Study on the ratio change measurement of 1000 kV HVDC divider based on improved DC voltage summation method

Determining the voltage ratio change is one of the core issues in the traceability of the DC voltage divider. Basing on the previous research results, this study proposes an improved DC voltage summation method to evaluate the voltage division ratio error of 1000 kV DC resistance divider. The principle of the method is to calibrate the voltage divider with rated voltage 2U by using two auxiliary voltage dividers which are with rated voltage U, wherein the high-voltage (HV) arm and the low-voltage arm of the auxiliary voltage divider can be separated. Research results show that compared with the conventional method, the method can reduce one measurement variable when determining the divider's ratio change, thus simplifying the calibration process. The voltage ratio of 100 kV measured by the method of this study was well-verified by the calibration results from the National Institute of Metrology (NIM, China) and Physikalisch-Technische Bundesanstalt (PTB, Germany). Using the proposed method, the ratio change of DC voltage divider at an applied voltage of 1000 kV was effectively obtained and the uncertainty of 2.5 μV/V was achieved. Research results can provide technical guarantee for the accurate measurement of HVDC magnitude.


Introduction
In the areas of long-distance large-capacity transmission, submarine cable transmission and power systems non-synchronous network, high-voltage direct current (HVDC) technology has unique advantages [1]. With the vigorously promoting of ultra-HV (UHV) strong smart grid in China, the voltage level of HVDC transmission keeps increasing. As of 2019, there are already ten ±800 kV UHVDC lines in operation, and one ±1100 kV UHVDC line is put into trial operation [2]. To measure the HVDC precisely and ensure the uniformity of the metrology is essential for the effective control and utilisation of UHVDC power transmission. Therefore, it is significant to study the traceability method of UHVDC voltage measurement devices [3].
Among the many HVDC measuring devices, the DC resistance voltage divider is the most accurate one [4,5]. HVDC measurement standards of 100 kV or higher have already been established at home and abroad, all of which use the resistance divider as the standard equipment for HVDC magnitude traceability [6][7][8]. However, due to the effects of resistance temperature coefficient, resistance voltage coefficient, leakage current, corona discharge, operating overvoltage etc., the divider ratio will change with the applied voltage.
In the engineering practise, the deviation of divider ratio that varies with the voltage change is defined as the ratio change, i.e. the ratio error. As mentioned above, the divider ratio of the HVDC divider has many influencing factors. Therefore, the determination of the ratio change is the core problem of the HV divider traceability [9][10][11]. However, the interaction of those effecting factors makes it difficult to achieve a perfect measurement of the ratio change, particularly for voltage levels above 500 kV. Methods of characterisation for voltage levels above 500 kV mainly include as follows.
Method I: Consider that the HV divider has low heat and leakage current at its 50% rated voltage, then the ratio change can be approximately neglected. Under this case, the voltage divider rated U can be calibrated by the auxiliary voltage divider formed by its cascade of two (rated voltage 2U). This method was used by the Australian National Measurement Institute of Australia (NMIA) [12] to determine the 150 kV voltage divider's ratio change, and the measurement uncertainty of 5 × 10 −6 was achieved during the calibration experiment. When there is no suitable HV divider for auxiliary tests if the calibrated DC voltage U reaches a certain level, the ratio change should be evaluated in combination with the total leakage current measurement. It is because that as the voltage level increases, the ratio error of the voltage divider rated U is unchanged with thermal effect, and the leakage current becomes the main factor affecting the voltage divider's ratio error. On the basis of this assumption, Australian NMIA [13] further measured the ratio change of a 1000 kV voltage divider which is made by seven pieces of 150 kV dividers connected in series. The ratio change at the applied voltage of 1000 kV was determined to be 100 μV/V through that method.
In the procedures of Method I, it is difficult to achieve the perfect balance of the resistor elements because the temperature coefficients of the resistors are quadratic. Consequently, the ratio of the HV reference divider may still change even when the applied voltage is small. Also, the screening of the resistors is performed individually or in small groups at their rated voltage. However, the resistor quantity of an HV divider is quite big, and the thermal effect will accumulate and interact. This makes the actual working condition of the assembled divider different from the screening process, particularly for UHV ones. These problems may lead to an inaccurate evaluation of the ratio change.
Method II: Calibrate the divider ratio of HV divider at a lower voltage to determine the ratio change. In [14], J. Hällström et al. first calibrated the 200 kV module divider at the applied voltage of 10 kV using a digital multi-meter and a 50 kV reference divider. Then, they used the calibrated 200 kV divider module to measure the ratio change of the 1000 kV divider modular at the applied voltage of 200 kV. An uncertainty of 17 μV/V was achieved by their calibration. This method does not comprehensively consider the influence of resistance temperature coefficient, resistance voltage coefficient and leakage current of the HV divider under operating voltage. A large number of supplementary experiments are still needed to quantitatively analyse the impact of various factors, and comprehensively evaluate the uncertainty of the divider ratio [15].
An improved DC voltage summation method was proposed in this paper to solve the problems above. Using this method, the ratio change of 1000 kV HV divider at full-scale voltage range is evaluated. This method has the advantage that it can evaluate the ratio change of HVDC divider caused by the parameter changes of the resistances with the applied voltage more accurately. Compared to the previous work in [16], the contributions of this paper are that first the proposed improved method well-simplified the calibration experiments procedure as well as the ratio change calculation process. Second, the previous work dealt with the DC voltage divider calibration up to 100 kV and provided discussion for the calibration up to 500 kV level or higher, but in this paper the ratio change determination of the self-made 1000 kV voltage divider using DC voltage summation method was successfully made. Verification experiments which can effectively validate the method were also presented in this paper. The results of this research can provide references for the accurate measurement and calibration of HVDC.

Expression of the divider ratio K
The working principle of the resistor divider is shown in Fig. 1, where U represents the input voltage of the voltage divider, u represents the output voltage, R H is the HV-arm resistance, I H is the total current flowing through the HV arm, R L is the low-voltage (LV)-arm resistance of the voltage divider and I L is the total current flowing through the LV arm.
If not considering the effects of leakage current and corona current, it has I H = I L . Under this case, the voltage divider ratio K should be Since the resistance value always changes with the applied voltage, the resistance R can be expressed by where R 0 is the resistance value when the applied voltage is U 0 ; α(U) is the ratio change of the resistance when the applied voltage is U. Substituting (2) into (1), we get Considering the effects of leakage current and corona current, there is I H ≠I L . Assuming that the leakage current, as well as the corona current, is constant given the same applied voltage, the ratio I H /I L will also be constant. So let I H /I L = 1 + Δi(U), then formula (1) can be written as . Although the divider ratio K(U) expressed by (4) is identical in form to (3), its physical meaning is different: the divider ratio K(U) expressed by (4) includes not only the influence of the HV-arm and LV-arm resistance changes, but also the influence of leakage current and corona current on the divider's voltage ratio.

Improved DC voltage summation method
The improved DC voltage summation method requires four HV dividers, as shown in Fig. 2, where 1#, 2# and 3# are the auxiliary voltage dividers, 4# is the calibrated voltage divider. The HV arm and LV arm of both dividers 1# and 2# can be separated. Divider 3# is combined by the HV arm of 1#, 2# (R 1 , R 3 in Fig. 2) and the LV arm of 1#, 2# (R 2 , R 4 in Fig. 2). Since 2R 1 = 2R 3 = R 5 , 2R 2 = 2R 4 = R 6 , the rated divider ratios of the 1#, 2#, 3# and 4# voltage dividers are the same, and the rated voltage of the 1#, 2# voltage dividers are 1/2 of the 3#, 4# voltage dividers. Using the improved method to evaluate the ratio change works in three steps: #1 calibrates #4 at the applied voltage of U (denoted as experiment A), #2 calibrates #4 at the applied voltage of U (denoted as experiment B) and #3 calibrates #4 at the applied voltage of 2U (denoted as experiment C). The circuit of experiments A, B and C is shown in Fig. 3.
Let the divider ratio of 1# at voltage U be K 1 (U), so it has Since R 20 ≪ R 10 , α' 1 , α 2 ≪ 1, then In (6), K 10 is the voltage ratio of 1# at voltage U 0 , and β 1 (U) is the ratio change of the 1# divider at voltage U against voltage U 0 as the reference point. The same is available Let ɛ a (U) represent the measurement result of experiment A, which shows the relative error of the #4 divider at voltage U against the #1 divider as the reference. Then, we have Similarly, the measurement results of experiment B at voltage U and experiment C at voltage 2U can be obtained Assume that According to (6)-(8), it has From (16) Adding (13) and (14) are together From (17) and (18), it has Recursive with (20) From (13)-(15), (21) can be simplified to Then rewritten (22) as If U 0 is small enough, we have β 4 (2U 0 ) = β 4 , then (23) can be written as When the applied voltage is raised from U to 2U, substitute the measurement results of experiments A, B and C into (24), and then the ratio change of the 4# divider can be calculated. Compared to the method in [16], the improvement of this proposed method is that the approximation processing steps are less, so the uncertainty of (24) should be smaller than 10 −7 ; it has only three variables needed to be measured when to determine the ratio change using (24); no additional auxiliary experiments are required (such as experiments D and E in method [16]). Therefore, it is preliminarily sure that using the optimised method by this paper to evaluate the ratio change of HV divider, not only the workload will be reduced greatly, but also a higher level of accuracy will be achieved.

Verification by the 100 kV divider calibration test
A calibration test was carried out on the self-made [16] 100 kV HVDC divider to verify the method proposed in this paper. During the calibration, U 0 was set as 10 kV. First, calibrate the 100 kV divider at the applied voltage of 10 kV with a 10 kV standard HVDC divider. The parameters of the 10 kV standard divider at 10 kV, the actual voltage ratio K 0 and the corresponding extended uncertainty U rel can be obtained by its valid calibration certificate. Given those, the voltage ratio K x of the 100 kV divider at 10 kV was then obtained through calibration, as shown in Table 1. On this basis, use the optimised DC voltage summation method to measure the ratio change the divider at the range of 10-100 kV. Test wiring is shown in Fig. 4.
Calibration test for 10-100 kV was carried out in three steps: (i) experiment A, calibrate the 100 kV DC divider with the 50 kV auxiliary voltage divider (lower segment) at the applied voltage of U, measure the relative error of the secondary output voltage; (ii) experiment B, calibrate the 100 kV DC divider with the 50 kV auxiliary voltage divider (upper segment) at the applied voltage of U, measure the relative error of the secondary output voltage; (iii) experiment C, connect the two HV arms of the two 50 kV auxiliary voltage dividers in series, connect the two LV arms of the two 50 kV auxiliary voltage dividers in series, connect the HV-arm series and LV-arm series together and get an auxiliary voltage divider with the ratio of (50 kV + 50 kV)/(5 V + 5 V), calibrate the 100 kV DC divider with the auxiliary voltage divider series at the applied voltage of 2U, measure the relative error of the secondary output voltage.
By calculating the test data, the ratio change of the self-made 100 kV HVDC voltage divider in the voltage range of 10-100 kV can be obtained using (24). Then, combined with the calibrated result of the voltage ratio at 10 kV, the actual voltage ratio of the 100 kV divider in the voltage range of 10-100 kV can be obtained, as is shown in Table 2. Table 2 shows the self-calibration results of the 100 kV HVDC divider using the improved DC voltage summation method proposed by this paper. It can be seen from Table 2 that the measurement uncertainty of the proposed method can achieve 1 μV/V at an applied voltage of 100 kV.
Another two calibration tests were made in the National Institute of Metrology (NIM, China) and Physikalisch-Technische Bundesanstalt (PTB, Germany) to validate the self-calibration test results. Two 100 kV standard HVDC dividers made by the two institutes were used to calibrate the self-made 100 kV HVDC divider. The calibration results were summarised in Fig. 5.
It can be seen from Fig. 5 that the calibration by the improved summation method gets similar results compared with the other    Table 3, where γ 1 − γ 2 is the maximum difference between the voltage ratio value obtained from the self-calibration and that obtained by the NIM or the PTB calibration at the same applied voltage in the range of 10-100 kV. It can be seen from Table 3 that the relative discrepancy of the 10-100 kV voltage ratio by selfcalibration compared with that by NIM and PTB is smaller than 10 −5 in magnitude, which means the self-calibration result is acceptable.

Determination of the ratio change up to 1000 kV
Evaluate the ratio change of the self-made 1000 kV DC voltage divider using the improved DC voltage summation method, and the wiring of the test is shown in Fig. 6. In Fig. 6, '1' is the 1000 kV calibrated voltage divider, '2' is the voltage divider series connected by two 500 kV auxiliary voltage dividers (denoted as upper segment and lower segment), '3' is the intermediate frequency transformer, '4' is the HV voltage doubler tower, '5' is the measuring voltage divider, the combination of '3', '4' and '5' works as a high-stability DC voltage source. The intermediate frequency transformer outputs the intermediate frequency voltage to the HV doubler tower, through which the output HVDC can reach up to several tens of kilovolts to 1000 kV. The measuring voltage divider measures the HVDC generated by the voltage doubler tower. The measured signal is fed back to the operator console. According to the feedback of the measurement signal, the control and adjustment module in the operation console controls and adjusts the output signal to improve the stability of the output voltage. Also, according to the feedback of the measurement signal, the console's screen shows the real-time output voltage level.
The two-dimensional model of 1000 kV DC resistance standard divider is shown in Fig. 7 ('1' in Fig. 6). In this figure, HV is applied to the measuring resistance layer and shielding resistance layer through HV conductive barrel. The electrical parameters of the calibrated 1000 kV divider and the 500 kV auxiliary divider are shown in Table 4.
The internal structure of the calibrated 1000 kV divider and the 500 kV auxiliary divider are the same: the measuring resistance layer and the shielding resistance layer are composed of a plurality of resistors connected in series, which are spirally and evenly distributed from top to bottom around the insulating inner cylinder. The measuring resistance layer is fixed on the insulated inner cylinder with insulated support rods, while the shielding resistance layer is fixed on the outer side of the measuring resistance layer by   an insulating support rod. The insulating inner cylinder is made of plexiglass, the insulating support rod is made of polytetrafluoroethylene and the inside is filled with nitrogen gas as the insulating medium. The absolute air pressure of the inner part is about 0.4 MPa during normal operation. The calibrated 1000 kV divider was made to have nominal voltage ratios of 1000/100 and 1000 kV/10 V, and the 500 kV auxiliary divider's voltage ratio is 500 kV/5 V. It should be noted that the HV and LV arms of the 500 kV auxiliary divider are separable. To carry out the calibration test using the optimised DC voltage summation method, the 500 kV auxiliary dividers (upper segment and lower segment) were designed with separable HV arm and LV arm. When conducting experiment A, shorten the HV and LV arms of the upper segment 500 kV auxiliary divider in Fig. 6, and use the lower segment 500 kV auxiliary voltage divider to calibrate the 1000 kV voltage divider at the voltage U. When conducting experiment B, the HV and LV arms of the upper segment 500 kV auxiliary divider are short circuited, and then calibrated the 1000 kV voltage divider with the lower segment 500 kV auxiliary divider at the voltage U. When conducting experiment C, connect the two HV arms of the two 500 kV auxiliary voltage dividers (upper segment and lower segment) in series, connect the two LV arms of the two 500 kV auxiliary voltage dividers in series, connect the HV-arm series and LV-arm series together, then calibrate the 1000 kV DC voltage divider by the auxiliary voltage divider series at the voltage of 2U. The applied voltage in experiments A, B and C was set as 50n kV, where n = 1 to 10, and each experiment repeated 30 times at each voltage point. Let U 0 = 50 kV, according to that ɛ' ɛ' c and the corresponding standard deviation of each measured voltage point can be calculated. Finally, according to (24), the results of [β(2U) −β(U)] and the corresponding standard deviation can be obtained, where β is just the ratio change of the 1000 kV divider. The test data of a certain group of the 1000 kV divider calibration test using the improved DC voltage summation method are shown in Table 5.
Repeat ten times the calibration test for the self-made 1000 kV divider using the proposed method, then the value of [β(2U)−β(U)] and its standard deviation for each test can be obtained. Then, calculate the average value of [β(2U)−β(U)] obtained from these ten tests, and according to its standard deviation, calculate its combined sample standard deviation s p , standard deviation between groups s b , and then synthesise s p and s b to obtain the standard deviation s c , as shown in Table 6.
According to the results shown in Table 6, with the voltage ratio under 100 kV as the zero point, the ratio change β and its standard deviation of the 1000 kV DC standard divider at each applied voltage value from 100 to 1000 kV can be obtained using interpolation method, as shown in Fig. 8. It can be seen from Fig. 8 that the ratio change of the 1000 kV HVDC divider at its full voltage scale can be effectively measured by the improved DC addition method of this paper, and the accuracy at 1000 kV is 2.5 μV/V.

Conclusion
In this paper, an improved DC voltage summation method is proposed, and the ratio change evaluation tests of HVDC divider were carried out by using this method. Compared to the existing methods, this method can accurately measure the ratio change caused by the change of the resistance parameter at the actual operating voltage. Furthermore, this method is advanced in reducing the approximate processing steps, reducing the computational variables and streamlining the auxiliary test steps, so the workload of the voltage divider calibration by this method can be greatly lowered.
By the use of the proposed method, the self-calibration test was conducted, and the uncertainty of 6 μV/V at 100 kV level was achieved. The relative discrepancy of the 10-100 kV voltage ratio by self-calibration compared with that by NIM and PTB is smaller than 10 −5 in magnitude, which validates the effectiveness of the proposed method well. The calibration for the self-made 1000 kV standard DC divider voltage using the improved DC voltage summation method shows that the accuracy of the measured ratio change at 1000 kV reaches 2.5 μV/V, which is acceptable. Research results will prove that the method can meet the traceability requirements of the 1000 kV voltage level HVDC measurement equipment in the HVDC transmission engineering.

Acknowledgment
This work was supported by the National Key Research and Development Programme (2017YFB0903705).