Topology Identification in Distribution Networks using Harmonic Synchrophasor Measurements

In general, dedicated meters are installed in DNs to monitor the harmonic current injections of large nonlinear loads such as large DGs. On this account, we assume that the harmonic current measurements are available for topology identification. Let $I_{i}^{h}$ denote the current of the $h^{\rm th}$ harmonic component associated with the harmonic source located at node $i$. According to the Kirchhoff’s Current Law (KCL) we have:

where $I^{h}_{(k,j)}$ is the $h^{\rm th}$ harmonic current on branch $(k,j)$; $\mathcal{K}$ is the set of nodes containing harmonic sources; and $\mathcal{N}_{k}$ is the set of nodes connected to the node $k$. For the nodes with no harmonic source, we have:

where $\mathcal{N}$ is the set of all nodes in the DN.

As was discussed in Section II, detecting the paths of harmonic currents is vital in identifying the status of the switches in the DN. To this end, one needs to examine whether or not a branch carries a harmonic current. This can be done by looking into the numerical value of $I_{(i,j)}^{h}$. More precisely, if $I^{h}_{(i,j)}$ is larger than the magnitude of the smallest-magnitude combination of all existing harmonic sources in the DN, i.e. the parameter $z$, then the branch must be on a harmonic current path. To simplify the analysis, we make two adjustments. First, instead of assessing the magnitude of harmonic current $I_{(i,j)}^{h}$ we choose to assess the absolute values of the real and imaginary parts of the harmonic current $I_{(i,j)}^{h}$. Second, instead of using the parameter $z$ in the comparisons, we use a threshold parameter $c$ that is less than the parameter $z$. To this end, we first obtain the absolute values of the real and imaginary parts of the harmonic current $I_{(i,j)}^{h}$ as follows:

where $\mathcal{B}$ is the set of all branches; $q^{h,r}_{(i,j)}$ and $q^{h,i}_{(i,j)}$ are binary variables; and $X^{h,r}_{(i,j)}$ and $X^{h,i}_{(i,j)}$ are non-negative variables. If the real or imaginary part of $I^{h}_{(i,j)}$ is positive, then $q^{h,r}_{(i,j)}$ or $q^{h,i}_{(i,j)}$ must be 1 and $X^{h,r}_{(i,j)}$ or $X^{h,i}_{(i,j)}$ must be equal to the real or imaginary part of $I^{h}_{(i,j)}$, respectively. If the real or imaginary part of $I^{h}_{(i,j)}$ is negative, then $q^{h,r}$ or $q^{h,i}$ must be 0 and $X^{h,r}_{(i,j)}$ or $X^{h,i}_{(i,j)}$ must be equal to the absolute value of the real or imaginary part of $I^{h}_{(i,j)}$, respectively. All in all, we see that the absolute values of the real and imaginary parts of $I^{h}_{(i,j)}$ are $X^{h,r}_{(i,j)}$ and $X^{h,i}_{(i,j)}$, respectively.

Next, we compare the summation of the variables $X^{h,r}_{(i,j)}$ and $X^{h,i}_{(i,j)}$ against the threshold $c$ to see whether or not the branch $(i,j)$ falls on any harmonic current path. The harmonic current flowing through a branch may be the combination of several harmonic sources at different nodes of the DN. If the branch $(i,j)$ is not on any harmonic current path, then we must have $X_{(i,j)}^{h,r}+X_{(i,j)}^{h,i}\leq c$. In contrast, if the branch $(i,j)$ is on a harmonic current path, we must have $X_{(i,j)}^{h,r}+X_{(i,j)}^{h,i}\geq c$. The comparison between $X_{(i,j)}^{h,r}+X_{(i,j)}^{h,i}$ and the threshold parameter $c$ can be mathematically formulated by the following constraint:

where $b^{h}_{(i,j)}$ is a binary variable. From (7), the numerical values of 0 and 1 for the binary variable $b_{(i,j)}^{h}$ translate to the inequalities of $X_{(i,j)}^{h,r}+X_{(i,j)}^{h,i}\leq c$ and $X_{(i,j)}^{h,r}+X_{(i,j)}^{h,i}\geq c$, respectively. Consequently, numerical values of $b_{(i,j)}^{h}=0$ and $b_{(i,j)}^{h}=1$ correspond to the branch $(i,j)$ not being on any harmonic current path and being on at least one harmonic current path, respectively.

From the discussion in section II, we know that a numerical value of $1$ for the binary variable $b_{(i,j)}^{h}$ testifies the in-service status of the branch $(i,j)$, but a numerical of $0$ for the binary variable $b_{(i,j)}^{h}$ doesn’t indicate the in-service or out-of-service status for the branch. As a result, additional information is needed to determine the status of all the branches. This can be done by looking into the fundamental currents measurements.

According to the KCL we have:

where $I_{(i,j)}$ is the fundamental current on branch $(i,j)$; and $I_{i}$ is the fundamental current injection into the node $i$, which can be obtained from the pseudo-measurements, i.e., the measurements estimated from the seasonal load curves. The status of the branch $(i,j)$ can be modeled by the following constraint:

where $M$ is a large number selected arbitrarily and $s_{(i,j)}$ is a binary variable. A numerical value of $1$ for the binary variable $s_{(i,j)}$ corresponds to the in-service status for the branch $(i,j)$ and requires the fundamental current $I_{(i,j)}$ to be within the interval of $[-M,M]$. In contrast, a numerical value of $0$ for the binary variable $s_{(i,j)}$ corresponds to the out-of-service status for the branch $(i,j)$ and requires the fundamental current $I_{(i,j)}$ to be 0.

So far, we have explained how measurements of fundamental and harmonic currents can be used to derive the status of the switches. However, to integrate both fundamental and harmonic into on TI problem, we need to derive the relationships between the binary variables $b^{h}_{(i,j)}$ and $s_{(i,j)}$. $b^{h}_{(i,j)}=1$ indicates that the switch is closed which enforces $s_{(i,j)}=1$. Also, $s^{(i,j)}=0$ indicates that the switch is open which enforces $b^{h}_{(i,j)}=0$. Finally, when $b^{h}_{(i,j)}$ is 0 the corresponding switch can be open or closed and accordingly $s_{(i,j)}$ can be 0 or 1. These relationships between the binary variables $b_{(i,j)}^{h}$ and $s_{(i,j)}$ can be modeled by the following constraint:

Besides the relationships between the fundamental and harmonic switch binary variables, there is a further relationship between the fundamental switch binary variables which should be taken into account. The following equation establishes another relationship between the binary variables $s_{(i,j)}$, using the fact that in every loop of a radial network at least one switch must be open:

where $\mathcal{L}$ is the set of all branches in an arbitrary loop of the DN; $\mathcal{P}$ is the set of all the possible loops in the DN, which can be formed using the algorithm provided in [24]; and $N_{l}$ is the number of branches in the loop $\mathcal{L}$. For (11), we notice that if in a loop $\mathcal{L}$ of the radial network $N_{l}-1$ switches turn out to be closed from the harmonic currents analysis, the binary variables $s_{(i,j)}$ for all these $N_{l}-1$ switches are enforced to be 1. Then the binary variable $s_{(i,j)}$ for the last remaining switch of the loop $\mathcal{L}$ must be 0, which requires the last switch to be open. Therefore, (11) addresses the said condition at the end of section II to identify the status of the switch with no harmonic current.

Finally, there is another relationship between all the binary variables $s_{(i,j)}$ which ensures the radial configuration of the DN [11]:

where $N$ is the total number of nodes in the DN.

Let $I^{h,m}_{(i,j)}$ and $I^{m}_{(i,j)}$ denote the measurements of harmonic and fundamental currents, respectively, collected by PMUs. The following optimization problem minimizes the error in the estimation of harmonic and fundamental currents in the DN [9]:

where $\mathcal{M}$ is the set of all branches equipped with PMUs. It should be noted that, in problem (13), we minimize the normalized values of the errors. If the accuracy is not the same for the fundamental and harmonic measurements, then we may include some coefficients in the objective function in (13) to further adjust normalization.

In problem (13), the constraints (4), (5), and (7) include non-linear terms. All the non-linear terms are formed by the product of a binary variable and a continuous variable. The non-linear constraint (4) in problem (13) can be replaced with the following linear constraints:

where $W_{(i,j)}^{h,r}$ is a new continuous optimization variable. A similar approach can be taken to replace the non-linear constraints (5) and (7) with linear ones. As a result, the constraint (5) can be replaced by the following constraints:

Similarly, the constraint (7) can be replaced by the following constraints:

The non-linearity of the objective function in problem (13) is because of the optimization variables that are complex numbers and the operators that calculate the absolute values. All these sources of non-linearity can be removed by adding the new variables $G_{(i,j)}^{h,r}$, $G_{(i,j)}^{h,i}$, $G_{(i,j)}^{r}$, and $G_{(i,j)}^{i}$ and the following new constraints to the optimization problem (13):

As a result, the problem in (13) can be reformulated to the following MILP:

As discussed in section I, the main source of error in the traditional TI scheme is the errors in pseudo-measurements. The pseudo-measurements of fundamental current injections are estimated using the seasonal load curves and the nodal voltages [12]. Compared to the pseudo-measurements of the loads, larger amounts of error could be carried by the pseudo current injections. When using only harmonic current measurements, the switches on the harmonic current paths are always identified correctly, which can be seen from the simulation results in Fig. 4. Fig. 5 shows the accuracy of the proposed and the traditional TI schemes for a various percentage of errors in the pseudo-measurements. From Fig. 5, the accuracy of the proposed scheme is always higher than that of the traditional scheme. As the percentage of error in pseudo-measurement increases, the accuracy of both schemes decreases. When the percentage of error in pseudo-measurement becomes $90\%$, the proposed scheme outperforms the traditional scheme by a $13\%$ improvement in accuracy.

Fig. 6 also shows that, for 40$\%$ or less percentage of error in pseudo-measurements, the accuracy of the proposed and the traditional TI schemes are almost the same. This result is justifiable by the fact that, the incorrect estimated status in the traditional TI scheme correspond to switches that are not on any harmonic current path and consequently cannot be corrected by the proposed TI scheme. The better performance of the proposed TI scheme compared to the traditional scheme is seen especially when the percentage of error in pseudo-measurements becomes more than $40\%$.

We repeat the same numerical case study discussed in Section VI-A with a minor change in the simulation setup, by considering erroneous PMUs measurements. The range of total vector error (TVE) in the PMUs data is set to $1\%$ to $3\%$, following the IEEE Std. C37.118.1-2011 [28]. For various percentage of pseudo-measurements, the accuracy of the proposed and the traditional TI schemes are almost identical to the case with errorless PMUs data. This again shows the outperformance of the proposed TI scheme compared to the traditional TI scheme for the case of erroneous PMU measurements.

This section assesses the impact of erroneous harmonic current measurements on the accuracy of the TI schemes. We consider a scenario where there are $5\%$ TVE in the PMUs data, $90\%$ error in the pseudo-measurements, and $3\%$ error in the measurements of fundamental currents flowing through the branches. From Fig. 7(a), for the case of erroneous harmonic current measurements all the branches on harmonic current paths can still be detected correctly. Also, the proposed TI scheme still comes with $13\%$ advantage in accuracy over the traditional TI scheme. Therefore, it can be observed that the proposed TI scheme is robust to the errors in harmonic currents measurements.

To justify this observation, first, we notice that the value of the threshold is less than the value of the contaminated harmonic current measurement. As a result, the harmonic current paths can be identified correctly. Second, the problem (27) minimizes the total error in the estimation of harmonic and fundamental currents. Since the harmonic currents are scattered all around the power system, incorrect identification of a harmonic current path is much more impactful on the objective function of problem (27), compared to incorrect identification of a switch status in the traditional TI scheme that uses fundamental current measurements. For instance, the incorrect identification of the harmonic current path generated by the current source at node 18 may increase the objective function of problem (27) by more than $1$. In contrast, incorrect identification of a switch status may increase the same objective function by only a few hundredth of $1$. In conclusion, the optimization problem (27) tends to be robust to the errors in harmonic current measurements.

The measurements of harmonic sources are performed by the monitoring sensors that are located at the loads. These sensors may produce a level of error larger than the error being produced by PMUs. Accordingly, in this section, we consider harmonic source measurements with up to $10\%$ error. The values of errors set in pseudo-measurements and fundamental current measurements are $90\%$ and $3\%$, respectively. Fig. 7(b) shows the accuracy of the TI schemes for a various percentage of harmonic source TVE. From Fig. 7(b), the proposed TI scheme is robust to the errors in harmonic source measurements. This observation is justifiable by a reasoning similar to the one provided in Section VI-C.

The threshold parameter $c$ should be set in a way that, in the proposed TI scheme the impact of errors in harmonic sources and harmonic currents are lowered. In this section, the threshold parameter is set to various levels from $5$ to $25$ percent of the parameter $z$ defined in section III-A. The errors in harmonic currents, harmonic sources, pseudo-measurements, and fundamental currents are $5\%$, $10\%$, $90\%$ and $3\%$, respectively.

Fig. 8 and Fig. 9 show the accuracy of the TI schemes for various values of the threshold parameter $c$. A low value for the threshold parameter, e.g. $c=5\%z$ may lead to incorrect detection of some harmonic current paths. For the TI scheme that uses only harmonic current measurements, this can result in incorrect identification of some branch status, e.g. the status of branches (8,21) and (9,15), as it can be seen from Fig. 9. In contrast, Fig. 8 shows that the accuracy of the proposed TI scheme is the same for various values of the threshold parameter $c$. This is due to the fact that, the proposed TI scheme in (28) utilizes the fundamental current measurements to identify the switch status correctly, even if the harmonic current paths are identified incorrectly. Consequently, we can observe that the proposed TI scheme is robust to the choice of the threshold parameter $c$ because of using both the harmonic and fundamental measurements together.

In this section, we study the impacts of the number of harmonic sources on TI accuracy. Here, we still assume that five PMUs are installed in our test system as shown in Fig. 3. It is assumed that when there are $n$ harmonic sources in the test network, then these harmonic sources are located at node 1 to node $n$. The harmonic source current at each node is the same as the harmonic source current at node 11 which is shown in Fig. 3. It is assumed that there are no harmonic measurement errors, and the threshold parameter $c$ is set to 10$\%$ of the parameter $z$. The errors in pseudo-measurements are set to $90\%$.

The accuracy of the TI schemes for different number of harmonic sources are shown in Fig. 10. As the number of harmonic sources increases, the accuracy of the proposed TI scheme increases. This is because by having more harmonic sources, more switches fall on harmonic current paths, which help the proposed TI scheme to correctly derive the status of the switches that would be identified incorrectly if one uses only the fundamental current measurements. Also, the proposed TI scheme works with $100\%$ accuracy when the number of harmonic sources is equal to the number of nodes in the DN, which is explained as a requirement for full observability in theorem 1.

There is a chance that the proposed TI scheme works with $100\%$ accuracy even with fewer sources. For instance, when there are harmonic sources at nodes 18, 22, 25, and 33, all the closed switches fall on at least one harmonic current path, and the proposed TI scheme also achieves $100\%$ accuracy. However, the theorem gives a sufficient condition to make sure that all closed switches fall on at least one harmonic current path, which consequently leads to 100$\%$ accuracy.