Stable Operation Domain Analysis of Back-to-Back Converter Based Flexible Multi-State Switches Considering PLL Dynamics

Abstract

Flexible multi-state switches (FMSSs), also called soft open point (SOP), are regarded as the core devices in flexible interconnected distribution network. However, due to their more complex control structure and nonlinear characteristics, there is a tight and complex coupling relationship between the transmission power limit and stability of two (or more) converter sides, which brings difficulties to the planning and power flow dispatching control of flexible interconnected distribution networks. In this paper, taking the back-to-back converter based FMSS (BTBC-FMSS) as the research object, the dynamic equivalent circuit connected to the distribution network is constructed, and the transmission active and reactive power expression of the converter on both sides considering PLL dynamics is deduced. The variation rules of transmission power limits on both sides of the device with various structures and control parameters are summarized. Furthermore, the stable operation domain of the BTBC-FMSS is constructed, and the effective measures to improve the transmission power stability of the device are given according to the analysis of the stable operation domain. Finally, a simulation model is built in Matlab/Simulink to verify the effectiveness of the analysis results.

1. Introduction

The traditional radial unidirectional power flow distributed network (DN), characterized by “closed-loop design and open-loop operation”, has had increasing difficulties in meeting the growing demand for flexible regulation [1]. To cope with the double challenges of high proportion of new energy resources’ access and sharp increase in load, in recent years, the academic community has proposed the concept of flexible interconnection of DN, where the core device is the FMSS, also known as the soft open point (SOP), to realize the controllability and power quality of the whole DN system [2,3]. An FMSS is usually composed of two (or more) voltage source inverters (VSCS) through sharing a common DC bus, which has flexible control ability and fast response characteristics, and can significantly improve the power flow control level and fault self-healing recovery ability of a traditional distribution network. Among all the configurations of FMSS, back-to-back converter (BTBC)-type FMSS (BTBC-FMSS) is the mainstream and basic equipment, which is also chosen as the research object of this paper.

Although BTBC-FMSS has unique advantages, as a highly electronic power transmission channel, its power transmission characteristics are different from the traditional line impedance, and its power transmission characteristics show obvious nonlinear controlled characteristics. The specific performance is as follows: (1) Similar to the new energy source grid-connected converter, when the dynamic state of the device control loop is not considered, there is a maximum value (limit power) of the VSC on both sides of the BTBC-FMSS device that is only related to the power grid strength [4]. (2) When considering the dynamic VSC control loop on both sides of the device, small signal instability may occur when the transmission power continues to increase; so, the actual transmission limit power is not equal to the static limit power [5,6]. (3) The control mode of the VSC on both sides of the BTBC-FMSS and the intensity of access to the power grid are different, the power transmission limit and dominant stability in different transmission directions are different [7,8], and the transmission power limit of the device itself should be the intersection of the transmission power limits of the two sides of the VSC.

It is worth noting that the transmission power limit and static stability analysis of BTBC-FMSS can learn from the related research results of grid-connected stability of new energy converters and high-voltage flexible DC transmission (VSC-HVDC), but BTBC-FMSS has its own unique characteristics. Firstly, in view of the back-to-back structure of the common DC side, the static stability of the VSC on both sides of the BTBC-FMSS is deeply coupled, the structure of the distribution network is weak, and there is no assumption of infinite power supply on either side of the power network (short-circuit capacity is comparable); so, it is unreasonable to analyze the static stability of the VSC alone. Secondly, in the distribution network, BTBC-FMSS is not only an active power transmission channel but also a two-way reactive power supply, and there is a four-quadrant operation requirement, which is far more complex than the task of the VSC-HVDC system only undertaking active power transmission channel in the transmission network. Finally, most of the existing VSC-HVDC studies derive its transmission power limit from the point of view in voltage control scheme, but this conclusion is only applicable to a scenario of a new energy station sent by flexible DC transmission, while under most application scenarios of BTBC-FMSS, the current control mode is adopted on both sides (one side is PQ control, the other side is V_dcQ control); so, the existing research results are not applicable.

Summing up the existing research, the research content of the following studies [4,5,6,7,8,9,10,11,12,13,14,15] involves the analysis and discussion of the above problems. Among them, references [4,9] study the influence of power grid strength and line impedance angle on the transmission power limit of VSC-HVDC system from the point of view of small signal stability analysis, and this study takes into account the bi-directional power transmission characteristics of sides, but the above literature does not derive the expression of transmission power from the point of view of internal current control of inverter; so, the analysis results are not accurate.

In [10], for the multi-feed power electronic power system, the generalized short-circuit ratio proposed is a static variable independent of the system oscillation frequency, but it can only reflect the small signal stability margin of the multi-feed system, and the static voltage stability is not considered. According to the single-machine infinite bus system model of a new energy source grid-connected converter under the condition of weak power grid, it is concluded that when constant DC voltage control (current control) is adopted in VSC, the positive or negative transmission power to the d-axis current partial derivative can be used as an important key criterion of power transmission stability, but the dynamic and stable operation point of phase-locked loop (Phase Locked Loop, PLL) control link is not taken into account in this study [11]. In [12], the static stability criterion of the single-machine infinite bus system of a new energy grid-connected converter is given under the time scale of DC voltage, and the relationship between static limit current, limit power angle and grid impedance and reactive current is analyzed, but the mode of PLL control link is not included in the static stability analysis. In [13,14,15], the static stability margin index is presented based on the analysis of static voltage equation for the synchronous stability of grid-following inverter controlled by PLL, which has important reference significance. However, refs. [13,14,15] mainly focus on the transient synchronization stability during LVRT, which is quite different from that of grid-connected VSC under normal operating conditions.

In summary, there have been many concerns and achievements on the single grid-connected system of the new energy converter, the transmission power limit of VSC-HVDC and the static and dynamic stability, but related research on BTBC-FMSS is still lacking. Most of the results focus on the small-signal stability analysis at a certain operating point, and the calculation method of the transmission power limit of the BTBC-FMSS device and stable operation domain is not presented.

Thus, the main contributions of this article are presented as follows:

(1) The transmission power limits of the two VSCs on both sides for BTBC-FMSS are derived. Through taking the existence of the stable working point of the PLL control ring as the constraint, the current amplitude variation range and transmission power expression based on current control under steady-state conditions are derived, and the variation rules of transmission power limits on both sides of the device with various structures and control parameters are summarized.

(2) The stable operation domain of BTBC-FMSS is constructed. Through adopting the equal-capacity transmission principle, the stable operation domain for active power transmission of BTBC-FMSS is constructed, the measures to improve the stability of transmission power are given, and a simulation model is built in Matlab/Simulink to verify the effectiveness of the theoretical analysis results.

The rest of this article is organized as follows: Section 2 introduces the equivalent model and control strategy of the flexible interconnected distribution network. Section 3 shows the transmission power expression of BTBC-FMSS. Section 4 analyzes the transmission power limit and stable operation domain of BTBC-FMSS. Section 5 presents the simulation results of the test system. Section 6 concludes the findings.

2. Equivalent Modeling and Control Strategy of Flexible Interconnected Distribution Network

2.1. System Modeling

The general structure of a BTBC-FMSS-based flexible interconnected DN is shown in Figure 1, which is composed of multiple feeders and multiple interconnected BTBC-FMSS devices. It is worth pointing out that the diversity of the FMSS topology leads to the diversity of flexible interconnected DNs. Generally, FMSS can be divided into a variety of types, such as double-terminal interconnection [16], multi-terminal interconnection [17], cross-voltage level interconnection and AC-DC hybrid interconnection [18]. Among them, the two-AC-DN system adopts the two-terminal interconnection (BTBC-FMSS) networking mode as its basic type. In this interconnection mode, the two ends of the distribution network can not only isolate each other and operate independently, but also dynamically adjust the interactive power when necessary, balance the load rate level of the feeders of the two sides of the DN, provide the contact channel at the same time, and effectively block the fault current propagation. Therefore, this paper takes the DN system of BTBC-FMSS interconnection as the research object.

In this paper, the transmission power limit of basic flexible interconnected DNs (two distribution networks connected by a single BTBC-FMSS) is studied. Firstly, the DN system on both sides is equivalent simplified to form a double-ended power supply system, as shown in Figure 2. After a series of star triangle transformations, the load and line are equivalent to the two impedance Z_s1 and Z_s2 of the AC-DN system, and the voltage sources at both ends are represented by U_g1 and U_g2. Z_s1 is the equivalent impedance between the converter VSC₁ side and AC system 1 (DN1), which is composed of line impedance Z_line, load Z_load on PCC and equivalent impedance Z_g1 of AC system 1. After star triangle transformation, it is expressed as Z_s1 = r_s1 + jx_s1 = |Z_s1|∠θ_z1. Similarly, Z_s2 is the equivalent impedance between the terminal side of the converter VSC₂ and the AC system 2 (DN2), expressed as Z_s2 = r_s2 + jx_s2 = |Z_s2|∠θ_z2. E₁ and E₂ indicate the side voltage amplitude of converter VSC₁ and VSC₂, respectively.

In addition, to further facilitate the analysis of the problem, the VSC₁ terminal side is set to adopt the inverter convention, and the output current is positive. The VSC₂ side adopts the rectifier convention, which considers the input current to be positive. Therefore, δ₁ is the phase difference of VSC₁ side voltage E₁ ahead of the grid voltage U_g1. δ₂ is the phase difference of the grid voltage U_g2 ahead of the VSC₂ side E₂. δ₁ and δ₂ can also be considered as the power angles of the VSC on both sides.

2.2. Control Strategy Analysis of Flexible Multi-State Switch

The detailed control architecture of VSC on both sides of BTBC-FMSS is shown in Figure 3. In normal operation, one end of the BTBC-FMSS uses power control strategy (PQ control), while the other end uses DC voltage control strategy (V_dcQ control). Since the positive direction of transmission power for the BTBC-FMSS determined in this paper is set as VSC₂→VSC₁, PQ control is adopted for the VSC₁ side and V_dcQ control is adopted for the VSC₂ side to facilitate the analysis of the problem. It is worth pointing out that the setting of the positive direction and the control mode on both sides does not affect the analysis result of the final power transmission relationship.

Both sides of BTBC-FMSS VSC adopt the current control mode; so, the inner loop of the controller only contains the current loop, which is expressed as follows [19,20]:

$$\left\{\begin{array}{l}{U}_{id}^{\ast }=K_P^{CC}\left(I_{id}^{\ast }-I_{id}\right)+K_I^{CC}\int \left(I_{id}^{\ast }-I_{id}\right)dt\hfill \\ U_{iq}^{\ast }=K_P^{CC}\left(I_{iq}^{\ast }-I_{iq}\right)+K_I^{CC}\int \left(I_{iq}^{\ast }-I_{iq}\right)dt\hfill \end{array}\right., i=1, 2$$

(1)

where $U_{id}^{\ast }$ and $U_{iq}^{\ast }$ are the dq axis voltage reference values of VSCs on both sides; $I_{id}$, $I_{id}^{\ast }$ and $I_{iq}$, $I_{iq}^{\ast }$ are, respectively, measured and reference values of the dq axis current of VSCs on both sides; $K_P^{CC}$ and $K_I^{CC}$ are the PI control parameters of the current inner loop; and i = 1, 2.

The external loop of VSC₁ uses PQ control to determine the active and reactive power output where the references are decided from the upper-level dispatching command or the local power grid status. The power loop is to make the difference between the reference value of the power and the actual value, and then through the PI control, to realize the error-free control of the power reference, as shown in Formula (2).

$$\left\{\begin{array}{l}{I}_{1d}^{\ast }=K_P^{PC}\left(P_1^{\ast }-P_1\right)+K_I^{PC}\int \left(P_1^{\ast }-P_1\right)dt\hfill \\ I_{1q}^{\ast }=K_P^{PC}\left(Q_1-Q_1^{\ast }\right)+K_I^{PC}\int \left(Q_1-Q_1^{\ast }\right)dt\hfill \end{array}\right.$$

(2)

where $P_1$, $Q_1$ and $P_1^{\ast }$, $Q_1^{\ast }$ are the measured and reference values of the output active and reactive power of VSC₁, respectively. $K_P^{PC}$ and $K_I^{PC}$ are the PI control parameters of the power outer loop.

The VSC₂ side is controlled by V_dcQ to maintain the voltage stability of the capacitor on the DC side of the BTBC-FMSS, and then realize the power balance of the two sides of the VSC. The corresponding controller expression is as follows:

$$\left\{\begin{array}{l}{I}_{2d}^{\ast }=K_P^{DVC}\left(V_{dc}^{\ast }-V_{dc}\right)+K_I^{DVC}\int \left(V_{dc}^{\ast }-V_{dc}\right)dt\hfill \\ I_{2q}^{\ast }=-\frac{2}{3}{Q}_2^{\ast }/E_{2d}\hfill \end{array}\right.$$

(3)

where E_2d is the d-axis measured value of the output voltage on the VSC₂ side. $Q_2^{\ast }$ is the reference value of the output reactive power of the VSC₂ side. V_dc and $V_{dc}^{\ast }$ are the measured values and reference values of the DC voltage of BTBC-FMSS $K_P^{DVC}$ and $K_I^{DVC}$ are the PI control parameters of the outer loop of the DC voltage.

3. Derivation of Transmission Power Expression of Flexible Multi-State Switch

According to the analysis in Section 1, current control modes are generally adopted at both ends of BTBC-FMSS under a quasi-steady state. Therefore, the transmission power limit analysis method given in the literature [9,10] based on the circuit correlation between the distribution network voltage and inverter terminal voltage is not accurate. In this section, the transmission power expressions on both sides of BTBC-FMSS are derived in detail according to the current source control circuit expression.

3.1. The Transmission Power Expression of VSC Controlled by Current under dq Transformation

Firstly, according to the Park transformation principle, the expression of the transmission power on both sides of VSC₁ and VSC₂ in the dq coordinate system is written, as shown in Equation (4).

$$\left\{\begin{array}{l}{P}_i=\frac{3}{2}\left(E_{id}{I}_{id}+E_{iq}{I}_{iq}\right)\hfill \\ Q_i=\frac{3}{2}\left(E_{iq}{I}_{id}-E_{id}{I}_{iq}\right)\hfill \end{array}\right., i=1, 2$$

(4)

where P_i and Q_i denote the active and reactive power output from the VSC_i side, respectively; E_id and E_iq denote the voltage components in the d and q axes of the VSC_i side, respectively; and I_id and I_iq denote the current components in the d and q axes of the VSC_i side, respectively.

In addition, converters generally adopt d-axis voltage orientation, that is, E_iq = 0; so, Equation (4) can be further written as follows:

$$\left\{\begin{array}{l}{P}_i=\frac{3}{2}{E}_{id}{I}_{id}\hfill \\ Q_i=-\frac{3}{2}{E}_{id}{I}_{iq}\hfill \end{array}\right., i=1, 2$$

(5)

Due to the d-axis voltage orientation, P_i and Q_i are approximately proportional to I_id and −I_iq when E_id does not change much, the approximate decoupling of active and reactive power control is realized, and the constant current control can be converted into constant power control. It is worth pointing out that Equation (5) cannot directly reflect the functional relationship between the output power P_i + jQ_i on the VSC_i side and its output current I_i and power angle δ_i, which needs to be further derived according to the vector diagram.

3.2. Derivation of Transmission Power Expression of VSC on Both Sides of BTBC-FMSS

Firstly, the vector relationship of voltage and current between VSC_i and distribution network on both sides in the dq coordinate system is drawn, as shown in Figure 4a,b.

As can be seen from Figure 4a, since VSC₁ adopts the inverter tradition, the VSC₁ terminal voltage E₁ is ahead of the power grid voltage U_g1. Based on the vector relationship, the expression of E₁ in d and q axis components can be derived as follows:

$$\left\{\begin{array}{l}{E}_{1d}=U_{g1}\cos {\delta }_1+I_1\left|Z_{s1}\right|\cos \left({\theta }_{z1}+{\mathit{\phi }}_1\right)\hfill \\ E_{1q}=-U_{g1}\sin {\delta }_1+I_1\left|Z_{s1}\right|\sin \left({\theta }_{z1}+{\mathit{\phi }}_1\right)\hfill \end{array}\right.$$

(6)

where δ₁ denotes the phase difference between the machine terminal voltage E₁ on the VSC₁ side and the grid voltage U_g1, I₁ denotes the current amplitude on the VSC₁ side, |Z_s1| denotes the amplitude of the equivalent impedance on the VSC₁ side, θ_z1 denotes the phase angle of the equivalent impedance on the VSC₁ side, and φ₁ denotes the power factor angle on the VSC₁ side.

By substituting E_1d in the above equation into Equation (5), the transmission power expression of VSC₁ side under d-axis voltage orientation can be written as follows:

$$\left\{\begin{array}{l}{P}_1=\frac{3}{2}\left[U_{g1}\cos {\delta }_1+I_1\left|Z_{s1}\right|\cos \left({\theta }_{z1}+{\mathit{\phi }}_1\right)\right]I_1\cos {\mathit{\phi }}_1\hfill \\ Q_1=-\frac{3}{2}\left[U_{g1}\cos {\delta }_1+I_1\left|Z_{s1}\right|\cos \left({\theta }_{z1}+{\mathit{\phi }}_1\right)\right]I_1\sin {\mathit{\phi }}_1\hfill \end{array}\right.$$

(7)

Similarly, by adopting the rectifier tradition on the VSC₂ side, the expression of E₂ in the d and q axes’ components can be obtained:

$$\left\{\begin{array}{l}{E}_{2d}=U_{g2}\cos {\delta }_2-I_2\left|Z_{s2}\right|\cos \left({\theta }_{z2}+{\mathit{\phi }}_2\right)\hfill \\ E_{2q}=U_{g2}\sin {\delta }_2-I_2\left|Z_{s2}\right|\sin \left({\theta }_{z2}+{\mathit{\phi }}_2\right)\hfill \end{array}\right.$$

(8)

where δ₂ denotes the phase difference between the machine terminal voltage E₂ on the VSC₂ side and the grid voltage U_g2, I₂ denotes the current amplitude on the VSC₂ side, |Z_s2| denotes the amplitude of the equivalent impedance on the VSC₂ side, θ_z2 denotes the phase angle of the equivalent impedance on the VSC₂ side, and φ₂ denotes the power factor angle on the VSC₂ side.

Similarly, by substituting E_2d into Equation (5), the transmission power expression of VSC₂ side under d-axis voltage orientation can be written:

$$\left\{\begin{array}{l}{P}_2=\frac{3}{2}\left[U_{g2}\cos {\delta }_2-I_2\left|Z_{s2}\right|\cos \left({\theta }_{z2}+{\mathit{\phi }}_2\right)\right]I_2\cos {\mathit{\phi }}_2\hfill \\ Q_2=-\frac{3}{2}\left[U_{g2}\cos {\delta }_2-I_2\left|Z_{s2}\right|\cos \left({\theta }_{z2}+{\mathit{\phi }}_2\right)\right]I_2\sin {\mathit{\phi }}_2\hfill \end{array}\right.$$

(9)

The expressions of active and reactive power transmission power shown in Equations (7) and (9) are related to the two controlled variables of output current amplitude I_i and power factor angle φ_i at the VSC_i side, but there is still the uncontrolled variable power angle δ_i to be further replaced.

Considering that the VSC on both sides adopts the d-axis voltage orientation and the PLL adopts the q-axis voltage E_iq phase locking, the E_iq = 0 element is valid under the steady-state condition of the system, and the relationships between E_1q and δ₁ and E_2q and δ₂ are plotted, respectively, as shown in Figure 5a,b.

In the above figure, there are two operating points for the δ_i angle of VSC on both sides under the condition that E_iq = 0 is met. According to the small-signal stability analysis, the static stable operating point of δ_i can be determined, as shown in Equation (10).

$${\delta }_i^s=\arcsin \frac{I_i\left|Z_{si}\right|\sin \left({\theta }_{zi}+{\mathit{\phi }}_i\right)}{U_{gi}}, i=1, 2$$

(10)

Obviously, according to the solution constraint, the condition $\left|{\delta }_i^s\right|$ ≤ π/2 is obtained.

By substituting the result of the above stable operating point ${\delta }_i^s$ into Equations (7) and (9), the transmission power expressions of VSC on both sides can be further written as follows:

(1)

VSC₁ side,

$$\begin{array}{ll}\hfill P_1& =\frac{3}{2}\left[U_{g1}\cos {\delta }_1^s+I_1\left|Z_{s1}\right|\cos \left({\theta }_{z1}+{\mathit{\phi }}_1\right)\right]I_1\cos {\mathit{\phi }}_1\hfill \\ & =\frac{3}{2}\left[\begin{array}{l}{U}_{g1}\cos \left(\arcsin \frac{I_1\left|Z_{s1}\right|\sin \left({\theta }_{z1}+{\mathit{\phi }}_1\right)}{U_{g1}}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+I_1\left|Z_{s1}\right|\cos \left({\theta }_{z1}+{\mathit{\phi }}_1\right)\hfill \end{array}\right]I_1\cos {\mathit{\phi }}_1\hfill \end{array}$$

(11)

$$\begin{array}{ll}\hfill Q_1& =-\frac{3}{2}\left[U_{g1}\cos {\delta }_1^s+I_1\left|Z_{s1}\right|\cos \left({\theta }_{z1}+{\mathit{\phi }}_1\right)\right]I_1\sin {\mathit{\phi }}_1\hfill \\ & =-\frac{3}{2}\left[\begin{array}{l}{U}_{g1}\cos \left(\arcsin \frac{I_1\left|Z_{s1}\right|\sin \left({\theta }_{z1}+{\mathit{\phi }}_1\right)}{U_{g1}}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+I_1\left|Z_{s1}\right|\cos \left({\theta }_{z1}+{\mathit{\phi }}_1\right)\hfill \end{array}\right]I_1\sin {\mathit{\phi }}_1\hfill \end{array}$$

(12)

(2)

VSC₂ side,

$$\begin{array}{ll}\hfill P_2& =\frac{3}{2}\left[U_{g2}\cos {\delta }_2^s-I_2\left|Z_{s2}\right|\cos \left({\theta }_{z2}+{\mathit{\phi }}_2\right)\right]I_2\cos {\mathit{\phi }}_2\hfill \\ & =\frac{3}{2}\left[\begin{array}{l}{U}_{g2}\cos \left(\arcsin \frac{I_2\left|Z_{s2}\right|\sin \left({\theta }_{z2}+{\mathit{\phi }}_2\right)}{U_{g2}}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-I_2\left|Z_{s2}\right|\cos \left({\theta }_{z2}+{\mathit{\phi }}_2\right)\hfill \end{array}\right]I_2\cos {\mathit{\phi }}_2\hfill \end{array}$$

(13)

$$\begin{array}{ll}\hfill Q_2& =-\frac{3}{2}\left[U_{g2}\cos {\delta }_2^s-I_2\left|Z_{s2}\right|\cos \left({\theta }_{z2}+{\mathit{\phi }}_2\right)\right]I_2\sin {\mathit{\phi }}_2\hfill \\ & =-\frac{3}{2}\left[\begin{array}{l}{U}_{g2}\cos \left(\arcsin \frac{I_2\left|Z_{s2}\right|\sin \left({\theta }_{z2}+{\mathit{\phi }}_2\right)}{U_{g2}}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-I_2\left|Z_{s2}\right|\cos \left({\theta }_{z2}+{\mathit{\phi }}_2\right)\hfill \end{array}\right]I_2\sin {\mathit{\phi }}_2\hfill \end{array}$$

(14)

It can be seen from Expressions (11)–(14) that when the VSC on both sides adopts the current control mode, its output power expression contains only two controlled variables: the output current amplitude I_i on both sides and the power factor angle φ_i, which is the transmission current vector ${\overrightarrow{I}}_i$. It can also be seen from the analysis of Equations (11) and (14) that the active and reactive transmission power are also related to the two structural parameters of the power network intensity U_gi/|Z_si| and the equivalent impedance angle of the power network φ_i.

In addition, the observation Formulas (11)–(14) show that the expressions of active and reactive power show some symmetry in form, that is, there is translation and axial symmetry in the active and reactive power characteristics of VSC_i, which is expressed as follows:

$${\left.P_i\left(I_i,{\mathit{\phi }}_i\right)\right|}_{{\theta }_{Zi}}={\left.Q_i\left(I_i,-{\mathit{\phi }}_i-\frac{\pi }{2}\right)\right|}_{\frac{\pi }{2}-{\theta }_{Zi}}$$

(15)

Through symmetry analysis, we can know that only the active power transmission characteristics are given, and then the reactive power transmission characteristics can be obtained by translation and axisymmetric transformation. Therefore, if not specified below, only the corresponding analysis of the output active power characteristics of the VSC_i on both sides of the BTBC-FMSS is carried out.

4. Transmission Power Limit and Stable Operation Domain Analysis of BTBC-FMSS

Section 3 derives the expression of the transmission power of the VSC on both sides of the BTBC-FMSS under the current control. This section will further analyze the transmission power limit and stable operation domain of the BTBC-FMSS equipment according to the table, and discuss its relationship with the BTBC-FMSS stability.

4.1. Analysis of Transmission Power Characteristics on Both Sides of BTBC-FMSS

BTBC-FMSS has the ability of four-quadrant operation, and there is also a potential demand for four-quadrant operation in DN. Therefore, if the output current amplitude I_i and power factor angle φ_i of the VSC on both sides are in the range of I_i ≥ 0, φ_i∈[−π, π]. Then, the output active and reactive power of VSC on both sides should be regarded as binary functions with respect to the above two variables, expressed as P_i(I_i, φ_i) and Q_i(I_i, φ_i), i = 1, 2.

Under the constraint of the existence of static stable working point of PLL, the input variable of PLL_i integrator of VSC_i on both sides of BTBC-FMSS must be strictly 0, that is, E_iq = 0. If the special angle α_i is further defined as the sum of the impedance angle θ_zi and the power factor angle φ_i, that is, α_i= θ_zi + φ_i, then using E_iq = 0 and according to the Formulas (5) and (10), the value range of I_i can be calculated, expressed as follows:

$$\begin{array}{c}\hfill I_i=\frac{U_{gi}\sin {\delta }_i}{\left|Z_{si}\right|\sin \left({\theta }_{zi}+{\mathit{\phi }}_i\right)}\le \frac{U_{gi}}{\left|Z_{si}\right|\left|\sin \left({\theta }_{zi}+{\mathit{\phi }}_i\right)\right|},\hfill \\ \hfill {\delta }_i\in \left[-\pi ,\pi \right],i=1,2\hfill \end{array}$$

(16)

What really affects the power transmission direction should be the power factor angle φ. When the range of the power factor angle is [−π/2, π/2], the direction of power transmission is forward, i.e., from VSC₂ to VSC₁; when the range of the power factor angle is [π/2, 3π/2], the direction of power transmission is inverted, i.e., from VSC₁ to VSC₂. Therefore, it is clear that the transmission direction has nothing to do with the impedance angle θ_z.

However, in this paper, in order to facilitate the analysis, we defined the special angle α to be the sum of power factor angle and impedance angle, that is α = θ_z + φ, which does not change the fact that the transmission direction is only related to the power factor Angle φ.

Considering the four-quadrant operation capability of VSC_i, there is I_i ≥ 0, φ_i∈[−π, π], and the equivalent impedance angle of the power grid on both sides θ_zi∈[0, π/2], i = 1, 2. Thus, θ_zi + φ_i∈[−π + θ_zi, π + θ_zi]. Under different values θ_zi + φ_i, the upper limit I_imax of I_i exists as follows:

$$\begin{array}{l}{\left.I_{i\max }\right|}_{i=1,2}=\hfill \\ \left\{\begin{array}{l}\frac{U_{gi}}{\left|Z_{si}\right|\left|\sin \left({\theta }_{zi}+{\mathit{\phi }}_i\right)\right|},{\theta }_{zi}+{\mathit{\phi }}_i\ne 0\cup {\theta }_{zi}+{\mathit{\phi }}_i\ne \pi \hfill \\ \hspace{1em}+\infty ,\hspace{1em}{\theta }_{zi}+{\mathit{\phi }}_i=0\cup {\theta }_{zi}+{\mathit{\phi }}_i=\pi \hfill \end{array}\right.\hfill \end{array}$$

(17)

As can be seen from the above formula, when θ_zi + φ_i = 0 and θ_zi + φ_i = π (that is, φ_i and θ_zi are opposite or complementary to each other), the output/input current on the VSC_i side has no upper limit, and is only influenced by the capacity of the VSC itself.

In addition to the inherent limit of VSC_i input/output current, it is known from Equations (11)–(14) that there is a complex non-linear relationship between VSC_i input/output power and input/output current and power factor angle. At the same time, because of the symmetry of Equations (11)–(14), this paper mainly focuses on the active power transmission capability of BTBC-FMSS. Therefore, taking the VSC₁ side of Equation (11) as an example, the expression of transmission power limit is derived.

Firstly, according to Formula (11), using the condition of dP₁/dI₁ = 0, the critical current of P₁ with extreme value is obtained, which is expressed as follows:

$$I_{1cr}=\left\{\begin{array}{l}\frac{\sqrt{2}{U}_{g1}}{2\left|Z_{s1}\right|\sqrt{1-\cos {\alpha }_1}},{\alpha }_1\ne 0\hfill \\ \hspace{1em}+\infty ,\hspace{1em}{\alpha }_1=0\hfill \end{array}\right.$$

(18)

According to the above formula, in the special case of α₁ = 0, the critical current tends to infinity because there is no constraint on the output current. By replacing Formula (18) with the Formula (11), the extreme value P_1cr of the transmitted active power on the VSC₁ side can be obtained, which is expressed as follows:

$$P_{1cr}=P_1\left(I_{1cr}\right)=\left\{\begin{array}{l}{K}_{1cr}\left({\alpha }_1\right)S_{ac1},{\alpha }_1\ne 0\hfill \\ \hspace{1em}+\infty \hspace{1em},{\alpha }_1=0\hfill \end{array}\right.$$

(19)

In the above formula, K_1cr(α₁) is a coefficient only related to the special angle α₁. In the quasi-steady state, S_ac1 is approximately the short-circuit capacity sent by the distribution network 1 to the VSC₁ side. K_1cr (α₁) and S_ac1 can be further expressed as follows:

$$\left\{\begin{array}{l}{K}_{1cr}\left({\alpha }_1\right)\hfill \\ =\frac{\frac{\sqrt{2}}{2}\cos \left({\alpha }_1-{\theta }_{z1}\right)\left(\sqrt{1-\frac{{\sin }^2{\alpha }_1}{2\left(1-\cos {\alpha }_1\right)}}+\frac{\sqrt{2}}{2}\frac{\cos {\alpha }_1}{\sqrt{1-\cos {\alpha }_1}}\right)}{\sqrt{1-\cos {\alpha }_1}}\hfill \\ S_{ac1}=\frac{3}{2}\frac{U_{g1}^2}{\left|Z_{s1}\right|}\hfill \end{array}\right.$$

(20)

According to the above formula, the magnitude of the extreme P_1cr of the transmitted active power on the VSC_i side is only related to the short-circuit capacity S_ac1 and the special angle α₁ on the grid side, and the transmission direction is only determined by the power factor angle φ₁ = α₁ − θ_Z1.

Formulas (18)–(20) give the expression of the transmission active power extreme value P_1cr and the corresponding critical current I_1cr on the BTBC-FMSS VSC₁ side. According to the maximum function theorem, the maximum value of a continuous function can only be obtained at the end of the interval or the extreme point. For VSC₁ side, according to Formula (17), the variation norm of the output current I₁ is [0, I_1max]; so. if (17) is substituted into (11), the expression of P₁ (I_1max) can be deduced as follows:

$$P_1\left(I_{1\max }\right)=\left\{\begin{array}{l}{K}_{1m}\left({\alpha }_1\right)S_{ac1}, {\alpha }_1\ne k\pi \hfill \\ \pm \infty , {\alpha }_1=k\pi \hfill \end{array}\right.,k=0,1,2$$

(21)

In the above formula, K_1m(α₁) can be expressed as follows:

$$K_{1m}\left({\alpha }_1\right)=\frac{\cos \left({\alpha }_1-{\theta }_{z1}\right)\left(\sqrt{1-\frac{{\sin }^2{\alpha }_1}{\left|{\sin }^2{\alpha }_1\right|}}+\frac{\cos {\alpha }_1}{\left|\sin {\alpha }_1\right|}\right)}{\left|\sin {\alpha }_1\right|}$$

(22)

Similarly, the critical current I_2cr of the transmitted active power P₂ on the VSC₂ side can be obtained, which is expressed as follows:

$$I_{2cr}=\left\{\begin{array}{l}\frac{\sqrt{2}{U}_{g2}}{2\left|Z_{s2}\right|\sqrt{1+\cos {\alpha }_2}},{\alpha }_2\ne \pi \hfill \\ \hspace{1em}+\infty ,\hspace{1em}{\alpha }_2=\pi \hfill \end{array}\right.$$

(23)

Correspondingly, the transmission active power extreme value P_2cr on the VSC₂ side is expressed as follows:

$$P_{2cr}=P_2\left(I_{2cr}\right)=\left\{\begin{array}{l}{K}_{2cr}\left({\alpha }_2\right)S_{ac2},{\alpha }_2\ne \pi \hfill \\ \hspace{1em}+\infty \hspace{1em},{\alpha }_2=\pi \hfill \end{array}\right.$$

(24)

In the above formula, K_2cr(α₂) and S_ac2 can be expressed as follows:

$$\left\{\begin{array}{l}{K}_{2cr}\left({\alpha }_2\right)\hfill \\ =\frac{\frac{\sqrt{2}}{2}\cos \left({\alpha }_2-{\theta }_{z2}\right)\left(\sqrt{1-\frac{{\sin }^2{\alpha }_2}{2\left(1+\cos {\alpha }_2\right)}}-\frac{\sqrt{2}}{2}\frac{\cos {\alpha }_2}{\sqrt{1+\cos {\alpha }_2}}\right)}{\sqrt{1+\cos {\alpha }_2}}\hfill \\ S_{ac2}=\frac{3}{2}\frac{U_{g2}^2}{\left|Z_{s2}\right|}\hfill \end{array}\right.$$

(25)

Similarly, by substituting (17) into (13), the expression of P₂ (I_2max) can be deduced as follows:

$$P_2\left(I_{2\max }\right)=\left\{\begin{array}{l}{K}_{2m}\left({\alpha }_2\right)S_{ac2},{\alpha }_2\ne k\pi \hfill \\ \hspace{1em}\pm \infty \hspace{1em},{\alpha }_2=k\pi \hfill \end{array}\right.,k=0,1,2$$

(26)

In the above formula, K_2m(α₂) can be expressed as follows:

$$K_{2m}\left({\alpha }_2\right)=\frac{\cos \left({\alpha }_2-{\theta }_{z2}\right)\left(\sqrt{1-\frac{{\sin }^2{\alpha }_2}{\left|{\sin }^2{\alpha }_2\right|}}-\frac{\cos {\alpha }_2}{\left|\sin {\alpha }_2\right|}\right)}{\left|\sin {\alpha }_2\right|}$$

(27)

Obviously, the comprehensive Formulas (18)–(27) show that the limit of the transmission power of the VSC on both sides of the BTBC-FMSS is mainly determined by the extreme value P_icr on each side and the interval endpoint values P_i(I_imax) and P_i(0). In addition, it is obvious that the expressions of P_icr and P_i (I_imax) are only related to the power factor angle φ_i(α_i) of the controlled variable. Therefore, if the curves of P_icr − α_i, P_i(I_imax) − α_i and P_i = 0 are drawn, the closed area surrounded by each curve is the static stable operation domain on each side for the VSC_i. The static stable operation domain of each side of BTBC-FMSS is intuitively analyzed with a specific example.

Table 1. The structure and control parameters of flexible interconnected distribution network.

Distribution Network
Structural Parameter	Numerical Value
	DN1	DN2
Bilateral voltage of distribution network Ug/V	380	380
Equivalent impedance of bilateral power grid of distribution network |Zg|/Ω	5	5
BTBC-FMSS device
VSC1 and VSC2 structure and control parameters	numerical value
RL filter Rfc/Lfc	0.1 Ω/0.8 mH
dc-link capacitor Cdc	1000 μF
PI parameters of inner current loop Kp_CC/Ki_CC	20/200
DC voltage ring PI parameters Kp_DVC/Ki_DVC	2/20
PLL parameters Kp_PLL/Ki_PLL	0.4/2

shows the typical specific values of the equivalent circuit of the BTBC-FMSS flexible interconnection DN as shown in Figure 2. Among them, to control the control variables to facilitate the problem analysis, it is assumed that the physical and control parameters of the VSC on both sides of the BTBC-FMSS are the same, and the equivalent impedance parameters and voltage levels of the flexible interconnected DN on both sides of the BTBC-FMSS are completely the same. In addition, the corresponding stable operation domains of the VSC on both sides are given.

The static stable operation domain of the VSC on both sides of the BTBC-FMSS at different impedance angles is shown in Figure 6. Figure 6a,b corresponds to two cases of VSC₁ side θ_Z1 = π/2 and θ_Z1 = π/3, respectively, and Figure 6c,d corresponds to two cases of VSC₂ side θ_Z2 = π/2 and θ_Z2 = π/3, respectively. The red line represents the P_icr − α_i curve, the blue line represents the P_i(I_imax) − α_i curve, and the black line represents the P_i = 0 curve. The gray shaded area represents the stable operation domain of active power forward transmission, which is expressed as SD₁, and the light red shadow represents the stable operation domain of active power reverse transmission, which is represented as SD₂. In addition, the realization indicates that the output current direction of the VSC_i side is the same as the active power direction, and the dotted line indicates that the two directions are opposite.

Looking carefully at Figure 6a–d, the following conclusions can be drawn by comparison:

(1) The P_icr curve takes 2π as the period, and the P_i (I_imax) curve takes π as the period, and the static stable operation domain surrounded by them and P_i (0) is unbounded when the conditions of α_i = 0 and α_i = π are satisfied.

(2) When the line impedance is approximately pure inductance (θ_zi = π/2), with α_i = π as the boundary, the static stable operation domain of each side of VSC with forward and reverse transmission is the same and the center is symmetrical, that is, the stable operation domain is the same in different transmission directions.

(3) In the inductive line (θ_zi = π/3), there is a difference in the stable operation domain of the forward and backward transmission power of each side of the VSC, that is, the feasible region of stable operation in different directions is different.

(4) Under the same impedance angle and short-circuit capacity, the static stable operation domain of forward transmission on the VSC₂ side is symmetrical with the static stable operation domain of the reverse transmission on the VSC₁ side, and vice versa.

(5) The static stable operation domain (solid line + dashed line region) under the current control mode is larger than that under the power control mode (solid line region). The reason is that the current control is stored in the operation point where the VSC current is negatively related to active power and can operate normally.

4.2. Stable Operation Domain Analysis of BTBC-FMSS

First, it is important to point out that the active and reactive power transmission characteristics of the BTBC-FMSS are fundamentally different: (1) For the active power transmission problem, since the BTBC-FMSS is equivalent to an active transmission channel, the equation P₁ = P₂ necessarily exists in the steady state while ignoring the power loss on the DC side. (2) For the reactive power transmission problem, due to the complete decoupling of the reactive power control on both sides, the BTBC-FMSS should be regarded as a bi-directional reactive power supply, and the size and direction of Q₁ and Q₂ are not necessarily related to each other but only to the reactive power control mode.

Secondly, it is considered that in a real flexible interconnected DN system, at the steady state, balancing the load ratio of feeders on both sides as an active transmission channel is the BTBC-FMSS’s core mode of operation. On the one hand, affected by the capacity, the proportion of reactive power transmission is too large to crowd the active transmission channel, and the reactive power transmission should be used as an auxiliary function without or with less transmission; on the other hand, the reactive power balancing in the grid should try to follow the principle of in situ, and a larger reactive power transmission will undoubtedly increase the active loss of the device. In addition, most studies also believe that even if the BTBC-FMSS can transmit reactive power, the reactive power control on both sides of it should establish a correlation relationship. Therefore, this paper further selects a special transmission mode, i.e., equal-capacity transmission mode (${\tilde{S}}_1={\tilde{S}}_2$). Obviously, in the equal capacity transmission mode, the intersection of the static stabilized operating areas of the VSCs on both sides is the static stabilized operating area of the whole device of the BTBC-FMSS. Considering the actual operation scenario, the equivalent impedance of each side of the BTBC-FMSS cannot be simply regarded as pure inductance, and the values and impedance angles of the equivalent impedance may be different, but the voltage levels of the feeder lines on both sides of the DN are generally the same. Therefore, to facilitate the analysis, the parameters of fixed DN1 remain unchanged, two scenarios of θ_Z1 = π/2 and θ_Z1 = π/3 are selected to change the short-circuit capacity (equivalent impedance module value|Zs₂|) and impedance angle θ_z2 of DN2, respectively, and the static stable operation domain of BTBC-FMSS as shown in Figure 7a–d is obtained.

As shown in Figure 7, the black solid line represents the P_1cr curve, and different colored solid lines represent the gradient of the P_2cr curve. The irregular closed area surrounded by P_1cr, P_2cr and P_i = 0 is the static stable operation domain of the BTBC-FMSS device, which is expressed as SD = SD₁∩SD₂. A careful analysis of the above two pictures shows that the following conclusions can be drawn:

(1) Different from the separate analysis of VSC static stable operation domain on each side, the static stable operation domain of BTBC-FMSS must be bounded under forward/reverse transmission, and there is a maximum upper limit of transmission active power, which is P_max+ in forward transmission and P_max− in reverse transmission.

(2) If the short-circuit capability and impedance angle of the DN on both sides are the same, that is, U_g1 = U_g2, |Z_s1| = |Z_s2|, when θ_Z1 = θ_Z2, the maximum transmission power limit of BTBC-FMSS forward and reverse must be obtained at π/2 and 3 π/2, and SD+ = SD−.

(3) With the increase in the short-circuit capability of the VSC₂ side (in Figure 7a), the overall transmission power limit of the BTBC-FMSS increases significantly, and the SD area expands significantly.

(4) When the equivalent circuit impedance angle of the VSC₂ side is reduced (in Figure 7b), the overall transmission power limit of the BTBC-FMSS is reduced, the SD decreases slightly under the forward transmission, and the SD decreases obviously under the reverse transmission.

According to conclusion 1, the maximum active power transmission limits of BTBC-FMSS in forward and reverse transmission directions are P_max+ and P_max−, respectively, and can be obtained by solving the intersection of P_1cr and P_2cr. However, when the parameters of the distribution network on both sides of the BTBC-FMSS are asymmetric, the analytical expression is difficult to obtain, and it can only be obtained by depicting the stable operation domain in practical application.

4.3. Transmission Active Power Limits Analysis and Enhancement Measures of BTBC-FMSS

(1) The analysis of BTBC-FMSS in the previous section shows that there must be a stability limit of transmission power in BTBC-FMSS in different transmission directions, and the limit is only related to the short-circuit capacity, line impedance angle and power factor angle of the power grid on both sides. The corresponding calculation flow chart can be shown as Algorithm 1:

Algorithm 1: BTBC-FMSS transmission power stability limit algorithm.

1: The Thevenin equivalent of the BTBC-FMSS based flexible interconnected DN on both sides is carried out.

2: Get the equivalent impedance on both sides:

3:    Zs1 = |Zs1| ∠ θ1

4:    Zs2 = |Zs2| ∠ θ2

5: Calculate, derive the maximum input/output current amplitude I1max and I2max.

6: Make drawings, derive the stable operation domain SD1 and SD2.

7: Take the intersection, SD = SD1∩SD2 (Figure 6)

8: Derive Pmax+ and Pmax− (the maximum upper limit of active power transmission).

end

(2) Enhancement measures to improve static stability

Measure 1: The VSC sides on both sides use the local reactive power compensation devices (capacitors) as the parallel reactive power compensator.

The advantage of local reactive power compensation is that it can support the voltage of VSC sides on both sides, improve the short-circuit capacity of AC system to a certain extent, and leave enough active power transmission channels.

Measure 2: The transmission of active and reactive power distribution ratio is adjusted according to the actual situation (adjust the power factor angle φ_i).

It can be seen from Figure 7 that the pure active power transmission mode often does not correspond to the maximum transmission active power limit, because the equivalent impedance angle of the DN on both sides is not strictly equal to π/2. In the absence of a local reactive power compensation device, properly adjusting the distribution ratio of active and reactive power is helpful to improve the transmission capacity of active power.

(3) Potential challenges and future research directions

Firstly, the potential challenges of the method proposed in this paper lie in the acquisition of grid equivalent impedance, which may be difficult to calculate when there are many nonlinear loads and distributed generators in the distribution network.

Secondly, this paper only investigates the BTBC-FMSS in conventional control scheme (PQ control for VSC₁ and V_dcQ control for VSC₂). However, in some cases, such as in fault condition, one of the VSC side can transfer into the virtual synchronous generator (VSG) control scheme. Thus, further discussion and research of its stable operation domain are needed.

Finally, in a multi-terminal (more than two VSC terminals) LVDC system, the constraints on power transmission for FMSS device can be presented as P_vsc1 + P _vsc2 + …+P _vscn = 0. Therefore, how to visualize the stable operating domain of FMSS device in this case is also one of the future research directions.

4.4. The Engineering Application Prospect of the Stable Operation Domain of BTBC-FMSS and the Specific Suggestions for the Actual Distribution Network

The flexible multi-state switching stable operation domain proposed in this paper can be applied to two typical scenarios: one is the BTBC-based interconnected microgrid clusters, and the other is the low-voltage direct current (LVDC) system.

Scenario I: Interconnecting multiple microgrids (MGs) to form a larger-scale interconnected MGs (IMGs) is an ideal way to increase the penetration rate of distributed generation (DG) and improve the reliability of power supply for the whole system. In addition to the conventional impedance lines interconnection scheme, nowadays, a new interlinking way, by using back-to-back converters (BTBCs), has become a novel interconnection scheme to solve the integration problems of IMGs. Usually, MG can be viewed as a weak grid since there is only a limited power supply and a weak grid structure. In such a system, the BTBC device is likely to have the transmission limit due to the constraint of stable operation domain.

Scenario II: More and more distributed energy sources, such as photovoltaic systems and wind power, are connected to the power grid through DC-DC or AC-DC converters, which, coupled with the wide application of DC devices such as electric vehicles, LEDs, and energy storage, has promoted the rapid development of low-voltage DC systems and AC-DC hybrid systems. In such a DN system, at the end of the feeders, the analysis of stable operation domain for the BTBC is necessary.

The specific suggestions for the actual distribution network about implementing BTBC-FMSS stable operation domain can be summarized in the following two points:

(1) The stable operation domain is both influenced by the short-circuit capacity and controller design on both sides of BTBC. Basically, the weaker the system, the more important it is to discuss the stable operating domain of BTBC. The two typical application scenarios are presented above.

(2) For a given capacity BTBC-FMSS device in DN, the main task is to evaluate its stable operation domain in real time comparing with its capacity. The intersection of the two is the actual stable operation domain for the BTBC-FMSS. In other words, the initial capacity planning for BTBC-FMSS should also refer to the verification of the stable operation domain, and the two issues are interactively coupled.

5. Simulation Results

To verify the validity of the proposed analysis method for BTBC-FMSS power transmission limit and static stability, the simulation verification model shown in Figure 8 is built in Matlab/Simulink. The structural and control parameters of distribution network and BTBC-FMSS devices are shown in

Table 2. The structure and control parameters of flexible interconnected DN.

Distribution Network
Structural Parameter	Numerical Value
	DN1	DN2
Bilateral voltage of distribution network Ug/V	380	380
Equivalent impedance of bilateral power grid of distribution network |Zg|/Ω	4.55	9.85
BTBC-FMSS device
VSC1 and VSC2 structure and control parameters	numerical value
RL filter Rfc/Lfc	0.15 Ω/0.85 mH
dc-link capacitor Cdc	1000 μF
PI parameters of inner current loop Kp_CC/Ki_CC	25/250
DC voltage ring PI parameters Kp_DVC/Ki_DVC	3/30
PLL parameters Kp_PLL/Ki_PLL	0.4/2

, and the impedance parameters of distribution lines and loads on both sides are shown in

Table 3. Structural parameters of flexible interconnected distribution grids before and after Davening equivalence.

Distribution Network 1 Side		Distribution Network 2 Side
Pre-equivalent
grid side voltage U1/V	380	grid side voltage U2/V	380
line impedance Ri + jXi/Ω (i = 1, 2, 3)	0.35 + j2	line impedance Ri + jXi/Ω (i = 4, 5)	1.058 + j6
	0.35 + j2		0.705 + j4
	0.18 + j1		/
load RLi + jXLi/Ω (i = 1, 2)	15 + j45	load RLi + jXLi/Ω (i = 3)	50 + j100
	10 + j30
Post-equivalent
equivalent voltage Ug1/V	324∠−1.27°	equivalent voltage Ug2/V	361∠−0.85°
equivalent impedance Zs1/Ω	4.55∠79.15°	equivalent impedance Zs2/Ω	9.85∠79.51°
Short-circuit capacity
23.072 kVA		13.231 kVA

.

According to Thevenin’s equivalent method, the equivalent impedance parameters of the DN system on both sides under full load access are Z_s1 = 4.55 Ω, Z_s2 = 9.85 Ω, U_g1 = 324, U_g2 = 361.1, and equivalent voltage.

According to the derivation process of this paper, the stable operation domain of BTBC-FMSS can be made under the simulation example, as shown in Figure 9.

Case 1: All loads are connected to the system at the beginning of the simulation, and the active and reactive power transmission references are set to 0. For active power control, the active power transmission references are changed into 2 kW and 4 kW at 2 s and 4 s, respectively; for reactive power control, the reactive power transmission references on both sides of VSC are changed into 1.5 kVar at 8 s time, and the simulation ends at 3 kVar at 12 s.

Figure 10 shows the dynamic response curve of the system under the setting of example 1. Through the gradual change in the transmission power reference, the operation state of the BTBC-FMSS is stable at the operating point 1, with P_VSC1 = P_VSC2 = 4 kW, Q_VSC1 = Q_VSC2 = 0 kVar at approximately 6 s, and the synchronization is identified in Figure 9. At 8 s, the reactive power reference of the VSC on both sides is changed to 1.5 kVar, and the operation point of the BTBC-FMSS immediately moves to the right, and finally stabilizes at the operation point 2, with P_VSC1 = P_VSC2 = 4 kW, Q_VSC1 = Q_VSC2 = 1.5 kVar. At 12 s, changing the reactive power reference of VSC on both sides to 3 kVar will cause operating point 2 to continue to move to the right. Due to crossing the stable boundary dominated by VSC₂. Looking at Figure 10, we can see that the system becomes unstable immediately after 12 s.

Case 2: All the loads are connected to the system at the beginning of the simulation, and the active and reactive power transmission references are set to 0. For active power control, the active power transmission references are changed to −4 kW and −8 kW at 2 s and 5 s, respectively, and for reactive power control, the reactive power transmission references on both sides of VSC are changed to −3 kVar at 12 s; the simulation ends at 13.5 s.

Figure 11 shows the dynamic response curve of each electric quantity of the system under the setting of example 2. Obviously, through the slow change in the active power transmission reference, the operation state of the BTBC-FMSS is stable at operation point 3, with P_VSC1 = P_VSC2 = −8 kW, Q_VSC1 = Q_VSC2 = 0 kVar at approximately 8 s, and the synchronization is identified in Figure 9. At 12 s time, when the reactive power reference is given to −3 kVar, the BTBC-FMSS becomes unstable immediately after 12 s because it crosses the stability boundary dominated by VSC₁.

Case 3: In order to verify the generality and scalability of the proposed method, based on the system in Figure 8, the photovoltaic system, energy storage device and DC load are further added to the DC bus side, and the line and load parameters of the AC side are changed at the same time. The varied system diagram is shown in Figure 12. Ignoring the dynamic of the DC/DC converter, and the equivalent DC component at DC bus can be equivalent to a constant power DC load. To facilitate the analysis, the equivalent constant power DC load is set to 1 kW in the simulation system. The structure and control parameters of the LVDC DN system and the BTBC-FMSS device are shown in

Table 4. The structure and control parameters of flexible interconnected DN in case 3.

Distribution Network
Structural Parameter	Numerical Value
	DN1	DN2
Bilateral voltage of distribution network Ug/V	380	380
Equivalent impedance of bilateral power grid of distribution network. |Zg|/Ω	5.06	8.68
BTBC-FMSS device
VSC1 and VSC2 structure and control parameters	numerical value
RL filter Rfc/Lfc	0.1 Ω/0.8 mH
dc-link capacitor Cdc	1000 μF
PI parameters of inner current loop Kp_CC/Ki_CC	20/200
DC voltage ring PI parameters Kp_DVC/Ki_DVC	2/20
PLL parameters Kp_PLL/Ki_PLL	0.4/2

, and the impedance parameters of the distribution lines and loads on both sides are shown in

Table 5. Structural parameters of flexible interconnected distribution grids before and after Davening equivalence in case 3.

Distribution Network 1 Side		Distribution Network 2 Side
Pre-equivalent
grid side voltage U1/V	380	grid side voltage U2/V	380
line impedance Ri + jXi/Ω (i = 1, 2, 3)	0.35 + j2	line impedance Ri + jXi/Ω (i = 4, 5)	1.058 + j6
	0.35 + j2		0.705+ j4
	0.26 + j1.5		/
load RLi + jXLi/Ω (i = 1, 2)	15 + j45	load RLi + jXLi/Ω (i = 3)	240 + j480
	10 + j30
Post-equivalent
equivalent voltage Ug1/V	324∠−1.27°	equivalent voltage Ug2/V	288∠−4.06°
equivalent impedance Zs1/Ω	5.06∠79.24°	equivalent impedance Zs2/Ω	8.68∠77.84°
Short-circuit capacity
31.119 kVA		14.333 kVA

.

After the load is added to the DC side, the operating states on both sides of BTBC-FMSS will no longer be consistent, and their stable operation domains will also change, but there is always an equation constraint relationship of P_VSC1 + P_dc = P_VSC2. When the power is transmitted forward and satisfies the inequality P_VSC1 + P_dc > P_VSC2, the stability boundary of VSC₂ side remains unchanged, while the stability operation domain of VSC₁ side becomes smaller due to the ‘pinning’ effect of VSC₂ side. The stability boundary of VSC₁ side is adjusted to the stability boundary of VSC₂ side minus the boundary of constant power of DC side. When the power on both sides satisfies the inequality P_VSC1 + P_dc < P_VSC2, the stable boundary on the VSC₁ side remains unchanged, and the stable operation domain on the VSC₂ side becomes larger. The specific performance is that the stable boundary on the VSC₂ side is adjusted to the stable boundary on the VSC₁ side and the boundary after the constant power on the DC side; the opposite is true when the power is transmitted in the reverse direction. The specific stable operation domains on both sides of BTBC-FMSS are shown in Figure 13. The area enclosed by the red curve in Figure 13a is the stable operation domain on the VSC₁ side, and the area enclosed by the blue curve in Figure 13b is the stable operation domain on the VSC₂ side.

According to Thevenin’s equivalent method, the equivalent impedance parameters of the distribution network on both sides under full load access are Z_s1 = 5.06 Ω∠79.24°, Z_s2 = 8.68 Ω∠77.84°, U_g1 = 324∠−1.27°, U_g2 = 288.6∠−4.06°, and equivalent voltage.

At the beginning of the simulation, all loads are connected to the system, and the active and reactive power transmission references are set to 0. For active power control, the active power transmission command is changed to 2 kW, 2.9 kW and 3.1 kW at 2 s, 7 s and 12 s, respectively. The reactive power reference is always 0; the simulation ends at 12.5 s.

Figure 14 shows the dynamic response curve of each electrical quantity of the system under the setting of example 3. Through the gradual change in the transmission power reference, roughly at 3 s, the VSC₁ side is stable at the operating state with P_VSC1 = 2 kW, Q_VSC1 = 0 kVar, and the VSC₂ side is stable at the operating state with P_VSC2 = 3 kW, Q_VSC2 = 0 kVar, which is synchronously identified in Figure 13a,b. At 7 s, the active command on the VSC₁ side is changed to 2.9 kW. At this time, the operating points on both sides will move upward, and finally stabilize at the operating point 2 of P_VSC1 = 2.9 kW, Q_VSC1 = 0 kVar, P_VSC2 = 3.9 kW, Q_VSC2 = 0 kVar. At 12 s, the active power command on the VSC₁ side is increased to 3.1 kW, which causes the operating points on both sides to continue to move upward due to the crossing of the VSC₂-dominated stability boundary. It can be seen from Figure 14 that the system is unstable immediately after 12 s. Therefore, the simulation verifies the effectiveness of the BTBC-FMSS stable operation domain under different network topologies, load/generation conditions and parameter settings.

6. Conclusions

This paper studies the transmission power limit and static stability of the ideal distribution system with BTBC-FMSS access, and draws the following conclusions:

(1) The transmission power limit of BTBC-FMSS is affected by the combined dynamic effects of structural parameters such as grid strength U_gi/|Z_si|, impedance angle θ_zi and other structural parameters, as well as control parameters such as current amplitude I_i and power factor angle φ_i on both sides of BTBC-FMSS.

(2) BTBC-FMSS’s two sides of the VSC transmission power limit itself by each side of the PLL static stabilization of the operating point of the existence of this condition constraint; there is a $P_{VSCi}^{cr}$ − α_i curve for each, and the actual BTBC-FMSS power transmission limit of the $P_{BTB}^{max}$ − α_i curve is the intersection of the two.

(3) The stable operating region of BTBC-FMSS proposed in this paper makes it easier to quantify the stable operating region of the device’s power transfer than the short-circuit ratio or the small-signal stability analysis, and it can guide the capacity configuration and optimal operation of BTBC-FMSS, which has good practical value.

Firstly, the small signal entails a large amount of work to determine the stable operating domain by determining whether each operation point is stable. Secondly, the small signal relies on linearization at the operation point. In contrast, the method proposed in this paper is a large-signal analysis method, which directly derives the stability boundary by mainly considering the dominant stable mode of the outer control loop (PLL control loop). Compared with the existing small-signal analysis methods, the method proposed in this paper is much more intuitive and accurate.