2.1. Establishment of Fault Diagnosis Equation
For a model of a grounding grid with N nodes and B branches, the entire grounding grid can be regarded as a real resistance network, ignoring the inductance, capacitance, and mutual coupling between each branch since the current is small and direct. The dispersion of the injected current is not considered. The equivalent figure of the grounding grid is shown in
Figure 1.
The following equations are given below according to the electrical network theory method and the model of the grounding grid established above.
where
is the incidence matrix of the network;
is the branch-admittance matrix;
is the node-admittance matrix;
is the branch-voltage matrix;
is the node-voltage matrix;
is the node-current matrix;
is the branch-current matrix; and
is the branch-resistance matrix.
In order to calculate the grounding resistance of the grounding grid after years of corrosion, the node-voltage part from the grounding grid is measured when direct current (DC) is injected into the conductor along the down-lead wires. Assuming that
is the measured value of the node-voltage and
is the calculated value, the branch resistance
remains unknown yet. As the number of accessible node-voltages is far less than the number of branches, this results in the number of unknowns being greater than the number of equations, which is the so-called underdetermined equation. Hence, it is necessary to find an iterative equation to calculate the ground resistance. For a real resistance network with unknown branch resistance, theoretically, by minimizing
, the branch resistance and its variation can be obtained when it is in accordance with the measured value. As a matter of fact, the measured value and the calculated value cannot be exactly equal due to the inverse problem, so the least square solution is established as follows:
The optimal solution of each branch resistance can be obtained when the solution of Equation (6) is at minimum. The original designed resistance of the branch can be calculated according to the length, cross-sectional area, and electric resistivity of the galvanized steel strap which is the material of most grounding grids in China. As the resistance of each branch is calculated, the state of corrosion can be analyzed by comparing the calculated resistance of the branches with their initial values to achieve the purpose of branch fault location. Since the calculated value of the branch resistance cannot equal to the real value as same because the number of the measured value is far less than the number needed, this results in the ill-posed problem. Considering that, this article minimized the errors between them by the L-curve regularization method. The derivation of the Newton-Raphson algorithm can be obtained by the least square method as follows:
The series of Taylor of (6) at
is as follows:
Calculate the extreme point of (8):
If
is reversible, then
Ignoring the higher-order terms, then
Take the derivative of (6) at
:
Thus,
where
is a Jacobi matrix.
Equation (13) is the Newton-Raphson (NR) algorithm iterative equation of the fault diagnosis equation. Nevertheless, an investigator who uses the NR algorithm alone will be unable to solve grounding grid fault diagnosis equation easily. The calculation of a Jacobi matrix is complex because of its morbidity, resulting in heavy computation in practice. Consequently, the resistance of branch needs to be calculated iteratively to be converged which may not be achieved when the initial value of the resistance is set inappropriately.
The regularization method is often used to improve the morbidity of the reconstruction process, which can stabilize the astringency of the solution.
2.2. The Tikhonov Regularization and the L-Curve Method
The forward and inverse problems of the fault diagnosis equation can be presented as follows:
Forward problem: .
Inverse problem: .
The under determinedness of the inverse problem can be described as
Figure 2.
For the Equation (4), the number of node-voltages is far less than the number of branches, so the Jacobi matrix is non-singular matrix, reflecting the underdeterminedness of the inverse problem. If the operators and are known, solving is the forward problem; otherwise, when and are given, the solution of may not be determined by only, named the inverse problem. The condition number of the Jacobi matrix is large, resulting in a larger condition number of the matrix in the process of solving the inverse problem, hence the solution of the inverse matrix cannot be obtained accurately. The process for solving the inverse problem needs to be optimized because of the high degree of morbidity when looking for the numerical solution. Ultimately, it comes down to a problem of optimization. There will be a significant large deviation from the real value when simply using the least squares method. Considering that, the Tikhonov regularization iteration method is applied in order to reduce the error due to the sickness.
The Tikhonov regularization algorithm, which is a common regularization algorithm, can achieve a damping effect on the solution by adding a penalty function to the objective function. It can more than stabilize the solution, and also ensure the spatial resolution of the solution to a certain extent. We can add a constraint (penalty function) to the least squares target to achieve the purpose of improving the stability of the solution in the Tikhonov regularization method. The penalty function is added to the original objective function to obtain a new objective function, as follows:
where
is the desired variable,
is the equation about
,
is the final iterative target,
is the regularization parameter, and
is the matrix of regularization.
Aiming at the objective function, a new iterative equation can be obtained by Newton-Raphson algorithm:
where
is the Jacobian matrix of
.
The inverse problem of solving matrix in the Newton-Raphson algorithm can be converted into solving matrix through regularization. The value of keeps changing to adjust the eigenvalue of the matrix in the process of iteration process, which is beneficial for meeting the requirement of matrix inversion.
The selection of regularization parameter is vital for the calculation’s result. For the ill-posed problem, the smaller is, the closer the results are to the real value. Nonetheless, the stability of the solution is better when the parameters should be bigger. So the parameter needs to be selected properly to make both the regularization solution and the residual term smaller. In this research, the regularization parameter is selected by the L-curve method.
The L-curve is made up of the regularization solution and the residual term with parameter variation in the logarithmic scale. The characteristic of the L-curve is that the scale curve is presented clearly as a capitalized “L”. The curve of the horizontal part is mainly dominated by the regularization error (
is selected too big), while the curve of the vertical part is mainly dominated by the error of
(
is selected too small). Hanke et al. [
23] take the maximum curvature position (L-corner) on the curve as the inflection point, whose corresponding parameter
is regarded as the appropriate regularization parameter.
Since the regularization parameter is continuous in the Tikhonov regularization, the L-curve is smooth and twice differentiable. So the curvature function
about parameter
can be defined as follows:
As the curvature is maximum, the corresponding point is the L-corner. The selection of the regularization parameter can be converted to the calculation of the L-corner, which is easier and more intuitive by a numerical analysis.
2.3. The Calculation of the Branch Resistance by L-Curve Regularization
Applying the regularization algorithm on the equation of fault location for grounding grids, Equations (14) and (15) can be materialized as follows:
where,
is the regularization term, and
is regularization parameter. Equation (19) is the fault diagnosis function of grounding grids based on regularization. Each branch resistance
can be obtained by solving (20) with the standard Tikhonov regularization iteration method, which is given below:
where
is the regularization term,
is the unit matrix, and
is regularization parameter. The selection method is carried out by the L-curve method, whose initial value is
, and
is the Jacobian matrix of
.
Calculate the Jacobian matrix as follows
Through Equation (21), solve Equation (19) and the actual value of the grounding grid can be obtained.
The flow diagram for fault location in a grounding grid based on the Tikhonov regularization algorithm is shown in the
Figure 3. The algorithm flow is as follows.
- (1)
Set and the accuracy . The initial value is selected as the branch resistance under normal conditions, .
- (2)
Calculate the Jacobian matrix and choose the regularization parameter from the L-curve method.
- (3)
Calculate the iteration step size .
- (4)
Calculate and the iteration error . If , let ; otherwise, make , and turn towards step (2).
- (5)
Export the optimal solution .
Through the solution method mentioned above, the value of the resistance of a grounding grid branch is calculated. Correspondingly, the corrosion status of the grounding grids can be graded easily. The feasibility and accuracy of the proposed method are verified by experiments below.
The initial branch resistance can be calculated accurately based on the design drawing. In addition, there is only a small part of the branches in the grounding grids where corrosion exists. At the same time, the resistance of the corroded branches is generally 1 to 10 times greater than the normal branches. Therefore, the initial values are close to the objective solution in the data set of the solution space, which is extremely helpful for the convergence of the iterative calculation. A broken branch in the solution space is reflected as an extreme point. The iteration step size from the initial value to the extreme point is greater than the initial value to the value of the corroded branch, which means that the iteration speed is faster and it is more easily converged at the extreme point. Consequently, the success of the solution sought can be guaranteed with a precise initial value and an appropriate threshold value.
2.4. Principle of Cycle Voltage Measurement
To solve Equation (19), massive data on the node-voltage must be obtained. However, there are a limited number of accessible down-lead wires to obtain the corresponding data. Consequently, a 16-channel cycle voltage measurement method [
15] is used to gain enough potential data with limited accessible down-lead wires.
We choose 16 accessible down-lead wires from the grounding grids to be the current channels f inflow and outflow, which are represented by N1, N2, …, N16. First of all, we maintain the N1 as the outflow node while change the inflow node from N2 to N16 in turn. We measure the voltage of the other 14 nodes when the DC current of 1A is injected into the inflow node and extract current from the outflow node at every turn. After the inflow node is N16, we change the outflow node to be N2 and the inflow node to be N3 to N16 in turn, repeat the steps above until N15 is the outflow node and N16 is the inflow node. The process can be described as follows:
i = 1, N1 is the outflow node, N2, N3, …, N16 is the inflow node in turn;
i = 2, N2 is the outflow node, N3, N3, …, N16 is the inflow node in turn;
……
i = 15, N15 is the outflow node, N16 is the inflow node.
Without changing the position of the wire, there are 120 sets of node-voltage in once measurement. The workload is greatly reduced with the massive data measured. The mode of cycle measurement is as shown in
Figure 4.
There will be 14 valid node-voltage data in a set of measurements. In addition, N16 is chosen to be the common zero potential reference node, hence there are 120 × 13 = 1560 node-voltage data can be used for the calculation through a single measurement with 16 down-lead wires.
For field measurements, 16 down-lead wires are not always necessary; it depends on the size of the grid. The length of each wire used to connect the device and the down-lead wire is 25 m, which means that a circle with a radius of 25 m can be covered in a single measurement. A regional measurement method is used when the area of the grid is out of the scope [
24]. For the small grounding grids, the number of channels can be changed from 4 to 16 when the number of the accessible down-lead wires is less than 16.
A 24-bit analog to digital conversion chip is used to a measurement’s accuracy. The effective resolution of the chip is
, which can completely satisfy the requirement of measurement. The switch of the 16-channel is managed by ADG1206 (Analog Devices, Norwood, MA, USA) to ensure the stability of the system. The DC current of 1A is generated by an Advanced RISC Machine (ARM) microcontroller STM32F103ZET6 (STMicroelectronics, Geneva, Switzerland), which is the major chip of the device. The structure of the channel switch module is shown
Figure 5.
There is a positive feedback circuit in the excitation current source module to make sure that the error of the output current is less than 0.15%. The output current curve is shown in
Figure 6 when the resistance load is 1 Ω. Normally, the resistance of a grounding grid is usually in the range of 50 mΩ to 500 mΩ.
The system must have a specific load capacity to deal with the different conditions of the grounding grids. The test diagram is shown in
Figure 7 and results of different load tests are shown in
Table 1.
The load test of the device shows that the output current can remain stable when the load varies between 0.05 Ω and 20 Ω, which corresponds with the actual situation of the grounding grid.
To verify the accuracy of the measuring device, a simulation model that was the same as the network in the laboratory was built in MATLAB (R2012a), shown in
Figure 8. The branch of a grounding grid is made by highly precise resistors of 1 Ω. The simulation diagram and the positions of the 16-channel are as follows. The results of the measurement and simulation are shown in
Table 2.
The results of the test between measurement and simulation shows that the percentage error of the measurement is less than 1.3%, which is appropriate for the fault diagnosis equation established above.