Abstract
Harmonic amplification phenomena could appear at the point of common connection (PCC) of the cable line terminal. However, the distributed parameter model of the cable line contains hyperbolic functions with plural variables, which makes it challenging to obtain the harmonic amplification factor (HAF). Hence, a time-domain method combining the Kalman filter and convolution inversion (KFCI) methods is proposed to address this problem. First, the Kalman filter method optimizes the square wave pulse response (SWPR) with measurement error. Then, the optimized SWPR data are used to get the HAF by the convolution inversion method. Next, the harmonic amplification characteristics of cable lines are explored. Finally, an experimental simulation model is built on the PSCAD software, verifying the optimization effectiveness of Kalman filter for the SWPR with error and the accuracy of the HAF calculated by the proposed method. The analysis rationality of harmonic amplification properties is also demonstrated.
IN modern electric energy systems such as wind farms and solar parks, the power generation and power grid could generate harmonics with wide-band frequency [
For cable lines in the high-voltage alternating current (HVAC) transmission power system, the resonance can be induced by the impedance interaction among the sending generation, cable line, and power grid [
In [
Inspired by the aforementioned methods and unconsidered scenarios, we propose a time-domain method to explore harmonic amplification characteristics considering the distributed parameters of cable lines. The main contributions of this research are as follows.
1) The convolution inversion method [
2) The harmonic amplification characteristics considering the distributed parameters of cable lines in a typical HVAC grid-connected (HVACG) system are explored.
3) The harmonic amplification characteristics obtained can be a reference for damper designing in practical engineering.
The rest of this paper is organized as follows. In Section II, a typical HVACG system model and the corresponding dynamic equations are established. In Section III, the basic theory based on KF and convolution inversion (KFCI) methods is presented. Then, the harmonic amplification characteristics are studied in Section IV. Next, simulation experiments are described in Section V. Finally, the conclusions of this paper are presented in Section VI.

Fig. 1 A typical HVACG system.

Fig. 2 Equivalent circuit of HVACG system.
As observed from
(1) |
The KF method is used to optimize the SWPR with error in this subsection.
The state space equation of the HVACG system can be expressed as:
(2) |
where , and y represent the state vector, input vector, and output vector, respectively; and , , and represent the coefficient matrices.
According to (1), is set and the matrices , , and can be obtained as:
(3) |
(4) |
(5) |
The state variable vector can be expressed as:
(6) |
where , and , represent the measured voltages and currents of the terminal of the cable line, respectively.
The excitation vector should be determined according to the actual injection situation: when square wave pulse excitation (SWPE) is injected; when SWPE is injected; and when SWPEs and are injected synchronously.
The reason is omitted from (1) to (3) is that the SWPE is a constant current source except for the period of the rising and falling edges. However, the period of rising and falling edges lasts briefly (only 2.5 ns in the experimental process), which would not affect the results seriously.
According to the state space
(7) |
(8) |
where represents the vector of measurements; and represent the discrete state vector and input vector, respectively; represents the observation error caused by the harmonics, instrumentation precision, or Gaussian noise, of which the covariance matrix is represented as ; and represents the model error that is caused by cable aging or the accuracy of the model establishment, of which the covariance matrix is represented as . For instance, the infinite distributed parameter model is approximated as a model with two sections in (3).
Equations (
1) The prediction vector of prior state variables and its covariance matrix can be expressed as:
(9) |
(10) |
where the superscripts + and represent the posteriori and priori estimation symbols, respectively; and the covariance matrix of the model noise is set as .
2) The KF gain can be expressed as:
(11) |
3) The vector of posterior state variables and its covariance matrix can be updated as:
(12) |
(13) |
where represents the identity matrix.
According to (12), the vector of optimization results can be obtained as:
(14) |
where represents the optimized vector of .
According to [
(15) |
where represents the SWPE; and . , , , and are the unit impulse responses (UIRs) from the SWPE to , , , and , respectively. Then the optimal UIR can be obtained as (16) by the convolution inversion method.
(16) |
where represents the excitation matrix, which can be found in the Appendix A.
The amplitude-frequency characteristics can be obtained by the discrete Fourier transform (DFT) using data . , , , and represent the amplitude-frequency characteristics from SWPE to , , , and , respectively.
The pseudocode of the KFCI for amplitude-frequency characteristics is given as
Algorithm 1 : pseudocode of the KFCI |
---|
Extract the SWPR data |
Set cable line as units (n=2) |
Create the state space matrices , , and |
Discretize the state space matrices , , and |
Calculate the prediction vector of prior state variables |
Update the prior covariance matrix |
Update the KF gain |
Update the posterior covariance matrix |
Update the vector of posterior state variables |
Output |
Deconvolve with and SWPE data |
Obtain UIR |
Obtain the amplitude-frequency characteristics by DFT |
Section III presents a method to estimate the amplitude-frequency characteristics. For exploring the harmonic amplification phenomena, the basic theory would be used to obtain the HAF in the actual scenarios.
The data extraction steps are provided as follows.
Step 1: before injecting the SWPE, the voltage (, ) and current (, ) data of the cable line terminals for a period of time are extracted first, as shown in
Step 2: the SWPE (, ) is injected into the HVACG system after an integer multiple of the fundamental wave period. Meanwhile, we extract the data (, , , ) with the same duration as Step 1 again, beginning with the injection time point of SWPE.
The injected and are SWPEs with the same amplitude and pulse width (PW). Meanwhile, is injected in series with , and is injected in parallel with . And the specific injection or would be introduced later according to the actual scenarios.
Step 3: the SWPR can be obtained by subtracting the two sets of data obtained in Steps 1 and 2, which can be represented as , , , and , respectively. After the subtraction process, the power sources and are replaced as and , respectively.
The SWPE data can be generated with the information of sampling frequency and PW directly by the software, which do not need to be extracted from the model.
The relationship of excitation-response in the frequency domain can be expressed as:
(17) |
(18) |
where represents the excitation ( or ); and represent the responses ( and or and ); and and represent the amplitude-frequency characteristics accordingly.
According to (17) and (18), the HAF from to can be expressed as:
(19) |
According to extraction processes of SWPR data, the injection of SWPE can be divided into two ways for HAF, considering the operation of different power sources.
1) Single power supply considered: only or needs to be injected for excitation when calculating the HAF.
2) Dual power supply considered: in the regular operation of an actual HVACG system, HAF can be obtained using the SWPR data of and injected synchronously.
To illustrate the equivalent circuit change due to SWPE injection and SWPR extraction processes, the HVACG system is re-depicted, as shown in

Fig. 3 Equivalent circuits before or after SWPE injection and SWPR extraction processes. (a) Before SWPE injection. (b) After SWPE injection. (c) After SWPR extraction.
By the superposition theorem, the relationship between the SWPE and SWPR can be obtained as:
(20) |
where and ; and represent the frequency domain expressions of and , respectively; and , , , and represent the supposed intermediate transfer function (TF) variables.
Generally, to obtain the HAF in different scenarios, the way of injecting SWPE and extracting SWPR is listed in
HAF | |||
---|---|---|---|
, | |||
, | |||
With the regulation for obtaining HAF and parameters listed in
Parameter | Value |
---|---|
Voltage of AC voltage source | 220 kV |
Sending resistance | 100 |
Sending inductance | 0.1 H |
Sending capacitance |
1 |
Cable resistance per cell | 0.031 |
Cable inductance per cell | 0.406 mH |
Cable capacitance per cell | 0.179 |
Current of AC current source | 0.5 kA |
Receiving resistance | 190 |
Receiving inductance |
1 |
PW of SWPE | 0.8 ms |
The HAF of the voltage at terminals of the cable line is represented as , which can be obtained as (21) according to (19) and (20).
(21) |
where and represent the frequency domain expressions of and , respectively; and and represent the frequency characteristics from SWPE ( and perform simultaneously) to and , respectively. To obtain or , the SWPR should be the response with and acting simultaneously. Meanwhile, the SWPE data can be either or , according to (21).

Fig. 4 HAF curves of voltage with different cable line lengths. (a) . (b) . (c) .
In

Fig. 5 with different distributed parameters. (a) . (b) . (c) .
Similar to the derivation process for HAF of voltage, the HAF of current from to () can be expressed as:
(22) |
where and represent the frequency domain expression of and , respectively; and and represent the frequency characteristics from SWPE ( and perform simultaneously) to and , respectively.

Fig. 6 HAF curves of current with different cable lengths. (a) . (b) . (c) .

Fig. 7 with different distributed parameters. (a) . (b) . (c) .
Compared with , decreases intensely from the fundamental frequency to the first resonant trough. Conversely, slightly increases from the fundamental frequency to the first resonance point.
In this subsection, we present a comparative test for exploring the effect of the injection of SWPE on HAF. And the HAF at cable line terminals is set as the discussing object. The test is implemented based on three injection schmes: ① scheme 1, only current SWPE is injected; ② scheme 2, only voltage SWPE is injected; and ③ scheme 3, both current SWPE and voltage SWPE are injected.
In

Fig. 8 HAF curves at cable line terminals obtained by different schemes. (a) . (b) .
A simulation model for HVACG system is carried out on PSCAD software, which is used to validate the effectiveness and rationality of the KFCI method on analyzing the harmonic amplification characteristics. The configuration and parameters of the model are shown in
As shown in
The parameter shift of cable aging can be a primary model error. The observation error is primarily caused by harmonics, instrumentation error, or Gaussian noise. Considering the effect of harmonics, instrumentation measuring shift, and Gaussian noise, the observation error can be represented by Gaussian noise with a mean value of 0.1 and a covariance value of 0.005. Two cases are disclosed as follows.
In

Fig. 9 Simulation curves of SWPR, UIR, and amplitude-frequency characteristics using KF algorithm. (a) SWPR. (b) UIR. (c) Amplitude-frequency characteristics using KF algorithm.
The actual parameters of cable line , , and shift to 0.85, 0.9, and 0.95 p.u., respectively. The measurement data ( and ) contain the error as case 1.

Fig. 10 Simulation curves of amplitude-frequency characteristics.
It can be concluded that the convolution inversion method is sensitive to measurement error, and the KF algorithm can optimize the SWPR data with error.
In this subsection, the TF method as a standard is used to verify the accuracy of HAF obtained by the KFCI method, and the magnitudes of and are chosen as the validation objects. According to (19), the HAF from current to a voltage obtained by the proposed method can be considered as a module of impedance obtained by the TF method.
The metric of root mean squared error (RMSE) is introduced to assess the performance of the KFCI method.
(23) |
where and represent the element of vectors and , respectively.

Fig. 11 HAF curves of and obtained by TF and KFCI methods. (a) . (b) .
represents the RMSE of the HAF obtained by scheme 2 and TF method, and represents the RMSE of the HAF obtained by scheme 3 and TF method.
In
This subsection only considers injecting harmonic voltage to validate the rationality of analysis of HAF for voltage presented in Section IV. And the cable length is set to be 24 km.
Case 1: and validation considering scheme 2. The current source is disconnected from the model built on PSCAD, as shown in

Fig. 12 Simulation waveforms and their frequency spectrums in case 1. (a) . (b) Frequency spectrum of . (c) . (d) Frequency spectrum of .
Case 2: validation considering scheme 3. The harmonic-voltage generator () and harmonic-current generator () emit the 2

Fig. 13 Simulation waveforms and their frequency spectrums in case 2. (a) . (b) Frequency spectrum of . (c) . (d) Frequency spectrum of .
It can be observed that harmonics could be amplified from the sending end to the grid end of the cable line. For instance, the THD of is , and the THD of is (already exceeding the THD standard of [
The convolution inversion method combining the KF algorithm can obtain the HAF accurately using the SWPR data with error. The main harmonic amplification characteristics are drawn as follows.
1) The harmonic voltage generated by the sending generator would decrease when it arrives at both terminals of the cable line. The harmonic voltage from the sending end to the grid end of the cable line could not be amplified seriously. Harmonic current from the sending end to the grid end of the cable line would be amplified in a wide frequency band.
2) The distributed capacitance and inductance of the cable line significantly influence harmonic amplification.
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