Harmonic Loss Calculation Method Based on Fourier Expansion for HVDC System Transformer

The harmonics generated by devices in HVDC (High Voltage Direct Current) system and the harmonics input at the grid side have a great impact on the loss assessment of the converter transformer. Therefore, this paper proposes a simplified and efficient harmonic loss algorithm based on the classical transformer harmonic model, and Fourier transform and generates the code in MATLAB software. Practical engineering examples have proved that the proposed algorithm can accurately calculate the load loss and harmonic loss of the rheology by filtering data in the HVDC system. Therefore, it can solve the problems such as unclear calculation of harmonic resistance in existing standard theoretical algorithms or difficulty in obtaining test data of stray loss and eddy current loss at the fundamental frequency.


Introduction
The harmonic wave of the power network is mainly caused by three aspects: power supply, transmission, and distribution equipment and the nonlinear load of the power system [1].Since the power system cannot provide the user with an ideal sinusoidal waveform voltage with constant power frequency, the integer multiples of the fundamental frequency are obtained by decomposing the periodic current or voltage Fourier [2][3][4].Electric energy in the ideal power system to provide the user with heat is a constant amplitude and frequency of the three-phase balanced positive sequence sinusoidal voltage.However, because of the load, the power system in the actual operation is random change, harmonic current in the power supply system has appeared for many years.
Harmonics are generated by electrical equipment in the converter station, such as rectifier thyristor equipment and transformer [5].The rectifier thyristor is widely used in switching power supplies, electromechanical control, charging devices, and many other aspects.It brings quite a lot of harmonics to the power grid [6].This paper focuses on the converter transformer power loss.The power transformer is the main source of harmonics in the process of transmission and distribution [7].Because the design of the transformer needs to consider the economy, the magnetization curve of its core is in a nonlinear saturation state, which makes the magnetization current at work as a corner top waveform, and thus produces odd harmonics [8].The high saturation degree of the transformer core makes its working point deviate from the linear curve, resulting in a large harmonic current.The proportion of the odd harmonic current can reach more than 0.5% of the rated current of the transformer [9].
Harmonic pollution will bring a large amount of harmonic loss to the transformer, leading to local overheating of the transformer, which not only reduces its service life but also poses a certain threat to the safe and stable operation of the system [10][11][12][13].Therefore, it is very important to calculate the harmonic loss of the transformer.

Theoretical calculation of harmonic loss of the transformer
In contrast to the traditional linear load, which has almost no harmonic component, the operation of nonlinear electrical equipment in the AC power grid distorts the voltage and current waveforms and generates harmonics [14].Non-sinusoidal waves are usually periodic electrical components that can be decomposed into fundamental and harmonic components [15].
Waveform distortion is caused by the superposition of fundamental and harmonic waves in a power system.The amplitude and frequency of harmonic current determine the degree of distortion [16].Nonlinear loads do not produce sine wave current but pulse-type current, which generates harmonic components and forms more harmonic currents than other loads in the grid.By analyzing the Fourier series, it can be concluded that the voltage and current at this time are composed of n harmonics and fundamental waves [17].
At present, the theoretical calculation methods of harmonic loss of converter transformers have not been unified, among which IEEE 1158-1991 and IEC6 1378-2 methods are the mainstream calculation methods [18].The calculation method from IEC 61378-2 by measuring the load loss P at power frequency and the load loss P at a low-frequency small current, the eddy current loss P , in the winding and the stray loss P , in the metal structure are deduced.Finally, the loss of each harmonic P and load loss P are obtained through the expression as follows:

e s e h h e h w e h h s e h h P I R P P P I R P I I f f P I I f f (1)
where I R , is the resistance loss; I is the effective value of the power frequency current; I is the Hth subharmonic current; f is fundamental frequency 50 Hz; and f is the Hth subharmonic frequency.
In addition to the method proposed above, it can also be solved according to the transformer harmonic circuit model.This model is currently used by the power system analysis synthesis program PSASP and IEEE, which has high recognition.The most classical model of the transformer harmonic thinks that the resistance of the transformer is proportional to the square root of the frequency: A. Read the Original Recorded Current Data First, it needs to take the current record wave at the valve side of the converter transformer as input and read it in ".mat" format.The matrix array for the Fourier transform is selected to be used in subsequent MATLAB.Taking the wave recording data of converter station A as an example, its current wave record diagram is shown in Figure 2. D3.After the current is converted into the frequency domain, according to the image and the base wave, the amplitude of the frequency is selected corresponding to the Nth harmonic.The current effective value of the corresponding harmonic number is calculated as follows and is output to the Excel table through the function "xlswrite".

Transformer harmonic loss calculation method based on MATLAB
where n is the number of harmonics, I is the current amplitude corresponding to the Nth harmonics, and I _ is the current effective value corresponding to the Nth harmonics.

E. Equivalent resistance calculation
The resistance of the single-phase transformer can be calculated by the transformer nameplate parameters, according to the rated voltage Unom, Snom, tested load loss Pload, rated current calculation is available from Transformer equivalent resistance at fundamental frequency: Equivalent resistance at the Nth harmonic: F. Harmonic loss calculation for transformer:

Harmonic loss calculation results and analysis for converter transformers
The conventional DC converter station A has 12 Y-connected transformers and 12 D-connected transformers.In the following table, Y is Y connected to the transformer, D is D connected to the transformer, H is high end, and L is low end.The Fourier harmonic component analytic algorithm described in Section 2 is used to perform Fourier expansion on the valve-side current recording data of station A. The current curve in the time domain is decomposed into the RMS of current in the frequency domain under the fundamental frequency component and each harmonic component.The following Figure 3 2) Fourier series expansion of A-phase current at valve side of D-connected converter transformer: It can be seen from the following Table 1 that whether the influence of commutation angle on harmonics is considered is greater.When the commutation angle is not considered, the current component of harmonics is larger, so the harmonic loss content is also higher.It can also be seen from the example that the harmonic loss of a D-type transformer is greatly affected by the commutating angle.When the commutating angle is taken into account, the harmonic loss of the transformer is about 8.81% of the fundamental frequency loss; when the commutating angle is not taken into account, the harmonic loss is about 27.07% of the fundamental frequency loss.
To verify the accuracy of the above calculation results, a Y-connected transformer at pole 1 of station A can be an example.The following loss calculation results are obtained from Formula 1, as shown in Table 2.The data in the table can verify that the calculation results of fundamental frequency and harmonic loss by using the recording current through the MATLAB program have high accuracy and are close to the theoretical losses.As can be seen from Table 3, the transformer selection of pole 1 and pole 2 is similar, and the harmonic resistance is similar, so the loss results of pole 1 and pole 2 are similar.The harmonic loss is 8.88% of the fundamental frequency loss, and the harmonic loss is relatively low.The fifth and seventh harmonic loss are obvious, accounting for 67.64% and 26.5% of the total harmonic loss.

Conclusions
In this paper, a simplified and fast algorithm for converter transformer load loss and harmonic loss is proposed by combining the classical transformer harmonic model and Fourier transform.The recorded current data of actual DC engineering is used for simulation calculation.The harmonic loss and load loss with and without commutation angle are calculated respectively and compared with the transformer standard theoretical loss to verify the accuracy of the algorithm.And it is clear that the effect of commutation angle on harmonic loss is larger and cannot be ignored.
The MATLAB code logic and steps to realize the current Fourier transform are shown in the figure below.The code is divided into four parts: A. The original recorded current data is read; B. Related parameters are set.C. Fourier expansion and calculation; D. Images and data are output.

Figure 1 .Figure 2 .
Figure 1.Matlab calculation flow chart B. Relevant parameters are set.After obtaining the input, relevant parameters are set, such as sampling frequency, sampling period, and the number of sampling points to provide data for the subsequent calculation and figure-plotting.C. Fourier expansion and related calculations The fast Fourier transfer function command is used: fft in MATLAB to analyze Station A transformer valve side current recording data to obtain the Fourier transform results.The correlation calculation is carried out to accurately obtain the current amplitude converted into the frequency domain and the frequency vector converted into the image abscissa.D. The images and the data are output D1.The horizontal and vertical coordinates are determined.The original signal map is output and drew, which is the time domain image.D2.According to the Fourier transformed data and calculation results, the Fourier expanded current map (signal amplitude map in the form of frequency function) is output which is the frequency domain image.
and Figure 4 are two examples of the Y-connected transformer and the D-connected transformer at pole 1 of station A. 1) Fourier series expansion of A-phase current at valve side of Y-connected converter transformer:

Figure 3 .
Y-type transformer recording/calculation current curve and RMS component in the frequency domain Figure 4. D-type recording/calculation current curve and RMS component in the frequency domain NESP-2023 Journal of Physics: Conference Series 2592 (2023) 012081

Table 1 .
Y-type transformer fundamental frequency and harmonic loss results.

Table 2 .
Station A pole 1 Y-type transformer load loss(W)

Table 3 .
Station A transformer harmonic loss analysis (kW).