Journal of Modern Power Systems and Clean Energy

ISSN 2196-5625 CN 32-1884/TK

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Harmonic Amplification Analysis of Cable Lines with Distributed Parameters Based on Kalman Filter and Convolution Inversion  PDF

  • Yang Pu
  • Qin Shu (Member, IEEE)
  • Fangwei Xu (Member, IEEE)
College of Electrical Engineering, Sichuan University, Chengdu 610065, China

Updated:2023-11-15

DOI:10.35833/MPCE.2022.000445

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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.

I. Introduction

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 [

1]-[4]. The wide-band resonance phenomenon would be induced due to numerous cable lines in the power system interacting with the impedance of the power grid or clean energy generation [5]-[7]. Likewise, the broadband frequency harmonics generated can be excited, leading to harmonic amplification phenomenon, which results in power quality problems [8]-[10]. Hence, the power quality problem caused by harmonic amplification becomes prominent with the rising proportion of clean energy and cable lines.

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 [

11]. In [12], the harmonic resonance problem of wind parks is explored using the sensitivity analysis method, and a resonance-suppressing plan is proposed based on the frequency offset method. In [13], it is suggested that the singularity of a network admittance matrix of modal sensitivity is related to system resonance. And the minimum eigenvalue of the matrix is used to identify harmonic resonance. In [14], based on the passivity-based analysis method, the grid-tied inverters are designed considering the capacitance of the cable line to get robustness characteristics. In [15], it is disclosed that harmonics generated from the grid-connected photovoltaic plant can be amplified with cable lines. Meanwhile, the harmonics have the feature of wide-band frequency. In [16], the cable line is modeled as multiple π units for high-frequency resonance analysis based on impedance method. Generally, when the distributed parameters of cable lines are not considered, the modal analysis and impedance methods are applicable. Otherwise, only the impedance method is applicable when the distributed parameters of cable lines are considered. However, it is arduous for calculation [17], [18].

In [

19], for scenarios where cable lines with distributed parameters are considered, the stability problem in the voltage source converter based high-voltage direct current (VSC-HVDC) transmission system is studied using the Nyquist criterion. The conclusion presents that distributed parameter models of cable lines in the high-frequency scene is more accurate. In [20], it is indicated that the capacitance of the cable line and the inductance of the power grid can induce system resonance. And the damping method is proposed for the resonance peaks. In [21], a practical circuit considering all phases of the asymmetry power line is established for the compensated distributed networks using impedance modeling method. In [22], the frequency-length factor is proposed to investigate the harmonic amplification considering the distributed parameters of ultra-long distance transmission lines. Generally, few deliberations have been paid into the scenario of the harmonic amplification of the cable line from one side terminal to another side terminal when the power at dual ends performs simultaneously.

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 [

23] integrated with the Kalman filter (KF) method can be applied to obtain the harmonic amplification factor (HAF) using the square wave pulse response (SWPR) data with error.

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.

II. HVACG System Model and Corresponding Dynamic Equations

Figure 1 shows a typical HVACG system consisting mainly of power generator, transformer, HVAC feeder, load, and power grid. The load consists of an active load P and a reactive load Q. The power grid covers a power supply in series with an impedance. A bus bar brings the load and grid together at the point of common connection (PCC).

Fig. 1  A typical HVACG system.

Figure 2 shows the equivalent circuit of the HVACG system. us and Z0 represent the power source and impedance that the power generator motor and transformer are equivalent to by the Thevenin’s theorem [

24], respectively. Z0 consists of the resistance rw, inductance lw, and capacitance cw. uw represents the voltage of capacitance cw. The power grid is equivalent to a current source ig in parallel with an impedance Zg that consists of resistance rg and inductance lg [25], [26]. For the distributed parameter model of the cable line, r0, l0, and c0 represent the per-unit-length resistance, per-unit-length inductance, and per-unit-length capacitance, respectively [27], [28]. u0(1), u0(2), , u0(n-1) and i0(1), i0(2), , i0(n) represent the node voltages and currents flowing in the unite π segment of the cable line, respectively. n represents the amount of equivalent π units of the cable line. i1 and i2 represent the currents flowing through the cable line. u1 and u2 represent the terminal voltages of the cable line.

Fig. 2  Equivalent circuit of HVACG system.

As observed from Fig. 2, the dynamic equations of the typical HVACG system with a l km length cable line can be expressed as [

23]:

u1=us+uw-i1rw-lwdi1dti1=-cwduwdti1-i0(1)=c02du1dtu1-u0(1)=l0di0(1)dt+r0i0(1)i0(1)-i0(2)=c0du0(1)dtu0(n-1)-u2=l0di0(n)dt+r0i0(n)i2=i0(n)-c02du2dtu2=lgdi2dt+digdt+rg(i2+ig) (1)

III. Basic Theory Based on KFCI

A. Optimization for State Variables Based on KF Method

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:

x˙=Ax+Buy=Cx (2)

where x, u, and y represent the state vector, input vector, and output vector, respectively; and A, B, and C represent the coefficient matrices.

According to (1), n=2 is set and the matrices A, B, and C can be obtained as:

A=0-4lc000004lc002ll0-r0l00-2ll0000000-r0l02ll0-2ll000002lc0-2lc000000004lc00000-4lc0000000-1cw0-1lw00001lw-rwlw000001lg00-rglg (3)
BT=0000001lw000002c0000 (4)
C=10000000000010000000001000000001 (5)

The state variable vector can be expressed as:

xT=u1ni0(1)i0(2)u0(1)u2nuwi1ni2n (6)

where u1n, u2n and i1n, i2n represent the measured voltages and currents of the terminal of the cable line, respectively.

The excitation vector u should be determined according to the actual injection situation: u=[u0,0] when square wave pulse excitation (SWPE) u0 is injected; u=[0,i0] when SWPE i0 is injected; and u=[u0,i0] when SWPEs u0 and i0 are injected synchronously.

The reason lgdig/dit is omitted from (1) to (3) is that the SWPE i0 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 equation (2), the state estimation equation and observation equation can be expressed as (7) and (8), respectively.

xk=A'xk-1+B'uk+Qk (7)
yk=C'xk+Rk (8)

where yk=[u1n,u2n,i1n,i2n] represents the vector of measurements; xk and uk represent the discrete state vector and input vector, respectively; Rk represents the observation error caused by the harmonics, instrumentation precision, or Gaussian noise, of which the covariance matrix is represented as R; and Qk 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 Q. For instance, the infinite π distributed parameter model is approximated as a model with two π sections in (3).

Equations (7) and (8) can be used to estimate the state variable vector x by the KF method as follows.

1) The prediction vector of prior state variables xk- and its covariance matrix Pk- can be expressed as:

xk-=A'xk-1++B'uk+ (9)
Pk-=APk-1+AT+Q (10)

where the superscripts + and - represent the posteriori and priori estimation symbols, respectively; and the covariance matrix of the model noise is set as Q=B'RBT.

2) The KF gain can be expressed as:

M=Pk-CT(C'Pk-CT+R)-1 (11)

3) The vector of posterior state variables xk+ and its covariance matrix Pk+ can be updated as:

xk+=xk-+M(yk-C'xk-) (12)
Pk+=(I-MC')Pk- (13)

where I represents the identity matrix.

According to (12), the vector of optimization results can be obtained as:

v=C'xk+ (14)

where v=[u1op,u2op,i1op,i2op] represents the optimized vector of yk.

B. Convolution Inversion Method for Amplitude-frequency Characteristics

According to [

23], the convolution inversion method can be used to calculate the amplitude-frequency characteristics, and the convolution relationship can be expressed as:

v(n)=ρ(n)h(n)nZ (15)

where ρ represents the SWPE; and h=[h1,h2,h3,h4]. h1, h2, h3, and h4 are the unit impulse responses (UIRs) from the SWPE to u1op, u2op, i1op, and i2op, respectively. Then the optimal UIR can be obtained as (16) by the convolution inversion method.

h^=(JTJ)-1JTv (16)

where J represents the excitation matrix, which can be found in the Appendix A.

The amplitude-frequency characteristics G(k)=[Gu1(k),Gu2(k),Gi1(k),Gi2(k)] can be obtained by the discrete Fourier transform (DFT) using data h^. Gu1(k), Gu2(k), Gi1(k), and Gi2(k) represent the amplitude-frequency characteristics from SWPE to u1op, u2op, i1op, and i2op, respectively.

The pseudocode of the KFCI for amplitude-frequency characteristics is given as Algorithm 1.

Algorithm 1  : pseudocode of the KFCI

Extract the SWPR data yk

Set cable line as n units (n=2)

Create the state space matrices A, B, and C

Discretize the state space matrices A', B', and C'

Calculate the prediction vector of prior state variables xk-

Update the prior covariance matrix Pk-

Update the KF gain M

Update the posterior covariance matrix Pk+

Update the vector of posterior state variables xk+

Output v

Deconvolve with v and SWPE data

Obtain UIR ĥ

Obtain the amplitude-frequency characteristics G(k) by DFT

IV. Harmonic Amplification Characteristic Analysis of Cable Lines

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.

A. Extraction of SWPR Data

The data extraction steps are provided as follows.

Step 1:   before injecting the SWPE, the voltage (u1, u2) and current (i1, i2) data of the cable line terminals for a period of time are extracted first, as shown in Fig. 2.

Step 2:   the SWPE (u0, i0) is injected into the HVACG system after an integer multiple of the fundamental wave period. Meanwhile, we extract the data (u1, u2, i1, i2) with the same duration as Step 1 again, beginning with the injection time point of SWPE.

The injected u0 and i0 are SWPEs with the same amplitude and pulse width (PW). Meanwhile, u0 is injected in series with us, and i0 is injected in parallel with ig. And the specific injection u0 or i0 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 u1n, u2n, i1n, and i2n, respectively. After the subtraction process, the power sources ug and ig are replaced as u0 and i0, 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.

B. Derivation Process of HAF

The relationship of excitation-response in the frequency domain can be expressed as:

X1(k)=U(k)Gx1(k) (17)
X2(k)=U(k)Gx2(k) (18)

where U(k) represents the excitation (u0 or i0); X1(k) and X2(k) represent the responses (u1 and u2 or i1 and i2); and Gx1(k) and Gx2(k) represent the amplitude-frequency characteristics accordingly.

According to (17) and (18), the HAF from X1(k) to X2(k) can be expressed as:

Hx2x1=Gx2(k)Gx1(k) (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 u0 or i0 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 u0 and i0 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. The circuits depicted in Fig. 3(a), (b) and (c) describe the processes of Steps 1, 2, and 3, respectively. Z and Y represent the impedance and admittance of the cable line, respectively. i0 and u0 are injected in parallel with the current source ig and in series with the voltage source us, respectively. The parameters (PW, sampling frequency) of the injected SWPE are the same. The relationship of the SWPE and SWPR can be depicted as the circuit shown in Fig. 3(c).

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:

X11=G1I0X12=G2U0X21=G3I0X22=G4U0 (20)

where X11+X12=X1 and X21+X22=X2; I0 and U0 represent the frequency domain expressions of i0 and u0, respectively; and G1, G2, G3, and G4 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 Table I. It needs to mention that the capital letters G and H represent the HAFs of single power supply and dual power supply, respectively.

Table I  HAF in Different Scenarios
HAFX1X2U
Huu u1 u2 u0, i0
Hii i1 i2 u0, i0
Gu1 u1 u0
Gu2 u2 u0
Gi1 i1 i0
Gi2 i2 i0

With the regulation for obtaining HAF and parameters listed in Table II, the analysis of HAF for voltage and current is represented as follows.

Table II  Parameters of Prototype
ParameterValue
Voltage of AC voltage source 220 kV
Sending resistance 100 Ω
Sending inductance 0.1 H
Sending capacitance 10-6 F
Cable resistance per cell 0.031 Ω
Cable inductance per cell 0.406 mH
Cable capacitance per cell 0.179 μF
Current of AC current source 0.5 kA
Receiving resistance 190 Ω
Receiving inductance 10-6 H
PW of SWPE 0.8 ms

C. HHAF of Voltage

The HAF of the voltage at terminals of the cable line is represented as Huu, which can be obtained as (21) according to (19) and (20).

Huu=U2U1=(G3+G4)I0(G1+G2)I0=(G3+G4)U0(G1+G2)U0 (21)

where U1 and U2 represent the frequency domain expressions of u1 and u2, respectively; and G3+G4 and G1+G2 represent the frequency characteristics from SWPE (u0 and i0 perform simultaneously) to u2 and u1, respectively. To obtain G3+G4 or G1+G2, the SWPR should be the response with u0 and i0 acting simultaneously. Meanwhile, the SWPE data can be either u0 or i0, according to (21).

Figure 4 illustrates the HAF curves of voltage with the cable line lengths of 24 km, 36 km, and 50 km. It can be observed from Fig. 4(a) that: when l=24 km, the curve has two resonance points impending the 10th and 50th harmonic orders; when l=36 km, it has two resonance points impending the 10th and 33rd harmonic orders; and when l=50 km, it has three resonance points impending the 10th, 24th, and 46th harmonic orders. The three curves all have resonance points around the 10th harmonic order. Meanwhile, Gu1 exceeding the resonance point around the 10th harmonic decreases gradually with the increase of the cable length. Gu1 at the resonance point with the high harmonic order is smaller than that with the low harmonic order. Gu1 is less than 1, and can be reduced to about 0. As observed in Fig. 4(b), Gu2 has the same property with Gu1. In Fig. 4(c), Huu at the first resonance points exceeds 1. And the longer the cable length is, the smaller the frequency interval between two adjacent resonance points is.

Fig. 4  HAF curves of voltage with different cable line lengths. (a) Gu1. (b) Gu2. (c) Huu.

In Fig. 5, for the case of l=24 km, the HAF from u1 to u2 is shown with different values of r0, l0, and c0. For exploring the influence of distributed parameters on harmonic amplification, one parameter is altered with other parameters fixed as 1 p.u.. For instance, when changing r0, l0 and c0 are both set to be 1 p.u., and the corresponding result is shown in Fig. 5(a). The resonance trough in Fig. 5(a) fixes at the same frequency with different r0. And the Huu decreases slightly with the increase of r0 at the resonance point. In Fig. 5(b), when l0 decreases, the resonance trough moves towards the high harmonic order. Meanwhile, the value of the resonance trough decreases with the increase of l0. As observed from Fig. 5(c), The change of c0 has similar harmonic amplification characteristics to that of l0.

Fig. 5  Huu with different distributed parameters. (a) l0=1 p.u. and c0=1 p.u.. (b) r0=1 p.u. and c0=1 p.u.. (c) r0=1 p.u. and l0=1 p.u..

D. HAF of Current

Similar to the derivation process for HAF of voltage, the HAF of current from i1 to i2 (Hii) can be expressed as:

Hii=I2I1=Gi2I0Gi1I0=Gi2U0Gi1U0 (22)

where I1 and I2 represent the frequency domain expression of i1 and i2, respectively; and Gi1 and Gi2 represent the frequency characteristics from SWPE (u0 and i0 perform simultaneously) to i1 and i2, respectively.

Figure 6 illustrates the HAF curves of current along with the cable lengths of 24 km, 36 km, and 50 km. In Fig. 6(a), the first resonance trough appears around the 10th harmonic order. Meanwhile, Gi1 increases with the cable line at the 10th harmonic order. As the harmonic order increases, Gi1 approaches 1. In Fig. 6(b), Gi2 decreases to about 0.2 and then the phenomenon of interval fluctuation occurs. Meanwhile, they oscillate more seriously along the increasing cable line, and the first resonance trough appears at the 10th harmonic order. In Fig. 6(c), the first resonance trough decreases to 0.57 at the 9th harmonic order, and the second resonance trough moves toward lower harmonic order along with increasing cable length. The HAF exceeds 1 except at the harmonic order near the resonance trough.

Fig. 6  HAF curves of current with different cable lengths. (a) Gi1. (b) Gi2. (c) Hii.

Figure 7 illustrates the HAF curves of current with different distributed parameters (r0, l0, and c0) of the cable line for the case of l=24 km. In Fig. 7(a), Hii curves coincide completely at different harmonic orders. The resonance troughs of Hii decrease slightly with the increase of r0. Meanwhile, the resonance trough is present at the same frequency. In Fig. 7(b), the Hii curves coincide well at about the 0th-35th harmonic orders. The second resonance trough appears only with l0 of 1 p.u., which means that the increase of l0 can induce the HAF oscillation more intensely. c0 and l0 have the same effect on the HAF, as observed in Fig. 7(b) and (c).

Fig. 7  Hii with different distributed parameters. (a) l0=1 p.u. and c0=1 p.u.. (b) r0=1 p.u. and c0=1 p.u.. (c) r0=1 p.u. and l0=1 p.u..

Compared with Huu, Hii decreases intensely from the fundamental frequency to the first resonant trough. Conversely, Huu slightly increases from the fundamental frequency to the first resonance point.

E. Results and Discussion

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 obtained by the three schemes are present. In Fig. 8(a), it can be observed that schemes 1 and 2 coincide well when exceeding the 30th harmonic order, and the HAF curves have a slight discrepancy at less than the 30th harmonic order. The scheme 3 significantly differs from schemes 1 and 2. In Fig. 8(b), it can be observed that the curves have huge differences except at the 11st and 49th harmonic orders. Hence, both current SWPE and voltage SWPE should be injected for obtaining the HAF at cable line terminals in the case that the voltage and current supplies act simultaneously.

Fig. 8  HAF curves at cable line terminals obtained by different schemes. (a) Huu. (b) Hii.

V. Experiment Verification

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 Fig. 2 and Table II, respectively. The cable line is equivalent to n π units, and each unit consists of c0, l0, and r0. Meanwhile, n equals the cable length, of which the unit is kilometer.

As shown in Fig. 2, with the SWPE injected, the SWPR data (u1n, i1n, u2n, and i2n) are measured in the case of n=24 (l=24 km).

A. Optimization Effectiveness Validation of KF Algorithm for SWPR with Error

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.

1) Case 1: Cable Line Parameter Fixed and SWPR Data with Error

In Fig. 9(a), u1, u1n, and u1op represent the SWPR without error, SWPR injected into the Gaussian noise, and the optimization result of u1n using the KF algorithm, respectively. It can be observed that u1 and u1op can overlap well, which means the error in u1n can be filtered out well. Figure 9(b) illustrates the UIR curves. UIR1, UIR1n, and UIR1op represent the UIR estimated using u1, u1n, and u1op data, respectively. It can be observed that UIR1 and UIR1op can coincide well. However, UIR1n deteriorates seriously due to the error in SWPR during the convolution inversion process. G1, G1n, and G1op represent the amplitude-frequency characteristics calculated using UIR1, UIR1n, and UIR1op data, respectively. It can be observed that G1 and G1op can coincide intensely with a slight difference in the high-frequency domain. For G1n, the resonance point deviation is serious, and a severe ripple appears on the curve.

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.

2) Case 2: Cable Line Parameter Shift and SWPR Data with Error

The actual parameters of cable line r0, l0, and c0 shift to 0.85, 0.9, and 0.95 p.u., respectively. The measurement data (u1n and u2n) contain the error as case 1. Figure 10 shows the simulation curves of amplitude-frequency characteristics. Gu1u2n, Gu1u2, and Gu1u2op represent the amplitude-frequency characteristics using u1n and u2n data, u1 and u2 data, and u1op and u2op data, respectively. It can be observed that Gu1u2op and Gu1u2 overlap well. Likewise, Gu1u2n and Gu1u2 have obvious deviation, which means that the model error and measurement error would lead to an inaccurate calculation for amplitude-frequency characteristics.

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.

B. Accuracy Validation for HAF

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 Z0 and Zg 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.

RMSE=1ni=1n(α(i)-β(i))2 (23)

where α(i) and β(i) represent the ith element of vectors α and β, respectively. Figure 11 illustrates the HAF curves of Z0 and Zg obtained by the TF and KFCI methods. And schemes 2 and 3 are considered when calculating HAF using the KFCI method.

Fig. 11  HAF curves of Z0 and Zg obtained by TF and KFCI methods. (a) Z0. (b) Zg.

RMSEV represents the RMSE of the HAF obtained by scheme 2 and TF method, and RMSED represents the RMSE of the HAF obtained by scheme 3 and TF method.

In Fig. 11(a), the HAF curves overlap well, and RMSEV and RMSED are 1.0247×10-5 and 1.2106×10-5, respectively. In Fig. 11(b), the HAF curves also overlap well, and RMSEV and RMSED are 0.3247×10-5 and 0.1072×10-5, respectively. Besides, the magnification is 190.4167 at the 3rd harmonic order, which coincides with the parameter of Zg listed in Table II. It can be concluded that schemes 2 and 3 can obtain the module of impedance, which verifies the accuracy for HAF using the two injection schemes.

C. Rationality Validation of Harmonic Amplification Characteristic Analysis

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: Gu1 and Gu2 validation considering scheme 2. The current source is disconnected from the model built on PSCAD, as shown in Fig. 2. Then, the harmonic-voltage generator (u0) emits the 12nd, 13rd, 14th, 15th, and 16th harmonics. As observed from Fig. 12, the total harmonic distortion (THD) of u1 and u2 is 4.65% and 6.67%, respectively. For the harmonic order range of 13rd to 16th, as shown in Fig. 4(a), Gu1 is about 0.19. Meanwhile, in Fig. 4(c), Gu2 is about 0.28. It can be found that the THD ratio (6.67%/4.65%=1.4344) of the experimental result matches with the HAF ratio (0.28/0.19=1.4737) of the analytical result.

Fig. 12  Simulation waveforms and their frequency spectrums in case 1. (a) u1. (b) Frequency spectrum of u1. (c) u2. (d) Frequency spectrum of u2.

Case 2: Huu validation considering scheme 3. The harmonic-voltage generator (u0) and harmonic-current generator (i0) emit the 23rd, 24th, 25th, 26th, and 27th harmonics synchronously. In Fig. 13, the THDs of u1 and u2 are 5.64% and 1.58%, respectively. For the harmonic order range of 23rd to 27th, as shown in Fig. 4(c), Huu is about 0.265. Using the same comparison method as the case 1, the THD ratio (1.58%/5.64%=0.28) matches with the value 0.265 of HAF.

Fig. 13  Simulation waveforms and their frequency spectrums in case 2. (a) u1. (b) Frequency spectrum of u1. (c) u2. (d) Frequency spectrum of u2.

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 u1 is 4.65%, and the THD of u2 is 6.67% (already exceeding the THD standard of 5% [

29]), as shown in Fig. 12.

VI. Conclusion

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.

Appendix

Appendix A

J=ρ(m)ρ(m-1)ρ(1)ρ(0)000ρ(m)ρ(2)ρ(1)ρ(0)000ρ(m-1)ρ(m-2)ρ(0) (A1)

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