Journal of Modern Power Systems and Clean Energy

ISSN 2196-5625 CN 32-1884/TK

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Harmonic State Space Based Impedance Modeling and Virtual Impedance Based Stability Enhancement Control for LCC-HVDC Systems  PDF

  • Yifeng Liu
  • Xiaoping Zhou
  • Quan Chen
  • Hanhang Yin
  • Lerong Hong
  • Hao Tian
  • Ying Chen
  • Siyuan Li
the College of Electrical and Information Engineering, Hunan University, Changsha 410082, China; the Zhejiang Magtron Intelligent Technology Limited Cooperation, Jiaxing 314000, China; the State Grid Qianjiang Electric Power Company, Qianjiang 433100, China

Updated:2024-01-22

DOI:10.35833/MPCE.2022.000722

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Abstract

Line commutated converter based high-voltage direct-current (LCC-HVDC) transmissions are prone to harmonic oscillation under weak grids. Impedance modeling is an effective method for assessing interaction stability. Firstly, this paper proposes an improved calculation method for the DC voltage and AC currents of commutation stations to address the complex linearization of the commutation process and constructs an overall harmonic state-space (HSS) model of an LCC-HVDC. Based on the HSS model, the closed-loop AC impedances on the LCC-HVDC sending and receiving ends are then derived and verified. The impedance characteristics of the LCC-HVDC are then analyzed to provide a physical explanation for the harmonic oscillation of the system. The effects of the grid strength and control parameters on system stability are also analyzed. To improve the impedance characteristics and operating stability of the LCC-HVDC system, a virtual impedance based stability enhancement control is proposed, and a parameter design method is considered to ensure satisfactory phase margins at both the sending and receiving ends. Finally, simulation results are presented to verify the validity of the impedance model and virtual impedance based stability enhancement control.

4, 2023.

A. Variable

α, μ, γ Firing, commutation, and extinction angles

Δx, x Perturbation and steady-state terms

θpll, θ1, φ Phase-locked loop (PLL) output phase, commutation voltage phase, and initial phase

ωp, ω1 Perturbation and fundamental angular frequencies

Γ Toeplitz transform

Cr1, Ci1 Shunt capacitor banks of sending end and receiving end

est Exponentially modulated periodic function

fp, f1 Perturbation and fundamental frequencies

ic AC current of commutation stations

idc DC line current of commutation stations

idcrm, vdcim Outputs of DC current and voltage filters

ig, vg Grid current and voltage

ilr1, ili1 Inductance currents of transmission line

ilr2, ilr3, ili2, ili3 Inductance currents of AC filters

it1yy, it2yy, vt2yy Primary current, secondary current, and secondary voltage of a Y/Y transformer

it1yd, it2yd, vt2yd Primary current, secondary current, and secondary voltage of a Y/D transformer

Idcref, Vdcref DC current and voltage reference

kt Line commutated converter (LCC) commutation transformer ratio

kpi, kii Proportional and integral gains of DC current controller

kpp, kip Proportional and integral gains of PLL

kpv, kiv Proportional and integral gains of DC voltage controller

Ld DC smoothing inductance

Lt Equivalent leakage reactance at the primary side of commutation transformer

Rd, Cd DC line resistance and capacitance

Rr1, Rr2 Transmission line resistances at sending end

Ri1, Ri2, Ri3 Transmission line resistances at receiving end

Lr1, Li1, Lgi Transmission line inductances

Rr3, Lr2, Cr2 Resistance, inductance, and capacitance of AC filter 1

Rr4, Rr5, Lr3, Resistances, inductance, and capacitances of Cr3, Cr4 AC filter 2

Ri4, Li2, Ci2 Resistance, inductance, and capacitance of AC filter 3

Ri5, Ri6, Li3, Resistances, inductance, and capacitances of

Ci3, Ci4 AC filter 4

Ti, Tv Time constants of DC current and voltage filters

vcr2, vcr3, vcr4 Capacitance voltages of AC filters 1and 2

vci2, vci3, vci4 Capacitance voltages of AC filters 3 and 4

vpcc Commutation bus voltage

vdc, vdc1 DC voltages of 12-pulse and 6-pulse commutation stations

vdcl DC line capacitance voltage

xidc, xvdc Integral outputs of DC current and voltage controllers

xpll, vpccqc Integral outputs and q-axis voltage of PLL controllers

Zgr, Zgi Grid impedances at sending and receiving ends

ZLCCp, ZLCCn Positive- and negative-sequence impedances of line commutated converter based high-voltage direct-current (LCC-HVDC)

B. Subscripts

d, q d- and q-axis components in the dq reference frame

k Three phases of a, b, and c

n Harmonic truncation order

p+n Perturbation at frequency fp+nf1

r, i Signals at sending and receiving ends

I. Introduction

LINE commutated converter based high-voltage direct-current (LCC-HVDC) transmissions are extensively used in power systems to realize long-distance and bulk-power transmissions [

1]. Because of the advantages of low cost and high efficiency, LCC-HVDC transmission projects have rapidly developed. However, with the increase of LCC-HVDC transmission capacity, the interaction between the LCC-HVDC and power grid can cause small-signal instability in power systems. For example, in 2016 and 2017, the interaction between the receiving-end commutation station of the Yongren-Funing LCC-HVDC project in China and weak grid led to several accidents related to 2.5 Hz to 97.5 Hz sub/super-synchronous oscillation in the receiving-end system [2], [3]. Moreover, in June 2015, the 20 Hz to 80 Hz sub/super-synchronous oscillations occurred several times in the sending-end system of the Hami-Zhengzhou LCC-HVDC project in China [4], [5]. Thus, conducting a small-signal stability analysis of the LCC-HVDC system and improving the corresponding stability control are urgent.

In recent years, small-signal models and stability analysis of LCC-HVDC transmissions have been reported [

6]-[8]. In [6], a small-signal model of an LCC-HVDC rectifier station is constructed using a time-domain state-space model. In [7], a time-domain state-space model of an LCC-HVDC inverter station is developed considering the 11th and 13th harmonics. Reference [8] comparatively studies the small-signal stability of an LCC-HVDC inverter station under constant extinction angle and DC voltage controls based on a time-domain state-space model. However, these models are established in time domains and an eigenvalue based stability analysis is adopted, which are not conducive to a high-order model [9]. The impedance based method is another stability analysis method that is performed by converting a non-linear time-varying periodical (NLTP) model in the time domain to a linear time-invariant (LTI) model in the frequency domain [10], [11]. The system stability can be conveniently analyzed using either the standard or generalized Nyquist stability criterion [12]-[14]. In [15], a sequence impedance model of an LCC is developed in which the switching function is modeled using a double-Fourier-series method. Reference [16] derives DC terminal impedance models of LCC-HVDC converters by applying a harmonic linearization technique. Reference [17] then adopts this sequence impedance modeling method in offshore wind farms via LCC-HVDC transmission and analyzes the interaction stability between an offshore wind farm that includes static synchronous compensator (STATCOM) and the LCC-HVDC. However, these impedance models do not consider the linearization of the commutation process of the LCC-HVDC converter. Later, a dq-frame impedance model of an LCC is established in [18] and [19] while considering the commutation process and avoiding the use of the complex switching function. For a typical 12-pulse LCC-HVDC system, an order of 12n±1 characteristic harmonics is used at the AC side [20]. Therefore, more frequency-coupling dynamics from characteristic harmonics must be considered to construct a small-signal model of the system. However, using the aforementioned modeling method is cumbersome when these frequency-coupling terms are included.

Harmonic state-space (HSS) modeling is a feasible method to build the model with multiple-harmonic and frequency-coupling dynamics [

21]-[23]. HSS modeling method is accomplished by applying a Fourier-series expansion and convolution operation to the state-space model. The HSS modeling method is widely applied to three- and single-phase grid-connected and modular multilevel converters (MMCs) and to LCC-HVDC systems [24]-[27]. In [26], a closed-loop impedance model of an MMC is developed based on its HSS model. This type of model considers grid impedance coupling and is useful in analyzing interaction stability. Reference [27] establishes an open-loop HSS model of an LCC-HVDC rectifier station but failes to analyze small-signal stability. To the best of our knowledge, no study has constructed an impedance model of a full LCC-HVDC system that includes both sending- and receiving-end systems. In addition, few studies have produced small-signal stability improvement methods for LCC-HVDC systems [28]-[30], and none has focused on improving LCC-HVDC systems.

LCC-HVDC system impedance modeling has three main obstacles: ① the commutation process of the LCC-HVDC system is difficult to linearize under small-signal perturbation; ② many characteristic harmonics occur during steady-state operation, which results in intricate frequency coupling behavior; and ③ numerous state variables of the LCC-HVDC system hinder the derivation of the impedance model. To solve these problems, this paper presents an improved calculation method for the DC voltage and AC current of commutation stations to address the complex linearization of the commutation process. Closed-loop AC impedance models at the LCC-HVDC sending and receiving ends are then derived based on the HSS model considering characteristic harmonics. Finally, virtual impedance based stability control for LCC-HVDC systems is proposed to improve impedance characteristics and operating stability.

The remainder of this paper is organized as follows. Section II presents LCC-HVDC model in time domain. Section III describes LCC-HVDC closed-loop impedance based on HSS. Section IV explains the verification and analysis. The virtual impedance based stability control for LCC-HVDC system is then proposed in Section V. Section VI presents simulation verification. Section VII draws the conclusions.

II. LCC-HVDC Model in Time Domain

Figure 1 shows the topology of a monopolar 12-pulse LCC-HVDC system, which consists of a sending-end AC system, rectifier station, DC line, inverter station, and receiving AC system. We first propose an improved calculation method for the DC voltage and AC current of a commutation station and then construct linearized time-domain models of the main circuit and controller of the LCC-HVDC system.

Fig. 1  Topology of a monopolar 12-pulse LCC-HVDC system.

A. Improved Calculation of AC Current and DC Voltage of LCC

Figure 2 shows the topology of an LCC connected to a commutation bus through a Y/Y transformer. When the commutation bus voltage vpcca, vpccb, vpccc and the switching function are used to calculate the DC voltage, the commutation process should be considered in the switching function. However, the voltage and current switching functions that are involved in the commutation process are complicated, where the small-signal linearization of commutation is difficult to achieve. Thus, an improved calculation method of the AC current and DC voltage of an LCC is proposed.

Fig. 2  Topology of an LCC connected to a commutation bus through a Y/Y transformer.

For the AC current calculation, the dq-frame current model that considers the commutation process proposed in [

19] is adopted. This calculation method can derive it1yyd and it1yyq of the Y/Y transformer as:

it1yyd=3πω1Lt(vpccdcosα-vpccqsinα)(cosα+cosγ)-23ktπidccosγ-0.75πω1Lt[vpccd(cos2α-cos2γ)-vpccq(2μ+sin2γ+sin2α)]it1yyq=3πω1Lt(vpccdcosα-vpccqsinα)(sinγ-sinα)-23ktπidcsinγ-0.75πω1Lt[vpccq(cos2γ-cos2α)+vpccd(2μ-sin2γ-sin2α)] (1)

Note that the perturbation response of DC voltage and line currents of the 12-pulse converter’s is twice that of the 6-pulse converter [

19]. Therefore, when (1) is linearized and the inverse Park transformation is applied, the linearized expressions of the AC current in the dq and abc frames can be obtained as:

Δic(d,q)=2Δit1yy(d,q)=Gα,i(d,q)(Δα-Δθpll)+Gidc,i(d,q)Δidc+Gvd,i(d,q)Δvpccd+Gvq,i(d,q)ΔvpccqΔica=sin(ω1t+φ¯)Δicd+cos(ω1t+φ¯)Δicq (2)

where Gα,id,q, Gidc,id,q, Gvd,id,q, and Gvq,id,q are expressed by (SA1)-(SA7) in Supplementary Material A, respectively; and the overline represents the steady-state terms.

For the DC voltage calculation, this paper adopts the neutral-point voltages of the bridge arms and the corresponding switching functions. As shown in Fig. 2, when the neutral-point voltages vt2yya, vt2yyb, vt2yyc of the bridge arms are used, the calculation of DC voltage vdc1 does not include Lt. Therefore, the corresponding switching functions of the voltage do not need to consider the commutation process, as shown in Fig. 3. Note that this calculation method does not ignore the commutation process. Instead, the effects of the commutation process are reflected in the notches of the arm neutral-point voltages..

Fig. 3  Voltage and switching function waveforms when neutral-point voltages of bridge arms are used.

Accordingly, the switching function model svyya of the phase-a voltage can be expressed by (3). The switching functions of phase-b voltage and phase-c voltage can be obtained by lagging and leading phases a 2π/3, respectively. Thus, vdc1 of the converter fed by the Y/Y transformer can be calculated using (4). The DC voltage and AC current of the converter fed by the Y/D transformer are calculated using the same method.

svyya(t)=-1α-φ-5π6<ωtα-φ-π60α-φ-π6<ωtα-φ+π61α-φ+π6<ωtα-φ+5π60α-φ+5π6<ωtα-φ+7π6 (3)
vdc1=k=a,b,csvyykvt2yyk (4)

To calculate the neutral-point voltages of the bridge arms, the commutation transformer model is derived. The magnetic saturation of the transformer can be neglected under normal operation, and the simplified equivalent model can be adopted. Thus, if we use the Y/Y transformer as an example, the relationship between the voltage and current of the transformer primary and secondary windings can be expressed as:

vt2yyk=vpcckkt-Ltkt2dit2yykdtit2yyk=ktit1yyk (5)

When (5) is substituted into (4), the linearized expression of the DC voltage can be expressed as:

Δvdc=2Δvdc1=2Δsvyykv¯pcckkt-Ltktdi¯t1yykdt+s¯vyykΔvpcckkt-LtktdΔit1yykdt (6)

B. Main Circuit and Controller Model in Time Domain

When a balanced three-phase system without a neutral connection is considered, no zero-sequence signal exists at the AC side. It is assumed that the positive/negative-sequence small-signal voltage perturbations Δvgr and Δvgi at fp are imposed on the sending-end and receiving-end AC systems, respectively. Let us use phase a as an example, where (t) is omitted for a simpler representation. According to Kirchhoff Laws and the linearization around periodic steady-state operating points of variables, the state-space equations in the time domain of a sending-end AC system, receiving-end AC filter, and DC line can be obtained as:

Δx˙srT=AsrTΔxsrT+ΔusrTΔx˙siT=AsiTΔxsiT+ΔusiTΔx˙dcT=AdcTΔxdcT+ΔudcT (7)

where ΔxsrT=[Δilr1,Δvpccr,Δvcr2,Δilr2,Δvcr3,Δilr3,Δvcr4]T; ΔxsiT= [Δigi,Δili1,Δvpcci,Δvci2,Δili2,Δvci3,Δili3,Δvci4]T; ΔxdcT=[Δidcr, Δidci,Δvdcl]T; ΔusrT=[Δvgr/(Lr1+Rr1Lr1/Rr2)-Δicr/Cr1,0,0,0, 0]T; ΔusiT=[Δvgi/Lgi,0,-Δici/Ci1,0,0,0,0,0]T; ΔudcT=[Δvdcr/Ld-Δvdci/Ld,0]T; and the coefficient matrixes AsrT, AsiT, and AdcT are expressed by (SA8)-(SA10) in Supplementary Material A, respectively.

Figure 4 presents a control scheme of an LCC-HVDC system under normal operation [

16]. The rectifier and inverter stations adopt the DC current controller and DC voltage controller, respectively. The phase-locked loop (PLL) is applied to capture the commutation voltage phase.

Fig. 4  Control scheme of LCC-HVDC system under normal operation.

Figure 4 shows that the linearized state-space model of the LCC-HVDC controller in the time domain can be obtained as:

Δx˙crT=AcrTΔxcrT+ΔucrTΔx˙ciT=AciTΔxciT+ΔuciTΔαr=-kiiΔxidcr+kpiΔidcrmΔαi=-kivΔxvdci-kpvΔvdcim (8)

where ΔxcrT=[Δxpllr,Δθpllr,Δidcrm,Δxidcr]T; ΔxciT=[Δxplli,Δθplli, Δvdcim,Δxvdci]T; ΔucrT=[Δvpccrqc,kppr,Δvpccrqc,Δidcr/Ti,0]T; ΔuciT=[Δvpcciqc,kppi,Δvpcciqc,Δvdci/Tv,0]T; and AcrT and AciT are expressed by (SA10) in Supplementary Material A.

III. LCC-HVDC Closed-loop Impedance Based on HSS

We next convert the time-domain model of the LCC-HVDC system into an LTI model and derive the overall HSS model. In addition, to facilitate small-signal analysis, the closed-loop AC impedance model of the LCC-HVDC sending and receiving ends is further derived.

A. LTI Model in Frequency Domain

To obtain an HSS model of the LCC-HVDC system, its linearized time-varying periodical (LTP) model should be converted to an LTI model in the frequency domain. According to the HSS modeling method [

21], the perturbation component Δx(t) is presented by Fourier coefficients as:

Δx(t)=estnZxp+nej(ωp+nω1)t=estnZxp+nej(p+n)ω1t (9)

Regarding the product of Δx(t) and steady-state component b(t), the results are expressed as a convolution operation in the frequency domain as:

b(t)Δx(t)Γ[b]Δx (10)

where the operator Γ indicates the Toeplitz transform Δx=[xp-n,,xp,,,xp+n]T.

When the differential of Δx(t) in the time domain is used in the frequency domain, the diagonal matrix is multiplied by the vector composed of the Fourier coefficient:

Δx˙(t)sΔx+NΔx (11)

where N=diag[j(p-n)ω1,, jpω1,, j(p+n)ω1].

To simplify the HSS model expression of (12), the following principles are applied [

21]: ① the complex variable s is equal to 0 under the system steady-state operation; ② the elements of the state and input vectors are expanded to (2n+1)-dimensional vectors, which is arranged according to the harmonic-order from fp-nf1 to fp+nf1; ③ the constant term elements of the coefficient matrices are multiplied by (2n+1)-dimensional identity matrix I, and the diagonal matrix N is subtracted from the diagonal elements of the coefficient matrices. Therefore, according to (9)-(11), the time-domain model (7) and (8) can be converted to an HSS model as:

0=AsrΔxsr+Δusr0=AsiΔxsi+Δusi (12)
0=AdcΔxdc+Δudc0=AcrΔxcr+Δucr0=AciΔxci+Δuci (13)

where Adc is expressed by (SA11) in Supplementary Material A; and the other coefficient matrices are addressed in the same manner.

Based on the frequency model of the Park and inverse Park transformations derived in [

26], Δvpcc(r,i)d and Δvpcc(r,i)q in the system dq frame and Δvpcc(r,i)qc in the PLL can be calculated as:

Δvpcc(r,i)d=T(r,i)d+Δvpcc(r,i)aΔvpcc(r,i)q=T(r,i)q+Δvpcc(r,i)a (14)
Δvpcc(r,i)qc=T(r,i)q+Δvpcc(r,i)-Γ[v¯pcc(r,i)d]Δθpll(r,i) (15)

where T(r,i)d+ and T(r,i)q+ are the transformation matrices expressed by (SA12) in Supplementary Material A.

Therefore, when (14) is substituted into (2) and HSS theory is applied, the LTP model of the AC currents of the rectifier and inverter stations can be converted to the LTI model as:

Δic(r,i)=Kpcc,i(r,i)Δvpcc(r,i)+Kidc,i(r,i)Δidc(r,i)+Kα,i(r,i)(Δα(r,i)-Δθpll(r,i)) (16)

where Kpcc,i(r,i), Kidc,i(r,i), and Kα,i(r,i) are expressed by (SA13) in Supplementary Material A.

To obtain Δvdc, the voltage switching function svyy(t) should first be shifted to the frequency domain. This paper applies the frequency-domain modeling method of switching function through double-Fourier series analysis [

31]. Under the assumption that the phase angle perturbation is Δα-Δθpll=Apcos((ωp-ω1)t+ψ), the switching function can be expressed as a 3D periodical function svyy(t)= f(x(t), y(t)) by variable substitution, i.e., x(t)=ω1t and y(t)=(ωp-ω1)t+ψ. Thus, svyy(t) can be expanded by a double-Fourier series as:

svyy(t)=n=-m=-P(n,m)ejnxejmy (17)

where P(n, m) is the complex Fourier coefficient of svyy(t) at frequency nf1+m(fp-f1), which can be obtained by integrating svyy(t) over the unit area. The calculation result is:

P(n,m)=0n=0j1-m2nπJm(nAp)e-jn(α¯-φ¯)e-jn2π3-1(1-e-jnπ)n0 (18)

where Jm(·) is the Bessel function of the first order.

For m=0, the steady-state components of svyy can be expressed in the frequency domain by (19). When the high-order perturbation components, i.e., m=1, are omitted, the linearized model of Δsvyy can be obtained in the frequency domain by (20).

s¯vyy[nf1]=j2nπe-jn(α¯-φ¯)e-jn2π3-1(1-e-jnπ) (19)
Δsvyy=Kα,sv(Δα-Δθpll) (20)

where Kα,sv is expressed by (SA14) in Supplementary Material A.

When (19) and (20) are substituted into (6) and HSS theory is applied, the DC voltages of the LTP model of the rectifier and inverter stations can be converted to the LTI model as:

Δvdc(r,i)=Kpcc,vdc(r,i)Δvpcc(r,i)+Kic,vdc(r,i)Δic(r,i)+Kα,vdc(r,i)(Δα(r,i)-Δθpll(r,i)) (21)

where Kpcc,vdc(r,i)=6/ktΓ(svyy(r,i)); Kic,vdc(r,i)=-3Lt/ktΓ(svyy(r,i))N; and Kα,vdc(r,i)=6/ktΓ(vt2yy(r,i))Kα,sv(r,i).

B. Overall HSS Model and Closed-loop Impedance Model

The overall HSS model of the LCC-HVDC system is obtained by combining (12), (13), (15), (16), and (21), which is illustrated as a block diagram, as shown in Fig. 5.

Fig. 5  Block diagram of overall HSS model of LCC-HVDC.

According to Fig. 5, to obtain the closed-loop AC impedance of the LCC-HVDC receiving end, Δvgr is set to be zero and the transfer function matrix from the imposed perturbation voltage Δvgia to the current response Δigia is solved by the overall HSS model, as shown in (22).

Δigia=Ygi[I-(Ygi+Yfi+YLi)-1Ygi]Δvgia (22)

where Yfi, Ygi, and YLi are expressed by (SA15) and (SA16) in Supplementary Material A.

Then, because Δvgia contains only the perturbation at fp, the current response Δigia[fp] can be solved by (22). Consequently, the closed-loop impedance of the LCC-HVDC receiving end can be obtained as:

ZLCCi(jωp)=Δvgia[fp]Δigia[fp]-Zgi(jωp)=Gii(jωp)-Zgi(jωp) (23)

Similarly, when Δvgi is set to be zero, the closed-loop impedance of the LCC-HVDC sending end can be solved by the overall HSS model as:

ZLCCr(jωp)=Δvgra[fp]Δigra[fp]-Zgr(jωp)=Gir(jωp)-Zgr(jωp) (24)

Note that the imposed perturbation voltage Δvgr[fp] or Δvgi[fp] will produce infinite AC current responses at frequency fp, fp-2f1, fp+12f1, fp-14f1, etc. In addition, the coupling current perturbations will flow across the grid impedance and cause the corresponding voltage perturbations and induce the LCC-HVDC system to affect the current at fp [

26]. Therefore, the frequency-coupling effects are reflected in the closed-loop impedance model. The derived closed-loop impedance can be easily measured and used to analyze system stability.

IV. Verification and Analysis

A. Verification of LCC-HVDC Impedance Model

To validate the precision of the LCC-HVDC sending- and receiving-end impedance model, the CIGRE benchmark model in PSCAD is used, where the system parameters are shown in Table I. The perturbation injection and signal measurements are performed in PSCAD. The measured data are then imported into MATLAB for impedance calculation. Both the positive- and negative-sequence impedances of LCC-HVDC system are calculated by MATLAB using (23) and (24).

TABLE I  System Parameters
SystemParameterValue

Sending-end

system

Rated power 1000 MW
Line-to-line RMS voltage 382.87 kV
Rr1, Rr2, Lr1 3.737 Ω, 2160.633 Ω, 0.151 H
Cr1 3.342 μF
Rr3, Lr2, Cr2 83.32 Ω, 0.0136 H, 6.685 μF
Rr4, Rr5, Lr3, Cr3, Cr4 261.87 Ω, 29.76 Ω, 0.1364 H, 6.685 μF, 74.28 μF
ktr 345/213.46
Ltr 0.113 H
Ti, kpi, kii 0.0012 s, 5.495×10-4, 0.0458
kppr, kipr 2.84×10-4, 3.55×10-4

Receiving-end

system

Line-to-line RMS voltage 215.05 kV
Ri1, Ri2, Ri3, Li1, Lgi 0.7406 Ω, 24.81 Ω, 0.7406 Ω, 36.5 mH, 36.5 mH
Ci1 7.522 μF
Ri4, Li2, Ci2 37.03 Ω, 0.0061 H, 15.04 μF
Ri5, Ri6, Li3, Ci3, Ci4 116.38 Ω, 13.23 Ω, 0.0606 H, 15.04 μF, 167.2 μF
kti 230/209.23
Lti 0.0512 H
Ti, kpv, kiv 0.02 s, 1.845×10-6, 1.367×10-4
kppi, kipi 4.26×10-4, 5.325×10-4
DC line Idcref 2 kA
Ld 0.5968 H
Rd, Cd 2.5 Ω, 26 μF

Because the main harmonic orders of the LCC-HVDC system are the 5th, 7th, 11th, and 13th at the AC line and the 6th and 12th at the DC line, the truncation order n is given as 13. Figure 6 shows the frequency response of the established impedance models and the measured results of the LCC-HVDC sending and receiving ends. Figure 7 shows the frequency response of the impedance models derived from the switching function without considering the commutation process. The derived positive- and negative-sequence impedance models of the LCC-HVDC system are represented by red and blue lines, respectively. It can be observed from Figs. 6 and 7 that, the established models of the LCC-HVDC system are more consistent with the simulation results, thus validating the correctness of the theoretical models.

Fig. 6  Frequency response of established impedance models and measured results of LCC-HVDC. (a) Sending end. (b) Receiving end.

Fig. 7  Frequency response of impedance models derived from switching function without considering commutation process. (a) Sending end. (b) Receiving end.

In addition, Fig. 6 shows that the positive-sequence impedance of the LCC-HVDC receiving end has negative-resistive and capacitive characteristics, i.e., phase angle ranges from -180° to -90° at approximately 65 Hz. Furthermore, the receiving-end grid model Zgi(s) indicates inductive characteristics, i.e., phase angle ranges from 0° to 90°. Therefore, the impedance characteristics of the LCC-HVDC receiving end easily lead to harmonic oscillation between the LCC-HVDC system and the weak receiving-end grid. The positive-sequence impedance of the sending end shows the negative-resistive and capacitive characteristics at approximately 65 Hz, whereas the corresponding impedance magnitude is relatively remote from the grid impedance magnitude. Thus, the stability of the LCC-HVDC sending end is relatively strong under a weak grid.

B. System Stability Analysis Under Different Parameters

According to [

32], [33], impedance-based stability analysis can be conducted through the impedance ratio of Nyquist curves or impedance Bode plots. Impedance Bode plots facilitate the analysis of the variations in system impedance characteristics with parameters. Once the phase margin (PM) is less than zero at the impedance magnitude crossover point, the system produces harmonic oscillation at the corresponding frequency [33]. In addition, Fig. 6 shows that small-signal instability cannot be easily induced by the negative-sequence model of the LCC-HVDC system because of its positive-resistive characteristics. Thus, we can use the positive-sequence model only to analyze the stability of the LCC-HVDC system.

Figure 8(a) and (b) shows the frequency response of the positive-sequence impedance of the LCC-HVDC sending end under different parameters, where the solid and dotted lines represent the LCC-HVDC and sending-end grid impedances, respectively. Figure 8(a) shows that the larger is kpi, the smaller is the PM at approximately 120 Hz. When kpi is increased by twice, the PM is less than zero at 118 Hz, which causes the system to oscillate at 118 Hz. Figure 8(b) shows the effects of the sending-end short-circuit ratio SESCR on the stability of the LCC-HVDC sending-end system. The figures show that the decrease in SESCR, i.e., weakening of the sending-end grid, has little effect on the stability of the LCC-HVDC sending-end system.

Fig. 8  Frequency response of positive-sequence impedance of LCC-HVDC sending end. (a) Changing with kpi. (b) Changing with SESCR.

Figure 9(a) and (b) shows the frequency response of positive-sequence impedance of LCC-HVDC receiving end changes with kpv and short-circuit ratio RESCR. As Fig. 9(a) and (b) shows, the smaller is kpv or RESCR, the more unstable is the LCC-HVDC receiving-end system. When kpv is decreased by four times, the PM is -4.6° at 61 Hz and the system oscillates at this frequency. When RESCR is set to be 2.0, the PM is -9.0° at 62 Hz and the LCC-HVDC receiving-end system oscillates at 62 Hz.

Fig. 9  Frequency response of positive-sequence impedance of LCC-HVDC receiving end. (a) Changing with kpv. (b) Changing with RESCR.

Figure 10 shows the effects of DC line parameters on the LCC-HVDC positive-sequence impedance characteristics. Figure 10(a) and (b) shows that the DC line resistance Rd has little effect on the LCC-HVDC sending- and receiving-end impedances. Figure 10(c) shows that when the DC-side capacitance Cdr of the rectifier station is connected in parallel and increased, the sending-end impedance phase increases to nearly 120 Hz, and the stability of the LCC-HVDC sending-end system is improved. Figure 10(d) shows the effects of Ld of the inverter station on the LCC-HVDC receiving-end impedance. The figure shows that the larger Ld is, the greater the receiving-end impedance phase at approximately 70 Hz and the stability margin of the LCC-HVDC receiving-end system will be.

Fig. 10  Effects of DC line parameters on LCC-HVDC positive-sequence impedance characteristics. (a) Sending-end impedance with the increase of Rd. (b) Receiving-end impedance with the increase of Rd. (c) Sending-end impedance with the increase of Cdr. (d) Receiving-end impedance with the increase of Ld.

V. Virtual Impedance Based Stability Control for LCC-HVDC

The above stability analysis shows that an increase in kpi reduces the phase of the LCC-HVDC sending-end impedance to approximately 120 Hz, whereas an increase in Cdr of the rectifier station increases the phase of the LCC-HVDC sending-end system in this frequency band. In addition, the decrease in kpv or RESCR reduces the PM of the LCC-HVDC receiving-end impedance to approximately 70 Hz, whereas an increase in Ld of the inverter station increases the phase of the LCC-HVDC receiving-end system in this frequency band. Therefore, the small-signal stability of the LCC-HVDC sending- and receiving-end systems can be improved by increasing Cdr and Ld of the DC line, respectively. However, adding capacitors and flat-wave reactors in practical projects increases the construction cost and operation loss of the system. Accordingly, a virtual impedance based stability control is proposed in this paper, and the principle of parameter design is further discussed.

A. Principle of Virtual Impedance

According to the stability analysis and control method of the LCC-HVDC system, parallel virtual capacitance Cv and series virtual inductance Lv are introduced in the rectifier and inverter station controls, respectively, as shown in Fig. 11. Therefore, the actual controlled DC current and voltage are calculated by (25).

idcr'=idcr-Cvdvcvdtvdci'=vdci-Lvdidcidt (25)

Fig. 11  Proposed virtual impedance based stability control.

After the virtual impedance control is introduced, the input vectors Δucr and Δuci in (13) can be re-expressed as Δucr=[Δvpccrqc, kpprΔvpccrqc, ((I-CvRdN)Δidcr-CvNΔvdcl)/Ti, 0]Tand Δuci  =  [Δvpcciqc ,   kppiΔvpcciqc ,   (Δvdci-LvNΔidci) / Tv  ,   0]T, respectively. Therefore, the proposed stability control can reshape the LCC-HVDC impedance characteristics by Δucr and Δuci.

B. Design of Virtual Impedance

To assign proper values for Cv and Lv, the effects of different Cv and Lv on the positive-sequence impedance characteristics of the LCC-HVDC must first be analyzed, as shown in Fig. 12. The dotted and solid lines correspond to the LCC-HVDC impedance characteristic curves varying with Cv and Lv, respectively. Figure 12(a) and (b) shows that with an increase in Cv, the phase of the sending-end impedance in the 110-130 Hz increases, whereas the phase of the receiving-end impedance in the 60-80 Hz range decreases. Nevertheless, Fig. 12(c) and (d) shows that the virtual inductance Lv has the opposite effect on the impedance characteristics of these two frequency bands. To ensure that the PMs of these two frequency bands are satisfactory, Cv and Lv must be carefully selected.

Fig. 12  Frequency responses of positive-sequence impedance of LCC-HVDC system. (a) Sending-end impedance with Lv=0 and the increase of Cv. (b) Receiving-end impedance with Lv=0 and the increase of Cv. (c) Sending-end impedance with Cv=0 and the increase of Lv. (d) Receiving-end impedance with Cv=0 and the increase of Cv.

Figure 13 shows the effects of Cv and Lv on PMs of ZLCCrp(s) in 105-130 Hz and ZLCCip(s) in 60-80 Hz. Figure 13 shows that the effects of Cv and Lv on the PM of ZLCCrp(s) are opposite to those on the PM of ZLCCip(s). Therefore, Cv and Lv are selected as the values corresponding to the maximum value of the PM on the intersection curve of the two planes, i.e., Cv=2×10-5 and Lv=1.5.

Fig. 13  Effects of Cv and Lv on PMs of ZLCCrp(s) and ZLCCip(s).

VI. Simulation Verification

To further verify the above theoretical analyses and stability control, an LCC-HVDC system simulation is performed in PSCAD. The system parameters are the same as those of the stability analysis listed in Table I.

Figure 14 presents the simulation waveforms with different parameters. Figure 14(a) shows that an increase in kpi causes system instability. Conversely, with an increase in SESCR, the LCC-HVDC system can remain stable, as shown in Fig. 14(b). Figure 14(c) and (d) shows that after kpv is reduced from three to four times or RESCR is reduced from 2.1 to 2.0, the LCC-HVDC system waveforms begin to diverge and oscillate. In addition, to analyze the harmonic oscillation frequency, fast Fourier transform (FFT) analyses of unstable AC currents are further conducted, as shown in Fig. 15. Figure 15(a) shows that after kpi is increased by twice, the dominant frequencies of the AC current oscillation are at 118 Hz and 18 Hz, respectively. The harmonics at 118 Hz coincide with the analytical results shown in Fig. 8(a), and the harmonics at 18 Hz represent the coupling result. Furthermore, the harmonic oscillation frequency forecasts shown in Fig. 9(a) and (b) match the harmonic spectra analyses shown in Fig. 15(b) and (c), respectively. Thus, the effectiveness of stability analysis as presented in Section IV is validated by the simulation.

Fig. 14  Simulation waveforms under different parameters. (a) Decreased kpi. (b) Increased SESCR. (c) Decreased kpv. (d) Decreased RESCR.

Fig. 15  FFT analysis of oscillation waveforms. (a) kpi is increased by twice. (b) kpv is decreased by four times. (c) RESCR is reduced to 2.0.

Figure 16 shows the simulation waveforms with the proposed stability control. Figure 16(a) shows that the proposed stability control has little effect on normal operation, whereas Fig. 16(b) shows that when the proposed stability control is adopted, the oscillation of the LCC-HVDC sending end caused by an increase in kpi disappears. As Fig. 16(c) and (d) shows, when the proposed stability control is added, the LCC-HVDC receiving end can maintain stable operation no matter whether the RESCR is reduced to 2.0 or kpv is decreased by four times. Therefore, the small-signal stability of the LCC-HVDC system is greatly enhanced with the proposed stability control.

Fig. 16  Simulation waveforms with proposed stability control. (a) Normal operation. (b) kpi is increased by twice. (c) RESCR is reduced to 2.0. (d) kpv is decreased by four times.

VII. Conclusion

This paper proposes an improved calculation method for the DC voltage and AC current of an LCC-HVDC commutation station to avoid the complex linearization of the commutation process. The paper also establishes an overall HSS model of the LCC-HVDC. To facilitate small-signal stability analysis and impedance measurements of the LCC-HVDC, closed-loop AC impedances of the LCC-HVDC sending and receiving ends are derived based on the HSS model. The positive-sequence impedance of the LCC-HVDC receiving end shows negative-resistive and capacitive characteristics, i.e., phase angle ranges from -180° to -90°, at 56-65 Hz, which easily led to the oscillation between the LCC-HVDC system and the weak receiving-end grid. In addition, the larger the proportional gain of the DC current controller or the smaller the proportional gain of the DC voltage controller, the more unstable the system is. Finally, a virtual impedance based stability control and parameter design method is proposed and verified to improve impedance characteristics and suppress the oscillation of the LCC-HVDC system.

References

1

R. Zhu, X. Zhou, H. Xia et al., “A commutation failure prediction and mitigation method,” Journal of Modern Power Systems and Clean Energy, vol. 10, no. 3, pp. 779-787, May 2022. [Baidu Scholar] 

2

D. Shu, X. Xie, H. Rao et al., “Sub- and super-synchronous interactions between STATCOMs and weak AC/DC transmissions with series compensations,” IEEE Transactions on Power Electronics, vol. 33, no. 9, pp. 7424-7437, Sept. 2018. [Baidu Scholar] 

3

C. Luo, Q. Guo, Y. Hu et al., “Analysis and countermeasures of harmonic instability in a practical HVDC system with fixed series compensation and STATCOM,” in Proceedings of IEEE 2021 4th CIEEC, Wuhan, China, May 2021, pp. 1-6. [Baidu Scholar] 

4

M. Zhang, X. Yuan, and J. Hu, “Mechanism analysis of subsynchronous torsional interaction with PMSG-based WTs and LCC-HVDC,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 9, no. 2, pp. 1708-1724, Apr. 2021. [Baidu Scholar] 

5

W. Du, Z. Zhen, and H. Wang, “The subsynchronous oscillations caused by an LCC HVDC line in a power system under the condition of near strong modal resonance,” IEEE Transactions on Power Delivery, vol. 34, no. 1, pp. 231-240, Feb. 2019. [Baidu Scholar] 

6

Y. Ma, L. Tong, D. Zhu et al., “Analysis of small-signal stability for LCC-HVDC system integrated to weak sending-end system,” in Proceedings of IEEE 2020 4th International Conference on HVDC, Xi’an, China, Nov. 2020, pp. 698-704. [Baidu Scholar] 

7

X. Lin, H. Li, W. Wei et al., “Impact of characteristic harmonics on the small-signal stability of LCC-HVDC station,” in Proceedings of IEEE 2020 4th International Conference on HVDC, Xi’an, China, Nov. 2020, pp. 705-711. [Baidu Scholar] 

8

A. Zheng, C. Guo, P. Cui et al., “Comparative study on small-signal stability of LCC-HVDC system with different control strategies at the inverter station,” IEEE Access, vol. 7, pp. 34946-34953, Jan. 2019. [Baidu Scholar] 

9

J. Sun, “Small-signal methods for AC distributed power systems: a review,” IEEE Transactions on Power Electronics, vol. 24, no. 11, pp. 2545-2554, Nov. 2009. [Baidu Scholar] 

10

V. Salis, A. Costabeber, S. M. Cox et al., “Stability assessment of power-converter-based AC systems by LTP theory: eigenvalue analysis and harmonic impedance estimation,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 5, no. 4, pp. 1513-1525, Dec. 2017. [Baidu Scholar] 

11

X. Wang and F. Blaabjerg, “Harmonic stability in power electronic-based power systems: concept, modeling, and analysis,” IEEE Transactions on Smart Grid, vol. 10, no. 3, pp. 2858-2870, May 2019. [Baidu Scholar] 

12

J. Sun, “Impedance-based stability criterion for grid-connected inverters,” IEEE Transactions on Power Electronics, vol. 26, no. 11, pp. 3075-3078, Mar. 2011. [Baidu Scholar] 

13

H. Xu, F. Nie, Z. Wang et al., “Impedance modeling and stability factor assessment of grid-connected converters based on linear active disturbance rejection control,” Journal of Modern Power Systems and Clean Energy, vol. 9, no. 6, pp. 1327-1338, Nov. 2021. [Baidu Scholar] 

14

A. Rygg, M. Molinas, C. Zhang et al., “On the equivalence and impact on stability of impedance modelling of power electronic converters in different domains,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 5, no. 4, pp. 1444-1454, Dec. 2017. [Baidu Scholar] 

15

H. Liu and J. Sun, “Analytical mapping of harmonics and impedance through phase-controlled converters,” in Proceedings of IEEE 13th COMPEL Workshop, Kyoto, Japan, Jul. 2012, pp. 1-8. [Baidu Scholar] 

16

H. Liu and J. Sun, “DC terminal impedance modeling of LCC-based HVDC converters,” in Proceedings of 14th IEEE Workshop on Control Model and Power Electronics, Salt Lake City, USA, Jun. 2013, pp. 1-9. [Baidu Scholar] 

17

H. Liu and J. Sun, “Small-signal stability analysis of offshore wind farms with LCC HVDC,” in Proceedings of IEEE Grenoble PowerTech, Grenoble, France, Jun. 2013, pp. 1-8. [Baidu Scholar] 

18

Y. Qi, H. Zhao, S. Fan et al., “Small signal frequency-domain model of a LCC-HVDC converter based on an infinite series-converter approach,” IEEE Transactions on Power Delivery, vol. 34, no. 1, pp. 95-106, Feb. 2019. [Baidu Scholar] 

19

X. Chen, J. Ma, S. Wang et al., “An accurate impedance model of line commutated converter with variable commutation overlap,” IEEE Transactions on Power Delivery, vol. 37, no. 1, pp. 562-572, Feb. 2022. [Baidu Scholar] 

20

M. Daryabak, S. Filizadeh, J. Jatskevich et al., “Modeling of LCC-HVDC systems using dynamic phasors,” IEEE Transactions on Power Delivery, vol. 29, no. 4, pp. 1989-1998, Aug. 2014. [Baidu Scholar] 

21

J. Kwon, X. Wang, F. Blaabjerg et al., “Harmonic interaction analysis in a grid-connected converter using harmonic state-space (HSS) modeling,” IEEE Transactions on Power Electronics, vol. 32, no. 9, pp. 6823-6835, Sept. 2017. [Baidu Scholar] 

22

X. Yue, X. Wang, and F. Blaabjerg, “Review of small-signal modeling methods including frequency-coupling dynamics of power converters,” IEEE Transactions on Power Electronics, vol. 34, no. 4, pp. 3313-3328, Apr. 2019. [Baidu Scholar] 

23

J. Lyu, X. Zhang, J. Huang et al., “Comparison of harmonic linearization and harmonic state space methods for impedance modeling of modular multilevel converter,” in Proceedings of IEEE IPEC-Niigata 2018-ECCE Asia, Niigata, Japan, May 2018, pp. 1004-1009. [Baidu Scholar] 

24

J. Kwon, X. Wang, F. Blaabjerg et al., “Linearized modeling methods of AC-DC converters for an accurate frequency response,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 5, no. 4, pp. 1526-1541, Dec. 2017. [Baidu Scholar] 

25

C. Zhang, M. Molinas, S. Føyen et al., “Harmonic-domain SISO equivalent impedance modeling and stability analysis of a single-phase grid-connected VSC,” IEEE Transactions on Power Electronics, vol. 35, no. 9, pp. 9772-9785, Sept. 2020. [Baidu Scholar] 

26

Z. Xu, B. Li, L. Han et al., “A complete HSS-based impedance model of MMC considering grid impedance coupling,” IEEE Transactions on Power Electronics, vol. 35, no. 12, pp. 12929-12948, Dec. 2020. [Baidu Scholar] 

27

L. Fan, P. Zhou, S. Wang et al., “Harmonic state space modeling of LCC-HVDC system with consideration of harmonic dynamics,” in Proceedings of IEEE 3rd CIEEC, Beijing, China, Sept. 2019, pp. 2076-2081. [Baidu Scholar] 

28

K. Sano, T. Kikuma, T. Nakajima et al., “A suppression method of harmonic instability in line-commutated converters applying active harmonic filters,” in Proceedings of IEEE IPECNiigata 2018ECCE Asia, Niigata, Japan, May 2018, pp. 3299-3303. [Baidu Scholar] 

29

B. Fang, Y. Jiao, L. Dong et al., “Using virtual synchronous generator control based energy storage to enhance the stability of sending terminal in LCC-HVDC system,” in Proceedings of IEEE 2020 4th International Conference on HVDC, Xi’an, China, Nov. 2020, pp. 304-309. [Baidu Scholar] 

30

G. Li, Y. Chen, A. Luo et al., “Wideband harmonic voltage feedforward control strategy of STATCOM for mitigating sub-synchronous resonance in wind farm connected to weak grid and LCC HVDC,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 9, no. 4, pp. 4546-4557, Aug. 2021. [Baidu Scholar] 

31

Z. Bing and J. Sun, “Frequency-domain modeling of multipulse converters by double-fourier series method,” IEEE Transactions on Power Electronics, vol. 26, pp. 3804-3809, Dec. 2011. [Baidu Scholar] 

32

Y. Liao and X. Wang, “Impedance-based stability analysis for interconnected converter systems with open-loop RHP poles,” IEEE Transactions on Power Electronics, vol. 35, no. 4, pp. 4388-4397, Apr. 2020. [Baidu Scholar] 

33

Y. Liu, L. Hong, X. Zhou et al., “HSS-based impedance modeling and oscillation suppression strategy for SDBC-STATCOM in LCC-HVDC receiving grid,” in Proceedings of IEEE CIYCEE, Chengdu, China, Dec. 2021, pp. 1-7. [Baidu Scholar]