Computational analysis of reconstructing current and sag of three-phase overhead line based on the TMR sensor array

Greek letters

α

catenary coefficient

δ

relative error of magnetic field intensity

ε

tolerance condition

μ ₀

vacuum permeability

ξ

coordinate direction

π

PI

2.1 Theoretical model of EMF for three-phase overhead lines

Bundled conductors are often used in overhead lines to reduce the electric field strength at the surface of the conductor and to limit the generation of corona. These conductors are fixed by spacer bars, which can be treated as a single transmission line of the same length and the same sag, considering the long transmission distance of the overhead line and the small sag relative to the overall height of the transmission line [24]. Figure 1(a) shows a schematic diagram of a 500 kV three-phase overhead line and a TMR sensor array. The overhead lines are, from left to right, phase A, phase B, and phase C. The distance from the lines’ suspension point to the ground is H, and the span of the lines is L. It should be noted that the suspension heights of the same-phase line are the same. The sag of phase A is marked as sag_A in the figure. In addition, a TMR sensor group composed of N vertically placed TMR sensors is installed below the line, and the vertical spacing of TMR sensors is d. The TMR sensor group is mounted on a detection rod, which is placed on the ground and has a horizontal spacing of p between neighboring detection rods. The TMR sensor array consists of these detection rods, and the jth sensor of each detection rod collectively constitutes the jth layer of sensors, e.g., T1 − S1, T2 − S1, and T3 − S1 collectively constitute the first layer of sensors. Figure 1(b) shows the schematic structure of the yz section where the suspension point of the 500 kV three-phase overhead line is located. The spacing between the phases of the line is 13.2 m. The suspension height of phase A and phase C is 22 m, and the suspension point of phase B is 2.35 m away from the suspension point of phase C. The sag of each phase in the figure is sag_A, sag_B, and sag_C, in turn. As shown in Figure 1(a), the geometrical shape formed by the overhead conductor under the action of both temperature and gravity can be described by using the catenary line equation. This equation requires that the suspension heights of the same-phase line are the same, as shown in the following equation:

where α is the coefficient of the catenary line, H is the suspended height of the overhead line, and L is the span.

Figure 1. The

500 kV three-phase overhead line model (ignoring tower): (a) diagram of three-phase overhead line and sensor array, and (b) yz section structure diagram of suspension point of three-phase overhead line.

In order to calculate the magnetic field generated by overhead lines, it is considered that in steady-state electromagnetism, the strength of the magnetic field excited by a charged infinitesimal conductor at any point in space can be derived from the Biot–Savart law. Note that in the overhead transmission line system, the EMF excited by the transmission line is generally a low-frequency EMF [25], and the wavelength of the electromagnetic wave is much larger than the characteristic scale of the conductor, so it can be approximated as a quasi-stationary EMF, and thus the Biot–Savart law can be applied to calculate the magnetic field strength excited by the overhead transmission line at an arbitrary point in the space as shown in the following equation:

where μ ₀ is the vacuum permeability, I is the current of the infinitesimal conductor, l is the integral path along the curve where the conductor is located, and r → is the position vector of the infinitesimal conductor to the field point.

As shown in Figure 1(a), combining Eqs. (1) and (2) yields the magnitude of the absolute value of the magnetic field strength excited by any infinitesimal conductor Q (x _m, y _m, z _m), where x _m, y _m, and z _m represent the coordinates of infinitesimal conductor, on a single overhead line at the spatial point P (x, y, z), as in the following equations:

where B _x , B _y , and B _z are the components of the magnetic field strength in the x, y, and z axes, respectively; and I is the current of a single overhead conductor.

For n overhead transmission lines, the magnetic field excited by each line interacts with each other, and the principle of vector superposition can be applied to write the magnitude of the absolute value of the magnetic field strength excited by multiple overhead transmission lines at the same point in space. Here, only the magnetic field components in each direction are considered in the following equations:

where

Considering that the physical model in this article is a three-phase overhead transmission line and multiple magnetic sensors (S ₁, S ₂, S ₃, …, S _N ) are required to measure the magnetic field strength under the overhead line, the magnitude of the magnetic field strength at the location of each sensor needs to be calculated simultaneously. To facilitate the calculation, the coordinates of each sensor are noted as S _j (x _j, y _j, z _j ), and the coordinates of the sensors are substituted into Eqs. (11)–(13), whereby Eqs. (8)–(10) are organized into a generalized matrix form as follows:

where

where ξ takes x, y, and z, depending on the magnetic field strength components in the direction to be calculated. The first subscript of the variable C represents the number of the conductor, and the second subscript represents the number of the sensor. Since the variable C contains only the geometric position information of the overhead conductor and the sensor, the matrix G _ξ is called the position matrix. Once the position matrix G _ξ is determined, the current of each conductor can be calculated.

According to the theoretical model of the electro-magnetic field of the overhead line established in the previous paper, EMF calculation is carried out for the 500 kV three-phase overhead line [26] shown in Figure 1(b). The geometrical parameters of the line are shown in Figure 1(b), taking the span of the line L = 400 m. Other parameters are shown in Table 1. The influence of the current in the other line spans has been analyzed in the literature [27], which shows that only the effect of the neighboring lines needs to be considered. The magnetic field strengths at the locations of vertical heights z = 0 m, z = 2 m, and z = 4 m are calculated, respectively, and the results are shown in Figure 2. Figure 2(a) shows the distribution of magnetic field strength (B) in the xy plane directly under the overhead line. As can be seen from the figure, the magnetic field strength (B) is strongest at the position where the center of the xy plane deviates a short distance from the negative direction of the y-axis. Since the current in phase A is the largest, the maximum value of the magnetic field strength after superposition should be located below phase A. Within the spacing range of the line y ∈ (−13.2 m, 13.2 m), the magnetic field strength (B) is greater than that at other locations outside the spacing, and the magnetic field strength within the spacing increases significantly with height. The elliptical distribution of the magnetic field strength is caused by the fact that the span of the overhead line is larger than the spacing of each phase of the line. Figure 2(b)–(d) shows the distribution of magnetic field strength components B _x , B _y , and B _z in the xy plane at different heights. In Figure 2(b), the magnetic field strength component B _x = 0 T at x = 0 m, and therefore, the value of B _x cannot be measured at that location; in Figure 2(c), the range y ∈ (−5 m, 0 m) spans B _y = 0 T, and therefore, the value of B _y cannot be measured at some locations in this range either; and in Figure 2(d), the locations spanning B _z = 0 T are located outside of the line spacing. Therefore, the B _z value can be measured in the line spacing range y ∈ (−13.2 m, 13.2 m).

Figure 2

Magnitude of the magnetic field strength and its components calculated by the theoretical model: (a) B, (b) B _x , (c) B _y , and (d) B _z .

2.2 Simplified model of EMF for three-phase overhead lines

For the theoretical model, multiple numerical integrations (Eqs. (11)–(13)) are required to calculate the variable C, which increases the computational complexity of current and sag reconstruction, so a simplified EMF model can be considered to compute the variable C. Budnik and Machczyński [27] assume that the magnetic field generated at a point on an overhead conductor is generated by an infinitely long straight conductor of the same height (the height of the point can be calculated by Eq. (1)). Therefore, the magnitude of the magnetic field and its components for each phase can be computed directly from Ampere’s law:

Budnik and Machczyński [27] used the above-simplified model to calculate the magnetic field of an infinitely long conductor located at the height of the sag and at the height of the suspension point, respectively, and compared it with the theoretical model. The results show that the error between the simplified model and the theoretical model in calculating the magnetic field at the mid-point of the line is about 0.05–0.2%, and at the suspension point, the error is about 3–4%. Note that when calculating the superimposed magnetic field of each conductor in this article, only the calculation of variable C needs to be modified. Since it is assumed that each phase conductor is composed of multiple infinite long straight conductors along the x-direction, the component of the magnetic field in the x-direction is 0. The variable C is calculated as follows:

In order to quantify the error between the simplified model and the theoretical model of the EMF, the magnetic field intensity at the height z = 0 m, z = 2 m, and z = 4 m is calculated with the simplified model in the xy plane (the calculation conditions are the same as those in Section 2.1). The relative error of the magnetic field intensity between the simplified model and the theoretical model is shown in Figure 3. As can be seen in Figure 3(a), the relative error of the magnetic field intensity calculated by the simplified model and the theoretical model in the x range of [−150 m, 150 m] is δ < 0.75% at different height planes, whereas near the suspension point, the error grows rapidly to more than 6%. Figure 3(b) shows that in the y = 0 m plane, the relative error δ < 0.4% between the simplified model and the theoretical model along the x direction. The above results show that using the simplified model to solve the magnetic field inverse problem can not only maintain high calculation accuracy but also greatly reduce the calculation complexity.

Figure 3

Relative errors between simplified and theoretical models of magnetic field: (a) xy planes of different heights and (b) y = 0 m plane.

3.1 Effect of condition number on TMR sensor array placement

First, consider the condition number of the simplified model built from a single-axis TMR sensor array. To be clear:

1. The calculations do not involve EMF. The sag values of each phase line are shown in Table 1. The horizontal distance p between sensors is 1 m.

2. When using the simplified model to calculate the magnetic field, there is no x-axis magnetic component, so there is no need to calculate the matrix condition number of the x-axis sensor.

3. Three single-axis TMR magnetic sensors located in the xy plane are used for the calculation (i.e., the first sensors of each group in Figure1(a): T ₁ − S ₁, T ₂ −S ₁, and T ₃ − S ₁), and the line where the sensors are located is parallel to the y-axis.

4. In order to shorten the calculation time, the step length Δx = Δy = 5 m is taken when calculating the condition number.

The condition numbers of the matrices G _y and G _z calculated at the z = 0 m plane, respectively, are shown in Figure 4, and the calculated condition numbers are converted to logarithm with base 10. As can be seen from this figure, the condition number of G _y is minimal at y = 0 m along the x coordinate in the range of the line’s span, and the condition number of G _z has two valley regions, in the positive half-axis valley region of the y-axis, the condition number is minimal at y = −5 m; in the negative half-axis valley region of the y-axis, the condition number is minimal at y = 10 m. On the xy plane at z = 0 m, the condition number of G _y takes the minimum value of about 721 at (0, 0); the condition number of G _z takes the minimum value of about 535 at (0, −5), and the condition number at (0, 10) is about 549. According to the calculation results of the condition numbers of matrices G _y and G _z , when using a single-axis TMR magnetic sensor, if it is required that lg(cond(G)) < 3.3 for matrices G _y and G _z , then the reasonable placement area of the y-axis sensor is {(x, y) | x ∈ (−200, 200), y ∈ (−5, 5)}, and the reasonable placement area of the z-axis sensor is {(x, y) | x ∈ (−200, 200), y ∈ (−10, 0) ∪ (0, 10)}.

Figure 4

The condition numbers of the three single-axis sensor coefficient matrices obtained by the simplified EMF model at z = 0 m: (a) G _y and (b) G _z .

From the above calculation results of condition number, it can be found that the reasonable placement positions of the y-axis and z-axis sensors have some overlaps. Therefore, it is possible to consider using a dual-axis magnetic sensor at the overlapping area to obtain as much magnetic field information as possible and improve the mathematical model. Using the same sag values of each phase line in Table 1, the condition numbers of the coefficient matrix G _yz constructed with the dual-axis magnetic sensor on the xy plane at different heights are calculated, and the calculation results are shown in Figure 5. As can be seen from the figure, at different heights, the condition number distribution of matrix G _yz is bag-shaped, reaching the minimum value at the line sag point, which is about 44, 33, and 25, respectively. In the area of y ∈ (−15 m, 15 m), the condition number of the dual-axis sensor coefficient matrix can be maintained at a relatively low level.

Figure 5

The condition numbers of the three dual-axis sensor coefficient matrices G _yz obtained by the simplified EMF model on the xy plane at different heights: (a) z = 0 m, (b) z = 2 m, and (c) z = 4 m.

3.2 Dual effects of magnetic field and condition number on TMR sensor array placement

By analyzing the condition number, the type and placement location of the TMR sensor can be basically determined. However, the condition number only affects the numerical calculation error, and it is also necessary to consider whether the sensor can obtain a sufficiently strong signal in the actual application process, which is the premise for subsequent calculations. At present, there is still no evaluation parameter for the sensor placement location under both influences of magnetic field and condition number in the literature. Therefore, this article defines the following sensor position sensitivity value (Eq. (23)) to evaluate the pros and cons of the sensor array placement location under both influences of magnetic field and condition number

where cond(G) is the condition number of the matrix formed by the sensor array. If it is the y-axis sensor array, then it is cond(G _y ). If it is the yz-axis sensor array, then it is cond(G _yz ); B _ξ,i is the ξ-direction magnetic field intensity component measured by the ith sensor; B _ξ,max is the maximum ξ-direction magnetic field intensity component measured by the sensor array; ξ takes x, y, and z. ps _ξ represents the ratio of the relative size of the magnetic field measured by a certain type of sensor in a certain sensor array to the condition number of that sensor array, which is a parameter to evaluate the pros and cons of the placement location of a single sensor in the sensor array. ps is a parameter to evaluate the pros and cons of the placement location of the entire sensor array. The larger the ps value, the stronger the magnetic field signal measured by the sensor, and the more stable the mathematical model corresponding to the placement location. It is worth noting that the condition number cond(G) ≥ 1, so dividing the magnetic field value by the condition number is equivalent to shrinking the measured magnetic field value, and the physical meaning of the ps value can be understood as the magnetic field intensity measured by the sensor array after being affected by its established mathematical model. In order to evaluate the placement location of the entire sensor array, it is necessary to dimensionless and sum up the magnetic field values measured by each sensor in the sensor array, so as to evaluate whether most sensors in this sensor array can measure relatively strong magnetic field values at a certain location.

Next, the ps values of the single-axis sensor array and dual-axis sensor array are calculated using the simplified magnetic field model (the calculation conditions are shown in Table 1). The position sensitivity values of the single-axis sensor array and the dual-axis sensor array calculated by the simplified magnetic field model on the xy plane at different heights are shown in Figure 6. As shown in Figure 6(a), the calculation results of ps _y indicate that the reasonable placement area of the y-axis sensor array is {(x, y) | x ∈ (−25, 25), y ∈ (0, 5)}, where ps _y > 2.7 × 10⁻³ in this area. Since the influence of the magnetic field is considered, {(x, y) | x ∈ (−200, 200), y ∈ (−5, 5)} (Figure 4(a)) is not the reasonable placement area for the y-axis sensor array anymore. In addition, as the height increases, not only does the ps _y value increase in the area, but also the reasonable placement area expands. Figure 6(b) shows that there are two areas suitable for arranging z-axis sensor arrays on the xy plane: {(x, y) | x ∈ (−125, 125), y ∈ (−13, −5) ∪ y ∈ (3, 10)}. In these two areas, ps _z > 3.6 × 10⁻³. Figure 6(b) is similar to Figure 4(b) in terms of condition number distribution, and there exist two reasonable placement areas, which is because within the spacing range under the overhead line y ∈ (−13.2 m, 13.2 m), B _z is strong (as shown in Figure 2(d)), so the magnetic field value has little influence on ps calculation, and the sensor position sensitivity value is mainly affected by condition number at this time. As shown in Figure 6(c), for the position sensitivity value of the dual-axis sensor array (yz-axis sensor), its distribution on the xy plane is similar to that of the condition number in Figure 5. This is because the magnetic field in Eq. (24) is the sum of measurement results of each axis sensor, and since B _y is smaller than B _z within the spacing range under overhead line y ∈ (−13.2 m, 13.2 m), B _z occupies a larger proportion in Eq. (24). In addition, the condition number cond(G _yz ) of the dual-axis sensor array in Eq. (24) is similar to B _z distribution, so under the dual effect of magnetic field and condition number, the position sensitivity value of the dual-axis sensor array is elliptical. The calculation results in Figure 6 show that the position sensitivity value of the dual-axis sensor array is significantly higher than that of the single-axis sensor array, so the dual-axis sensor array should be used for subsequent inverse problem calculations.

Figure 6

The position sensitivity values of the single-axis sensor array and the dual-axis sensor array obtained by the simplified EMF model on the xy plane at different heights: (a) ps_y of the y-axis sensor array, (b) ps_z of the z-axis sensor array, and (c) ps of the yz-axis sensor array.

4.1 Current-sag inversion algorithm

Based on the analysis of the sensor array placement in Section 3, this article proposes a line current-sag inversion calculation algorithm. As previously mentioned, the suspension heights of the same phase line are the same. The TMR sensor array is placed at the midpoint (x = 0 m) of the overhead line span and on the ground within the line spacing range, as shown in Figure 1(b). Such placement has the following advantages:

1. The magnetic field intensity at the midpoint of the line span (x = 0 m) is less affected by the neighboring lines, thus minimizing the magnetic field error measured by the sensors.

2. According to the analysis in the previous section, the midpoint of the span is a position with both strong magnetic field intensity and a better mathematical model, thus reducing the error in the inversion calculation.

3. The sensor array placed on the ground is not only easy to install but also faster and safer for subsequent inspection and maintenance.

When performing the inverse problem calculation of the mathematical model, the number of unknown parameters needs to be determined first, and then the specific parameters of the TMR sensor array are determined, including the number of sensors, the horizontal distance between adjacent detection rods, and the vertical distance between sensors on the same detection rod. This article considers the line current and sag, and there exist six unknown parameters: I _a, I _b, I _c, sag_A, sag_B, and sag_C. For this case, at least six equations are required for calculations. Because of using a dual-axis TMR sensor, one sensor can provide two independent equations, so three sensors can meet the minimum requirements. Thus, the specific parameters of the TMR sensor array under the minimum requirements can be determined, as shown in the first row of Table 2. The value of sensor horizontal spacing p is not given in the table. This value can be adjusted according to the sensor position sensitivity value. It is not required that the horizontal spacing between each detection rod be the same.

Note that the magnetic field intensity components measured by the dual-axis sensor can be approximated as the same point. According to Eq. (14), Eq. (25) can be obtained by eliminating the current value in the equation. The equation represents a nonlinear equation group only related to three unknown sags. Thus, the sag can be calculated first, and then the current value of the line can be determined, thereby reducing the number of unknowns

Eq. (26) is the mathematical model established in this article, where B _y is the measured magnetic field intensity in the y-direction; B ^cal _y is the calculated magnetic field intensity in the y-direction; e is the Euclidean distance between B _y and B ^cal _y , which is also the target function used for the optimization; S _set is a manually set range of sag, where the variation of sag should be limited within S _set. Then, a current-sag inversion algorithm is proposed, as shown in Figure 7. The algorithm mainly includes four parts: signal acquisition module, sag calculation module (SCM), data storage module (DSM), and current calculation module (CCM). The calculation process of the algorithm is described as follows:

1. Use sensors to sample B _y and B _z at each moment;

2. Initialize the values of the three unknown sags; according to the collected magnetic field data B _y and B _z at the t _n moment, calculate the magnetic field component B _y at the t _n moment, that is, calculate the y-direction magnetic field intensity component with Eq. (25);

3. Judge with Eq. (26) whether e satisfies the tolerance condition ε. If not, update the sags and repeat steps (2) and (3); if the tolerance condition is satisfied, store the sags at the current moment, turn to (1), and continue to calculate the sag value at the next moment t _{n + 1};

4. After finishing the calculation of the magnetic field B _y at all moments, obtain the sags at each moment and enter the CCM. Remove outliers in the line sag data set;

5. Calculate the average sags of each phase line, that is, obtain the final sag value of each phase line;

6. Merge the coefficient matrix to obtain [G _y ; G _z ], and calculate the current value of each phase line at each moment according to the measured [B _y ; B _z ].

Figure 7

Current-sag inversion algorithm.

In the SCM, nonlinear least squares (NLS) method is chosen for the calculation. In order to reduce the adverse effect of the least squares method falling into the local optimum, a random optimization algorithm is used to globally search for a better sag solution at first, and then further calculate it using NLS. Considering that the time spent on solving should be minimized as much as possible, it is more appropriate to use the surrogate optimization algorithm for global search. In the CCM, since the sag has been obtained, the linear system Eq. (14) can be solved directly to obtain the current value. The reason why the position coefficient matrix G _ξ is merged is that this can minimize the condition number of the matrix (as shown in Figure 6) and reduce the numerical calculation error. It should be pointed out that in the SCM, the sensors located in the same xy plane are selected for the calculation (such as T ₁ − S ₁, T ₂ − S ₁, T ₃ − S ₁), and the simplified model (Eqs. (19)–(21)) is used; in the CCM, all sensors are selected for the calculation, and the theoretical model (Eqs. (11)–(13)) is used to calculate the current value.

4.2 Verification and analysis of current-sag prediction method

To verify the rationality of the sensor placement position proposed in this article and the effectiveness of the algorithm, numerical calculations are carried out with MATLAB. The line parameters are shown in Figure 1(b). To improve the reliability of the calculation results, an array consisting of multiple vertical sensors is used. The parameters are shown in the second row of Table 2, where the first layer of sensors is located on the horizontal ground. The coordinates of the detection rods in the xy plane are as follows: T ₁ (0, 7 m), T ₂ (0, 6 m), T ₃ (0, −6 m), T ₄ (0, −7 m). If the number of sensor layers needs to be increased, the vertical spacing d of the sensors should be reduced.

First, the calculation and verification analysis of the balanced three-phase current under different sag is carried out. The sinusoidal current amplitude of the line is 1 kA, and the phases are 0, −120°, and 120°, respectively. In one current cycle, 20 magnetic field values are sampled with Δt = 1 ms, and the sag of each phase line is calculated based on these data points. The calculation results are shown in Table 3. The average sag in Table 3 is the average of the sag values calculated by the sensors of each layer, and the relative error is calculated by comparing the average sag with the real value. It can be seen that the relative error of the sag calculated with the algorithm proposed in this article is less than 3%. After calculating the sag, the DSM collects the sag values at each moment, and the CCM calculates the current values at each moment. For the case of balanced current in the line, the root mean square error (RMSE) of the current calculated for the sensors of each layer in each case is shown in Figure 8. It can be seen from the figure that the maximum RMSE of the current does not exceed 14 A, and the minimum RMSE is lower than 1 A. The RMSE of each phase current does not depend on the number of sensor layers, while the average RMSE of each layer of sensors gradually decreases. In Case 2 (as shown in Figure 8(b)), the current error calculated for the first layer of sensors is the largest, while the current error calculated for the third layer of sensors is the smallest. Figure 9 shows the current values calculated for the first and third layer sensors in one cycle in Case 2, where the solid line represents the real current value of each phase line. It can be seen that although the RMSE of the current value calculated for the first layer sensor is relatively large, its calculation result still agrees well with the real current value. In Case 2, the maximum RMSE of each phase current value to the measurement current value range are 1.07, 1.14, and 1.19%, respectively, indicating that the algorithm proposed in this article is accurate.

Figure 8

The RMSE and the mean RMSE calculated from each layer sensor in each case under balanced current: (a) Case 1, (b) Case 2, and (c) Case 3.

Figure 9

The calculated current of Case 2 in one cycle: (a) first layer and (b) third layer.

It is worth noting that the current numerical verification only samples 20 magnetic field values, which may affect the final calculation accuracy. Since increasing too many sampling points will increase the calculation time, in this article, 40 magnetic field values are sampled in one cycle with Δt = 0.5 ms, and the current-sag values of each phase line are calculated accordingly. Taking Case 2 as an example, the calculated sag of each phase line is shown in Table 4. It can be seen from the table that the relative error of the line sag does not improve significantly after increasing the sampling points. The RMSE of each layer of sensors is shown in Figure 10(a). It can be seen from the figure that the variation of the mean RMSE is different from Figure 8. The mean RMSE of the third layer sensor is the largest, which is 8.20 A, while the value of the first layer is 8.19 A. The current value calculated by the third layer sensors is shown in Figure 10(b). Compared with Figure 9(b), it can be seen that whether increasing the magnetic field sampling points or not, the calculated current value agrees well with the real value. According to the above analysis, it can be seen that the algorithm proposed in this article can reconstruct the balanced three-phase current and sag of the three-phase overhead line with fewer sampling points, and the calculation results are not sensitive to the sampling frequency.

Figure 10

The calculated error from each layer sensor in Case 2 and the calculated current of maximum RMSE using 40 magnetic field values: (a) RMSE and mean RMSE of each layer sensor and (b) third layer in Case 2.

In practical applications, most of the three-phase overhead lines have unbalanced currents, so it is necessary to carry out the verification and analysis of the unbalanced currents of the three-phase overhead lines. Three kinds of unbalanced currents are randomly selected for calculation. The sinusoidal current amplitudes and phases are shown in Table 5. The sags of each phase line are set to 7.4, 7.8, and 7.2 m, respectively. In one current cycle, 20 magnetic field values are sampled with Δt = 1 ms, and the sag of the line for the three cases is calculated based on these data points. The calculation results of the line sag for the unbalanced three-phase current are shown in Table 6. Compared with Case 2 of the balanced three-phase current (Table 3), it can be seen that the maximum relative error of the line sag calculated when the three-phase current is unbalanced has increased slightly, but the maximum relative error is less than 4%.

After calculating the sag, the DSM collects the sag values at each moment, and the CCM calculates the current values at each moment. Under the same sag of the line, the RMSE of the current of each layer sensor in the three cases is calculated, as shown in Figure 11. It can be seen from the figure that the RMSE of each phase current in each case does not depend on the number of sensor layers, and the variation of the maximum RMSE is random. The maximum RMSEs in each case are 14.74, 14.61, and 13.02 A, respectively. Compared with Case 2 of the balanced three-phase current condition, the error slightly increases when calculating the unbalanced three-phase current. In one cycle of current change, the current values of the sensor layer with the largest mean RMSE error in each case are calculated, and the calculation results are shown in Figure 12. It can be seen from the figure that the unbalanced current calculated by the algorithm proposed in this article still agrees well with the real current value and can reflect the waveform of the current change.

Figure 11

The RMSE and the mean RMSE calculated from each layer sensor in each case under unbalanced current: (a) Case 1, (b) Case 2, and (c) Case 3.

Figure 12

The calculated current of the sensor layer with maximum mean RMSE in each case under unbalanced current: (a) second layer in Case 1, (b) third layer in Case 2, and (c) third layer in Case 3.

To make the current and phases obtained by the calculation more intuitive, the calculation results of the amplitude of the unbalanced three-phase currents are listed in Table 7, and the calculation results of the initial phases of the unbalanced three-phase currents are listed in Table 8. By comparing with the real values, it is found that the maximum relative error of the unbalanced current amplitude obtained by the calculation is 2.53%, and the maximum absolute error of the current initial phase is 0.5°. Therefore, the TMR sensor array and corresponding algorithm proposed in this article can accurately predict the sag and current of the three-phase overhead line.