# A Parameterization Approach for the Dielectric Response Model of Oil Paper Insulation Using FDS Measurements

^{1}

^{2}

^{*}

## Abstract

**:**

^{−4}Hz–10

^{3}Hz) with goodness of fit over 0.99. Deviations between the tested and modelled PDC/RVM from FDS are then discussed. Compared with the previous studies to parameterize the model using time domain dielectric responses, the proposed method solves the problematic matching between EDM and FDS especially in a wide frequency band, and therefore assures a basis for quantitative insulation condition assessment of OIP-insulated apparatus in energy systems.

## 1. Introduction

^{−4}Hz–10 Hz) [34] and a tendency of larger deviation starting from 1 Hz is observed. With this in mind, the Authors have set out to solve the unsatisfying time domain parameterization by using FDS data in the present study.

^{−4}Hz–10

^{3}Hz); and (b) rather than using the default model orders of EDM conventionally, Akaike’s information criterion (AIC) is employed to indicate the optimal branch numbers in the EDM. The procedure to identify the circuit parameters will be detailed in the remainder of this paper.

## 2. Equivalent Model of OIP Insulation

_{i}and C

_{i}(i = 1,…,n) indicate the polarization resistance and capacitance, respectively. R

_{0}and C

_{0}indicate the geometric resistance and capacitance, respectively.

_{n}and C

_{n}so that the resulting terminal responses Equations (3)–(5) conform to the experimentally measured FDS data. Meanwhile, the obtained values of R

_{n}and C

_{n}should be physical. Please note that in our case, the two terms nonlinear “optimization” and “fitting” will not be strictly distinguished, and we shall use both terms hereafter, as they actually have the same essential meaning in this study.

## 3. Methodology of Model Parameterization

#### 3.1. Problem Statement

^{−11}–10

^{−7}). Common residue conditions worsen and only large-valued capacitance undergoes significant changes and achieves good approximation after several iterations. Therefore, the residue should be weighed. The capacitance and resistance in the model also extend over many magnitudes. Another similar maladaptation was encountered wherein small-valued capacitance goes through insignificant changes. Thus, the variables should be normalized.

_{i}and C

_{i}are 10

^{9}Ω and 10

^{−9}F, respectively, which have been validated by the work using the two time domain responses [19,25,26,27]. C

_{0}and R

_{0}should conform to nameplates or power frequency test results and the DC insulation resistance measurement. However, it is not recommended to apply any further constraints explicitly other than merely a fundamental nonnegativity restriction for each capacitor and resistor. Compulsory boundaries are very likely to significantly decrease the goodness of fit once unknown optimal solutions are excluded. As we shall see, fitting results that match the physical essence can be expected as long as the above two adaptations are correctly applied.

#### 3.2. Algorithm

_{k})) satisfies |fitness(x

_{k}

_{+1}) − fitness(x

_{k})|

_{n}≤ ε.

#### 3.3. Akaike Information Criterion (AIC)

## 4. Case Study

#### 4.1. Dielectric Response Measurement

#### 4.2. Parameter Identification Results

_{0}and R

_{0}should be noted first as the two variables are essentially the geometry capacitance and resistance. The values are ratings which can be directly revealed by equipment nameplates or routing tests, such as the DC resistance test. To be physical, C

_{0}in EDM conforms to the rated capacitance of the tested bushing 330 pF. It is noteworthy that all the resistances and capacitances are obtained in the order of 10

^{9}Ω and 10

^{−9}F using the proposed algorithm, although no explicit constraints are applied to the variables. This range of polarization capacitance and resistance is widely accepted and in line with reported EDM parameters that are calculated from PDC or RVM [19,25,26,27].

_{0}is observed to decrease with the increase of temperature. This phenomenon can be formulated by the well-known Arrhenius relationship: μ = μ

_{0}exp(−E

_{μ}/kT), where μ is the mobility; E

_{μ}is the activation energy; K is the Boltzmann constant; and T is the temperature. The time constant of each polarization branch that can be calculated by τ

_{i}= R

_{i}× C

_{i}is also given and sorted in descending order. It is revealed that the largest time constant also decreases with the increase of temperature. The time constant can be understood as the time taken for dipole groups to establish the polarization, and the following equation helps to explain this phenomenon: τ = πexp(U/kT)/ω

_{0}, where U is the barrier height in double-well potential; and ω

_{0}is the angular frequency of particle vibration [43]. The equation indicates that the time constant decreases exponentially as the temperature increases. Besides this, it is observed that as the temperature decreases, the capacitance in the maximum and submaximal time constant branches significantly decreases. This is because the mobility of charge and polar particles as well as the relaxation of dipole groups are retarded, which reduces the stored energy in the dipole, shown as a decline of capacitance in the polarization branches.

#### 4.3. Polarization Current Modelled from FDS

_{p}. During the charging, the current flowing through the dielectric is recorded. The excitation voltage is then removed, and the test object is short-circuited to ground. The depolarization current that flows in the opposite direction due to the previously activated polarization process is recorded.

_{0}is the applied DC voltage for polarization, and τ

_{i}presents the time constant of each polarization branch. Likewise, the depolarization current can be calculated as

_{i}is given by

_{0}and the largest time constant branch (denoted herein as B

^{1st}), whereas the other exponential currents in Equations (9) and (10) attenuate and vanish well before that time (see the sum of multiple negative exponential terms in Equations (9) and (10)). Saha et al. explored the dependence of PDC and RVM on the R

_{i}–C

_{i}in each branch of EDM, and approached the same conclusion by comparing the changes of the two time domain responses, while varying the R

_{i}–C

_{i}in larger time constant branches [16]. Then, we rewrite C″(ω) Equation (4) in the form as Equation (12). This indicates that the spectrum of imaginary capacitance comprises n + 1 subspectra, among which the peak of the ith (i = 1,…,n) subspectrum appears at the angular frequency w = 1/τ

_{i}. Therefore, R

_{0}and B

^{1st}again dominate the low-frequency (LF) band of the imaginary capacitance. The LF C″(ω), in turn, is governing data to determine the value of R

_{0}and B

^{1st}parameters if the FDS curves are precisely fitted.

_{0}and B

^{1st}values from LF FDS are imprecise, which governs the difference between the measured and modelled PDC.

#### 4.4. RVM Modelled from FDS

_{1}and switch off S

_{2}(see Figure 9).

_{0}is U

_{0}and the voltage across C

_{i}after a charging period t

_{c}is

_{1}and switch on S

_{2}. The dielectric discharge then. After a discharging period t

_{d}, the voltage of C

_{i}is

_{2}, and start the measurement of return voltage. The voltage across C

_{0}can be calculated as a superposition of contributions from C

_{i}. In this case, the resulting U

_{ri}from U

_{ci}satisfies the following equation:

_{ci}in the time domain can be obtained by applying inverse Laplace transform regarding U

_{ci}(s). The total U

_{r}can be calculated as

_{i}is the pole of the transfer function; z

_{i}

_{,l}is the zero of the transfer function under C

_{i}; and P

_{i}is the time when the return voltage reaches its peak.

_{c}˂ 1 s). However, more significant discrepancy is observed in the long time range. The modelling error in low frequencies from FDS data governs these differences.

_{i}–C

_{i}in larger time constants and R

_{0}will not prominently affect the front portion of the RVM spectra, but a noticeable shift in the subsidiary peak of RVM spectra is observed with the change of R

_{0}and large time constants R

_{i}–C

_{i}. Besides this, another influential factor is the nonlinear characteristic in the dielectric response because 500 V DC is applied in RVM, which is higher than the 200 V (peak value) in the FDS tests.

## 5. Conclusions

^{−4}Hz–10

^{3}Hz) where FDS is recorded, with the goodness of fit of the real/imaginary capacitance and power loss factor all over 0.99. Simulated PDC approximates measured results at the start while deviation was found in the current tail (t > 1000 s). A similar discrepancy was observed in RVM tests. The difference is considered to be due to the noise susceptibility of FDS measurement, especially at low frequency.

## Author Contributions

## Conflicts of Interest

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**Figure 2.**The flow chart of the combination of the genetic algorithm (GA) and the Levenberg–Marquardt Algorithm (LMA).

**Figure 3.**Illustration of an oil-impregnated paper (OIP) bushing (not to scale): (

**a**) overall structure; (

**b**) the condenser body.

**Figure 4.**(

**a**) OIP condenser bushing placed in an oast house for tests; (

**b**) dielectric response analyzer and recovery voltage meter.

**Figure 6.**The residual sum of squares (RSS) and Akaike information criterion (AIC) values of the parametrization results with different branch numbers under the three controlled temperatures: (

**a**) 30 °C; (

**b**) 70 °C; (

**c**) 80 °C.

**Figure 7.**The reconstructed FDS from the parametrized model and the measured FDS: (

**a**) complex capacitance; (

**b**) tanδ.

**Figure 8.**(

**a**) Polarization current; (

**b**) depolarization currents modelled from the parameterized EDM using FDS measurements.

**Figure 10.**Measured vs. reconstructed RVM spectra modelled from the parameterized EDM using FDS measurements.

Temperature (°C) | C′ | C″ | tan δ |
---|---|---|---|

30 | 0.9938 | 0.9985 | 0.9970 |

70 | 0.9992 | 0.9990 | 0.9929 |

80 | 0.9974 | 0.9996 | 0.9928 |

Branch No. | R_{i} (GΩ) | C_{i} (nF) | τ_{i} (s) |
---|---|---|---|

0 | 52.16955 | 0.32593 | 17.00362 |

1 | 850.57341 | 1.86212 | 1583.86976 |

2 | 12.60626 | 0.00324 | 0.04084 |

3 | 0.63505 | 0.00211 | 0.00134 |

4 | 0.01399 | 0.00334 | 4.67266 × 10^{−5} |

Branch No. | R_{i} (GΩ) | C_{i} (nF) | τ_{i} (s) |
---|---|---|---|

0 | 6.1842 | 0.32893 | 2.03417 |

1 | 23.89544 | 38.74998 | 925.94782 |

2 | 28.02376 | 1.82357 | 51.10329 |

3 | 5.36016 | 0.00994 | 0.05328 |

4 | 0.48943 | 0.00206 | 0.00101 |

5 | 0.01946 | 0.00232 | 4.51472 × 10^{−5} |

Branch No. | R_{i} (GΩ) | C_{i} (nF) | τ_{i} (s) |
---|---|---|---|

0 | 3.84327 | 0.33044 | 1.26997 |

1 | 15.23857 | 60.5265 | 922.33731 |

2 | 14.98666 | 3.24172 | 48.58256 |

3 | 4.63316 | 0.01374 | 0.06366 |

4 | 0.47013 | 0.00222 | 0.00104 |

5 | 0.02076 | 0.00219 | 4.54644 × 10^{−5} |

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## Share and Cite

**MDPI and ACS Style**

Yang, F.; Du, L.; Yang, L.; Wei, C.; Wang, Y.; Ran, L.; He, P.
A Parameterization Approach for the Dielectric Response Model of Oil Paper Insulation Using FDS Measurements. *Energies* **2018**, *11*, 622.
https://doi.org/10.3390/en11030622

**AMA Style**

Yang F, Du L, Yang L, Wei C, Wang Y, Ran L, He P.
A Parameterization Approach for the Dielectric Response Model of Oil Paper Insulation Using FDS Measurements. *Energies*. 2018; 11(3):622.
https://doi.org/10.3390/en11030622

**Chicago/Turabian Style**

Yang, Feng, Lin Du, Lijun Yang, Chao Wei, Youyuan Wang, Liman Ran, and Peng He.
2018. "A Parameterization Approach for the Dielectric Response Model of Oil Paper Insulation Using FDS Measurements" *Energies* 11, no. 3: 622.
https://doi.org/10.3390/en11030622