nonumber \\
\end{equation}Where: Dset2 is the setting threshold value. For the selection of Dset2
, it should be able to reliably identify external metallic ground faults
and internal high resistance faults. Due to the difference between
internal and external faults occurring, a reasonable value of Dset2 is
1.1, based on a large number of simulations and taking into account
certain margins. If equation (14) is satisfied, the fault is judged as
an internal fault, i.e., the fault occurs on the DC transmission line.
Otherwise, it is an external fault, i.e., the fault does not occur on
the transmission line.
5.2.3 Fault Pole Identification Criteria: After judging that the
fault occurred in the DC line interval, the fault pole was further
identified.
Based on the difference between positive and negative pole faults, the
fault waveform distances of the two poles are extracted respectively and
the ratio of the positive and negative fault waveform distances is
further calculated as equation (15).
\begin{equation}
K_{2}=\frac{\left|D_{\text{allP}}\right|}{\left|D_{\text{allN}}\right|}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (15)\nonumber \\
\end{equation}Due to the coupling effect, the fault pole will cause the not-fault pole
to induce a certain fault component, and there will be a certain change
in the fault current waveform and an increase in the waveform distance
for both positive and negative poles. However, due to boundary effects
and the relatively small waveform distance of the non-faulted pole, the
faulted pole can be reliably determined by selecting a reasonable
setting value of K2. The constructed fault pole criterion is shown in
(16).
\begin{equation}
\left\{\begin{matrix}K_{2}\leq 0.6\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{The}\ \text{positive}\ \text{fault}\\
0.6<K_{2}\leq 1.5\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{Bipolar}\ \text{fault}\\
K_{2}>1.5\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{The}\ \text{negative}\ \text{fault}\\
\end{matrix}\right.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (16)\nonumber \\
\end{equation}Where: the setting values 1.5 and 0.6 are adjusted according to the
reliability and sensitivity requirements.
5.3 Protection process flowchart
Combined with the above fault characteristics analysis and protection
criteria, the specific protection process is as follows:
(1) Run the system model, debug the installed protection measuring
devices, monitor the electrical quantity signals during normal operation
in real time, and observe whether there is abnormal.
(2) Collect the transient electrical quantity signals at the converter
stations, and extract the current fault component of the positive and
negative DC transmission line. Then the fault can be judged by the
protection start criterion. If the criterion is satisfied, it can be
known that a fault occurs on the DC line.
(3) Calculate the comprehensive distances of the current fault component
waveform of positive and negative poles between the two converter
stations, which are denoted as DPX and DNX. Then compare them with the
protection setting value Dset1 to determine the fault occurrence
interval. When DPX is smaller than the protection setting value Dset1
and DNX is also smaller than the protection setting value Dset1, the
fault occurs between the two converter stations, otherwise it occurs
between other converter stations.
(4) Compare the minimum value D1 of the comprehensive distance of the
positive and negative current fault component waveform between the fault
stations with the value of Dset2 to further determine the internal and
external faults. When D1 is smaller than the value of Dset2, the current
fault component waveform similarity between the two stations is high and
the internal fault occurs. When D1 is larger than the value of Dset2,
the current fault component waveform similarity between the two stations
is low and the external fault occurs.
(5) Calculate the ratio K2 of the comprehensive distance between the
positive and negative current waveform, then compare it with the setting
value to further identify the fault poles.
When K2 is less than or equal to 0.6, the positive fault occurs;
When K2 is greater than 1.5, the negative fault occurs;
When K2 is larger than 0.6 and less than or equal to 1.5 bipolar fault
occurs.
The overall protection process is shown in Fig.5.