Impedance Analysis of Grid Forming Control Based Modular Multilevel Converters

Rongcai Pan; Guangfu Tang; Shan Liu; Zhiyuan He

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Impedance Analysis of Grid Forming Control Based Modular Multilevel Converters PDF

- ORCID：
Rongcai Pan
✉
- ORCID：
Guangfu Tang
✉
- ORCID：
Shan Liu
✉
- ORCID：
Zhiyuan He
✉

China Electric Power Research Institute, Beijing 100192, China； State Grid Smart Grid Research Institute, Beijing 102209, China； State Grid Smart Grid Research Institute, Beijing 102209, China

Updated：2023-05-22

DOI：10.35833/MPCE.2021.000649

OUTLINE

Abstract

Grid-forming control (GFC) is promising for power electronics based power systems with high renewable energy penetration. Naturally, the impedance modeling for GFC is necessary and has gained significant attention recently. However, most of the impedance analyses for GFC are based on a two-level converter (TLC) rather than a modular multilevel converter (MMC). MMC differs from TLC with respect to its dominant multi-frequency response. It is necessary to analyze the impedance of GFC-based MMC owing to its superiority in high-voltage direct current (HVDC) transmission to interlink two weak AC systems with high renewable energy penetration. As the main contribution, this paper presents the AC- and DC-side impedance analyses for the GFC-based MMC with both power and DC voltage control using the harmonic transfer function (HTF), and compares the impedances of GFC-based MMC and TLC. It is inferred that although the impedance is mainly influenced within 200 Hz, the instability still could occur owing to negative resistance triggered by relatively larger parameters. The difference in AC-side impedance with power and DC voltage control is not apparent with proper parameters, while the DC-side impedance differs significantly. The generalized Nyquist criterion is necessary for AC-side stability owing to the relatively large coupling terms under GFC. Moreover, the coupling between AC- and DC-side impedances is noneligible, especially considering the DC-side resonance around the system resonant peak. The effects of parameters, system strength, and virtual impedance on the impedance shaping are analyzed and verified through simulations.

Keywords

Sequence impedance; modular multilevel converter (MMC); grid-forming control (GFC); harmonic transfer function; generalized Nyquist criterion

I. Introduction

MODULAR multilevel converters (MMCs) [

1] are gaining more and more applications with the demand for high power conversion, including the integration of renewable energy and high-voltage direct current (HVDC) transmission [2]-[4], large motor drive for its superiorities in modularity, flexibility, voltage level scalability, and low harmonics compared with conventional two-level converters (TLCs) [5]. Although an MMC has several advantages, it remains limited, especially owing to its control complexity and harmonic stability due to the large number of submodules (SMs) in the upper/lower arm and the multi-frequency coupling characteristics.

There are two main stability analysis methods, i.e. eigenvalue-based and impedance-based, for converter modeling [

6]-[8]. The eigenvalue-based method usually adopts the state-space and dynamic phasor models [9]. To fully elaborate the system dynamics, the order of state-space model and computation burden could be a predominant shortage [10]. Moreover, the state-space model needs to be updated with system structure modifications, which makes it inappropriate for practical analysis. In contrast, the impedance-based method becomes prevalent, which can avoid complicated elaboration and is easy to develop with black-box property for stability assessment [11], [12].

There are different forms of impedance considering different modeling methods. The state-space averaging and harmonic linearization could obtain the dq small-signal impedance model and sequence impedance model, respectively [

13]-[15]. The relationship between these two methods is demonstrated in [14]. These methods could lead to the misjudgment of MMC stability with only low-frequency harmonics considered. To comprehensively characterize the multi-frequency response of MMC, the harmonic state space (HSS) and harmonic transfer function (HTF) methods are considered for MMC, which are effective and belong to the typical linear time periodical (LTP) system [2]-[4], [12], [16]-[20]. In essence, these methods are the same and termed as HTF in this paper. Reference [16] presents an example for the HTF analysis, where AC-side impedance is deduced considering the phase-locked loop (PLL), and phase and circulating current controllers, but without considering outer-loop controllers. In addition, [18] and [19] adopt HTF to obtain the modified AC-side impedance of MMC with voltage control and vector-based power control for stability analysis in islanded and grid-connected modes. To assess the DC-side impedance for the stability analysis of HVDC system, [2] proposes the DC-side impedance of MMC with vector-based power and DC voltage control. Accordingly, the AC-side impedance of MMC without DC coupling is introduced in [17]. Compared with [17], the AC-side impedance considering DC coupling is further proposed in [3]. Furthermore, [4] analyzes the coupling between AC- and DC-side impedances and introduces the quantitative coupling terms. Similarly, [21] analyzes the influence of grid impedance on the AC- and DC-side impedances of MMC. To obtain an accurate impedance model, [20] proposes the relatively complete DC-side impedance analysis including conventional power loops.

Nevertheless, the grid-forming control (GFC), belonging to inertial support methods for power electronics based power system, operates with power loop synchronization without PLL [

22]-[25]. The influence of GFC without PLL on the TLC has been revealed in [26], [27]. Recently, GFC is proposed for HVDC systems [28], [29]. However, most of these studies focus on the TLC analysis and neglect the multi-harmonics property of MMC. Therefore, it is necessary to build a complete GFC-based HTF model to elaborate the AC- and DC-side impedances for stability analysis, which is the primary contribution of this paper. In this paper, the AC- and DC-side impedance models of GFC-based MMC with power and DC voltage control are analyzed and established. As observed later, the impedance shaping effect of GFC without inner cascaded voltage and current loop is relatively more limited than that with the cascaded voltage and current loop. The influence of GFC itself would be overshadowed with cascaded voltage and current loop. Moreover, the impedance of GFC with inner cascaded voltageand current loop can be analyzed in a way similar to the impedance deduced in [2]-[4].

Therefore, the main work and contributions of this paper are listed as follows. The AC- and DC-side impedances of GFC-based MMC without inner cascaded voltage/current loop are established in a uniform formation with DC voltage and power control, to better elucidate the impedance shaping effect of GFC itself. Moreover, the comparisons of GFC-based MMC and TLC are presented. Like the GFC-based TLC [

26], the negative resistance is also introduced near the fundamental frequency for MMC. It is found that the impedance is influenced within 200 Hz and the instability could happen owing to the negative resistance if parameters are not suitable. The difference in AC-side impedance of MMC with power and DC voltage control is not apparent with proper parameters, while the DC-side impedance with power and DC voltage control differs in the low frequency range. The coupling terms are not well suppressed; hence, the generalized Nyquist criterion (GNC) is necessary for the AC-side stability analysis. Moreover, the DC- and AC-side impedance shaping effects of GFC are relatively weak, and the coupling between AC- and DC-side (from AC to DC or DC to AC) impedances is noneligible considering the interconnected MMC in AC/DC grid. The influences of parameters, system strength, and virtual impedance on the impedance shaping effect are analyzed and verified via simulations.

The remainder of this paper is organized as follows. Section II presents the small-signal model with detailed analysis. Section III presents the modeling method of the AC- and DC-side impedances of the GFC-based MMC. Section IV presents the impedance verification and stability analysis. Section V demonstrates the AC- and DC-side simulations, and Section VI presents the conclusion.

II. Small-signal Model

Figure 1 presents a simplified single-phase diagram of the MMC. Each phase consists of upper and lower arms, both of which comprise $N$ SMs as well as arm inductor $L_{a r m}$ and arm resistance $R_{a r m}$ . Each SM is of half-bridge structure with the capacitor $C_{m}$ . The sum of each SM capacitor voltage for each arm is denoted as $u_{x k}^{Σ}$ , where the subscript $x = a, b, c$ represents the three phases, respectively; and $k = u, l$ represents the upper arm and lower arm, respectively. Similarly, $u_{x k}$ , $m_{x k}$ , $i_{x k}$ , $i_{s x}$ , and $u_{s x}$ represent the equivalent arm voltage, arm modulation index, arm current, grid current, and point of common connection (PCC) voltage, respectively; i_cir is the circulation current; $v_{o}$ is the voltage from the DC midpoint to the neutral point O; and $u_{d c}$ and i_dc are the DC voltage and current, respectively. Then, the MMC is connected with AC and DC grids with AC transmission lines $L_{g}$ and $R_{g}$ and a DC transmission line, respectively.

Fig. 1 Single-phase diagram of MMC.

A. Power Stage of MMC

From Fig. 1, according to the averaged model of MMC and Kirchhoff’s law, the power stage equation can be obtained as:

(\begin{array}{l} L_{a r m} \frac{d i_{x u}}{d t} + R_{a r m} i_{x u} = \frac{u_{d c}}{2} - u_{x u} - u_{s x} + v_{o} \\ L_{a r m} \frac{d i_{x l}}{d t} + R_{a r m} i_{x l} = \frac{u_{d c}}{2} - u_{x l} + u_{s x} - v_{o} \\ u_{x u} = m_{x u} u_{x u}^{Σ} \\ u_{x l} = m_{x l} u_{x l}^{Σ} \\ \frac{C_{m}}{N} \frac{d u_{x u}^{Σ}}{d t} = m_{x u} i_{x u} \\ \frac{C_{m}}{N} \frac{d u_{x l}^{Σ}}{d t} = m_{x l} i_{x l} \end{array}

(1)

In the steady state, there are mainly fundamental, second-order, and third-order harmonics in the arm variables, which are denoted as $f_{1}$ , $2 f_{1}$ , and $3 f_{1}$ , respectively, and they can be expressed in the complex vector form in the frequency domain. Therefore, the third-order harmonic modeling is sufficient to accurately capture the steady-state characteristics of MMC. Moreover, because the notation in different phases and the upper/lower arms share the same principle, the notations with phases are omitted and only the upper-arm variables in complex vector form are provided owing to the sequence relationship in Table I, where $f_{p}$ represents the frequency of perturbation signal. For the steady state, $h = f_{p} = 0$ .

Table I Small-signal Components Under AC and DC Perturbations

Perturbation	Positive AC perturbation		DC perturbation
Perturbation	Sequence	Sign	Sequence	Sign
$f_{p} + 6 h f_{1}$	Positive	DM	Zero	CM
$f_{p} + (6 h + 1) f_{1}$	Negative	CM	Positive	DM
$f_{p} + (6 h + 2) f_{1}$	Zero	DM	Negative	CM
$f_{p} + (6 h + 3) f_{1}$	Positive	CM	Zero	DM
$f_{p} + (6 h + 4) f_{1}$	Negative	DM	Positive	CM
$f_{p} + (6 h + 5) f_{1}$	Zero	CM	Negative	DM

Note: DM and CM indicate that the harmonic is in the differential mode and common mode, respectively.

The complex vector for upper-arm current in steady state $i_{u 0}^{}$ is expressed as (2) with each element as the Fourier coefficient of the corresponding frequency.

i_{u 0}^{} = [I_{u}^{- 3} I_{u}^{- 2} I_{u}^{- 1} I_{u}^{0} I_{u}^{1} I_{u}^{2} I_{u}^{3}]^{T}

(2)

where the subscript 0 represents the steady state; and the superscripts $\pm 3$ , $\pm 2$ , $\pm 1$ , and $0$ represent the variables corresponding to the frequecies of $\pm 3 f_{1}, \pm 2 f_{1}, \pm f_{1}$ , and 0, respectively.

Other related variables can also be expressed in this manner. When considering the impedance deduction, a positive-sequence sinusoidal perturbation current is presented at the PCC or a sinusoidal perturbation voltage is presented at the DC side. The positive-sequence sinusoidal perturbation currents $i_{p a}$ , $i_{p b}$ , and $i_{p c}$ can be expressed as:

\{\begin{array}{l} i_{p a} = I_{p} c o s (2 π f_{p} + ϕ_{i_{p}}) \\ i_{p b} = I_{p} c o s (2 π f_{p} + ϕ_{i_{p}} - 2 π / 3) \\ i_{p c} = I_{p} c o s (2 π f_{p} + ϕ_{i_{p}} + 2 π / 3) \end{array}

(3)

where $I_{p}$ and $ϕ_{i_{p}}$ are the magnitude and phase of perturbation signals, respectively. With perturbations, the small-signal response of harmonic orders would be infinite. However, the magnitude would decrease sharply with the increase of the order. In this paper, the harmonic orders of $f_{p} \pm 3 f_{1}$ , $f_{p} \pm 2 f_{1}$ , $f_{p} \pm f_{1}$ , and $f_{p}$ are considered for both accuracy and simplification. For example, the complex vector for the small-signal upper-arm current $i_{u}$ can be expressed as:

i_{u} = [I_{u}^{p - 3} I_{u}^{p - 2} I_{u}^{p - 1} I_{u}^{p} I_{u}^{p + 1} I_{u}^{p + 2} I_{u}^{p + 3}]^{T}

(4)

where the superscripts $p \pm 3$ , $p \pm 2$ , and $p \pm 1$ represent the variables corresponding to the harmonic orders of $f_{p} \pm 3 f_{1}$ , $f_{p} \pm 2 f_{1}$ , and $f_{p} \pm f_{1}$ , respectively.

Similar to the elements in the steady-state complex vector, each element in the complex vector with perturbations represents the magnitude and phase of the corresponding harmonic order. Other related variables can also be described in this manner. For the symmetrical three-phase circuit of MMC, based on the relationship between the lower and upper arms with harmonics of different orders considered in Table I, the small-signal model can be expressed by one phase with the CM-DM [

3] notations for modulation index, currents, and voltages as:

\{\begin{array}{l} y_{x c} = (y_{x u} + y_{x l}) / 2 \\ y_{x d} = (y_{x u} - y_{x l}) / 2 \end{array}

(5)

where the subscripts c and d represent the CM and DM, respectively; and y could be the vectors of modulation index, current, or voltage, i.e., $y = m, i, o r u$ .

Therefore, with the index of upper/lower arm omitted, the small-signal model of (1) can be transferred into the vector form in frequency domain as:

(\begin{array}{l} \begin{array}{l} Z_{l r} i_{c} = u_{d c} / 2 - u_{c} \\ Z_{l r} i_{d} = v_{o} - u_{d} - u_{s} \end{array} \\ u_{c} = M_{c} u_{c}^{Σ} + M_{d} u_{d}^{Σ} + U_{c}^{Σ} m_{c} + U_{d}^{Σ} m_{d} \\ u_{d} = M_{c} u_{d}^{Σ} + M_{d} u_{c}^{Σ} + U_{c}^{Σ} m_{d} + U_{d}^{Σ} m_{c} \\ Y_{c} u_{c}^{Σ} = M_{c} i_{c} + I_{c} m_{c} + M_{d} i_{d} + I_{d} m_{d} \\ Y_{c} u_{d}^{Σ} = M_{d} i_{c} + I_{c} m_{d} + M_{c} i_{d} + I_{d} m_{c} \end{array}

(6)

where $v_{o} = G_{o} u_{d}$ with $G_{o} (6,6) = 1$ ; $M_{c}$ , $M_{d}$ , $U_{c}^{Σ}$ , $U_{d}^{Σ}$ , $I_{c}$ , and $I_{d}$ are the Toeplitz matrices of the steady-state complex vectors of $m_{c 0}$ , $m_{d 0}$ , $u_{c 0}^{Σ}$ , $u_{d 0}^{Σ}$ , $i_{c 0}$ , and $i_{d 0}$ , respectively [

16], and the matrices

I_{c}

Z_{l r}

, and

Y_{c}

can be expressed as:

I_{c} = [\begin{matrix} I_{c}^{0} & I_{c}^{1} & I_{c}^{2} & I_{c}^{3} & 0 & 0 & 0 \\ I_{c}^{- 1} & I_{c}^{0} & I_{c}^{1} & I_{c}^{2} & I_{c}^{3} & 0 & 0 \\ I_{c}^{- 2} & I_{c}^{- 1} & I_{c}^{0} & I_{c}^{1} & I_{c}^{2} & I_{c}^{3} & 0 \\ I_{c}^{- 3} & I_{c}^{- 2} & I_{c}^{- 1} & I_{c}^{0} & I_{c}^{1} & I_{c}^{2} & I_{c}^{3} \\ 0 & I_{c}^{- 3} & I_{c}^{- 2} & I_{c}^{- 1} & I_{c}^{0} & I_{c}^{1} & I_{c}^{2} \\ 0 & 0 & I_{c}^{- 3} & I_{c}^{- 2} & I_{c}^{- 1} & I_{c}^{0} & I_{c}^{1} \\ 0 & 0 & 0 & I_{c}^{- 3} & I_{c}^{- 2} & I_{c}^{- 1} & I_{c}^{0} \end{matrix}]

(7)

\{\begin{array}{l} Z_{l r} = j 2 π L_{a r m} d i a g (f_{p} + n f_{1} + R_{a r m}) \\ Y_{c} = j 2 π \frac{C_{m}}{N} d i a g (f_{p} + n f_{1}) \end{array}

(8)

Other Toeplitz matrices of the related steady-state variables can be expressed in the same manner without repetition. To obtain the AC-side impedance, the vector of modulation can be expressed as:

\{\begin{array}{l} m_{c} = Q_{c} i_{c} \\ m_{d} = \underset{Q_{i d}}{\underset{︸}{(Q_{p i} + Q_{q i} + Q_{v i})}} i_{d} + Q_{d c} u_{d c} + \underset{Q_{v}}{\underset{︸}{(Q_{p v} + Q_{q v})}} u_{s} \end{array}

(9)

where $Q_{c}$ is the transfer matrix of CM current perturbation for circulating current suppression and DC-side virtual impedance control; $Q_{p i}$ is the transfer matrix of AC (DM) current perturbation for APC; $Q_{p v}$ is the transfer matrix of AC voltage perturbation for APC; $Q_{q i}$ is the transfer matrix of AC (DM) current perturbation for APC; $Q_{q v}$ is the transfer matrix of AC voltage perturbation for RPC; $Q_{v i}$ is the transfer matrix of AC (DM) current perturbation for AC-side virtual impedance control; and $Q_{d c}$ is the transfer matrix of the DC voltage perturbation for DC voltage control. In this section, the AC impedance would be deduced, while DC impedance would be briefly discussed in the next section.

B. GFC

The GFC structure of MMC is illustrated in Fig. 2, where $P_{r e f}$ , $Q_{r e f}$ , $ω_{r e f}$ , $U_{d c r e f}$ , $u_{d r e f}$ , and $u_{q r e f}$ are the reference values of active power, reactive power, frequency, DC voltage, d-axis voltage, and q-axis voltage, respectively; $P_{o u t}$ , $Q_{o u t}$ , and $U_{d c}$ are the output active power, reactive power, and DC voltage, respectively; $i_{s}$ , $u_{s}$ , and $i_{c}$ are the PCC current, PCC voltage, and CM current of MMC, respectively. There are two types of GFC, which are with power control and DC voltage control, respectively. The GFC can include power loop and inner cascaded voltage and current loops, including the virtual impedance method. However, for simplicity, in this paper, the focus is placed on the GFC power loop itself without inner cascaded voltage and current loops.

Fig. 2 GFC structure of MMC. (a) Power control. (b) Cascaded voltage and current control.

In the dq reference frame, the calculated power perturbation could be expressed as:

\{\begin{array}{l} p = 1.5 (u_{d} i_{d} + u_{q} i_{q}) \\ q = 1.5 (u_{q} i_{d} - u_{d} i_{q}) \end{array}

(10)

where $u_{d}$ and $u_{q}$ are the d- and q-axis PCC voltages, respectively; and $i_{d}$ and $i_{q}$ are the d- and q-axis PCC currents, respectively. The power perturbations after linearizing the equations around the equilibrium point are mainly in the order of $f_{p} - f_{1}$ , which can be denoted as $f_{p - 1}$ . And it is easy to obtain (11) for the positive perturbations.

\{\begin{array}{l} p_{p - 1} = 1.5 (U_{d} - j U_{q}) i_{p} + 1.5 (I_{d} - j I_{q}) u_{p} \\ q_{p - 1} = 1.5 (U_{q} + j U_{d}) i_{p} - 1.5 (I_{q} + j I_{d}) u_{p} \end{array}

(11)

where the subscript $p - 1$ represents the variables of frequency $f_{p - 1}$ in the time domain; $U_{d}$ and $U_{q}$ are the magnitudes of d- and q-axis steady-state PCC voltages, respectively; and $I_{d}$ and $I_{q}$ are the magnitudes of d- and q-axis steady-state grid currents, respectively. Therefore, the small-signal complex vectors of active and reactive power can be further expressed as:

\{\begin{array}{l} p = P_{i} i_{d} + P_{u} u_{s} + P_{d c} u_{d c} \\ q = R_{i} i_{d} + R_{u} u_{s} \end{array}

(12)

The PCC currents are twice the arm currents in the DM. Only a few elements in the matrices $P_{i}$ , $P_{u}$ , $R_{i}$ , and $R_{u}$ are non-zero, so the related matrices in the power small-signal equation can be expressed as:

\{\begin{array}{l} P_{i}^{*} (3,2) = P_{i} (3,4) = 3 (U_{d} - j U_{q}) \\ P_{u}^{*} (3,2) = P_{u} (3,4) = 1.5 (I_{d} - j I_{q}) \\ R_{i}^{*} (3,2) = R_{i} (3,4) = 3 (U_{q} + j U_{d}) \\ R_{u}^{*} (3,2) = R_{u} (3,4) = - 1.5 (I_{q} + j I_{d}) \end{array}

(13)

where the superscript * is the conjugate operator. When considering the perturbation from active power to phase $θ$ , two points need to be considered. The first is the power feedback, and the second is the power reference. The second point is only in the DC voltage control. Therefore, the transfer matrix $P_{d c}$ from $u_{d c}$ to reference active power $P_{r e f}$ can be expressed as:

P_{d c} (3,3) = - k_{p d c} - k_{i d c} / s^{'}

(14)

where $s^{'} = s - j 2 π f_{1}$ ; and $k_{p d c}$ and $k_{i d c}$ are the proportional and integral coefficients, respectively. Further, when considering the active power control (APC), for the control structure in Fig. 2, the phase angle perturbation $θ_{p}$ can be expressed as:

θ_{p} = G_{p} (s^{'}) p = G_{p} (s^{'}) (P_{d c} u_{d c} + P_{i} i_{d} + P_{u} u_{s})

(15)

where the control transfer function is $G_{p} (s) = - 1 / [s (D_{p} + J s)]$ with $D_{p}$ and $J$ as the damping coefficient and virtual inertia, respectively.

Similarly, for the reactive power control (RPC), the power perturbation term expressed by matrices $R_{i}$ and $R_{u}$ can also be obtained. The small-signal complex vector of $u_{d r e f}$ can be expressed as (16) with $K_{q}$ as the integral coefficient of reactive power controller.

u_{d r e f} = - \frac{K_{q}}{s^{'}} q = G_{q} (s^{'}) (R_{i} i_{d} + R_{u} u_{s})

(16)

C. Shaping Effect of APC and RPC

The power perturbations and the perturbations caused by DC voltage control are all in the order of $f_{p - 1}$ , which significantly simplifies the analysis. The effect of APC on the impedance comes from two aspects. The first is the effect of Park transformation (PT) on the reference voltage, and the second is the influence on the AC-side virtual impedance shaping.

For the first aspect, the steady-state reference voltage in dq reference frame is $[U_{d r e f}^{0} {, 0]}^{T}$ , and after being transformed back to the abc frame, the reference voltage perturbation in abc frame $[u_{a, r e f}, u_{b, r e f}, u_{c, r e f}]^{T}$ and the modulation perturbation $m_{d p v}$ caused by the phase perturbation can be expressed by inverse Park transformation (IPT) matrix $T_{d q}^{T}$ with phase shift $π / 2$ as:

[\begin{matrix} u_{a, r e f} \\ u_{b, r e f} \\ u_{c, r e f} \end{matrix}] = T_{d q}^{T} (2 π f_{1} t + π / 2) [\begin{matrix} U_{d r e f}^{0} \\ 0 \end{matrix}] θ_{p - 1}

(17)

m_{d p v} = \frac{U_{d r e f}^{0}}{U_{d c}} s i n (2 π f_{1} t) θ_{p - 1}

(18)

where $θ_{p - 1}$ and $U_{d r e f}^{0}$ are the phase angle perturbation of frequency $f_{p - 1}$ in the time domain and the d-axis steady-state reference voltage, respectively. The perturbation triggered by the phase perturbation is expressed in the complex vector form as:

m_{d p v} = G_{p} (s^{'}) G_{p v} (P_{d c} u_{d c} + P_{i} i_{d} + P_{u} u_{s})

(19)

where the element of the matrix $G_{p v}$ is defined as:

G_{p v} (2,3) = G_{p v}^{*} (4,3) = 0.5 j U_{d r e f}^{0} / U_{d c}

(20)

As for the second aspect, either the AC-side virtual resistance or inductance can be implemented, as illustrated in Fig. 2. When switches SWA and SWB in Fig. 2 are connected to 1 or 2, the AC-side virtual inductance or resistance is activated, respectively. Here, the AC-side virtual inductance is analyzed and implemented as (21), and the AC-side virtual resistance can be analyzed in a similar way.

(\begin{array}{l} v_{d v i} (ω) = - \frac{k_{v d} τ_{d} s}{1 + τ_{d} s} i_{q} (ω) = - G_{h d} (s) i_{q} (ω) \\ v_{q v i} (ω) = \frac{k_{v d} τ_{d} s}{1 + τ_{d} s} i_{d} (ω) = G_{h d} (s) i_{d} (ω) \end{array}

(21)

where $i_{d} (ω)$ and $i_{q} (ω)$ are the d- and q-axis small-signal perturbation currents in the frequency domain, respectively; $k_{v d}$ and $τ_{d}$ are the gain and time constant of AC-side virtual impedance controller, respectively; and $v_{d v i} (ω)$ and $v_{q v i} (ω)$ are the voltage drop caused by $i_{q} (ω)$ and $i_{d} (ω)$ , respectively. The d- and q-axis perturbation currents $i_{d θ}$ and $i_{q θ}$ caused by the PT with phase shift $π / 2$ are expressed as:

[\begin{matrix} i_{d θ} \\ i_{q θ} \end{matrix}] = [\begin{matrix} I_{q} \\ - I_{d} \end{matrix}] θ_{p - 1} = T_{d q} (2 π f_{1} t + π / 2) [\begin{matrix} i_{a} \\ i_{b} \\ i_{c} \end{matrix}] θ_{p - 1}

(22)

where $T_{d q}$ is the PT matrix.

Therefore, (21) can be further simplified as:

[\begin{matrix} v_{d v i} (ω) \\ v_{q v i} (ω) \end{matrix}] = G_{h d} (s^{'}) [\begin{matrix} I_{d} \\ I_{q} \end{matrix}] θ_{p - 1} (ω)

(23)

After the IPT, the modulation perturbation of virtual impedance triggered by the APC $m_{d p v i}$ can be obtained as:

m_{d p v i} = G_{h d} (s^{'}) G_{p} (s^{'}) G_{p v i} (P_{d c} u_{d c} + P_{i} i_{d} + P_{u} u_{s})

(24)

where $G_{p v i} (4,3) = G_{p v i}^{*} (2,3) = (I_{d} + j I_{q}) / (2 U_{d c})$ .

Similarly, for the RPC, only one aspect is introduced compared with APC, and the modulation perturbation $m_{d q v}$ caused by the RPC can be obtained as:

m_{d q v} = - \frac{c o s (2 π f_{1} t) u_{d r e f}^{p - 1}}{U_{d c}}

(25)

where $u_{d r e f}^{p - 1}$ is the reference of d-axis small-signal voltage perturbation of frequency $f_{p - 1}$ in time domain. Therefore, the modulation perturbations triggered by RPC can be expressed as:

m_{d q v} = G_{q} (s^{'}) G_{q v} (R_{i} i_{d} + R_{u} u_{s})

(26)

where the matrix $G_{q v}$ can be defined as $G_{q v} (2,3) = G_{q v} (4,3) = - 0.5 / U_{d c}$ .

Considering the AC and DC coupling, the DC- and AC-side perturbations $u_{d c}$ and $u_{s}$ can be expressed with the CM and DM currents as:

\{\begin{array}{l} u_{d c} = Z_{d c} G_{d c} i_{c} \\ u_{s} = Z_{a c} G_{a c} i_{d} \end{array}

(27)

where $Z_{d c}$ and $Z_{a c}$ are the DC- and AC-side impedance matrices in the frequency domain, respectively; $G_{d c} (3,3) = - 3$ ; and $G_{a c} (2,2) = G_{a c} (4,4) = 2$ .

D. Shaping Effect of Virtual Impedance

The current perturbations of the positive- and negative-sequences in the dq frame after the PT are in the order of $f_{p - 1}$ . Therefore, the modulation perturbation $m_{d i}$ caused by AC-side virtual inductance can be expressed as:

m_{d i} = G_{h d} (s^{'}) G_{d i} i_{d}

(28)

where $G_{d i}^{*} (2,2) = G_{d i} (4,4) = 0.5 j / U_{d c}$ .

The DC-side virtual impedance can also be adopted, as illustrated in Fig. 2. Note that the DC-side virtual impedance is implemented in each phase in the abc frame. The DC-side virtual impedance for the CM current can be expressed as (29) with $v_{c v i} (ω)$ being the DC voltage drop across the DC-side virtual impedance by CM currents $i_{c} (ω)$ .

v_{c v i} (ω) = \frac{k_{v c} τ_{c} s}{1 + τ_{c} s} i_{c} (ω) = G_{h c} i_{c} (ω)

(29)

where $k_{v c}$ and $τ_{c}$ are the gain and time constant of the DC-side virtual impedance controller, respectively. The modulation perturbation $m_{c v i}$ caused by DC-side virtual impedance can be obtained as:

m_{c v i} = G_{c v i} i_{c}

(30)

where $G_{c v i} (1,1) = G_{h c} (s - j 2 π \times 3 f_{1}) / U_{d c}$ , $G_{c v i} (3,3) = G_{h c} (s - j 2 π f_{1}) / U_{d c}$ , $G_{c v i} (5,5) = G_{h c} (s + j 2 π f_{1}) / U_{d c}$ , and $G_{c v i} (7,7) = G_{h c} (s + j 2 π \times 3 f_{1}) / U_{d c}$ .

E. Shaping Effect of Circulating Current Suppressing Control

The modulation perturbation $m_{c i}$ with circulating current suppressing control (CCSC) and the PI controller $P I_{c}$ can be expressed as:

m_{c i} = G_{c i} i_{c}

(31)

where $G_{c i}$ is the circulating current controller, $G_{c i} (1,1) = G_{c i} (5,5) = [k_{c p} + k_{c i} / (s - 2 π f_{1})] / U_{d c}$ , and $G_{c i} (7,7) = [k_{c p} + k_{c i} / (s + 2 π 5 f_{1})] / U_{d c}$ .

III. Modeling Method for AC- and DC-side Impedances of GFC-based MMC

A. AC-side Impedance of MMC

With the supplementary definitions in (32) and (33), the detailed expression of the AC-side impedance $Z_{m m c, a c}$ is expressed as (34) with (6) and (9) [

3].

\{\begin{array}{l} \begin{array}{l} Q_{c} = G_{c i} + G_{c v i} \\ Q_{v i} = G_{d i} G_{h d} (s^{'}) \end{array} \\ Q_{d c} = G_{p} (s^{'}) P_{d c} Z_{d c} G_{d c} (G_{p v} + G_{h d} (s^{'}) G_{p v i}) \\ Q_{p i} = G_{p} (s^{'}) P_{i} (G_{h d} (s^{'}) G_{p v i} + G_{p v}) \\ Q_{p v} = G_{p} (s^{'}) P_{u} (G_{h d} (s^{'}) G_{p v i} + G_{p v}) \\ \begin{array}{l} Q_{q i} = G_{q} (s^{'}) G_{q v} R_{i} \\ Q_{q v} = G_{q} (s^{'}) G_{q v} R_{u} \end{array} \end{array}

(32)

\{\begin{array}{l} G_{i c d} = 0.5 Z_{d c} G_{d c} - Ζ_{l r} - B_{d} - C_{d} Q_{c} - C_{c} Q_{d c} \\ Β_{c} = M_{c} Y_{c}^{- 1} M_{d} + M_{d} Y_{c}^{- 1} M_{c} \\ B_{d} = M_{c} Y_{c}^{- 1} M_{c} + M_{d} Y_{c}^{- 1} M_{d} \\ C_{c} = M_{c} Y_{c}^{- 1} I_{d} + M_{d} Y_{c}^{- 1} I_{c} + U_{d}^{Σ} \\ C_{d} = M_{c} Y_{c}^{- 1} I_{c} + M_{d} Y_{c}^{- 1} I_{d} + U_{c}^{Σ} \end{array}

(33)

\begin{array}{l} Z_{m m c, a c} = - 0.5 \frac{Z_{l r} - (G_{o} - E) (B_{d} + C_{d} Q_{i d}) -}{(G_{o} - E) (B_{c} + C_{c} Q_{c} + C_{d} Q_{d c}) G_{i c d}^{- 1} C_{c} Q_{v} +} \to \\ \leftarrow \frac{(G_{o} - E) (B_{c} + C_{c} Q_{c} + C_{d} Q_{d c}) G_{i c d}^{- 1} (B_{c} + C_{c} Q_{i d})}{(G_{o} - E) C_{d} Q_{v} - E} \end{array}

(34)

In summary, CCSC plays the same role as the conventional control. The APC generates $θ$ , while influencing the voltage and AC-side (differential) virtual impedance control. Because the power perturbation includes both current and voltage perturbations, the modulation index perturbation caused by $θ$ includes current and voltage perturbations, which is the main difference between the PLL-based following control. Regarding the RPC, it is the same as conventional control, and the reference voltage includes both current and voltage perturbations. It is worth mentioning that the DC voltage control makes the DM modulation perturbation comprise CM current perturbation, which couples the AC and DC sides. In addition, the DC-side virtual impedance implemented in the abc frame causing CM modulation perturbations can provide damping for CM current. The impedance shaping coupling effect of GFC can be explained and summarized in Table II.

Table II Impedance Shaping Coulping Effect of GFC

Impedance	Control strategy	Perturbation feedback	Detailed expression	Modulation perturbation
AC-side impedance	RPC	u_s, i_d	Q_qvu_s+Q_qii_d	m_d
	APC (voltage reference)	u_s, i_d, i_c	$Q_{p v} u_{s} + Q_{p i} i_{d} + Q_{d c} i_{c}$
	APC (AC-side virtual impedance coupling)	u_s, i_d, i_c	$Q_{p v} u_{s} + Q_{p i} i_{d} + Q_{d c} i_{c}$
	AC-side virtual impedance	i_d	Q_vii_d
	DC-side virtual impedance	i_c	Q_ci_c	m_c
	CCSC	i_c	Q_ci_c	m_c
DC-side impedance	RPC	i_d	Q_qii_d	m_d
	APC (voltage reference)	i_d, u_dc, i_c	Q_pii_d+Q_dcu_dc
	APC (AC-side virtual impedance coupling)	i_d, u_dc, i_c	Q_pii_d+Q_dcu_dc
	AC-side virtual impedance	i_d	Q_vii_d
	DC-side virtual impedance	i_c	Q_ci_c	m_c
	CCSC	i_c	Q_ci_c	m_c

B. DC-side Impedance of MMC

Similarly, the DC-side impedance can also be obtained in a way similar to the DC voltage perturbations. According to Table II, the DC-side impedance would be simpler because it comprises less elements caused by the perturbations. For DC-side impedance, (9) can be revised as:

\{\begin{array}{l} m_{c} = Q_{c} i_{c} \\ m_{d} = \underset{Q_{i d}}{\underset{︸}{(Q_{p i} + Q_{q i} + Q_{v i})}} i_{d} + Q_{d c} u_{d c} \end{array}

(35)

As expressed in (27), the AC voltage perturbations are dependent on the AC-side impedance and DM current. Therefore, only the DM current is required for the power calculations. Similarly, the reactive power feedback introduces the AC current perturbations. Moreover, the phase perturbation includes the DC voltage perturbation and AC current perturbation if DC voltage control is adopted. The corresponding matrices $Q_{c}$ , $Q_{p i}$ , $Q_{q i}$ , $Q_{v i}$ , and $Q_{d c}$ for DC-side impedance can be expressed as:

\{\begin{array}{l} \begin{array}{l} Q_{c} = G_{c i} + G_{c v i} \\ Q_{v i} = G_{h d} (s) G_{d i} \end{array} \\ Q_{d c} = G_{p} (s) P_{d c} (G_{p v} + G_{h d} (s) G_{p v i}) \\ Q_{p i} = G_{p} (s) (G_{p v} + G_{h d} (s) G_{p v i}) (P_{i} + P_{u} Z_{a c}) \\ Q_{q i} = G_{q} (s) G_{q v} (R_{i} + R_{u} Z_{a c}) \end{array}

(36)

The above definitions are similar to those of AC-side impedance with few modifications and are no longer repeated. The DC-side impedance can be calculated using (37) with (35) and (36).

Y_{m m c, d c} = 3 \frac{(0.5 E - C_{c} Q_{d c}) - (B_{c} + C_{c} Q_{i d}) G_{i c c}^{- 1} (G_{o} - E) C_{d} Q_{d c}}{Z_{l r} + B_{d} + C_{d} Q_{c} + (B_{c} + C_{c} Q_{i d}) G_{i c c}^{- 1} (G_{o} - E) (B_{c} + C_{c} Q_{c})}

(37)

where the matrix $G_{i c c}$ is defined as:

G_{i c c} = Z_{l r} + Z_{a c} G_{a c} - (G_{o} - E) (B_{d} + C_{d} Q_{i d})

(38)

Presently, the DC-side impedance of MMC with power or DC voltage control is obtained.

IV. Impedance Validation and Stability Analysis

In this section, the AC- and DC-side impedances are measured via simulation and compared with the analysis results. The comparison of impedances between MMC and TLC is also provided. To obtain a clear understanding with the impedance shaping effect of the control system, the incremental impedance analysis and comparison are conducted in this paper. The detailed impedance measurement procedure can be found in [

3]. The system parameters for the analysis are presented in Table III. Here, the droop control (

J = 0

) is adopted for the brief analysis and

J \neq 0

can be analyzed in the same way.

Table III System Parameters

Quantity	Value
AC system line voltage $U_{s}$	$320 k V$
$f_{1}$	$50 H z$
$U_{d c}$	$640 k V$
$P_{r e f}$	$800 M W$
$C_{m}$	$7000 μ F$
$L_{a r m}$	$40 m H$
$N$	$288$
Circulating current controller	$k_{i c} = 2000, k_{p c} = 20$
DC voltage controller	$k_{i d c} = 50, k_{p d c} = 20$
RPC	$k_{q} = 4$
APC of power control mode	$D_{p} = 40$ , $J = 0$
APC of DC voltage control mode	$D_{p} = 100$ , $J = 0$
AC-side virtual impedance controller	$k_{v d} = 50, τ_{d} = 0.1 s$
DC-side virtual impedance controller	$k_{v c} = 10, τ_{c} = 0.1 s$
Equivalent resistance in each arm $R_{a r m}$	$1 Ω$

A. AC-side Impedance

The AC-side impedance of open-loop control with CCSC is presented in Fig. 3(a).

Fig. 3 AC-side impedance of MMC (1 p.u. active power towards AC grid). (a) Open-loop control with CCSC. (b) Impedance of GFC-based MMC with APC.

Note that $Z_{n p} = Z_{m m c, a c} (2,4)$ , $Z_{p n} = Z_{m m c, a c} (4,2)$ , and $Z_{n p} (s) = Z_{p n}^{*} (j 2 ω_{1} - s)$ are the coupling terms [

3]. The CCSC can suppress the open-loop resonances, and no apparent negative resistance and magnitude peaks are increased with CCSC [3].

However, in Fig. 3(b), the impedance of GFC-based MMC with the APC activated and RPC disabled is demonstrated. As shown in Fig. 3, the APC causes the negative resistance around the fundamental frequency, which might make the stability worse than the open-loop control with CCSC. In addition, the APC behaves as the current source with a relatively large impedance around the fundamental frequency, which is not really the voltage source type owing to current perturbations in the active power perturbations compared with the voltage-frequency (VF) control in [

3]. Moreover, the impedance shaping effect of APC is weaker than that of the conventional vector control [3]. The main affected frequency range is within 200 Hz. The coupling terms

Z_{n p}

are apparent and not properly suppressed under GFC without inner voltage or current loop. Therefore, GNC is necessary for stability analysis. RPC can also make the stability worse than the constant voltage control and the effect of RPC would be demonstrated later in the Nyquist analysis for brevity.

The instability caused by the APC and RPC can be made up by the proper virtual impedance design. Figure 4(a) presents the AC-side impedance of MMC under open-loop control with both CCSC and DC-side virtual impedance. As illustrated in Fig. 4(a), the DC-side virtual impedance can effectively suppress the resonances. In addition, the AC-side virtual inductance can effectively shape the impedance, as illustrated in Fig. 4(b). As for the AC-side virtual inductance coupled with the phase angle generated by APC, it is significantly smaller than the AC-side virtual inductance shaping effect itself. Here, $R_{a r m} = 0.5 Ω$ , while other parameters are the same as Table III.

Fig. 4 AC-side impedance of MMC (1 p.u. active power towards AC grid). (a) Under open-loop control with CCSC and DC-side virtual impedance. (b) With AC-side virtual inductance and RPC disabled. (c) With DC-side virtual inductor and RPC disabled.

As for the DC voltage control strategy, the main difference from the power control is that the power reference is generated by the DC voltage controller. However, the perturbation caused by the DC voltage controller is significantly smaller when the DC-side impedance is not so large. To verify this, the DC side is equipped with the typical $0.1 H$ inductor.

It can be observed from Fig. 4(c) that the difference between the DC voltage control and power control is small. The DC voltage control mainly influences the DC-side impedance of MMC. In addition, as shown in Figs. 3(b) and 4(c), the DC-side impedance can have an influence on the AC-side impedance, and the AC and DC couplings are noneligible.

B. DC-side Impedance

The DC-side impedance with CCSC, APC, and RPC under power control and DC voltage control modes is presented in Fig. 5(a).

Fig. 5 DC-side impedance of MMC (1 p.u. active power towards AC grid). (a) Under power control and DC voltage control with DC inductor and RPC disabled. (b) Considering virtual impedance.

Apparently, the impedance shaping effect of GFC on the DC-side impedance is also weak, and the main influences are in low-frequency ranges (e.g., $<$ 10 Hz) and around a system resonance peak (e.g., 80 Hz). Here, $D_{p} = 100, L_{g} = 0.1 H$ , and others are the same as Table III.

Compared with the GFC, the virtual impedance can effectively influence the DC-side impedance. Figure 5(b) presents the DC-side impedance considering the virtual impedance. It is apparent that the DC-side virtual impedance can effectively suppress the resonances around 60 Hz and 100 Hz, while the AC-side virtual impedance can suppress the resonance around 80 Hz.

C. Comparison Between Impedances of GFC-based TLC and MMC

To demonstrate the difference between impedances of GFC-based TLC and MMC, power control mode is presented for simplicity. First, the control parameters are the same and the AC-side filter impedance of TLC is half of the arm impedance of MMC, and the arm impedance of MMC is equivalent to half of the AC-side impedance. Therefore, the open-loop positive-sequence impedance and the coupling term of TLC are $Z_{o p p} = 0.5 (s L_{a r m} + R_{a r m})$ and $Z_{o n p} = 0$ , respectively, while $Z_{o p p} = Z_{a c, m m c} (4,4)$ , $Z_{o n p} = Z_{a c, m m c} (2,4)$ with $Q_{i d} = Q_{v} = Q_{d c} = 0$ for MMC using HTF. As illustrated in Fig. 6(b), the open-loop positive-sequence impedance of TLC is very simple and there is no coupling term ( $Z_{o n p} = 0$ ) compared with that of MMC with CCSC.

Fig. 6 Impedance comparison of open-loop and GFC-based TLC and MMC. (a) Unified block based on harmonic linearization. (b) Detailed impedance comparison of TLC and MMC under open-loop control and GFC with RPC disabled.

After the open-loop comparison, the impedance of GFC-based MMC can be obtained in a uniform way according to the impedance shaping circuit in Fig. 6(a) based on harmonic linearization in [

12]. Here, the GFC with APC and RPC disabled is implemented for simplicity to demonstrate the shaping effect of GFC on open-loop impedance. The impedances of GFC-based TLC and MMC share the same formations as (39). The parameters of

A_{p}

A_{n}

B_{p}

B_{n}

C_{p}

, and

C_{n}

for GFC-based MMC could be obtained from the previous analysis, while

A_{p}

A_{n}

B_{p}

B_{n}

C_{p}

, and

C_{n}

for GFC-based TLC share similar formations and are not presented for simplicity, which can refer to the impedance modeling in [12]. Apparently, the GFC shaping effects are the same in a degree, as shown by the yellow blocks in Fig. 6(b). Only TLC measurements are presented in Fig. 6(a) because the MMC measurements are already shown in Fig. 3. Therefore, the main difference of GFC-based MMC and TLC is the open-loop characteristic, which reflects the inherent difference, as marked in the green blocks in Fig. 6.

\{\begin{array}{l} u_{p} = Z_{o p p} i_{p} + A_{p} i_{p} + B_{p} u_{p} + C_{p} u_{n} \\ u_{n} = Z_{o n p} i_{p} + A_{n} i_{p} + B_{n} u_{p} + C_{n} u_{n} \\ Z_{p p} = \frac{u_{p}}{i_{p}} = \frac{(Z_{o p p} + A_{p}) (1 - C_{n}) + C_{p} (Z_{o n p} + A_{n})}{(1 - B_{p}) (1 - C_{n}) - C_{p} B_{n}} \\ Z_{n p} = \frac{u_{n}}{i_{p}} = \frac{Z_{o n p} + A_{n} + Z_{p p} B_{n}}{1 - C_{n}} \end{array}

(39)

D. AC-side Nyquist Stability Analysis

The power control station is used for the AC-side Nyquist stability analysis, and the DC power control is adopted for the DC-side stability analysis. The main purpose for the AC-side stability analysis here is to reveal the influence of the APC and RPC as well as the virtual impedance control on the AC-side stability. The AC-side stability with GFC can be accessed by GNC on $Z_{m m c, a c} / Z_{g}$ with the main coupled positive- and negative-sequence considered [

26].

In this analysis, groups 1 and 2 listed in Table IV with different grid impedances are tested. The DC side of MMC is connected with the idea DC voltage, and the parameters of MMC are the same as those in Table IV. It is common sense that weak grid helps stabilize the system, as shown in Fig. 7(a) that the system is stable with $L_{g} \geq 0.11 m H$ , $S C R \leq 3.7$ (case 2). In contrast, when the grid-side impedance is $L_{g} = 0.095 H$ , i.e., in case 1, the system is unstable and the AC-side oscillation frequency is around 37.6 Hz with the APC and RPC but without virtual impedance, which can be observed from Fig. 7. In this situation, increasing the damping coefficient or decreasing the reactive power integral coefficient can stabilize the system, as observed from cases 6-8 shown in Fig. 7. In addition, the implementation of DC- or AC-side virtual impedance (either inductance or resistance) can also help the stabilization of the system, as observed from cases 3-5 in Fig. 7, which alleviates the parameter design of GFC under strong grid conditions.

Table IV Test Cases and Parameters

Test group	Case	Condition	State
1	Case 1	$L_{g} = 0.095 H$	Unstable
	Case 2	$L_{g} = 0.11 H$	Stable
	Case 3	$L_{g} = 0.095 H$ , AC-side virtual inductance ( $k_{v d} = 10$ )	Stable
	Case 4	$L_{g} = 0.095 H$ , AC-side virtual resistance ( $k_{v d} = 10$ )	Stable
	Case 5	$L_{g} = 0.095 H$ , DC-side virtual resistance ( $k_{v c} = 15$ )	Stable
	Case 6	$L_{g} = 0.095 H, D_{p} = 100$	Stable
	Case 7	$L_{g} = 0.095 H, D_{p} = 50, K_{q} = 4$	Unstable
	Case 8	$L_{g} = 0.095 H, D_{p} = 50, K_{q} = 1$	Stable
2	Case 9	$L_{g} = 0.04 H$ , without AC-side virtual impedance	Unstable
	Case 10	$L_{g} = 0.04 H$ , with AC-side virtual inductance ( $k_{v d} = 20$ )	Stable
	Case 11	$L_{g} = 0.04 H$ , with AC-side virtual resistance ( $k_{v c} = 20$ )	Unstable
	Case 12	$L_{g} = 0.04 H$ , with DC-side virtual impedance ( $k_{v c} = 20$ )	Unstable
3	Case 13	$L_{g i n v} = 0.16 H, D_{p i n v} = 100$ , $L_{g r e c} = 0.15 H, D_{p r e c} = 25, k_{p d c} = 1$	Unstable
	Case 14	$L_{g i n v} = 0.16 H, D_{p i n v} = 100$ , $L_{g r e c} = 0.17 H, D_{p r e c} = 25, k_{p d c} = 1$	Unstable
	Case 15	$L_{g i n v} = 0.16 H, D_{p i n v} = 100$ , $L_{g r e c} = 0.21 H, D_{p r e c} = 25, k_{p d c} = 1$	Unstable

Fig. 7 Generalized Nyquist analysis $Z_{m m c, a c} / Z_{g}$ . (a) Nyquist plot of cases 1-8. (b) Nyquist plot of cases 9-12.

The second group is the very strong grid, with the grid impedance as $L_{g} = 0.04 H$ , $S C R = 10$ , in case 9, the system is unstable. With the AC-side virtual inductance, the system is stable again as observed from case 10 in Fig. 7(b), while the system is unstable with AC-side virtual resistance, as observed from case 11 in Fig. 7(b).

This is mainly because the inductive impedance is the prerequisite for the GFC, while larger resistive impedance would result in the instability and power coupling, although some resistance is necessary for better stability, e.g., impedance angle is 80°. Besides, with the implementation of DC-side virtual impedance, the system remains unstable, as observed from case 12 in Fig. 7(b), which is not that effective in the stronger grid.

E. DC-side Nyquist Stability Analysis

Owing to the single-input single-output (SISO) property, the DC-side stability can be analyzed using the bode plot. The grid inductance is selected sufficiently large to ensure the AC-side system stability. The back-to-back system is used here for the demonstration, and the power control station operates as the inverter (1 p.u. active power towards AC grid), while the DC-side voltage control works as the rectifier (-1 p.u. active power towards AC grid). Cases 13-15 in the test group 3 are presented in Table IV with the modified parameters. Other parameters are the same as in Table III. The proportional gain $k_{p d c}$ is decreased and damping coefficient for rectifier station $D_{p r e c}$ is decreased (subscript rec or inv is used for rectifier ( $D_{p r e c} / L_{g r e c}$ ) or inverter station ( $D_{p i n v} / L_{g i n v}$ ), respectively, when the parameters of two stations are not the same).

Figure 8 presents the bode plot of cases 13-15 with $L_{g r e c} = 0.15 H, 0.17 H, 0.21 H$ , respectively.

Fig. 8 Bode plot for DC-side impedances of MMC with power control and DC voltage control.

As shown in Fig. 8, the increase of $L_{g r e c}$ makes bode plots of two stations intersect at lower frequency in the range of 70-80 Hz, which is termed as system resonant peak and the resonant peak may vary depending on specific configurations. Moreover, the closer the grid impedance is, the more serious the resonance around system resonant peak is, and the less phase margin is. In addition, the decrease of the proportional gain $k_{p d c}$ would also pose frequency resonances within 5 Hz, and the resonance would be more serious with larger inductance. As discussed in Section IV-B, the resonances around 77 Hz can be better suppressed with the AC-side virtual impedance rather than the DC-side virtual impedance. Moreover, the resonance in the range of 3-5 Hz mainly caused by the improper DC controller parameters and large inductance cannot be well suppressed by either the AC- or DC-side virtual impedance.

V. Simulation

In this section, two cases are tested to demonstrate the AC- and DC-side stability phenomena predicted by the impedance model, respectively. First, for the AC-side stability, cases 1 and 3 are used for the demonstration of AC-side stability test. For the AC-side stability, the DC side of MMC is connected with the ideal DC voltage. At the beginning, the AC-side virtual inductance is not implemented, as illustrated in case 1. After $t = 3$ s, the AC-side virtual inductance is enabled as illustrated in case 3, after that the system begins to be stable, which verifies the AC-side stability analysis. As shown in Fig. 9(a), the AC-side oscillation (approximately 12.4 Hz reflected at the DC power oscillation) is quickly attenuated once the AC-side virtual inductance is activated.

Fig. 9 Simulation results of AC- and DC-side stability. (a) AC-side oscillation from case 3 to case 1. (b) DC-side oscillation within 5 Hz in case 14. (c) DC-side oscillation within 70-80 Hz in case 13.

Regarding the DC-side stability simulation, cases 13 and 15 are used for demonstration. For case 15 with DC-side virtual impedance, the system is stable, and the resonance around 77 Hz is well attenuated with DC-side virtual impedance ( $k_{v c} = 50$ ) in Fig. 9(b). In addition, there is no resonance around 3 Hz with $k_{p d c} = 20$ before $t = 3$ s. However, when $k_{p d c} = 1$ after $t = 3$ s, the resonance emerges at approximately 3 Hz, which is the same as the previous analysis. However, in case 13 with the DC-side virtual impedance ( $k_{v c} = 50$ , $k_{p d c} = 1$ ), the system is unstable with resonance around 76 Hz, and the system is stable only after the activation of AC-side virtual impedance ( $k_{v d} = 20$ ), which implies closer grid impedances result in serious resonance. For this case, the AC-side virtual impedance is more effective.

VI. Conclusion

This paper establishes the AC- and DC-side impedance models of the GFC-based MMC without inner cascaded voltage and current loop. Like TLC, GFC would inevitably introduce negative resistance. However, the impedance shaping effect is relatively weaker than the inner cascaded voltage and current control.

Only the narrow frequency range behaves as the negative resistance under GFC, while the remaining frequency range remains the same as that under open-loop control with CCSC characteristics. Therefore, GFC keeps the open-loop characteristic for both MMC and TLC to a degree. The difference mainly lies in the open-loop sequence impedance characteristic of MMC and TLC. GFC has no suppression on the coupling terms and the negative resistance is introduced, such that GNC is necessary for the AC-side stability analysis.

Regarding the DC-side impedance, GFC also has the limited effect on the impedance shaping effect. The main resonance within 5 Hz is identified owing to the DC voltage controller and grid strength while the resonance around system resonant peak is mainly owing to the grid-side impedance of the system. Moreover, the closer grid impedances at both sides result in more serious resonance and it is better attenuated by the AC-side virtual impedance rather than the DC-side virtual impedance. The corresponding attenuation strategies with virtual impedance are also analyzed.

Finally, the improper parameter could result in the poor performance of GFC. Moreover, the resonances triggered under various grid strength might not be well addressed by the GFC and active damping method such as AC- and DC-side virtual impedances (such as frequency oscillation in case 16 with small $k_{p d c}$ and large grid impedance). More research on this is needed in the future. In addition, simulations are conducted to verify the proposed model.

References

J. Fang, F. Blaabjerg, S. Liu et al., “A review of multilevel converterswith parallel connectivity,” IEEE Transactions on Power Electronics, vol. 36, no. 11, pp. 12468-12489, Nov. 2021. [Baidu Scholar]

K. Ji, G. Tang, J. Yang et al., “Harmonic stability analysis of MMC-based DC system using DC impedance model,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 8, no. 2, pp. 1152-1163, Jun. 2020. [Baidu Scholar]

K. Ji, G. Tang, H. Pang et al., “Impedance modeling and analysis of MMC-HVDC for offshore wind farm integration,” IEEE Transactions on Power Delivery, vol. 35, no. 3, pp. 1488-1501, Jun. 2020. [Baidu Scholar]

K. Ji, H. Pang, J. Yang et al., “DC side harmonic resonance analysis of MMC-HVDC considering wind farm integration,” IEEE Transactions on Power Delivery, vol. 36, no. 1, pp. 254-266, Feb. 2021. [Baidu Scholar]

L. Harnefors, A. Antonopoulos, S. Norrga et al., “Dynamic analysis of modular multilevel converters,” IEEE Transactions on Industrial Electronics, vol. 60, no. 7, pp. 2526-2537, Jul. 2013. [Baidu Scholar]

D. Yang and X. Wang, “Unified modular state-space modeling of grid-connected voltage-source converters,” IEEE Transactions on Power Electronics, vol. 35, no. 9, pp. 9700-9715, Sept. 2020. [Baidu Scholar]

R. Pan and P. Sun, “Wireless fundamental frequency-based variable mode power sharing strategy for autonomous microgrid,” Electrical Engineering, vol. 104, no. 3, pp. 1473-1486, Jun. 2022. [Baidu Scholar]

A. Jamshidifar and D. Jovcic, “Small-signal dynamic dq model of modular multilevel converter for system studies,” IEEE Transactions on Power Delivery, vol. 31, no. 1, pp. 191-199, Feb. 2016. [Baidu Scholar]

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]

M. Amin and M. Molinas, “Small-signal stability assessment of power electronics based power systems: a discussion of impedance- and eigenvalue-based methods,” IEEE Transactions on Industry Applications, vol. 53, no. 5, pp. 5014-5030, Sept.-Oct. 2017. [Baidu Scholar]

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

M. Cespedes and J. Sun, “Impedance modeling and analysis of grid-connected voltage-source converters,” IEEE Transactions on Power Electronics, vol. 29, no. 3, pp. 1254-1261, Mar. 2014. [Baidu Scholar]

B. Wen, D. Boroyevich, R. Burgos et al., “Analysis of d-q small-signal impedance of grid-tied inverters,” IEEE Transactions on Power Electronics, vol. 31, no. 1, pp. 675-687, Jan. 2016. [Baidu Scholar]

A. Rygg, M. Molinas, C. Zhang et al., “On the equivalence and impact on stability of impedance modeling 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]

L. Xu, L. Fan, and Z. Miao, “DC impedance-model-based resonance analysis of a VSC-HVDC system,” IEEE Transactions on Power Delivery, vol. 30, no. 3, pp. 1221-1230, Jun. 2015. [Baidu Scholar]

J. Sun and H. Liu, “Sequence impedance modeling of modular multilevel converters,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 5, no. 4, pp. 1427-1443, Dec. 2017. [Baidu Scholar]

J. Lyu, X. Zhang, X. Cai et al., “Harmonic state-space based small-signal impedance modeling of a modular multilevel converter with consideration of internal harmonic dynamics,” IEEE Transactions on Power Electronics, vol. 34, no. 3, pp. 2134-2148, Mar. 2019. [Baidu Scholar]

H. Wu, X. Wang, and L. H. Kocewiak, “Impedance-based stability analysis of voltage-controlled MMCs feeding linear AC systems,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 8, no. 4, pp. 4060-4074, Dec. 2020. [Baidu Scholar]

H. Wu and X. Wang, “Dynamic impact of zero-sequence circulating current on modular multilevel converters: complex-valued AC impedance modeling and analysis,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 8, no. 2, pp. 1947-1963, Jun. 2020. [Baidu Scholar]

Z. Li, Z. Wang, Y. Wang et al., “Accurate impedance modeling and control strategy for improving the stability of DC system in multiterminal MMC-based DC grid,” IEEE Transactions on Power Electronics, vol. 35, no. 10, pp. 10026-10049, Oct. 2020. [Baidu Scholar]

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]

Q. Zhong, F. Blaabjerg, and C. Cecati, “Power-electronics-enabled autonomous power systems,” IEEE Transactions on Industrial Electronics, vol. 64, no. 7, pp. 5904-5906, Jul. 2017. [Baidu Scholar]

R. Pan and P. Sun, “Extra transient block for virtual synchronous machine with better performance,” IET Generation, Transmission & Distribution, vol. 14, no. 7, pp. 1186-1196, Apr. 2020. [Baidu Scholar]

R. Pan and P. Sun, “Multifunctional inverter based on virtual synchronous machine implemented in synchronous reference frame,” Electrical Engineering, vol. 103, no. 4, pp. 2093-2101, Feb. 2021. [Baidu Scholar]

R. Pan and P. Sun, “Microgrid power sharing using variable droop coefficient control,” in Proceedings of 2019 IEEE 10th International Symposium on Power Electronics for Distributed Generation Systems (PEDG), Xi’an, China, Jun. 2019, pp. 659-664. [Baidu Scholar]

W. Wu, L. Zhou, Y. Chen et al., “Sequence-impedance-based stability comparison between VSGS and traditional grid-connected inverters,” IEEE Transactions on Power Electronics, vol. 34, no. 1, pp. 46-52, Jan. 2019. [Baidu Scholar]

Y. Liu, X. Zhou, Y. Chen et al., “Sequence impedance modeling and stability analysis for load converters with inertial support,” IEEE Transactions on Power Electronics, vol. 35, no. 12, pp. 13031-13041, Dec. 2020. [Baidu Scholar]

L. Huang, H. Xin, and Z. Wang, “Damping low-frequency oscillations through VSC-HVDC stations operated as virtual synchronous machines,” IEEE Transactions on Power Electronics, vol. 34, no. 6, pp. 5803-5818, Jun. 2019. [Baidu Scholar]

H. Zhang, W. Xiang, W. Lin et al., “Grid forming converters in renewable energy sources dominated power grid: control strategy, stability, application, and challenges,” Journal of Modern Power Systems and Clean Energy, vol. 9, no.6, pp. 1239-1256, Nov. 2021. [Baidu Scholar]

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Impedance Analysis of Grid Forming Control Based Modular Multilevel Converters PDF

Abstract

Keywords

I. Introduction

II. Small-signal Model

A. Power Stage of MMC

B. GFC

C. Shaping Effect of APC and RPC

D. Shaping Effect of Virtual Impedance

E. Shaping Effect of Circulating Current Suppressing Control

III. Modeling Method for AC- and DC-side Impedances of GFC-based MMC

A. AC-side Impedance of MMC

B. DC-side Impedance of MMC

IV. Impedance Validation and Stability Analysis

A. AC-side Impedance

B. DC-side Impedance

C. Comparison Between Impedances of GFC-based TLC and MMC

D. AC-side Nyquist Stability Analysis

E. DC-side Nyquist Stability Analysis

V. Simulation

VI. Conclusion

References