The Journal of Engineering
The 14th IET International Conference on AC and DC Power Transmission (ACDC 2018)
Parameter identification of dc black-box arc model using non-linear least squares
eISSN 2051-3305
Received on 21st August 2018 Accepted on 19th September 2018 E-First on 10th December 2018 doi: 10.1049/joe.2018.8369 www.ietdl.org
Kyu-Hoon Park
1, Ho-Yun Lee
1, Mansoor Asif
1, Bang-Wook Lee
11Department of Electronic Engineering, Hanyang University, Ansan, Republic of Korea E-mail: [email protected]
Abstract: The black-box arc model is known to be a useful technique to simulate dynamic arc-circuit interaction by reflecting arc characteristics. Existing researches have shown that the black-box model has been widely used to analyse the ac arc characteristics of SF6 and air circuit breakers. Due to the enormous energy and steep rise rate (di/dt) during dc fault, it is important to consider dc arc characteristics. However, there are no examples of black-box models for dc circuit breakers utilised in railway systems and dc microgrid. In this study, the applicability of the black-box model, which was applied to the existing ac arc analysis, was verified for the dc arc analysis. Black-box modelling is applied to datasheet of industrial low-voltage circuit breakers and parametric sweep method was used to select Schwarz model parameters considering the tendency of dc arc current waveform for the dc pole-to-pole fault. The authors also applied the Levenberg–Marquardt algorithm, which is the most extensively used for the optimisation of functional parameters, to the Schwarz model for accurate and reliable arc modelling. As a result, the dc arc feasibility of the black-box model was analysed through simulation results, and a model optimisation technique was proposed.
1 Introduction
Recently, dc load such as smart grid-based data centre and communication facilities, and dc power sources like solar power and wind power are increasing due to the trend of environment- friendly government policy. Since these loads must be located within a stable and sustainable grid, the application of a dc microgrid, which can be operated independently from the main grid in which distributed power sources are connected, has been discussed [1].
For the stable and reliable implementation of the dc system, many manufacturers are constantly striving to improve the interrupting performance of their products. This has led to the requirement of improved arc modelling techniques due to the difficulty of conducting circuit breaker experiments.
Unlike ac, dc arc has the large magnitude of fault current and rapid rate of rise (di/dt). It is difficult to interrupt the fault current because the required arc extinguishing energy is very high in the dc interrupting process. Therefore, the transient analysis, which can be realised through the design of a precise and reliable dc circuit breakers (DCCB) reflecting the actual dc arc characteristics, must be established. The most widely utilised arc analysis techniques are physical, black-box, graphics and diagrams and parameter models.
Among them, black-box model is the useful tool to simulate transient analysis of the power system and is optimised for dynamic arc-circuit interaction analysis.
Most of the dc transient simulation that have been performed are applied to the breaker model that does not reflect the arc characteristics. In the case of dc system, it is necessary to simulate the arc characteristics because the arc characteristics greatly affect the transient analysis of the system. However, in most researches, the black-box model has been mainly applied to ac circuit breakers such as air circuit breakers and SF6 gas circuit breakers.
In this paper, we designed the DCCB model using Schwarz model which is known to represent the highest accuracy among many derivative models. The model parameter values were designed using a parametric sweep technique based on the datasheet of commercially available Eaton's DCCB.
A mathematical algorithm is proposed for model parameter optimisation that involves solving a non-linear least squares optimisation problem by suitably coupling gradient moves with the investigation of approximate value by heuristic search methods.
The Levenberg–Marquardt algorithm (LMA) was applied to the Schwarz model to perform reliable and precise arc modelling [2, 3].A simple dc grid was designed using Matlab/Simulink, and model parameter optimisation was performed through analysis of the current–voltage characteristics of the designed DCCB. From the results, we evaluated the applicability of the dc breaker to the ac black-box arc model and proposed a model optimisation technique.
2 Methodology 2.1 Simulation setup
SimPowerSystems (Matlab/Simulink) was used to simulate the interruption capability of DCCB for faults in dc grid. For simulation, simple dc grid is designed and the differential equation editor (DEE) was used to implement a black-box model which is composed of first-order differential equations.
2.1.1 Low-voltage dc grid model: Fig. 1 shows a simulation system for performance evaluation and analysis of a DCCB based on black-box model.
The dc grid model uses an ac voltage of 600 V using a 22.9 kV/
600 V transformer from the standard 22.9, 60 Hz standard power distribution system of the main power grid. It is converted to dc 400 V through a converter. Also, it is assumed that the dc grid has a considerable distance from the existing system because the cables designed by using distributed parameter lines which are arranged at intervals of 100 m for each element. The dc grid has a load of 90 kW under steady-state circumstances.
The converter consists of a smoothing reactor (L) of 0.65 mH and smoothing capacitors (C1, C2) of 15 mF. When fault occurs, 0.3 Ω snubber resistors (R1, R2) are connected in series with C1 and C2, respectively, to discharge the capacitor smoothly [4].
The interrupting capacity of DCCB was tested for the most severe condition of short-circuit faults. Fault occurs at 20 ms, and after 0.4 ms, the two DCCBs open simultaneously under the assumption that DCCB performs interruption operation normally.
2.2 Black-box arc model
The black-box model is a very useful tool for analysing the interaction between the switching arc and the corresponding system during the fault current interruption. Despite the great advances in physical arc models using finite-element methods [5, 6], the black-box arc model is still extensively used to simulate dynamic arc-circuit interactions due to simple computational processes [7, 8]. The fundamental purpose of the black-box arc model is to deduce a mathematical model of the arc by applying differential equations to the voltage and current traces obtained from the circuit breaker test. The model obtained by this process can be used to predict interruption characteristics of circuit breaker under different system condition. Most of the published models, such as the basic models Cassie, Mayr, and the derived models Schwarz, Urbanek and Schavemaker, are based on the first-order differential equation, which can be transformed into the general form. Various black-box models depend on the type of parameter function [9]
1 g⋅dg
dt = 1
T(i,G)⋅ ui
P(i,G)− 1 (1)
where G is the arc conductance, u is the arc voltage, i is the arc current and P, T are black-box model parameters.
2.2.1 Schwarz arc model: To overcome the disadvantages of the Cassie and Mayr models, which are not suitable for representing the arc characteristics for the very low and very high current regions, the Schwarz model has been proposed
1 g
dg dt = 1
τ0gα gu2
P0gβ− 1 (2)
The Schwarz model has four parameters, where P is the cooling power, which is the ability to extinguish the arc, is the arc time constant, and α and β are the exponential components of the time constant and cooling power.
2.2.2 Low-voltage dc circuit breaker model: The Schwarz differential equation is designed by using a DEE. The DCCB parameter values were selected using the parametric sweep method based on the dc breaker datasheet commercially available from Eaton's corporation as shown in Fig. 2 and Table 1. The following criteria were applied to the Schwarz model:
• Considering the current breaking capacity of the Eaton model, the maximum value of the cooling power P was assumed 3.5 MW [10].
• Considering the air insulation of the DCCB, the value of was set to <0.8 ms.
• The constants α and β were supposed to be between 0 and 2 [9].
2.3 Fitting procedure
The main principle of the fitting procedure is to define the residual rk(x) according to the appropriate time steps. As described above, the arc characteristic evaluation simulation was performed by the black-box model that consists of conductivity based differential equation. Therefore, in this paper, we have optimised the Schwarz model using the LMA, which is the most representative method for solving the non-linear least-squares problem to select more accurate and elaborate parameters of the black-box model. The LMA was implemented using Matlab code and applied Schwarz differential equations including the experimental data and parameters P, , α, β to the LMA.
2.3.1 σon-linear least squares: The least square method is a solution determining parameters of a model to minimise the sum or average of squares of residuals with data as a means of obtaining parameters of a model. Residual indicates that the measurement data is far from the deduced model's values. That is, there is an error between the experimental data and the function derived from the experimental data. This error is called residual, and the residual of an arbitrary data xi, yi can be expressed as
ri=yi−f(xi) (3)
where yi is the observed value and f(xi) is the value of mode.
Fig. 1 Power system in low-voltage dc grid
Fig. 2 Eaton's trip curve for 125 A continuous current and 800-2265 A instantaneous trip [10]
Table 1 Electrical rating of Eaton's DCCB
Electrical characteristic Capacity
Value Unit
maximum continuous current 225 A
maximum interrupted capacity 42 kA
insulation withstand voltage 750 V
rated voltage 600 V
Using the least squares method minimises the sum of the squares of residuals which is the difference between the observed and model values. Finally, the parameter of f(x) should be determined to minimise the following equation:
∑
i= 1 n
ri2=
∑
i= 1 n
(yi−f(xi))2 (4)
For a least squares problem, if the model f (x, p) is linear with respect to the model parameter, it is called a linear least squares problem, otherwise it is called a non-linear least squares problem.
2.3.2 Levenberg–Marquardt algorithm: The LMA is a combination of the Gauss–Newton method and the gradient descent method. When the data is far from the specific value, it operates in a gradient descent method, otherwise, Gauss–Newton method. There are two gradient descent methods. The gradient ascent method is used to move the gradient direction at the current position to find the maximum point of a certain function, and the gradient descent method is used to move the gradient direction to find the minimum point. On the other hand, the main principle of the Gauss–Newton method is to approximate a non-linear function locally with a linear function to obtain a solution.
However, the LMA is more stable than the Gauss–Newton method. The LMA is used for non-linear least squares problems because it converges to a relatively fast solution.
The LMA is the complement of Levenberg's algorithm in 1963 by Marquardt [11]. The optimisation tool, i.e. widely used gradient descent method, the Gauss–Newton method and the LMA combining both methods, are as follows.
Gradient descent method
Pk+ 1=Pk− 2λkJrT(pk)r(pk), k≥ 0 (5) Gauss–σewton method
Pk+ 1=Pk− (JrTJr)−1JrTr(pk), k≥ 0 (6) Levenberg–Marquardt algorithm
Pk+ 1=Pk− (JrTJr+μkdiag(JrTJr))−1JrTr(pk), k≥ 0 (7) First, the gradient descent method moves to the opposite direction of the gradient but approaches the minimum point that minimises the error function in proportion to the gradient size. On the other hand, the Gauss–Newton method is a method of finding solutions by simultaneously considering the gradient and the curvature. In (6) and (7), JrTJr is an approximate matrix for the second derivative Hessian and represents the curvature of the function.
Even though gradient is large, it moves a little when the curvature is large, and move more when the curvature is small, to find the minimum point. Thus, the Gauss–Newton method can find a solution much more accurately and faster than the gradient descent method. However, since the inverse matrix calculation of JrTJr is required in the calculation process, it is unstable when the solution is difficult to obtain and solution may divert.
Unlike the conventional method, LMA is a method of adding diag (JrTJr) in lieu of the identity matrix I. As JrTJr originally has the meaning of an approximate matrix for Hessian, the diagonal elements of JrTJr represent the curvature for each parameter component pi. Therefore, LMA avoids the singular problem of the Gauss–Newton method, so that even if is large, it reflects the curvature and can find the solution effectively.
3 Results
3.1 Results of parametric sweep method
The fitting procedure was performed on the reference data values using the parametric sweep method. The tendency of each parameter of parametric sweep data of Schwarz model used in this paper was studied in previous research [12]. Based on previous studies on the characteristics of each parameter of the Schwarz model, fitting procedure was performed to the reference data. The reference data were selected considering the Eaton's DCCB datasheet that is referred in this paper and the selected reference data are included considering the maximum interrupting current and maximum insulation withstand capability of the DCCB.
The results of the fitting according to the change of four parameters are shown in Figs. 3 and 4, and the four sweep values for the time constant , the cooling power P, and the constant α and β parameters are shown in Table 2. The optimal fitting of the reference data is performed by parametric sweep iteration. The selected Schwarz model parameter values are 0.6 for , 2 MW for P0, 0.015 for α and 1 for β. The selected value is used as the actual value of the parameter in the fitting procedure, i.e. the Ptrue value.
3.2 Results of fitting procedure
The fitting procedure was performed using the LMA iteration algorithm implemented with Matlab [13]. To realise the noise generated during the measurement, noise of a standard deviation of 0.2 of the actual value expressed as (8) is added to the reference data value
Fig. 3 Curve fitting for Schwarz model parameter selection by (a) Time constant, (b) α
Fig. 4 Curve fitting for Schwarz model parameter selection by (a) Cooling power, (b) β
yi=y_dat +N(0, 0.2) (8)
The initial value Pinit of the Schwarz model parameter and the actual value Ptrue value were investigated to check whether the parameter values converged. The number of measured values is set to 120, which is the value obtained by adding the noise of the reference data value.
The squares of residual χ were calculated for each parameter value. In the algorithm, the gradient descent method is applied to approximate the parameter value in the down-hill direction to
minimise χ2. The Gauss–Newton method in this algorithm was also used to approximate a parameter optimisation value by moving it to the minimum in small steps. This method is effective when the parameter is closest to the optimal value. The LMA combining these two methods retains the best properties of both methods.
In the fitting procedure, we plotted changes in parameter values, squares of residual and lambda values, and a 99% confidence interval between reference data and curve fitting. The parameter values are expressed such that P1 is , P2 is cooling power P0, P3 is α and P4 is β, respectively, and squares of residuals are represented by χv2. is a damping parameter, and the initial value is set to 1 × 10−2. As the value increases, the LMA performs the optimisation process using the gradient descent method which is downhill direction solution. On the contrary, when the value becomes smaller, the LMA derives the optimal parameter value using the Gauss–Newton method which approximate to the optimisation value for linearisation of non-linear curve.
The ‘true’ parameter values ptrue, the initial parameter values pinit, resulting curve-fit parameter values pfit and standard errors of the fit parameters p are shown in Table 3. The fit criterion of R2, which represents the correlation between the reference data and the model values, is 89% and the squares of residuals = 1.004. The standard parameter error is up to 11% of the parameter value.
The bowl shape of the χv2 objective function is shown in Fig. 5.
This graph shows that the logarithm of the error criterion of reduced χv2 according to the values of P2 and P4 is as small as −0.6 to 0.2 and that the shape is nearly quadratic and has one solution for each value. Therefore, it can be confirmed that the error is not large depending on the correlation between P2 and P4.
The convergence of the parameters and the evolution of χv2 and are shown in Fig. 6. The value of χv2 is 1 until the end of the fitting procedure from the beginning, and the value of λ starts from the initial value of 1 × 10−2 and becomes 1 when it ends. Therefore, considering that the value of increased from the initial value, the optimisation was performed using the gradient descent method during the fitting process.
Data points, curve fits and curve fit confidence bands are shown in Fig. 7. The y-axis represents the arc current and x-axis represents the number of data. The standard error of fit is smaller near the centre of the fit region and larger at the edge of the region.
Therefore, we could identify that the fitting curve is in the 99%
confidence region and the consistency that how the fitting value is to fit well between reference data and model values.
4 Conclusion
In this paper, the applicability of the dc arc to the Schwarz black- box arc model is evaluated by using the parametric sweep method.
Schwarz model parameter optimisation procedure uses the LMA combining the gradient descent method and the Gauss–Newton method. The DCCB was designed based on Eaton's DCCB. To Table 2 Sweep values for iterative simulation in Schwarz
model
Model Parameter Sweep value Fitting value
Schwarz T, ms 0.55, 0.6, 0.65, 0.7 0.6
P0, MW 1, 1.5, 2, 2.5 2
0.01, 0.015, 0.02, 0.025 0.015 0.5, 0.75, 1, 1.25 1
Table 3 Parameter values and standard errors
P Pinit Ptrue Pfit p p/Pfit, %
0.55 0.6 0.6124 0.0325 5.3
P0 1.5 2 1.9872 0.0213 1.07
0.01 0.015 0.0185 0.0021 11.35
0.75 1 0.9974 0.0038 0.38
Fig. 5 Sum of squared errors as a function of P2 and P4
Fig. 6 Convergence of the parameters with each iteration values of χv2 (upper) and (bottom) each iteration
Fig. 7 Traces of data y, curve-fit y, curve-fit + error
investigate the performance of the DCCB, we analysed the tendency of the fault duration, peak value and waveform variation by simulating the most severe condition such as short circuit fault.
The parameter values of Schwarz model selected using the parametric sweep method were used as the input values of the LMA implemented through Matlab. The elaborate and reliable parameter values of Schwarz model, which could not be obtained by the parametric sweep method, were optimised.
The precise and reliable parameter values of a DCCB derived through the fitting procedure can be used to analyse the interrupting performance and characteristics of the DCCB produced by specialised manufacturers in an arbitrary dc system. In addition, by improving the precision of the simulation, it is possible to overcome the disadvantages of the development of circuit breakers which are difficult to conduct actual experiments and verification tests.
In future research, it is considered that the study to apply the optimisation technique by performing the parameter value fitting using the actual dc breaker data should be performed. In addition, in this paper, we considered only the current waveform of dc arc.
However, it is expected that it will be a good research to conduct graph fitting by optimising current and voltage waveform of the dc arc including the electrical current and voltage rating of the DCCB using various black-box arc models and error range setting.
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