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Modelling and analysis of half‐/full‐bridge hybrid MMC when riding through DC‐side pole‐to‐ground fault

2021; Institution of Engineering and Technology; Volume: 7; Issue: 3 Linguagem: Inglês

10.1049/hve2.12144

2096-9813

Zhen He, Jiabing Hu, Lei Lin, Pingliang Zeng,

Cardiac Structural Anomalies and Repair

High VoltageEarly View ORIGINAL RESEARCH PAPEROpen Access Modelling and analysis of half-/full-bridge hybrid MMC when riding through DC-side pole-to-ground fault Zhen He, Zhen He orcid.org/0000-0002-0684-6440 School of Automation, Hangzhou Dianzi University, Hangzhou, ChinaSearch for more papers by this authorJiabing Hu, Corresponding Author Jiabing Hu j.hu@hust.edu.cn State Key Laboratory of Advanced Electromagnetic Engineering and Technology, School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan, China Correspondence Jiabing Hu, State Key Laboratory of Advanced Electromagnetic Engineering and Technology, School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan, China. Email: j.hu@hust.edu.cnSearch for more papers by this authorLei Lin, Lei Lin State Key Laboratory of Advanced Electromagnetic Engineering and Technology, School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan, ChinaSearch for more papers by this authorPingliang Zeng, Pingliang Zeng School of Automation, Hangzhou Dianzi University, Hangzhou, ChinaSearch for more papers by this author Zhen He, Zhen He orcid.org/0000-0002-0684-6440 School of Automation, Hangzhou Dianzi University, Hangzhou, ChinaSearch for more papers by this authorJiabing Hu, Corresponding Author Jiabing Hu j.hu@hust.edu.cn State Key Laboratory of Advanced Electromagnetic Engineering and Technology, School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan, China Correspondence Jiabing Hu, State Key Laboratory of Advanced Electromagnetic Engineering and Technology, School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan, China. Email: j.hu@hust.edu.cnSearch for more papers by this authorLei Lin, Lei Lin State Key Laboratory of Advanced Electromagnetic Engineering and Technology, School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan, ChinaSearch for more papers by this authorPingliang Zeng, Pingliang Zeng School of Automation, Hangzhou Dianzi University, Hangzhou, ChinaSearch for more papers by this author First published: 19 September 2021 https://doi.org/10.1049/hve2.12144 Associate Editor: Zixin Li AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract In a modular multilevel converter-based high-voltage direct current (MMC-HVDC) system, the dc fault ride through (FRT) control is an effective way to deal with a dc-side pole-to-ground (PTG) fault. However, the setting of FRT duration brings potential hazards: 1) MMC will face the risk of "secondary short circuit" if FRT duration is short; 2) ac grid may have power angle stability issue if FRT duration is long. To avoid these hazards and provide theoretical guidance for the FRT duration setting, transient behaviour of dc fault current during the PTG FRT stage is explored in this work. Firstly, challenges of the transient analysis are summarised as non-linearity and high-order issues. In light of this, numerical and Hilbert-Huang Transformation methods are introduced to evaluate the non-linearity issue. It is found that the path of MMC from dc current to dc internal voltage is weakly non-linear. Hence, a linear transient model is built to analyse the dc fault current. By participation factor analysis, the order of the proposed model is further reduced, so that an analytical expression of the dc fault current is approximately derived. Based on the analytical expression, regularities and mechanism of dc fault current are fully revealed. Application of the transient analysis to the setting of FRT duration is elaborated in detail. Finally, the PSCAD/EMTDC simulation verifies the validity of the proposed model and analytical expression. 1 INTRODUCTION With growing demands for long-distance and bulk-power transmission, the technology of modular multilevel converter-based high-voltage direct current (MMC-HVDC) develops rapidly [1-3]. However, the vulnerability against dc short-circuit faults is a main obstacle that restricts the wider application of MMC-HVDC [3-6]. Of the various protection methods, the fault ride through (FRT) control that is based on half-/full-bridge hybrid MMC is a promising solution [4-6]. By this method, the dc fault current can be suppressed to zero without relying on extra protection equipment. Meanwhile, MMC is capable of providing a reactive power support to the ac grid during dc faults. Due to these advantages, the hybrid MMC-based FRT control draws much attention, and has already been applied in real projects, for example the Wudongde Project in China [7, 8]. Figure 1 shows a typical waveform of the dc current, which contains fault detecting, riding through and recovery stages. Once a dc short-circuit fault is detected, the FRT control is enabled, and the dc fault current will decay to zero gradually. However, even under the FRT control, the HVDC system will still face some potential hazards. More specifically, if pre-set FRT duration is short while the fault current decays relatively slow, the dc system will recover before dc lines have been fully deionised. At this point, even though the dc fault has been cleared, the system will still have the risk of 'secondary short circuit' [9]. Contrary to that, if pre-set FRT duration is too long, since the dc system cannot transmit active power during the FRT stage, long time power interruption may result in power angle instability of the ac grid [10]. To avoid these hazards, understanding the transient behaviour of the dc fault current during the FRT stage is crucial [11]. Considering that pole-to-ground (PTG) fault is the most common dc fault [3, 4], and symmetrical bipolar is the most typical configuration in HVDC system [12, 13], this work will concentrate on the symmetrical bipolar-based hybrid MMC, and analyse the transient behaviour of the dc fault current during the PTG FRT stage. FIGURE 1Open in figure viewer Motivation of the transient behaviour research Aiming at the transient behaviours study, two challenges exist. Firstly, MMC generally contains some non-linear elements, such as the sub-module (SM) capacitor modulation and phase-locked loop (PLL). Due to these elements, transient behaviour shows a feature of non-linearity, rendering the mature linear analysis methods invalid. Secondly, various states, namely the states of energy storage elements and integral controllers, are included in MMC. With these states, the mathematical relation of MMC has an obvious high-order feature, which complicates the transient behaviour analysis. Many excellent studies are performed to investigate the transient behaviour of the dc fault current, where numerical simulation and differential-algebraic equation-based analysis are two effective ways. With numerical simulation, accurate transient behaviours can be acquired [14-16]. But, with this method alone, the mechanism underlying the transient behaviours cannot be revealed. The differential-algebraic equation-based way inherently has the advantage of understanding the fault mechanism. Until now, most of the analytical calculations are focussed on the fault-detecting stage [17-19]. Building on the infinite switching frequency assumption, the RLC model becomes the most classical and satisfactory transient model, where the calculation error can be maintained within 5% [17]. As for the FRT stage, to the best known of authors, limited attentions are paid. The authors in [20, 21] derived an RL model for dc fault current calculation, which is quite simple and accurate. However, the RL model is based on compensated modulation while practical projects normally adopt direct modulation [22, 23]. Due to the great difference between the two modulation schemes, the transient analysis in [20, 21] cannot fully reveal the fault mechanism either. The author in [24] analysed the SM imbalance issue with consideration of direct modulation. But, due to the non-linearity and high-order challenges, the dc fault current analysis in [24] can only be carried out in qualitative ways while no quantitative conclusions can be drawn. In summary, existing studies are deficient in understanding the fault mechanism underlying the transient behaviours of the dc fault current. Meanwhile, for the setting of FRT duration, theoretical guidance cannot be provided either. In this work, to further explore the dc fault current behaviour of hybrid MMC during the PTG FRT stage, three contributions are made: 1) With numerical and Hilbert-Huang Transformation (HHT) methods, the path of MMC that is from dc current to dc internal voltage is found to be weakly non-linear, so that a linear transient model is established for the dc fault current analysis (Sections 2 and 3). 2) By the participation factor analysis, the order of the established model is reduced. Hence, an analytical expression of the dc fault current is derived (Section 4). 3) Based on the analytical expression, regularities of the dc fault current and the underlying fault mechanism are adequately revealed (Section 5). Finally, application of the transient analysis to the setting of FRT duration is presented in detail (Section 6). Potential hazards, therefore, can be avoided fundamentally. 2 MATHEMATICAL RELATIONS OF HYBRID MMC 2.1 Topology and control structure of hybrid MMC With symmetrical bipolar configuration, MMC station is grounded at the connection point of positive and negative poles. Figure 2 only shows the topology of faulty pole MMC, where the ac-side connects to an infinite ac bus, and each arm consists of half-bridge SMs (HBSMs) and full-bridge SMs (FBSMs). A PTG fault is assumed to occur at the dc-side. C, L and R are SM capacitance, arm inductance and resistance, respectively. Idc and ij (j = a, b, c) are dc and ac currents. utj is the ac terminal voltage. vkj and ikj (k = p, n) are the upper and lower arm voltage and current, respectively. P and Q are active and reactive powers of MMC, respectively. FIGURE 2Open in figure viewer Configuration of faulty pole MMC With the number of FBSM (Nf) equalling or larger than that of the HBSM (Nh), MMC has the capacity of dc fault riding through. Note that without considering ac voltage boosting [25, 26], hybrid MMCs will present the same FRT performance as long as the ratio of FBSMs and HBSMs meets the above constraint (Nf ≥ Nh). As a result, the following study will not take the ratio of Nf and Nh into consideration. Figure 3 shows the PTG FRT control diagram. It contains the SM capacitor voltage control (SCVC), reactive power control (RPC), ac current control (ACC), circulating current suppression control (CCSC) and PLL. vc_avg is the average value of SM capacitor voltages. Skj and icirj are modulation signals and circulating current, ω and θpll are the fundamental angular velocity and output of PLL, respectively. Id and Iq are d- and q-axis components of the ac current, respectively. I2d and I2q are d- and q-axis components of the circulating current, respectively. Superscript * represents the reference value. Due to space limit, more control details can refer to [4-6], and are not repeated here. FIGURE 3Open in figure viewer DC-side PTG FRT control 2.2 Equivalent model of the DC transmission line Figure 4 shows two typical models for the dc transmission line, viz. the lumped RL model and the cascade π model. Before exploring the transient behaviour of Idc, selecting an appropriate model for the dc line is important. ZRL and Zπ are impedance of the two models. With the parameters listed in Table A1 of Appendix A, a comparison between ZRL and Zπ is presented in Figure 4c. The two models have almost the same impedance at a low frequency range (below 100 Hz). However, Zπ differs largely from ZRL at a high frequency range: 1) several resonance points exist; 2) cascade π model appears to be capacitive while the lumped RL model is inductive. FIGURE 4Open in figure viewer DC transmission line model. (a) Lumped RL model, (b) Cascade π model, (c) Impedance comparison between the two models In this work, to avoid the potential hazards mentioned in Introduction and provide theoretical guidance for FRT duration setting, approximate attenuation characteristics of Idc become the main research focus. Accurate analytical analysis of Idc is not necessary for these purposes. From the perspective of the frequency domain, this research focus is closely associated with the low-frequency characteristics of MMC and dc lines. Although some high-frequency components will occur during dc fault (due to stray capacitance), they are generally small, and decay rapidly [27, 28]. Without considering the high-frequency characteristics of the dc line, the research focus will not be significantly influenced. Hence, for simplicity, the following study will adopt the lumped RL model for the dc line, where Ldc and Rdc represent the inductance and resistance, respectively. 2.3 Mathematical description of MMC Take phase A as an example. During the FRT stage, basic mathematical relations can be expressed as follows: { S p a = − m sin ( ω t + φ v ) / 2 S n a = + m sin ( ω t + φ v ) / 2 (1) { i p a = − I d c / 3 − I m sin ( ω t + φ i ) / 2 + I m 2 sin ( 2 ω t + φ i 2 ) i n a = − I d c / 3 + I m sin ( ω t + φ i ) / 2 + I m 2 sin ( 2 ω t + φ i 2 ) (2) { v c p a = V c 0 + V c m 1 sin ( ω t + φ v 1 ) + V c m 2 sin ( 2 ω t + φ v 2 ) v c n a = V c 0 − V c m 1 sin ( ω t + φ v 1 ) + V c m 2 sin ( 2 ω t + φ v 2 ) (3) { C p v c p a = S p a i p a C p v c n a = S n a i n a (4) { v p a = N S p a v c p a v n a = N S n a v c n a (5)where p is the differential operator, N is the total SM number m and φv are the amplitude and initial phase angle of the modulation index, respectively, Im and φi are amplitude and initial phase angle of the ac current, respectively, Im2 and φi2 are amplitude and initial phase angle of the circulating current, respectively, vckj is the SM capacitor voltage, Vc0 is the dc component of vckj, Vcm1 and φv1 are the amplitude and initial phase angle of the fundamental frequency component in vckj, respectively, and Vcm2 and φv2 are the amplitude and initial phase angle of the double fundamental frequency component in vckj, respectively. When FRT control is enabled, the output of CCSC will not immediately follow the step variation of the circulating current's excitation voltage. Hence, the circulating current that is generally suppressed at a steady state will reoccur at this stage. Equation (2), as a result, contains the double fundamental frequency component. Following are the mathematical descriptions of MMC's dc and circulating current loops, which are the two crucial paths that MMC interacts with the dc system. 2.3.1 DC current loop MMC's dc current loop mainly obtains current from the dc system as input, and provides an internal voltage to the dc system as output. From input to output, the dc current loop contains the following: 1) Arm current modulation SM capacitor current is generated by the arm current modulation. Among the three components in Equation (3), only the fundamental frequency component will contribute to the dc internal voltage Vdc. Hence, the fundamental frequency current in the SM capacitor (marked as Ic1d and Ic1q) is derived. { I c 1 d = m d I d c / 6 + ( m d I 2 q − m q I 2 d ) / 4 (6a) { I c 1 q = m q I d c / 6 − ( m d I 2 d + m q I 2 q ) / 4 (6b)where md and mq are d- and q-axis components of modulation index, respectively. With dq transformation, md, mq, I2d and I2q satisfy the following: { m d = m cos φ v m q = m sin φ v (7) { I 2 d = I m 2 cos φ i 2 I 2 q = I m 2 sin φ i 2 (8) It needs to be pointed out that since the ac-side of MMC connects to an infinite ac bus, the dynamics of the PLL is omitted. The dq transformations in Equations (7) and (8), as a result, do not contain θpll. Based on the expressions of Ic1d and Ic1q, the fundamental frequency voltage in the SM capacitor (marked as Vc1d and Vc1q) can be further expressed as follows: { V c 1 d = ( ω I c 1 q + p I c 1 d ) / ( C p 2 + C ω 2 ) V c 1 q = ( p I c 1 q − ω I c 1 d ) / ( C p 2 + C ω 2 ) (9) 2) SM capacitor voltage modulation Arm voltage is generated by the SM capacitor voltage modulation. Combining the upper and lower arm voltages, the dc internal voltage Vdc is obtained as follows: V d c = − N ( m d V c 1 d + m q V c 1 q ) / 2 (10) 2.3.2 Circulating current loop The circulating current forms a close loop inside the MMC. Within the close loop, two elements are included. 1) Excitation voltage of circulating current Focusing on the double fundamental frequency component in Equation (5), the excitation voltage of the circulating current (marked as V2d and V2q) is shown as follows: { V 2 d = − N ( m d V c 1 q + m q V c 1 d ) / 2 (11a) { V 2 q = − N ( m q V c 1 q − m d V c 1 d ) / 2 (11b) 2) Circulating current suppression control Considering CCSC, dynamics of the circulating current loop can be expressed as follows: { ( L p + R ) I 2 d + ( K p 4 + K i 4 / p ) I 2 d + V 2 d / 2 = 0 ( L p + R ) I 2 q + ( K p 4 + K i 4 / p ) I 2 q + V 2 q / 2 = 0 (12)where Kp4 and Ki4 are the proportional and integral parameters of CCSC, respectively. 2.4 DC-side input-output relation of MMC Combining the above description, the two current loops that are respectively represented by red and blue lines in Figure 5 constitute the dc-side input-output relation. For simplicity, they are called the dc part of MMC in this work. Equations (6, 10 and 11) stand for the SM capacitor modulation, while the rest elements correspond to the dynamic of state variables. mdq is the output of the controllers, which is time-variable. Hence, Equations (6, 10 and 11) are non-linear algebraic equations. Compared with the state variable dynamic elements, Equations (6, 10 and 11) are the challenges of the transient analysis. FIGURE 5Open in figure viewer The input–output relation of MMC during the PTG FRT stage 3 TRANSIENT MODELLING OF HYBRID MMC The input–output relation obtained above contains non-linear elements, which cannot be used to analyse the dc fault current directly. In this section, with numerical and HHT methods, the nonlinear characteristics of MMC are analysed at first. Then, a transient model is established for MMC. 3.1 Numerical-simulation-based transient analysis There is no mature analysis method for a non-linear system. This part will, firstly, observe the transient characteristics of MMC with PSCAD/EMTDC. Simulation model is shown in Figure 1. Parameters are presented in Table A2 of Appendix A. Fault detecting period is set to 1 ms. Generally, the characteristics of the non-linear system are closely related not only to the system structure, but also to the specific forms of transient signals. In Figure 5, the non-linear elements are mainly caused by the SM capacitor modulation. The type of these nonlinear elements, from the system structural point of view, is the product of the modulation index (md, mq) and the state variables. The modulation index is the output of the inner control loops, which is time-variable. Hence, the specific forms of md and mq determine the non-linear characteristics of the dc part. Figure 6a shows the simulation waveforms of the modulation index during the FRT stage. It can be seen that at the initial period of the FRT stage, md and mq obviously vary with time. However, from the time scale of the entire FRT stage (hundreds of milliseconds), md and mq reach their fault steady-state values rapidly after experiencing a short fluctuation, and remain constant thereafter. Hence, in the authors' point of view, the modulation index does not have an obvious time-variable feature during the FRT stage. Hence, it is speculated that the path in Figure 5, that is, from the dc fault current to the dc internal voltage is weak non-linear during the FRT stage. FIGURE 6Open in figure viewer Simulated transient characteristics of MMC during the FRT stage. (a) Modulation index, (b) dc fault current 3.2 Verification of the weak-non-linear speculation To verify the weak-non-linear speculation, this part will, on one hand, perform an HHT analysis for the transient signals and, on the other hand, conduct a modal analysis based on a linear assumption. 3.2.1 Results of HHT analysis HHT is a data-based method to analyse non-stationary (transient) signals [29]. It is a combination of the Empirical Mode Decomposition (EMD) and Hilbert Transformation (HT). With the EMD, finite intrinsic mode functions (IMFs) can be decomposed from a transient data sequence. By applying the HT, instantaneous amplitude and frequency of each decomposed IMF can be further obtained. Figure 6b shows the waveform of Idc. Performed HHT analysis for Idc. Results are shown in Figure 7. From Figure 7a, two IMFs exist in Idc. The blue line is a natural attenuation component, marked as IMF1. The orange line is an oscillation component, marked as IMF2. Applying HT to the IMF2, the instantaneous frequency is obtained, which is around 51.55 Hz. FIGURE 7Open in figure viewer The HHT analysis results for Idc during the FRT stage. (a) EMD decomposing, (b) analysis of IMF1, (c) analysis of IMF2 Figure 7b and c are the envelopes of IMF1 and IMF2, respectively. Since there exists inherent errors when performing HHT to analyse the terminals of transient signals (namely the end effect of EMD) [30], this work only concentrates on the intermediate segment of IMF1 and IMF2 (from 1.6 to 1.9 s). It can be found that the two envelops are approximately the zero-input responses of first-order systems. Theoretically, in the zero-input response (marked as h(t)) of a typical first-order system, the time interval between Points M and N is exactly equal to the time constant of the first-order system, where M is any point on h(t), and N is the intersection of Point M's tangent and the steady-state value of h(t). This feature is employed to estimate the attenuation speed of the two IMFs. In Figure 7b and c, the orange solid lines are tangents of any point on the two envelops, and the black dashed lines are steady-state values. With observation, the time intervals, viz. the attenuation time constants of the two envelops, are 0.1008 and 0.0895 s, respectively. 3.2.2 Results of modal analysis The modal analysis is a model-based analysis method for linear systems [31]. When the FRT control is enabled, this part assumes that md and mq reach their steady-state value immediately and remain constant. Hence, the dc part of MMC can be expressed with a set of linear differential-algebraic equation (namely a linear model) as follows: { d ξ I 2 d / d t = I 2 d d ξ I 2 q / d t = I 2 q d I 2 d / d t = [ N ( m d V c 1 q + m q V c 1 d ) / 4 − ( K p 4 + R ) I 2 d − K i 4 ξ I 2 d ] / L d I 2 q / d t = [ N ( m q V c 1 q − m d V c 1 d ) / 4 − ( K p 4 + R ) I 2 q − K i 4 ξ I 2 q ] / L d V c 1 d / d t = [ m d I d c / 6 + m d I 2 q / 4 − m q I 2 d / 4 ] / C + ω V c 1 q d V c 1 q / d t = [ m q I d c / 6 − m d I 2 d / 4 − m q I 2 q / 4 ] / C − ω V c 1 d d I d c / d t = − N ( m d V c 1 d + m q V c 1 q ) / 2 L e q − R e q I d c / L e q (13)where Req = Rdc + 2 R/3, Leq = Ldc + 2L/3, ξI2d and ξI2q are the integrals of I2d and I2q, respectively. Modal analysis results is shown in Table 1. Four modes exist in MMC. For Idc, the former three modes are dominant, whereas the last mode is non-dominant. The attenuation time constant of Mode 1 is 0.102 s, which basically matches IMF1 in Figure 7a. The attenuation time constant and oscillation frequency of Mode 2 are 0.0869 s and 51.49 Hz, respectively. It is approximately consistent with the IMF2 in Figure 7a. However, another dominant mode (Mode 3) cannot be found in HHT-based analysis results. This is because the oscillation frequency of Mode 3 is very low, and compared with Mode 1, its residue is relatively small. Hence, EMD cannot identify this component. TABLE 1. Modal analysis results of the dc part under linear assumption Mode No. Characteristics root Residue Type 1 −9.83 2.85 Dominant 2 −11.51 ± 323.33i 0.138 ∓ 0.015i Dominant 3 −24.49 ± 0.74i 0.188 ± 0.152i Dominant 4 −1244.41 ± 2.95i 0.00006 ± 0.00018i Non-dominant Table 2 shows the comparison of HHT and modal analysis under another three sets of parameters (shown in Table A3 of Appendix A), where only the dominated components are presented. With different parameters, IMF1 and IMF2 still basically match Mode 1 and Mode 2, respectively. Therefore, the data-based HHT analysis and linear model-based modal analysis have a good consistency. The path of MMC that is from the dc fault current to the dc internal voltage, as a result, is considered to be weakly non-linear during the FRT stage. TABLE 2. Comparison of HHT and modal analysis results Parameter 1 Parameter 2 Parameter 3 HHT analysis IMF1 0.145 s 0.102 s 0.146s IMF2 0.0896 s/51.7 Hz 0.0581s/52.3 Hz 0.0557 s/52.5 Hz Modal analysis Mode 1 −6.848 −9.839 −6.835 Mode 2 −11.4 ± 324.88i −18.3 ± 327.49i −18.2 ± 329.03i Mode 3 −24.5 ± 0.75i −36.3 ± 2.31i −36.4 ± 2.28i 3.3 Proposed transient model of MMC Based on the above conclusion, this part directly adopts the assumption in Section 3.2.2. A linear transient model is established for MMC, which is helpful for the analytical analysis of the dc fault current. The established linear model is the same as Equation (13), where md and mq can be calculated based on the mathematical relation at the steady state of the FRT stage. To verify the validity of the proposed model, the calculated Idc, Vdc, I2d and I2q are compared with the simulation results, respectively. In Figure 8, comparison results are presented in per unit value, where the current base is 1.875 kA, and the voltage base is 400 kV. Orange dashed line represents the simulation model while blue solid line stands for the proposed model. Through comparison, the blue curve matches highly with the orange curve during the entire FRT stage, which fully verifies the correctness of the proposed linear transient model. FIGURE 8Open in figure viewer Verification of the proposed transient model. (a) Dc current, (b) dc internal voltage, (c) circulating current in d-axis, (d) circulating current in q-axis 4 ANALYTICAL EXPRESSION OF THE DC FAULT CURRENT The high-order feature (seventh order) of the established linear transient model, however, still challenges the analytical analysis. To address this issue, this section will simplify the proposed transient model at first. Then, an approximate analytical expression of Idc will be derived. 4.1 Simplification of the proposed transient model As shown in Table 1, Mode 2 is the smallest among the three dominant modes in Idc. Neglecting mode 2 will not significantly affect the approximate attenuation characteristics of Idc while the order of the proposed transient model can be lowered. In order to build the relevancy between Mode 2 and the state variables in MMC, a participation factor analysis is conducted [31]. Results are shown in Table 3, where the participation factors that are larger than 0.1 are marked with red. It can be observed that the participation factors of characteristic roots λ2 and λ3 in the state variables Vc1d and Vc1q are larger than 0.9. This means that the relevancy between λ2, λ3 and Vc1d, Vc1q is the highest. Therefore, by ignoring the dynamics of Vc1d and Vc1q, Mode 2 in the proposed transient model can be eliminated. Additionally, the participation factor of λ2 and λ3 in Idc is 0.04, which also implies that ignoring the dynamic of Vc1d and Vc1q will not significantly affect the attenuation characteristics of Idc. TABLE 3. The participation factor analysis of the proposed transient model λ1 λ2, λ3 λ4, λ5 λ6, λ7 −9.83 −11.51 ± 323.33i −24.49 ± 0.74i −1244.41 ± 2.95i ξI2d 0.059703 0.002538 1.12374 0.020021 ξI2q 0.000193 0.002654 1.199732 0.020021 I2d 0.000191 0.008811 0.022378 1.028260 I2q 6.17 × 10−07 0.009212 0.023891 1.028232 Idc 0.899785 0.043813 0.074694 2.79 × 10−05 Vc1d 0.000136 1.010925 0.003163 0.00937 Vc1q 0.041032