Impedance Model for Instability Analysis of LCC-HVDCs Considering Transformer Saturation

Qin Jiang; Ruiting Xu; Baohong Li; Xiang Chen; Yue Yin; Tianqi Liu; Frede Blaabjerg

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Impedance Model for Instability Analysis of LCC-HVDCs Considering Transformer Saturation PDF

- ORCID：
Qin Jiang ^1,2
- ORCID：
Ruiting Xu ^1,2
- ORCID：
Baohong Li ^1,2
✉
- ORCID：
Xiang Chen ^1,2
- ORCID：
Yue Yin ^1,2
- ORCID：
Tianqi Liu ^1,2
- ORCID：
Frede Blaabjerg ³

1. College of Electrical Engineering, Sichuan University, Chengdu 610065, China, and they are also with； 2. Smart Grid Key Laboratory of Sichuan Province, Chengdu 610065, China； 3. Department of Energy Technology, Aalborg University, Aalborg 9920, Denmark

Updated：2024-07-26

DOI：10.35833/MPCE.2023.000340

OUTLINE

Abstract

In line commutated converter based high-voltage direct current (LCC-HVDC) transmission systems, the transformer saturation can induce harmonic instability, which poses a serious threat to the safe operation of the power system. However, the nonlinear characteristics of the power grids introduced by the transformer saturation considerably limit the application of the conventional analysis methods. To address the issue, this paper derives a linear model for the transformer saturation caused by the DC current due to the converter modulation. Afterwards, the nonlinear characteristics of power grids with the transformer saturation is described by a complex valued impedance matrix. Based on the derived impedance matrix, the system harmonic stability is analyzed and the mechanism of the transformer saturation induced harmonic instability is revealed. Finally, the sensitivity analysis is conducted to find the key factors that influence the system core saturation instability. The proposed impedance model is verified by the electromagnetic transient simulation, and the simulation results corroborate the effectiveness of the proposed impedance model.

Keywords

Line commutated converter based high voltage direct current (LCC-HVDC); transformer saturation; harmonic instability; impedance model

I. Introduction

THE line commutated converter based high-voltage direct current (LCC-HVDC) technologies have been widely used for large-scale and long-distance power transmission [

1], [2]. To increase electricity transmission capacity with lower environment impacts, it is common to build high-voltage direct current (HVDC) transmission lines in the existing corridors of high-voltage alternative current (HVAC) transmission lines, like the Hydro-Quebec-New England system [3]-[6].

In the HVAC and HVDC hydro system, the HVAC line may induce the fundamental (1^st-harmonic) current on the HVDC line [

7], [8]. This coupled fundamental current would then be converted to DC components on the AC side of the converter [9], which consequently causes the transformer saturation and generates positive-sequence 2^nd-harmonic current. This 2^nd-harmonic component would be modulated back to the DC side as the fundamental component. Therefore, the dynamic interaction between the converter and the transformer saturation may trigger a type of instability, which is called the core saturation instability. For the safe operation of the LCC-HVDC, it is necessary to analyze the core saturation instability [10].

The harmonic instability of the LCC-HVDC has been studied for many years and a typical analysis method is the impedance modeling method [

11]. In the impedance analysis, the LCC-HVDC is divided into the line commutated converter (LCC) subsystem and the grid subsystem from the point of common coupling (PCC). The instability of the LCC-HVDC is thus described by the interactivity between a linear system (power grids) and a nonlinear system (LCC), as shown in Fig. 1(a). By linearizing the LCC, the system stability is analyzed. For instance, the sequence-domain impedance models have been developed in [12] and [13] for the LCC to predict the instability of the LCC-HVDC. Considering the phase-locked loop (PLL), the dq-frame impedance models are derived for LCC in [14] and [15]. Reference [16] develops a complex

α β

-frame impedance model from the dq-frame impedance model to clearly analyze the frequency coupling effect of the LCC-HVDC. Coupling effects of the characteristic harmonics are taken into consideration with Harmonic state space (HSS) method in [17]-[19]. However, all the above studies only treat the grid side as linear, which may not be the case for the core saturation instability. In practice, the transformer can exhibit a strong nonlinear characteristic under the saturation condition, making the power grids to be nonlinear, as shown in Fig. 1(b). As a result, the conventional methods cannot be applied directly, and the transformer saturation should be taken into consideration for grid impedance modeling.

Fig. 1 Interactive system. (a) Without transformer saturation. (b) With transformer saturation.

Although a wide body of research has been reported on the analysis of transformer saturation, mostly focused on the transformer saturation characteristics. In [

20], the electrical circuit and the magnetic circuit are combined to predict the distorted excitation current in the saturated transformer. Using the Fourier analysis, [21] and [22] find out that the main component of the excitation current in the single-phase saturated transformer is the 2^nd-harmonic. Reference [9] reveals that a three-phase transformer saturated by the converter-modulated DC current can generate positive-sequence 2^nd-harmonic current. However, the previous studies lack elaboration of how to incorporate the transformer saturation in the grid impedance modeling process.

The criterion for the core saturation instability of LCC-HVDC has been analyzed in the existing literature. In [

9], the circulating loop of the coupled fundamental current in the LCC-HVDC is investigated to explain the mechanism of the instability. A criterion based on the increase of the current in a circulating loop is derived in [9], however, the simulation results show that the criterion is conservative. According to the equivalent impedance change of the system with transformer saturation, a mathematical method is derived to analyze the core saturation instability of the LCC-HVDC in [23]. However, the interaction between the power grids and the converter is not discussed. Reference [11] proposes a frequency-scanning method to assess the core saturation instability of the LCC-HVDC under different operations. It shows that the transformer saturation will induce variations to the power grid impedances, which leads to the core saturation instability. Nonetheless, [11] only provides a rough explanation without any rigorous mathematical analysis. Therefore, it is necessary to provide an accurate and in-depth method to analyze the core saturation instability.

In view of the above gaps, a complex valued impedance modeling method is developed to analyze the core saturation instability of the LCC-HVDC. The main contributions of this paper are summarized as follows.

1) A complex valued impedance model is proposed to describe the nonlinear characteristics of power grids with the transformer saturation. The proposed impedance model can be used to effectively analyze the core saturation instability of LCC-HVDC.

2) The mechanism of core saturation instability for the LCC-HVDC is theoretically and quantitatively explained.

3) The sensitivity study is carried out to find the key factors that affect the core saturation instability of the LCC-HVDC. The results indicate that the knee voltage of the transformer and the impedance of power grids predominantly contribute to the core saturation instability, while the control parameters have very limited impact.

The remainder of this paper is organized as follows. Section II provides the system and variable description. Section III derives the admittance model of the LCC. The impedance model is derived for power grids with the transformer saturation in Section IV. Section V analyzes the system stability and the instability mechanisms. Section VI presents the sensitivity analysis, and Section VII draws conclusions.

II. System and Variable Description

The analysis system is single-pole and 12-pulse, which is constructed by AC system, converter transformer, LCC, and DC transmission line. To simplify the analysis, the receiving end is replaced with an ideal voltage source, but the analysis system contains all the components of a real system, and the derived theory still has migration capabilities.

A. Description of System Under Study

The detail of the analysis system under study is shown in Fig. 2, where $i_{d c}$ and $u_{d c}$ are the current and voltage on the DC line, respectively; $u_{c o u p}$ is the voltage source employed to imitate the harmonic induced by the parallel AC transmission line; $v_{a c}$ is the voltage at the PCC; $u_{s}$ is the AC grid voltage; $α$ is the firing angle; $θ = ω_{1} t$ is the phase angle measured by the PLL, and $ω_{1}$ is the fundamental frequency; and R₁-R₃, C₁-C₃, and L₁-L₃ are the resistances, capacitances, and inductances, respectively. The control system is composed of a PLL and a DC-side current control block. The detailed control system is shown in Fig. 3, where $k_{p p}$ and $k_{p i}$ are the control parameters of the PLL; $G$ , $T$ , $k_{c p}$ , and $k_{c i}$ are the control parameters of the DC-side current control; and $i_{d c, r e f}$ is the reference value of $i_{d c}$ . As shown in Fig. 2, the system is divided into AC side and converter side from the PCC.

Fig. 2 Detail of analysis system.

Fig. 3 Control system of LCC-HVDC.

The transfer functions of the control system are expressed as:

G_{p l l} = (k_{p p} + \frac{k_{p i}}{s}) \frac{1}{s} \frac{1}{s + (k_{p p} + \frac{k_{p i}}{s}) \frac{V_{d}}{s}}

(1)

G_{c} = (k_{c p} + \frac{k_{c i}}{s}) \frac{G}{1 + s T}

(2)

where $V_{d}$ is the magnitude of $v_{a c}$ ; G_pll is the transfer function of the PLL; and G_c is the transfer function of the DC-side current control.

B. Complex Described Variable

The stationary-frame, i.e., $α β$ -frame, variables with the subscript “ $α β$ ” can be interchangeably represented by the real-space vector and the complex, as shown in (3).

v = [\begin{matrix} v_{α} \\ v_{β} \end{matrix}] \leftrightarrow v = v_{α} + j v_{β}

(3)

The dq-frame variables with subscripts “dq” are given as:

v_{d q} = [\begin{matrix} v_{d} \\ v_{q} \end{matrix}] \leftrightarrow v_{d q} = v_{d} + j v_{q}

(4)

The variables in one reference frame can be transformed to the other as:

v_{d q} = e^{- j θ} v \leftrightarrow v_{d q} (s) = v (s + j θ)

(5)

In addition, we have the complex conjugation as:

v_{d q}^{*} (s) = v_{d} (s) - j v_{q} (s)

(6)

III. Admittance Model of LCC

Figure 4 shows the closed-loop diagram of the converter side in Fig. 2, where linearized transfer functions can be calculated according to the linearized modeling method in the previous work [

17].

Fig. 4 Closed-loop diagram of converter side.

In Fig. 4, the input small signals are the phase voltage $Δ v_{d q}$ , the firing angle $Δ α$ , and the DC current $Δ i_{d c}$ . The output signals are the DC-side voltage $Δ u_{d c}$ and the AC-side current $Δ i_{d q}$ . The extinction delay angle $Δ δ$ is employed to describe commutation overlap process of the LCC, which is the sum of $Δ α$ and the angle of the commutation overlap $Δ μ$ , i.e., $Δ δ = Δ α + Δ μ$ . $Z_{d c}$ is the equivalent impedance of the DC lines. $G_{α, u_{d c}}$ , $G_{δ, u_{d c}}$ , $G_{i_{d c}, u_{d c}}$ , and $G_{v_{d q}, u_{d c}}$ in the green blocks are the linearized transfer functions between the output $Δ u_{d c}$ and the inputs $Δ α$ , $Δ δ$ , $Δ i_{d c}$ , and $Δ v_{d q}$ , respectively. The transfer functions in the purple blocks ( $G_{α, i_{d q}}$ , $G_{δ, i_{d q}}$ , $G_{i_{d c}, i_{d q}}$ , and $G_{v_{d q}, i_{d q}}$ ) and blue blocks ( $G_{α, δ}$ , $G_{i_{d c}, δ}$ , and $G_{v_{d q}, δ}$ ) are defined in the similar way for the output $Δ i_{d q}$ and the input $Δ δ$ , respectively. As an example, the complex transfer function $G_{v_{d q}, u_{d c}}$ is expressed as:

G_{v_{d q}, u_{d c}} = G_{v_{d}, u_{d c}} + j G_{v_{q}, u_{d c}}

(7)

By rearranging the closed-loop diagram in Fig. 4, the output admittance matrix of the converter side can be calculated, as shown in (8).

[\begin{matrix} I_{d} (s) \\ I_{q} (s) \end{matrix}] = [\begin{matrix} Y_{d d} & Y_{d q} \\ Y_{q d} & Y_{q q} \end{matrix}] [\begin{matrix} V_{d} (s) \\ V_{q} (s) \end{matrix}]

(8)

where $I_{d} (s)$ and $I_{q} (s)$ are the output currents in the dq-frame; $V_{d} (s)$ and $V_{q} (s)$ are the input voltages in the dq-frame; and the expressions of $Y_{d d}$ , $Y_{d q}$ , $Y_{q d}$ , and $Y_{q q}$ are given in Appendix A.

The mathematical relationship between the impedance models in the dq-frame and $α β$ -frame is derived in [

17]. Based on this mathematical relationship, the dq-frame admittance in (8) is transformed to the

α β

-frame as:

[\begin{matrix} I (s) \\ I^{*} (s - 2 j ω_{1}) \end{matrix}] = [\begin{matrix} Y_{s} (s - j ω_{1}) & Y_{c} (s - j ω_{1}) \\ Y_{c}^{*} (s - j ω_{1}) & Y_{s}^{*} (s - j ω_{1}) \end{matrix}] [\begin{matrix} V (s) \\ V^{*} (s - 2 j ω_{1}) \end{matrix}]

(9)

where $s - 2 j ω_{1}$ is the coupled frequency of frequency s; and $Y_{s} (s - j ω_{1})$ and $Y_{c} (s - j ω_{1})$ are the self-admittance and the coupling-admittance, respectively.

In (9), the positive-sequence and negative-sequence variables can be represented by the positive frequency ( $s \geq 0$ ) and negative frequency ( $s < 0$ ), respectively. Specifically, the admittances in (9) are calculated as:

\{\begin{array}{l} Y_{s} = \frac{1}{2} [(Y_{d d} + Y_{q q}) + j (Y_{q d} - Y_{d q})] \\ Y_{s}^{*} = \frac{1}{2} [(Y_{d d} + Y_{q q}) - j (Y_{q d} - Y_{d q})] \\ Y_{c} = \frac{1}{2} [(Y_{d d} - Y_{q q}) + j (Y_{q d} + Y_{d q})] \\ Y_{c}^{*} = \frac{1}{2} [(Y_{d d} - Y_{q q}) - j (Y_{q d} + Y_{d q})] \end{array}

(10)

IV. Impedance Model of Power Grids with Transformer saturations

When the DC current flows into the transformer, the asymmetrical flux $ϕ$ is induced, as shown in Fig. 5(a). Owing to the nonlinear magnetic characteristics of the transformer in Fig. 5(b), the asymmetrical flux further induces the distorted excitation current $i_{e x c i}$ , as illustrated in Fig. 5(c).

Fig. 5 Excitation current under DC bias. (a) Asymmetrical flux. (b) Nonlinear magnetic curve. (c) Distorted excitation current.

It can be found in [

22] that the distorted excitation current is mainly composed of the 2^nd-harmonic component. The linear relationship between the amplitude of the 2^nd-harmonic excitation current and the input DC current is specified in (11) [9], [23].

i_{e x c i, 2}^{} = - k i_{m, 0} c o s (2 ω_{1} t)

(11)

where k is the constant saturation coefficient; $i_{e x c i, 2}$ is the 2^nd-harmonic current; and $i_{m, 0}$ is the input DC current.

Directly influenced by $i_{m, 0}$ , the 2^nd-harmonic components in the three-phase transformer can be calculated using (12).

\{\begin{array}{l} i_{e x c i, a, 2}^{} = - k i_{m, a, 0} c o s 2 ω_{1} t \\ i_{e x c i, b, 2}^{} = - k i_{m, b, 0} c o s 2 (ω_{1} t - \frac{2 π}{3}) \\ i_{e x c i, c, 2}^{} = - k i_{m, c, 0} c o s 2 (ω_{1} t + \frac{2 π}{3}) \end{array}

(12)

where the subscripts “abc” represent the three phases.

Based on (12), the linearized model of the three-phase transformer saturated by the converter-modulated DC currents will be derived in Section IV-A. The impedance model of power grids with the transformer saturation will then be presented in Section IV-B.

A. Linearized Relation Between Input DC Current and the 2^nd-harmonic Current

According to the modulation theory [

12], the coupled fundamental current on the DC transmission line can be modulated into the AC side of the converter as:

\{\begin{array}{l} i_{a} = i_{1} S_{i, a} = I_{1} c o s (ω_{1} t + φ_{1}) \cdot A_{i, a, 1} c o s ω_{1} t \\ i_{b} = i_{1} S_{i, b} = I_{1} c o s (ω_{1} t + φ_{1}) \cdot A_{i, b, 1} c o s (ω_{1} t - 2 π / 3) \\ i_{c} = i_{1} S_{i, c} = I_{1} c o s (ω_{1} t + φ_{1}) \cdot A_{i, c, 1} c o s (ω_{1} t + 2 π / 3) \end{array}

(13)

where $S_{i, a}, S_{i, b},$ and $S_{i, c}$ are the current modulation functions; $A_{i, a, 1}, A_{i, b, 1}$ , and $A_{i, c, 1}$ are the constant coefficients at the fundamental frequency; $φ_{1}$ is the initial phase angle; and $I_{1}$ is the amplitude of the coupled fundamental current. Alternatively, (13) can be rewritten as:

\{\begin{array}{l} i_{a} = 0.5 I_{1} A_{i, a, 1} (c o s φ_{1} + c o s (2 ω_{1} t + φ_{1})) \\ i_{b} = 0.5 I_{1} A_{i, b, 1} (c o s (φ_{1} + 2 π / 3) + c o s (2 ω_{1} t + φ_{1} - 2 π / 3)) \\ i_{c} = 0.5 I_{1} A_{i, c, 1} (c o s (φ_{1} - 2 π / 3) + c o s (2 ω_{1} t + φ_{1} + 2 π / 3)) \end{array}

(14)

In (14), the 2^nd-harmonic currents flow into the power grids directly, and their effect will be analyzed in the next subsection. In addition, the converter-modulated DC currents in (14) are shown as:

\{\begin{array}{l} i_{c o n v, a, d c} = I_{c o n v, a, d c} c o s φ_{1} \\ i_{c o n v, b, d c} = I_{c o n v, b, d c} c o s (φ_{1} + 2 π / 3) \\ i_{c o n v, c, d c} = I_{c o n v, c, d c} c o s (φ_{1} - 2 π / 3) \end{array}

(15)

where $I_{c o n v, a, d c}, I_{c o n v, b, d c}$ , and $I_{c o n v, c, d c}$ are the amplitudes of the converter-modulated DC currents. Equation (15) shows that the three-phase DC currents are different and their phase angle shifts obey a negative sequence. Therefore, $i_{c o n v, a, d c}$ , $i_{c o n v, b, d c}$ , and $i_{c o n v, c, d c}$ are called the negative-sequence DC currents in this paper.

Substituting (15) into (12) yields the 2^nd-harmonic excitation currents in the three-phase transformer ( $i_{t r a n s, a, 2}^{}$ , $i_{t r a n s, b, 2}^{}$ , and $i_{t r a n s, c, 2}^{}$ ) under the converter-modulated negative-sequence DC currents:

\{\begin{array}{l} i_{t r a n s, a, 2}^{} = - k I_{c o n v, a, d c} c o s φ_{1} c o s 2 ω_{1} t \\ i_{t r a n s, b, 2}^{} = - k I_{c o n v, b, d c} c o s (φ_{1} + 2 π / 3) c o s (2 ω_{1} t + 2 π / 3) \\ i_{t r a n s, c, 2}^{} = - k I_{c o n v, c, d c} c o s (φ_{1} - 2 π / 3) c o s (2 ω_{1} t - 2 π / 3) \end{array}

(16)

Equation (16) can be further rearranged as:

\{\begin{array}{l} i_{t r a n s, a, 2}^{} = - 0.5 k I_{c o n v, a, d c} (c o s (2 ω_{1} t + φ_{1}) + c o s (2 ω_{1} t - φ_{1})) \\ i_{t r a n s, b, 2}^{} = - 0.5 k I_{c o n v, b, d c} (c o s (2 ω_{1} t + φ_{1} - 2 π / 3) + c o s (2 ω_{1} t - φ_{1})) \\ i_{t r a n s, c, 2}^{} = - 0.5 k I_{c o n v, c, d c} (c o s (2 ω_{1} t + φ_{1} + 2 π / 3) + c o s (2 ω_{1} t - φ_{1})) \end{array}

(17)

The currents $i_{t r a n s, a, 2}$ , $i_{t r a n s, b, 2}$ , and $i_{t r a n s, c, 2}$ in (17) can be decomposed into zero-sequence and positive-sequence components. The zero-sequence currents that flow into the power grids will induce the zero-sequence voltages u_zs_,_a, u_zs_,_b, and u_zs_,_c on the AC side of the converter. According to the modulation theory [

12], the zero-sequence voltages are then modulated back to the DC side of the converter as:

\begin{array}{l} u_{d c} = u_{z s, a} S_{u, a} + u_{z s, b} S_{u, b} + u_{z s, c} S_{u, c} = \\ U_{z s} c o s (2 ω_{1} t + φ_{1}) \cdot A_{u, a, 1} c o s ω_{1} t + \\ U_{z s} c o s (2 ω_{1} t + φ_{1}) \cdot A_{u, b, 1} c o s (ω_{1} t - 2 π / 3) + \\ U_{z s} c o s (2 ω_{1} t + φ_{1}) \cdot A_{u, c, 1} c o s (ω_{1} t + 2 π / 3) = 0 \end{array}

(18)

where $S_{u, a}, S_{u, b}$ , and $S_{u, c}$ are the voltage modulation functions of the voltages; $A_{u, a, 1}$ , $A_{u, b, 1}$ , and $A_{u, c, 1}$ are the fundamental coefficients of the voltage modulation functions; and $U_{z s}$ is the amplitude of the zero-sequence voltage.

Equation (18) indicates that the zero-sequence voltages on the AC side of the converter correspond to zero-sequence voltage after being modulated back to the DC side. This means the zero-sequence currents in (17) cause no interaction between the power grids and the converter. Thus, the zero-sequence currents in (17) can be neglected in the instability analysis. Then, (17) is simplified as:

\{\begin{array}{l} i_{t r a n s, a, 2}^{} = - 0.5 k I_{c o n v, a, d c} c o s (2 ω_{1} t + φ_{1}) \\ i_{t r a n s, b, 2}^{} = - 0.5 k I_{c o n v, b, d c} c o s (2 ω_{1} t + φ_{1} - 2 π / 3) \\ i_{t r a n s, c, 2}^{} = - 0.5 k I_{c o n v, c, d c} c o s (2 ω_{1} t + φ_{1} + 2 π / 3) \end{array}

(19)

It is clear that $I_{c o n v, a, d c}$ , $I_{c o n v, b, d c}$ , and $I_{c o n v, c, d c}$ induce the positive-sequence 2^nd-harmonic excitation currents in (19). Equation (19) can be transformed into the frequency domain and described by:

I_{t r a n s} (s) = - 0.5 k I_{c o n v, d c}^{*} (s - 2 j ω_{1}) |_{s = 2 j ω_{1}}

(20)

where $I_{c o n v, d c}^{*} (s - 2 j ω_{1})$ is the negative-sequence DC current; and $I_{t r a n s} (s)$ is the 2^nd-harmonic component in the excitation current. It is clear that (19) describes the linear relationship between the input DC current and the output 2^nd-harmonic current.

B. Impedance Model of Power Grids with Transformer Saturations

In Fig. 6, the transformer is represented as a T equivalent circuit. $I_{c o n v} (s)$ is the current modulated by the converter; $I_{c o n v, m} (s)$ and $I_{c o n v, a c} (s)$ are the parts of $I_{c o n v} (s)$ that flow into the magnetizing winding and the power grids, respectively; $V_{c o n v} (s)$ is the voltage induced by $I_{c o n v} (s)$ ; z_ac(s) is the impedance of power grids without transformer saturation; $z_{d c} (s)$ is the impedance of the DC transmission line; $Y_{d c} (s)$ is the admittance model of the LCC; L_m and L_c are the magnetizing inductance and the leakage inductance of the transformer, respectively; and $Z_{a c} (s)$ and $Z_{a c + s t} (s)$ are defined at the end of this subsection.

Fig. 6 Current flows in power grids.

With reference to Fig. 6, $I_{c o n v, a c} (s)$ and $I_{c o n v, m} (s)$ can be expressed as:

I_{c o n v, a c} (s) = \frac{s L_{m} I_{c o n v} (s)}{s L_{m} + z_{a c} (s)}

(21)

I_{c o n v, m} (s) = \frac{z_{a c} (s) I_{c o n v} (s)}{s L_{m} + z_{a c} (s)}

(22)

Considering the frequency coupling effect of the converter shown in (9), $I_{c o n v, a c} (s)$ and $I_{c o n v, m} (s)$ are extended to the $α β$ -frame as:

[\begin{matrix} I_{c o n v, a c} (s) \\ I_{c o n v, a c}^{*} (s - 2 j ω_{1}) \end{matrix}] = [\begin{matrix} \frac{s L_{m} I_{c o n v} (s)}{s L_{m} + z_{a c} (s)} \\ {[\frac{(s - 2 j ω_{1}) L_{m} I_{c o n v} (s - 2 j ω_{1})}{(s - 2 j ω_{1}) L_{m} + z_{a c} (s - 2 j ω_{1})}]}^{*} \end{matrix}]

(23)

[\begin{matrix} I_{c o n v, m} (s) \\ I_{c o n v, m}^{*} (s - 2 j ω_{1}) \end{matrix}] = [\begin{matrix} \frac{z_{a c} (s) I_{c o n v} (s)}{s L_{m} + z_{a c} (s)} \\ {[\frac{z_{a c} (s - 2 j ω_{1}) I_{c o n v} (s - 2 j ω_{1})}{(s - 2 j ω_{1}) L_{m} + z_{a c} (s - 2 j ω_{1})}]}^{*} \end{matrix}]

(24)

Because sL_m>>z_ac(s), only the negative-sequence DC current ( $I_{c o n v, m}^{*} (s - 2 j ω_{1}) |_{s = 2 j ω_{1}} i n (24)$ ) can flow into the magnetizing winding. After substituting (24) into (20), the positive-sequence 2^nd-harmonic excitation current $I_{e x c i, g} (s)$ $(I_{e x c i, g} (s) = - I_{t r a n s} (s))$ induced by the converter-modulated negative-sequence DC current can be obtained as:

[\begin{matrix} I_{e x c i, g} (s) \\ I_{e x c i, g}^{*} (s - 2 j ω_{1}) \end{matrix}] = [\begin{matrix} \frac{k}{2} {[\frac{z_{a c} (s - 2 j ω_{1}) I_{c o n v} (s - 2 j ω_{1})}{(s - 2 j ω_{1}) L_{m} + z_{a c} (s - 2 j ω_{1})}]}^{*} \\ 0 \end{matrix}]

(25)

Given the transformer saturation, the equivalent circuit in Fig. 6 can be changed into Fig. 7. In Fig. 7, the distorted $I_{e x c i} (s)$ flows into the transformer, while $I_{c o n v, a c} (s)$ flows out of the transformer. Then, current flowing into the power grids ( $I_{a c} (s)$ ) and inducing the voltage ( $V_{a c} (s)$ ) can be mathematically described as (26). Since $V_{a c} (s) = V_{c o n v} (s)$ in Fig. 7, (26) can be equivalently written as (27). The matrix $Z_{a c + s t} (s)$ in (27) depicts the relationship between the input current and the output voltage of the power grids.

Fig. 7 Current flows in power grids considering 2^nd-harmonic excitation current.

[\begin{matrix} V_{a c} (s) \\ V_{a c}^{*} (s - 2 j ω_{1}) \end{matrix}] = \underset{Z_{a c} (s)}{\underset{︸}{[\begin{matrix} z_{a c} (s) & 0 \\ 0 & z_{a c} (s - j 2 ω_{1}) \end{matrix}]}} \cdot [\begin{matrix} \frac{s L_{m}}{s L_{m} + z_{a c} (s)} I_{c o n v} (s) + \frac{k}{2} {[\frac{z_{a c} (s - 2 j ω_{1}) I_{c o n v} (s - 2 j ω_{1})}{(s - 2 j ω_{1}) L_{m} + z_{a c} (s - 2 j ω_{1})}]}^{*} \\ {[\frac{(s - 2 j ω_{1}) L_{m} I_{c o n v} (s - 2 j ω_{1})}{(s - 2 j ω_{1}) L_{m} + z_{a c} (s - 2 j ω_{1})}]}^{*} \end{matrix}]

(26)

\begin{array}{l} [\begin{matrix} V_{c o n v} (s) \\ V_{c o n v}^{*} (s - 2 j ω_{1}) \end{matrix}] = \\ \underset{Z_{a c + s t} (s)}{\underset{⏟}{[\begin{matrix} \frac{s L_{m} z_{a c} (s)}{s L_{m} + z_{a c} (s)} & \underset{\begin{matrix} C o u p l i n g & i m p e d a n c e \end{matrix}}{\underset{︸}{\frac{k}{2} z_{a c} (s) {[\frac{z_{a c} (s - 2 j ω_{1})}{(s - 2 j ω_{1}) L_{m} + z_{a c} (s - 2 j ω_{1})}]}^{*}}} \\ 0 & {[\frac{(s - 2 j ω_{1}) L_{m}}{(s - 2 j ω_{1}) L_{m} + z_{a c} (s - 2 j ω_{1})}]}^{*} z_{a c} (s - 2 j ω_{1}) \end{matrix}]}} \cdot \\ [\begin{matrix} I_{c o n v} (s) \\ I_{c o n v}^{*} (s - 2 j ω_{1}) \end{matrix}] = [\begin{matrix} Z_{a c + s t, 11} (s) & Z_{a c + s t, 12} (s) \\ 0 & Z_{a c + s t, 22} (s) \end{matrix}] [\begin{matrix} I_{c o n v} (s) \\ I_{c o n v}^{*} (s - 2 j ω_{1}) \end{matrix}] \end{array}

(27)

Comparing the impedance models of the power grids in (26) and (27), the converter-modulated DC current that flows into the transformer introduces a coupling impedance for the power grids.

V. Simulation Verification

To verify the effectiveness and correctness of the proposed impedance model, the simulation tests are carried out in PSCAD. The system in Fig. 2 is simulated to verify the impedance model of the power grids and the transformer saturation harmonic instability of the LCC-HVDC. The parameters of the LCC-HVDC and the transformer are given in Tables I and II, respectively. In Sections IV-A and IV-B, the derived equivalent admittance models for the LCC-HVDC and power grids with transformer saturation are verified compared with the measured results. In Section IV-C, the correctness of the system harmonic stability analysis method is verified compared with the simulation results.

TABLE I Parameters of LCC-HVDC

Parameter	Value	Parameter	Value
u_dc	100 kV	K_ci	277.8
R₁	10 Ω	K_pp	10
R₂	1030 Ω	K_pi	50
R₃	1 Ω	G	1
L₁	6.33 mH	i_dc	1 kA
L₂	600 mH	u_coup	8 kV
L₃	918 mH	$V_{d}$	100 kV
C₁	1200 μF	u_s	100 kV
C₂	397.31 μF	T	0.0012 s
C₃	22 μF	R_a, R_b, R_c	0.5 Ω
K_cp	0.4540	α	0.28 rad

TABLE II Parameters of Transformer

Parameter	Value
Line-line voltage ratio	100 kV/41 kV
Transformer capacity	80.118 MVA
Leakage reactance	0.13 p.u.
Knee voltage	1.33 p.u.
Magnetizing current	1%

A. Verification of Equivalent Admittance Model for LCC-HVDC

Figures 8 and 9 show the self- and coupling-admittances for the LCC-HVDC under different frequencies, respectively. The calculated results are obtained from (9), while the measured results are obtained via frequency scanning. In Figs. 8 and 9, the calculated results match well with the measured results. Hence, it can be concluded that the admittance model for the LCC-HVDC is accurately derived.

Fig. 8 Frequency response of self-admittance $Y_{s} (s)$ .

Fig. 9 Frequency response of coupling-admittance $Y_{c} (s)$ .

B. Verification of Impedance Model for Power Grids with Transformer Saturation

As the impedance $Z_{a c + s t, 22} (s)$ lags only $2 j ω_{1}$ behind $Z_{a c + s t, 11} (s)$ in (27), only $Z_{a c + s t, 11} (s)$ and $Z_{a c + s t, 12} (s)$ are verified in this subsection. According to Fig. 2, z_ac(s) can be calculated as:

z_{a c} (s) = R_{1} ∥ \frac{1}{s C_{1}} + (R_{2} ∥ s L_{1} ∥ \frac{1}{s C_{2}})

(28)

According to the method in [

18], the constant saturation coefficient k in (27) is measured to be 0.5. In Figs. 10 and 11, the calculated results are obtained from (27), while the measured results are obtained based on [17]. The calculated results match well with the measured results in Figs. 10 and 11, which verify the correctness of the proposed impedance model for the power grids with the transformer saturation in (27). In Fig. 11, it can be found that a spike is introduced to

Z_{a c + s t, 12} (s)

at 100 Hz owing to the transformer saturation. The value of

Z_{a c + s t, 11} (s)

around 0 Hz (highlighted with the green box in Fig. 10) is presented in Fig. 12. It is obvious that

Z_{a c + s t, 11} (s)

at 0 Hz is zero owing to the transformer saturation.

Fig. 10 Frequency response of self-impedance $Z_{a c + s t, 11} (s)$ .

Fig. 11 Frequency response of coupling-impedance $Z_{a c + s t, 12} (s)$ .

Fig. 12 Frequency response of self-impedance $Z_{a c + s t, 11} (s)$ around 0 Hz.

C. Stability Analysis and Mechanism of Core Saturation Instability for LCC-HVDC

Utilizing the admittance model for the LCC-HVDC in Section III-A and the impedance model of the power grids in (26) and (27), the Nyquist contours of the LCC-HVDC are drawn according to (29) [

24].

d e t [λ I (s) - Z_{a c + s t} (s) Y_{d c} (s)] = 0

(29)

where $I (s)$ is the unit matrix.

The Nyquist contours when $k = 0.5$ with and without the transformer saturation are shown in Fig. 13. In Fig. 13(a), the brown contour does not encircle the point $(- 1, 0)$ , which means the system without transformer saturation is stable. However, the blue contour encircles the point $(- 1, 0)$ . According to the generalized Nyquist stability criterion [

24], the system with transformer saturation is predicted to be unstable. According to [9] and [10], the corresponding criterion values with and without transformer saturation are shown in Table III. In Table III, two methods in [9] and [10] predict that the system are unstable whenever the system is with transformer saturation or not.

Fig. 13 Nyquist contour when $k = 0.5$ . (a) Nyquist contour of the first eigenvalue. (b) Nyquist contour of the second eigenvalue.

TABLE III Corresponding Criterion Values with and Without Transformer Saturation

Reference	Criterion value
Reference	With transformer saturation	Without transformer saturation
[9]	-0.6 (unstable)	-1.3 (unstable)
[10]	-0.2 (unstable)	-0.4 (unstable)

In Fig. 13(a), the real part of the eigenvalue for the system without transformer saturation is large than $- 1$ at 100 Hz, suggesting the system has positive damping ( $- 0.01 + 1 > 0$ ) [

12], [24]. However, the real part of the eigenvalue for the system with transformer saturation is less than

- 1

at 100 Hz, and thus the system has negative damping (

- 1.2 + 1 < 0

). The mechanisms of the core saturation instability for the LCC-HVDC can be summarized as follows.

1) The converter-modulated DC current saturates the transformer and induces the positive-sequence 2^nd-harmonic current, which is described in (20).

2) The induced positive-sequence 2^nd-harmonic current changes the equivalent impedance of the power grids, which is described as a coupling impedance shown in (27).

3) According to (29), the coupling impedance degrades the damping of the system from positive to negative, thereby triggering the core saturation instability of the LCC-HVDC.

Based on the model in Fig. 2 and the parameters in Tables I and II, the electromagnetic transient (EMT) simulation is conducted to verify the stability analysis. The fundamental voltage source u_coup is set to be 8 kV during the time period of $3.0 - 3.5 s$ to induce the fundamental current. The measured DC flux of transformer and current on DC transmission line with $k = 0.5$ are shown in Fig. 14.

Fig. 14 Measured DC flux of transformer and current on DC transmission line with $k = 0.5$ . (a) Measured DC flux of transformer. (b) Measured current on DC transmission line.

Before the fundamental voltage is injected, it is clear that the DC flux of the transformer is zero in Fig. 14(a) and the current on the DC transmission line is stable in the Fig. 14(b). After the fundamental voltage is injected, the DC flux rises and does not decay to zero even though the fundamental voltage source is removed at $t = 3.5$ s. As observed from Fig. 14(b), the current on the DC transmission line keeps oscillating after the fundamental voltage source is removed. Therefore, it is concluded that the core saturation instability is triggered, which verifies the prediction results of the proposed impedance model. From Fig. 14, the proposed impedance model is in accordance with the simulation results, which means the proposed impedance model can effectively find core saturation instability of the system. According to Table III, the simulation results show that the methods in [

9] and [10] can effectively find the instability of the system with transformer saturation. However, the two criteria cannot effectively find the stability of the system and tend to be conservative when the transformer of the system is not saturated. It can conclude that the proposed impedance model is accurate than those in [9] and [10].

VI. Sensitivity Analysis

The influences of the knee voltage of transformer, the impedance of power grids, and the control parameters of the LCC-HVDC on the core saturation stability will be analyzed in this section.

A. Sensitivity of Core Saturation Stability to Knee Voltage of Transformer

The knee voltage of transformer is the key factor to decide the saturation coefficient. Thus, the system core saturation stabilities under different knee voltages are analyzed in this subsection. The knee voltages and the corresponding constant saturation coefficients k are shown in Table IV. In addition, the corresponding Nyquist contours are presented in Fig. 15.

TABLE IV Knee Voltages and Corresponding Constant Saturation Coefficients

Knee voltage (p.u.)	k
1.330	0.500
1.463	0.438
1.490	0.400

Fig. 15 Nyquist contours with different k. (a) Nyquist contour of the first eigenvalue. (b) Nyquist contour of the second eigenvalue.

In Fig. 15(a), the blue contour ( $k = 0.5$ ) encircles the point $(- 1,0)$ and the brown contour ( $k = 0.438$ ) passes through this point, which means the systems are predicted to be unstable. However, the green contour ( $k = 0.4$ ) does not encircle the point $(- 1,0)$ , thus the system is predicted to be stable. It indicates that the system tends to be stable with a decreased k.

The measured current on the DC transmission line with $k = 0.438$ and $k = 0.4$ are displayed in Fig. 16. Clearly, the system with $k = 0.5$ is unstable, which has been verified in Fig. 14. In Fig. 16(a), with $k = 0.438$ , the current keeps oscillating with a constant amplitude right after the fundamental voltage source is removed at $t = 3.5$ s, which demonstrates the system is in critical state. In Fig. 16(b), with $k = 0.4$ , the oscillating current converges after the fundamental voltage is removed, indicating that the system is stable. The simulation results in Figs. 14 and 16 are in accordance with the predicted results in Fig. 15.

Fig. 16 Measured current on DC transmission line. (a) $k = 0.438$ . (b) $k = 0.4$ .

B. Sensitivity of Core Saturation Stability to Impedances of Power Grids

As shown in (27), the impedances of power grids at 100 Hz directly affects the coupling impedance. In this subsection, the inductance $L_{1}$ is changed to analyze its influence on the core saturation stability. The corresponding impedances of power grids at 100 Hz are shown in Table V. As can be observed from Table V, the impedance amplitude of power grids decreases with a reduction in $L_{1}$ , while the phase angle remains approximately constant. Figure 15 presents the Nyquist contours with different $L_{1}$ . It is clear from Fig. 17(a) that the blue contour encircles the point $(- 1, 0)$ and brown contour passes through that point, which means the system is unstable. In contrast, the green line does not encircle the point $(- 1, 0)$ , thereby the system is predicted to be stable. From Fig. 17(a), it can be found that the system tends to be stable with a reduction in the impedances of power grids at 100 Hz.

TABLE V Impedances of Power Grids with Different

L_{1}

L₁ (mH)	Impedance (p.u.)
6.330	3.916∠1.073
6.323	3.489∠1.130
6.310	2.883∠1.210

Fig. 17 Nyquist contours with different L₁. (a) Nyquist contour of the first eigenvalue. (b) Nyquist contour of the second eigenvalue.

The measured current on the DC transmission line with different $L_{1}$ is shown in Fig. 18. It can be observed that the systems with $L_{1} = 6.33$ mH (Fig. 14(b)) and $L_{1} = 6.323$ mH (Fig. 18(a)) are unstable, while the system with $L_{1} = 6.31$ mH (Fig. 18(b)) is stable. The simulation results are in agreement with the predicted results in Fig. 17. Thus, it can be concluded that the core saturation stability of the LCC-HVDC is improved by reducing the impedance amplitudes of power grids at 100 Hz.

Fig. 18 Measured current on DC transmission line with different L₁. (a) $L_{1} = 6.323$ mH. (b) $L_{1} = 6.31$ mH.

C. Sensitivity of Core Saturation Stability to Controller Parameter

Equation (29) shows that the DC-side equivalent impedance of the LCC-HVDC can directly influence the stability of the system. Among all the control parameters in Fig. 3, this paper chooses the parameter $k_{c p}$ to shape the impedance of the LCC-HVDC systems. Thus, this subsection investigates the influence of the controller by varying the parameter $k_{c p}$ . The Nyquist contours with different $k_{c p}$ are plotted in Fig. 19.

Fig. 19 Nyquist contours with different $k_{c p}$ . (a) Nyquist contour of the first eigenvalue. (b) Nyquist contour of the second eigenvalue.

In Fig. 19(a), the three Nyquist contours encircle point $(- 1, 0)$ , which means the system under these three states is unstable. The measured currents on the DC transmission line with different $k_{c p}$ are shown in Fig. 20. It is clear that the system is unstable, which are in accordance with the predicted results in Fig. 19. The simulation results also indicate that the control parameter k_cp has a negligible effect on the core saturation instability, and the reasons can be explained as follows.

Fig. 20 Measured current on DC transmission line with different $k_{c p}$ . (a) $k_{c p} = 0.354$ . (b) $k_{c p} = 0.654$ .

With reference to Fig. 3, the oscillation frequency in the DC-side current control or PLL is equal to the frequency of the DC-side harmonic current (50 Hz) or the q-axis harmonic voltage (50 Hz). The LCC-HVDC employs a fixed duration firing strategy to reduce the harmonic components shown in Fig. 21 [

25]. In Fig. 21, only the first firing signal is triggered by the control system in Fig. 3 and the other eleven pulses are sequentially triggered after a fixed duration of

π / 6

for the twelve-pulse converter. Therefore, there is only one control firing pulse in a fundamental circle and the control frequency for the LCC-HVDC systems is 50 Hz. According to the sampling theorem [26], [27], to rebuild the continuous signal, the frequency of the discrete signal should be twice of the continuous signal at least. It can derive that the LCC-HVDC systems can only effectively generate control signals whose frequency are lower than 25 Hz. Namely, the cut-off frequency for the LCC-HVDC systems is 25 Hz at most, which is lower than the transformer saturation induced oscillation (50 Hz). Therefore, changing the control parameters of the PLL or DC-side current control has little effect on the stability of the system.

Fig. 21 Firing pulse of LCC-HVDC.

VII. Conclusion

This paper employs the impedance model to analyze the core saturation instability of the LCC-HVDC systems. Through the theoretical analysis and simulation comparisons, the effectiveness of the proposed impedance model is verified.

1) The nonlinear characteristics of the power grids with the transformer saturation can be described by the complex valued impedance matrix, and the impedance model is thus able to analyze the core saturation instability of the LCC-HVDC systems.

2) From the Nyquist-contour-based stability analysis, the results indicate that the transformer saturation degrades the damping of the system from positive to negative by introducing a coupling impedance at 100 Hz. Hence, the core saturation instability can be triggered in the LCC-HVDC systems.

3) The core saturation instability of the LCC-HVDC systems can be enhanced by increasing the knee voltage of the transformer or reducing the impedances of the power grids at 100 Hz. However, the control parameter has little effect on the core saturation instability.

Appendix

Appendix A

\{\begin{array}{l} Y_{d d} = G_{v_{d}, δ} G_{δ, i_{d}} + G_{v_{d}, i_{d}} + (G_{α, i_{d}} G_{c} + G_{δ, i_{d}} G_{i_{d c}, δ} + G_{i_{d c}, i_{d}}) (G_{δ, u_{d c}} G_{v_{d}, δ} + \\ G_{v_{d}, u_{d c}}) / (z_{d c} - G_{i_{d c}, u_{d c}} - G_{δ, u_{d c}} G_{i_{d c}, δ} - G_{c} G_{α, u_{d c}}) \\ Y_{q d} = G_{v_{d}, δ} G_{δ, i_{q}} + G_{v_{d}, i_{q}} + (G_{α, i_{q}} G_{c} + G_{δ, i_{q}} G_{i_{d c}, δ} + G_{i_{d c}, i_{q}}) (G_{δ, u_{d c}} G_{v_{d}, δ} + \\ G_{v_{d}, u_{d c}}) / (z_{d c} - G_{i_{d c}, u_{d c}} - G_{δ, u_{d c}} G_{i_{d c}, δ} - G_{c} G_{α, u_{d c}}) \\ Y_{d q} = G_{v_{q}, δ} G_{δ, i_{d}} + G_{v_{q}, i_{d}} + G_{α, i_{d}} G_{p l l} + (G_{α, i_{d}} G_{c} + G_{i_{d c}, i_{d}} + \\ G_{δ, i_{d}} G_{i_{d c}, δ}) \frac{G_{v_{q}, u_{d c}} + G_{δ, u_{d c}} G_{v_{q}, δ} - G_{α, u_{d c}} G_{p l l}}{z_{d c} - G_{i_{d c}, u_{d c}} - G_{δ, u_{d c}} G_{i_{d c}, δ} - G_{c} G_{α, u_{d c}}} \\ Y_{q q} = G_{v_{q}, δ} G_{δ, i_{q}} + G_{v_{q}, i_{q}} - G_{α, i_{q}} G_{p l l} + (G_{α, i_{q}} G_{c} + G_{δ, i_{q}} G_{i_{d c}, δ} + \\ G_{i_{d c}, i_{q}}) \frac{G_{δ, u_{d c}} G_{v_{q}, δ} + G_{v_{q}, u_{d c}} - G_{α, u_{d c}} G_{p l l}}{z_{d c} - G_{i_{d c}, u_{d c}} - G_{δ, u_{d c}} G_{i_{d c}, δ} - G_{c} G_{α, u_{d c}}} \end{array}

(A1)

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Introduction

Editorial Board

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Call For Papers

APC

Sponsor & Publisher

Impedance Model for Instability Analysis of LCC-HVDCs Considering Transformer Saturation PDF

Abstract

Keywords

I. Introduction

II. System and Variable Description

A. Description of System Under Study

B. Complex Described Variable

III. Admittance Model of LCC

IV. Impedance Model of Power Grids with Transformer saturations

A. Linearized Relation Between Input DC Current and the 2^nd-harmonic Current

B. Impedance Model of Power Grids with Transformer Saturations

V. Simulation Verification

A. Verification of Equivalent Admittance Model for LCC-HVDC

B. Verification of Impedance Model for Power Grids with Transformer Saturation

C. Stability Analysis and Mechanism of Core Saturation Instability for LCC-HVDC

VI. Sensitivity Analysis

A. Sensitivity of Core Saturation Stability to Knee Voltage of Transformer

B. Sensitivity of Core Saturation Stability to Impedances of Power Grids

C. Sensitivity of Core Saturation Stability to Controller Parameter

VII. Conclusion

Appendix

Appendix A

References

Home

Introduction

Editorial Board

For Author

Call For Papers

APC

Sponsor & Publisher

Impedance Model for Instability Analysis of LCC-HVDCs Considering Transformer Saturation PDF

Abstract

Keywords

I. Introduction

II. System and Variable Description

A. Description of System Under Study

B. Complex Described Variable

III. Admittance Model of LCC

IV. Impedance Model of Power Grids with Transformer saturations

A. Linearized Relation Between Input DC Current and the 2nd-harmonic Current

B. Impedance Model of Power Grids with Transformer Saturations

V. Simulation Verification

A. Verification of Equivalent Admittance Model for LCC-HVDC

B. Verification of Impedance Model for Power Grids with Transformer Saturation

C. Stability Analysis and Mechanism of Core Saturation Instability for LCC-HVDC

VI. Sensitivity Analysis

A. Sensitivity of Core Saturation Stability to Knee Voltage of Transformer

B. Sensitivity of Core Saturation Stability to Impedances of Power Grids

C. Sensitivity of Core Saturation Stability to Controller Parameter

VII. Conclusion

Appendix

Appendix A

References

A. Linearized Relation Between Input DC Current and the 2^nd-harmonic Current