Weak Grid-Induced Stability Problems and Solutions of Distributed Static Compensators with Voltage Droop Support

Abstract

Distributed static compensators (DSTATCOMs) are grid-connected power electronic equipment dedicated to compensating reactive power as well as improving voltage regulation in distribution networks. They eclipse conventional compensation approaches, such as capacitor banks, in terms of flexibility and effectiveness. Despite their identified advantages, STATCOMs with voltage droop are subject to weak grid-induced stability problems, as first revealed by this paper. Specifically, the voltage droop controller that couples the amplitude of point of common coupling (PCC) voltages to the reactive current reference creates a local control loop. Such a loop greatly deteriorates system stability in weak grids, which feature large and variable grid impedances. To address such stability problems, we propose a novel virtual resistance control scheme, which improves system stability through mitigation of local control loop gains in the low-frequency band. Experimental results obtained from a DSTATCOM prototype clearly demonstrate the correctness of stability analysis and the effectiveness of stability improvement.

1. Introduction

The global energy sector witnesses a fast-increasing penetration level of renewable energy resources, such as wind and solar photovoltaics (PVs), due to a strong desire for clean electricity and decarbonization. Along with the growth of renewable energy resources is the widespread deployment of grid-tied power converters. However, the switching feature of power converters, together with the intermittent and stochastic behaviors of renewable energy sources, challenge the power quality (PQ) of modern power systems [1,2,3,4,5,6]. Typical PQ issues, such as voltage sags/swells, reactive currents, voltage/current harmonics, and voltage/current imbalances, may cause undesirable power losses, equipment malfunction, and even blackouts [7,8,9,10]. As such, grid codes, e.g., IEEE Standard 519-2014 [11], set clear requirements on current and voltage qualities of grid-tied power converters.

To resolve PQ issues and comply with grid codes, various stand-alone power electronic equipment, including active power filters (APFs) [12,13], dynamic voltage restorers (DVRs) [14,15], uninterruptible power supplies (UPSs) [16], unified power quality conditioners (UPQCs) [17], and distributed static compensators (DSTATCOMs), have been proposed. Among them, DSTATCOMs aim to compensate reactive power and improve voltage regulation in distribution networks [18]. Specifically, the voltage droop, which links the grid voltage to the current control loop, allows the DSTATCOM to regulate the PCC voltage in an acceptable range by absorbing and generating reactive power. As compared to their passive alternatives, such as capacitor banks, DSTATCOMs benefit mostly from their flexibility and effectiveness. As a result, DSTATCOMs are increasingly employed in the replacement of passive approaches for reactive power and voltage compensation [19,20].

Despite obvious benefits, DSTATCOMs are subject to stability problems due to the interaction with other power electronic converters [21,22]. The interaction among multiple STATCOMs may also deteriorate system stability [23].

Notably, DSTATCOMs are frequently installed at the point of common coupling (PCC) for better voltage regulation in weak grids. Characterized by large and variable grid impedances, weak grids may interact with current control, LCL filter resonances, phase-locked-loops (PLLs), and virtual inertia control, resulting in stability degradation of grid-tied power conversion systems [24,25,26,27]. It should be noted that the large grid impedance can also interact with the voltage droop control. However, the effects of such interactions on the stability of DSTATCOMs have so far not been investigated.

To bridge the research gap, we investigated the stability of DSTATCOMs with voltage droop in weak grids. First, the instability mechanisms of DSTATCOMs with voltage droop in weak grids are revealed. As we introduce in the following sections, the voltage droop control, on top of reactive power compensation, may possibly introduce stability issues, particularly when exposed to weak grids. Specifically, the increment in voltage droop coefficients, paired with large grid impedances, degrades system stability. Second, to address such instability problems, we put forward a novel virtual resistance control scheme that ensures stable operations of DSTATCOMs. It improves system stability without any additional hardware. Therefore, no power losses are generated.

The remainder of this paper is organized as follows. Section 2 presents the fundamental operating principle of DSTATCOMs. In Section 3, we investigate the mechanisms behind instabilities of DSTATCOMs with voltage droop. Section 4 details the proposed control scheme for stability improvement. Section 5 provides experimental results for verification purposes. Finally, Section 6 provides concluding remarks.

2. Fundamental Principles of DSTATCOMs

As shown in Figure 1, the DSTATCOM system consists of a three-phase grid-connected inverter, three filter inductors L_c, and a DC capacitor. Notably, three-phase quantities are differentiated by the subscripts a, b, and c. The DSTATCOM is connected through the PCC to a weak grid, which is modeled as a series connection of the grid inductors L_g and an ideal three-phase voltage source v_abc. To yield the worst stability case, equivalent series resistors (ESRs) of inductors are all ignored. Figure 1 presents the control scheme of DSTATCOMs, which is implemented in the synchronous dq0-frame, where a standard cosine signal in phase A will be transformed into a constant in the d-axis. Correspondingly, d-axis or q-axis quantities are denoted by the subscript d or q, respectively. In additional, the subscript “ref” designates the reference notation. As shown, the control scheme features an outer voltage controller and an inner current controller. In Figure 1, a phase-lock loop (PLL) captures the phase of PCC voltages v_gabc for abc/dq and dq/abc transformations. It is worth mentioning that the voltage control not only regulates the DC-link voltage v_dc but also offers voltage droop control by linking the PCC voltage magnitude V_gd to the q-axis current reference i_{cq_ref}. Subsequently, the output signals of the voltage controller are sent to the current controllers as the d- and q-axes’ current references. In what follows, the current controller generates duty cycles for the PWM block that drives semiconductor switches.

Figure 2 provides the vector diagrams of DSTATCOMs. In most scenarios, DSTATCOMs supply (inductive) reactive power to power grids, where the reactive current I_cq lags the PCC voltage V_gd by 90 degrees. However, DSTATCOMs may also absorb reactive power by making I_cq lead V_cd. For illustration, Figure 2a shows a condition where the converter voltage magnitude V_cd is greater than V_gd (i.e., V_cd > V_gd). In this case, DSTATCOMs act as capacitors and inject reactive power. Alternatively, DSTATCOMs serve as inductors and absorb reactive power from the grid if V_cd < V_gd, as shown in Figure 2b. Figure 2c demonstrates a case where no exchange of reactive power is expected, as V_cd and V_gd are identical.

The fundamental operating principle of DSTATCOMs with voltage support can be explained as follows. Given that the demanded reactive power exceeds the generated reactive power, the grid voltage amplitude drops (known as voltage sags). Accordingly, the PCC voltage amplitude V_gd decreases. DSTATCOMs should then respond promptly to increase their current I_cq and inject more reactive power to the grid. In this way, the injected reactive power helps mitigate voltage drops at PCC. The opposite is true in the case of voltage swells.

3. Weak Grid-Induced Stability Problems of DSTATCOMs with Voltage Droop

This section investigates the influence of voltage droop and weak grids on the stability of DSTATCOMs. It is worthwhile to note that voltage droop mainly affects current control through a link between the PCC voltage amplitude and the reactive current reference. As such, we focus on the modeling of current controller and voltage droop in this section.

For illustration, the detailed control block diagrams of DSTATCOMs without and with voltage droop are shown in Figure 3, where ∆ represents the small-signal perturbation, the capital letter X stands for the steady-state value, and the current regulator G_i(s) is implemented as a PI regulator:

$$G_i\left(s\right)=K_{cp}+\frac{K_{ci}}{s},$$

(1)

where K_cp and K_ci are the proportional and integral coefficients of the current regulator, respectively. G_d(s) represents the linearized control delay, which is given as [28]:

$$G_d\left(s\right)=\frac{1-T_{d}s}{1+T_{d}s},$$

(2)

where T_d = 0.75/f_s, and f_s refers to the sampling frequency.

According to [26], the interactions between power converters and grids are mathematically described by

$$\begin{array}{l}\left[\mathrm{\Delta }{v}_{cd}\left(s\right)-\mathrm{\Delta }{v}_d\left(s\right)+\mathrm{\Delta }{i}_{cq}\left(s\right){\omega }_0{L}_t\right]G_{\mathrm{plant}}\left(s\right)=\mathrm{\Delta }{i}_{cd}\left(s\right)\\ \left[\mathrm{\Delta }{v}_{cq}\left(s\right)-\mathrm{\Delta }{v}_q\left(s\right)-\mathrm{\Delta }{i}_{cd}\left(s\right){\omega }_0{L}_t\right]G_{\mathrm{plant}}\left(s\right)=\mathrm{\Delta }{i}_{cq}\left(s\right)\end{array}$$

(3)

where L_t = L_c + L_g. ω₀ represents the fundamental angular frequency. ∆i_cqω₀L_t and −∆i_cdω₀L_t are cross-coupling terms introduced by frame transformations. Additionally, the plant transfer function G_plant(s) is

$$G_{\mathrm{plant}}\left(s\right)=\frac{1}{L_{t}s}.$$

(4)

In Figure 3, G_pll(s) models the small-signal PLL transfer function from the PCC voltage q-axis component ∆v_gq to the phase angle ∆θ_pll, which can be derived as [26]:

$$G_{\mathrm{pll}}\left(s\right)=\frac{\mathrm{\Delta }{\theta }_{\mathrm{pll}}\left(s\right)}{\mathrm{\Delta }{v}_{gq}\left(s\right)}=\frac{K_{\mathrm{pll}\_p}s+K_{\mathrm{pll}\_i}}{s^2+V_d{K}_{\mathrm{pll}\_p}s+V_d{K}_{\mathrm{pll}\_i}}.$$

(5)

where K_{pll_p} and K_{pll_i} are the proportional and integral coefficients of the PI controller with PLL, respectively.

In Figure 3a, the PLL introduces the terms I_cq∆θ_pll, I_cd∆θ_pll, and V_d∆θ_pll [26]. Since the main purpose of DSTATCOMs is to compensate reactive power, the d-axis current reference can be designed as zero (i_{cd_ref} = 0), and the steady-state d-axis current I_cd is also 0. In this case, the converter voltage and PCC voltage are almost in alignment with identical phase angles. Hence, there is only reactive power exchange between DSTATCOMs and the grid.

Moreover, the grid voltages v_abc are assumed to be ideal sinusoidal voltages. Therefore, the small-signal perturbations of grid voltages ∆v_d and ∆v_q are ignored, i.e., ∆v_d = ∆v_q = 0. Under this assumption, there are only two control inputs (i.e., ∆i_{cd_ref} and ∆i_{cq_ref}) in Figure 3a. Correspondingly, ∆i_cd and ∆i_cq stand as two control outputs. System stability can be evaluated by the transfer matrix given in (6). Notably, the system will be stable only when all the four matrix elements are with stable poles. Otherwise, unstable pole(s) result in system instabilities. Comparing Figure 3a and Figure 3b, one can find that the major influence of voltage droop is reflected on the q-axis current control loop. Therefore, we will analyze the effect of voltage droop mainly through G_{icq_cl}(s) (without and with voltage droop) in the following subsections.

$$\left[\begin{array}{l}\mathrm{\Delta }{i}_{cd}\left(s\right)\\ \mathrm{\Delta }{i}_{cq}\left(s\right)\end{array}\right]=\left[\begin{array}{cc}{G}_{icd\_\mathrm{cl}}\left(s\right)& G_{icq\mathrm{ref}\_icd}\left(s\right)\\ G_{icd\mathrm{ref}\_icq}\left(s\right)& G_{icq\_\mathrm{cl}}\left(s\right)\end{array}\right]\left[\begin{array}{l}\mathrm{\Delta }{i}_{cd\_\mathrm{ref}}\left(s\right)\\ \mathrm{\Delta }{i}_{cq\_\mathrm{ref}}\left(s\right)\end{array}\right].$$

(6)

3.1. Control without Voltage Droop

Recapping Figure 3a, we first derive the transfer function of the local control loop from ∆v_{cq_pll}(s) to ∆v_cq(s) as

$$G_{q\_\mathrm{pll}}\left(s\right)=\frac{\mathrm{\Delta }{v}_{cq}\left(s\right)}{\mathrm{\Delta }{v}_{cq\_\mathrm{pll}}\left(s\right)}=\frac{L_t{G}_d\left(s\right)}{L_t-L_g{V}_d{G}_d\left(s\right)G_{\mathrm{pll}}\left(s\right)}.$$

(7)

Furthermore, we derive the closed-loop transfer function of the q-axis current control G_{q_cl}(s) as

$$G_{q\_\mathrm{cl}}\left(s\right)=\frac{G_i\left(s\right)G_{q\_\mathrm{pll}}\left(s\right)G_{\mathrm{plant}}\left(s\right)}{1+G_i\left(s\right)G_{q\_\mathrm{pll}}\left(s\right)G_{\mathrm{plant}}\left(s\right)}.$$

(8)

In contrast, the closed-loop transfer function of the d-axis current control G_{d_cl}(s) is derived as

$$G_{d\_\mathrm{cl}}\left(s\right)=\frac{G_i\left(s\right)G_d\left(s\right)G_{\mathrm{plant}}\left(s\right)}{1+G_i\left(s\right)G_d\left(s\right)G_{\mathrm{plant}}\left(s\right)}.$$

(9)

Referring to Figure 3a, we can express ∆i_cq as

$$\mathrm{\Delta }{i}_{cq}\left(s\right)=G_{q\_\mathrm{cl}}\left(s\right)\mathrm{\Delta }{i}_{cq\_\mathrm{ref}}\left(s\right)+G_{icd\_icq}\left(s\right)\mathrm{\Delta }{i}_{cd}\left(s\right)$$

(10)

where G_{icd_icq}(s) denotes the transfer function from ∆i_cd to ∆i_cq, which is derived as

$$G_{icd\_icq}\left(s\right)=\frac{\mathrm{\Delta }{i}_{cq}}{\mathrm{\Delta }{i}_{cd}}=-{\omega }_0{L}_t\frac{G_{\mathrm{plant}}\left(s\right)}{1+G_i\left(s\right)G_{q\_\mathrm{pll}}\left(s\right)G_{\mathrm{plant}}\left(s\right)}$$

(11)

when deriving G_{icq_cl}(s), we should keep in mind that ∆i_cd is an intermediate variable and ∆i_{cd_ref} equals zero. As a step further, ∆i_cd can be expressed as

$$\mathrm{\Delta }{i}_{cd}\left(s\right)=G_{icq\_icd}\left(s\right)\mathrm{\Delta }{i}_{cq}\left(s\right)-I_{cq}{G}_{d\_\mathrm{cl}}\left(s\right)\mathrm{\Delta }{\theta }_{\mathrm{pll}}\left(s\right)$$

(12)

where G_{icq_icd}(s) denotes the transfer functions from ∆i_cq to ∆i_cd, which is derived as

$$G_{icq\_icd}\left(s\right)=\frac{\mathrm{\Delta }{i}_{cd}\left(s\right)}{\mathrm{\Delta }{i}_{cq}\left(s\right)}={\omega }_0{L}_t\frac{G_{\mathrm{plant}}\left(s\right)}{1+G_i\left(s\right)G_d\left(s\right)G_{\mathrm{plant}}\left(s\right)}$$

(13)

To eliminate ∆i_cd in (10), we first express ∆θ_pll as

$$\mathrm{\Delta }{\theta }_{\mathrm{pll}}\left(s\right)=\frac{L_g}{L_t}{G}_{\mathrm{pll}}\left(s\right)\mathrm{\Delta }{v}_{cq}\left(s\right)$$

(14)

Next, by substitution of (14) into (12), we obtain

$$\begin{array}{ll}\mathrm{\Delta }{i}_{cd}\left(s\right)& =G_{icq\_icd}\left(s\right)\mathrm{\Delta }{i}_{cq}\left(s\right)-I_{cq}{G}_{d\_\mathrm{cl}}\left(s\right)\mathrm{\Delta }{\theta }_{\mathrm{pll}}\left(s\right)\\ & =G_{icq\_icd}\left(s\right)\mathrm{\Delta }{i}_{cq}\left(s\right)-\frac{L_g}{L_t}{I}_{cq}{G}_{\mathrm{pll}}\left(s\right)G_{d\_\mathrm{cl}}\left(s\right)\mathrm{\Delta }{v}_{cq}\left(s\right)\\ & =G_{icq\_icd}\left(s\right)\mathrm{\Delta }{i}_{cq}\left(s\right)+G_{vcq\_icd}\left(s\right)\mathrm{\Delta }{v}_{cq}\left(s\right)\end{array}$$

(15)

where G_{vcq_icd}(s) designates the transfer function from ∆v_cq to ∆i_cd. Specifically, we represent G_{vcq_icd}(s) as

$$G_{vcq\_icd}\left(s\right)=\frac{\mathrm{\Delta }{i}_{cd}\left(s\right)}{\mathrm{\Delta }{v}_{cq}\left(s\right)}=-\frac{L_g}{L_t}{I}_{cq}{G}_{\mathrm{pll}}\left(s\right)G_{d\_\mathrm{cl}}\left(s\right)$$

(16)

For clarity, the q-axis control block diagram is redrawn as Figure 4 based on (10) and (15). As observed from Figure 4, we can readily derive the overall loop gain of the q-axis current control as

$$G_{icq\_\mathrm{ol}}\left(s\right)=\frac{\mathrm{\Delta }{i}_{cq}\left(s\right)}{\mathrm{\Delta }{i}_{cq\_\mathrm{err}}\left(s\right)}=\frac{G_i\left(s\right)G_{q\_\mathrm{pll}}\left(s\right)G_{\mathrm{plant}}\left(s\right)}{1+{\omega }_0{L}_t{G}_{icq\_icd}\left(s\right)G_{\mathrm{plant}}\left(s\right)}\left[1-{\omega }_0{L}_t{G}_{vcq\_icd}\left(s\right)\right].$$

(17)

Finally, we express the closed-loop transfer function from ∆i_{cq_ref} to ∆i_cq as

$$G_{icq\_\mathrm{cl}}\left(s\right)=\frac{\mathrm{\Delta }{i}_{cq}\left(s\right)}{\mathrm{\Delta }{i}_{cq\_\mathrm{ref}}\left(s\right)}=\frac{G_{icq\_\mathrm{ol}}\left(s\right)}{1+G_{icq\_\mathrm{ol}}\left(s\right)}.$$

(18)

3.2. Control with Voltage Droop

To achieve the objective of reactive power compensation, the PCC voltage amplitude ∆v_gd is linked to the q-axis current reference ∆i_{cq_dr} through a voltage droop gain K_vq, as visualized in Figure 3b.

Due to the employment of voltage droop, ∆i_cq in (16) should be reorganized as

$$\mathrm{\Delta }{i}_{cq}\left(s\right)=\mathrm{\Delta }{i}_{cq\_\mathrm{ref}}\left(s\right)G_{q\_\mathrm{cl}}\left(s\right)-\mathrm{\Delta }{v}_{gd}\left(s\right)K_{vq}{G}_{q\_\mathrm{cl}}\left(s\right)+\mathrm{\Delta }{i}_{cd}\left(s\right)G_{icd\_icq}\left(s\right)$$

(19)

where ∆v_gd is an additional intermediate variable that is influenced by ∆i_cq and ∆v_cq. Recapping Figure 3b, we calculate the transfer function from ∆i_cq to ∆v_gd as

$$G_{icq\_vgd}\left(s\right)=-{\omega }_0{L}_g{G}_{d\_\mathrm{cl}}\left(s\right).$$

(20)

In addition, the transfer function from ∆v_cq to ∆v_gd is derived from Figure 3b as

$$G_{vcq\_vgd}\left(s\right)=-{\left(\frac{L_g}{L_t}\right)}^2{I}_{cq}{G}_{\mathrm{pll}}\left(s\right)\frac{G_i\left(s\right)G_d\left(s\right)}{1+G_i\left(s\right)G_d\left(s\right)G_{\mathrm{plant}}\left(s\right)}.$$

(21)

For illustration, the q-axis current control with voltage droop is redrawn as Figure 5, where the modifications introduced by voltage droop are highlighted. In Figure 5, ∆i_{cq_dr} models the effect of ∆v_gd on the q-axis current control. In this case, the sum of ∆i_{cq_dr} and ∆i_cq should be regarded as the output of the q-axis control loop. Correspondingly, the loop gain of the q-axis control with voltage droop is derived as

$$\begin{array}{l}{G}_{icq\_\mathrm{ol}\_\mathrm{dr}}\left(s\right)=\frac{\mathrm{\Delta }{i}_{cq}\left(s\right)-\mathrm{\Delta }{i}_{cq\_\mathrm{dr}}\left(s\right)}{\mathrm{\Delta }{i}_{cq\_\mathrm{err}}\left(s\right)}=G_{icq\_\mathrm{ol}}\left(s\right)-K_{vq}{G}_{icq\_vgd}\left(s\right)G_{icq\_\mathrm{ol}}\left(s\right)\\ -K_{vq}{G}_i\left(s\right)G_{q\_\mathrm{pll}}\left(s\right)G_{vcq\_vgd}\left(s\right).\end{array}$$

(22)

Finally, we present the closed-loop transfer function from ∆i_{cq_ref} to ∆i_cq as

$$G_{icq\_\mathrm{cl}\_\mathrm{dr}}\left(s\right)=\frac{\mathrm{\Delta }{i}_{cq}\left(s\right)}{\mathrm{\Delta }{i}_{cq\_\mathrm{ref}}\left(s\right)}=\frac{G_{icq\_\mathrm{ol}}\left(s\right)}{1+G_{icq\_\mathrm{ol}\_\mathrm{dr}}\left(s\right)}.$$

(23)

3.3. Stability Analysis

Based on the previous derivations, we present the pole-zero maps of G_{icq_cl}(s) and G_{icq_cl_dr}(s) in Figure 6 and Figure 7, respectively, where the insets highlight the poles and zeros near the imaginary axis. The corresponding system and control parameters are tabulated in

Table 1. System and control parameters.

Symbol	Description	Value
Vdc	DC-link voltage	250 V
Lc	Filter inductance	2 mH
Vd	Grid voltage amplitude	100 V
Lg	Grid inductance	10 mH
fs	Sampling/switching frequency	10 kHz
S	Power rating	600 VA
Kpll_p	PLL proportional gain	3
Kpll_i	PLL integral gain	300
Kcp	Current regulator proportional gain	15
Kci	Current regulator integral gain	300
icd_ref	Current reference of d-axis	0 A
icq_ref	Current reference of q-axis	5 A
Kvq	Voltage droop gain	1.8

. It is clear from Figure 6 that G_{icq_cl}(s) is stable since all the poles are located at the left-hand plane in Figure 6. However, right-hand plane (RHP) poles appear in Figure 7, indicating that G_{icq_cl_dr}(s) registers instabilities due to voltage droop.

Similarly, the other three matrix elements in (6) can also be derived. However, the detailed deriving procedures are excluded here due to page limits. For demonstration, their pole-zero maps without and with voltage droop are shown in Figure 8 and Figure 9, respectively. The subscript “dr” denotes the matrix elements in (6) that are modified by voltage droop. Clearly, there is no RHP pole of the transfer functions without voltage droop. In contrast, the presence of RHP poles in Figure 9 confirms that the employment of voltage droop causes stability problems. This is as expected by noting the strong coupling effects between the d- and q-axes.

To investigate the root cause of stability problems, Figure 10 compares the Bode plots of G_{icq_ol}(s) and G_{icq_cl_dr}(s). As seen, the considerable magnitude amplification, together with the phase decrease of G_{icq_cl_dr}(s), can destabilize DSTATCOM systems. The mechanism behind instabilities can be found in Figure 5, where two additional loops from ∆i_cq and ∆v_cq to ∆i_{cq_dr} are introduced due to voltage droop. The transfer functions from ∆i_cq and ∆v_cq to ∆i_{cq_dr} can be obtained from Figure 5 as

$$G_{icq\_icq\mathrm{dr}}\left(s\right)=\frac{\mathrm{\Delta }{i}_{cq\_\mathrm{dr}}\left(s\right)}{\mathrm{\Delta }{i}_{cq}\left(s\right)}=K_{vq}{G}_{icq\_vgd}\left(s\right)$$

(24)

$$G_{vcq\_icq\mathrm{dr}}\left(s\right)=\frac{\mathrm{\Delta }{i}_{cq\_\mathrm{dr}}\left(s\right)}{\mathrm{\Delta }{v}_{cq}\left(s\right)}=K_{vq}{G}_{vcq\_vgd}\left(s\right).$$

(25)

The magnitudes of the two transfer functions G_{icq_icqdr}(s) and G_{vcq_icqdr}(s) are both proportionally related to the droop coefficient K_vq. In addition, according to (20) and (21), they are also positively related to the grid inductance L_g. Figure 11 shows the Bode plots of G_{icq_icqdr}(s) and G_{vcq_icqdr}(s) with various K_vq and L_g. In Figure 11a, K_vq is swept from 1.2 to 2 with a step size of 0.2. In Figure 11b, L_g is gradually changed from 2 mH to 10 mH. Figure 11 implies that G_{icq_icqdr}(s) should be responsible for the magnitude amplification of G_{icq_cl_dr}(s). Moreover, the increment in K_vq or L_g further amplifies the magnitude of G_{icq_icqdr}(s), which means that ∆i_{cq_dr} becomes more sensitive to ∆i_cq. This tends to degrade the stability of DSTATCOM systems.

Furthermore, the influences of K_vq and L_g are visualized by the root loci of G_{icq_cl_dr}(s) shown in Figure 12, where K_vq ranges from 0 to 10. Figure 12 demonstrates the negative effect of voltage droop on system stability. For a fixed L_g, the closed-loop poles are gradually shifted rightwards as K_vq increases. Furthermore, the increment in L_g will lower the critical value of K_vq. These observations can be explained by the fact that the negative effect of voltage droop is essentially caused by the multiplication of K_vq and a gain of L_g/L_t (see Figure 3b). The above analysis discloses the necessity of stability improvement for DSTATCOMs under weak grid conditions.

4. Proposed Control Scheme for Stability Improvement

According to the previous analysis, the magnitude amplification of G_{icq_icqdr}(s) by the droop controller is responsible for instabilities. This amplification effect is positively related to the product of K_vq and L_g/L_t that links ∆v_cd to ∆v_gd. This implies that stability problems can be solved by decreasing this voltage gain. For instance, the instability phenomenon will disappear in stiff grids, where L_g = 0 mH. Moreover, Figure 11a illustrates that the G_{icq_icqdr}(s) shows the low-pass filter characteristics. Referring to (20) and (24), the cut-off frequency ω_c of G_{icq_icqdr}(s) is essentially the bandwidth of G_{d_cl}(s), which is designed to be 1600 rad/s in this paper.

Thus, we can improve system stability by decreasing the gain from ∆v_cd to ∆v_gd in the low-frequency band. A straightforward method is adding resistors R in series with the filter inductors L_c, as shown in Figure 13.

With the employment of resistors, the relationship between ∆v_gd and ∆v_cd is modified as

$$\mathrm{\Delta }{v}_{gd}=\frac{L_{g}s}{L_{t}s+R}\mathrm{\Delta }{v}_{cd}.$$

(26)

Similarly, the relationship between ∆v_gq and ∆v_cq with resistors R should be rewritten as

$$\mathrm{\Delta }{v}_{gq}=\frac{L_{g}s}{L_{t}s+R}\mathrm{\Delta }{v}_{cq}.$$

(27)

Correspondingly, the system plant is changed into

$$G_{\mathrm{plant}\_\mathrm{pd}}\left(s\right)=\frac{1}{L_{t}s+R}.$$

(28)

The remaining control blocks in Figure 3b remain unchanged.

Equation (26) essentially replaces the gain L_g/L_t in G_{icq_icqdr}(s) by a high-pass filter:

$$G_{hp}\left(s\right)=\frac{L_{g}s}{L_{t}s+R}$$

(29)

whose cut-off frequency is derived as

$${\omega }_{c\_hp}=\frac{R}{L_t}.$$

(30)

The magnitude of G_{icq_icqdr}(s) can be attenuated at low frequencies by setting ω_{c_hp} = ω_c, which gives R = 19.2 Ω. However, high resistance generally deteriorates the dynamic response. In fact, it is conservative to set ω_{c_hp} as the same as ω_c. Note that ω_{c_hp} denotes the frequency at which the magnitude of G_hp(s) falls 3 dB from its high-frequency value, and that ω_c denotes the frequency where the magnitude of G_{d_cl}(s) declines 3 dB from its low-frequency value. Since the magnitude of G_{icq_icqdr}(s) is positively related to the multiplication of G_hp(s) and G_{d_cl}(s), setting ω_{c_hp} = ω_c actually makes the magnitude of G_{icq_icqdr}(s) drop 6 dB at ω_c. Therefore, ω_{c_hp} is lowered to 0.2 ω_c. Considering the inductances listed in Table 1, the value of R can then be determined as

$$R=0.2{\omega }_c{L}_t=3.84\Omega$$

(31)

The effectiveness of R is evaluated by the pole-zero maps of G_{icq_cl_dr}(s) shown in Figure 14. As seen, the stability of DSTATCOMs can be improved by increasing R. Specifically, RHP poles disappear when R is greater than or equal to 5 Ω, indicating that DSTATCOM systems are stabilized. This agrees well with (31).

Although adding extra resistors is effective, the associated power loss deteriorates system efficiency. Therefore, an active control scheme is further proposed to mimic the behavior of filter resistors. The proposed virtual resistance control scheme is shown in Figure 15, where G_ad(s) and G_aq(s) stand for the proposed controllers on the d-axis and q-axis, respectively.

For better illustration, the control block diagrams that incorporate resistors are given in Figure 16. To fully emulate resistors, G_ad(s) and G_aq(s) should be designed as

$$G_{ad}\left(s\right)=\frac{R}{G_d\left(s\right)}$$

(32)

$$G_{aq}\left(s\right)=\frac{R}{G_{q\_\mathrm{pll}}\left(s\right)}.$$

(33)

When the proposed controller is designed as (32) and (33), the systems in Figure 15 and Figure 16 will be identical. However, note that the inverse transfer functions of time delay G_d(s) and G_{q_pll}(s) are not necessarily available in practice. Since G_d(s) and G_{q_pll}(s) both exhibit a unity gain in the low-frequency band, as proved by Figure 17, G_ad(s) and G_aq(s) can be designed as pure gains to approximate the virtual resistance behavior, i.e.,

$$G_{ad}\left(s\right)=G_{aq}\left(s\right)=K_{ad}.$$

(34)

5. Experimental Results

To validate the theoretical analysis and the effectiveness of the proposed control scheme, a laboratory prototype was developed based on the schematic diagram shown in Figure 1 and parameters listed in Table 1. A three-phase inverter, together with filter inductors, operated as a DSTATCOM, while three additional inductors were inserted between the inverter and an AC programable power supply to emulate weak grids. All the control algorithms were implemented by a dSPACE controller (Microlabbox).

Figure 18 shows the experimental waveforms of the DSTATCOMs without and with voltage droop (of different droop coefficients). Clearly, the DSTATCOM system is stable without voltage droop, as proved by Figure 18b. As seen in Figure 18c, when the voltage droop with K_vq = 1.2 is applied, the inductor currents i_cabc and PCC voltages v_gabc remain stable shortly. However, when K_vq is increased to 1.8, the currents i_cabc diverge rapidly, indicating that instability occurs. This is as expected, since K_vq exceeds the upper limit 1.65 according to the previous analysis.

When K_vq is designed as 1.8, the stability problems of STATCOMs can be successfully addressed by the proposed control scheme, as demonstrated in Figure 19. With K_ad = 7, the PCC voltages v_gabc and the converter currents i_cabc are all stable, as shown in Figure 19b. When the proposed controller is disabled, i.e., K_ad = 0, the system becomes unstable again. These experimental results demonstrate the effectiveness of the proposed control scheme in terms of stability improvement.

6. Conclusions

We identified the potential stability problems of DSTATCOMs with voltage droop support to weak power grids. Through analyzing system stability, it is revealed that the increment in voltage droop coefficients tends to destabilize DSTATCOMs in weak grids. The mechanism of instabilities lies in the additional local control loop that links the PCC voltage amplitude to the reactive current. To mitigate the adverse effect of voltage droop, we proposed a novel control scheme that can improve system stability by emulating resistors and making PCC voltages less sensitive to converter control. Finally, experimental results demonstrate that DSTATCOMs with the proposed control scheme can operate stably even under weak grid conditions.