3.3 Fault Pole Selection
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.
4.1 Euclidean Distance
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.
4.2 DTW distance
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 Protection criteria
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.