Damping Characteristic Analysis of Wind-thermal-bundled Systems

Shiying Ma; Liwen Zheng

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OUTLINE

Abstract

Wind-thermal-bundled system has emerged as the predominant type of power system, incorporating a significant proportion of renewable energy. The dynamic interaction mechanism of the system is complex, and the issue of oscillation stability is significant. In this paper, the damping characteristics of the direct current (DC) capacitance oscillation mode are analyzed using the path analysis method (PAM). This method combines the transfer-function block diagram with the damping torque analysis (DTA). Firstly, the linear models of the permanent magnet synchronous generator (PMSG), the synchronous generator (SG), and the alternating current (AC) grid are established based on the transfer functions. The closed-loop transfer-function block diagram of the wind-thermal-bundled systems is derived. Secondly, the block diagram reveals the damping path and the dynamic interaction mechanism of the system. According to the superposition principle, the transfer-function block diagram is reconstructed to achieve the damping separation. The damping coefficient of the DTA is used to quantify the effect of the interaction between the subsystems on the damping characteristics of the oscillation mode. Then, the eigenvalue analysis is used to analyze the system stability. Finally, the damping characteristic analysis is validated by time-domain simulations.

Keywords

Wind-thermal-bundled system; path analysis method (PAM); transfer-function block diagram; damping

I. INTRODUCTION

IN China, wind farms are mainly concentrated in the western, northern, and eastern coastal areas. The wind power and thermal power are transmitted to central load centers through high-capacity, long-distance transmission systems [

1]. In addition, the permanent magnet synchronous generator (PMSG) has gradually become the dominant model for wind farms due to its high efficiency and low failure rate [2]. Thus, the wind-thermal-bundled system is gradually emerging as one of the primary methods for wind energy supply, offering the advantages of low cost and high efficiency [3]. However, the grid-connected PMSG system consists of multiple power electronic devices and controllers, which can lead to oscillation problems [4]-[6]. It is of great scientific and engineering importance to investigate the damping characteristics of wind-thermal-bundled systems.

Currently, several methods are available to study system oscillations, including the eigenvalue method, impedance model analysis (IMA) method, open-loop mode resonance method, damping torque analysis (DTA) method, and time-domain simulation method [

7]-[9].

1) The eigenvalue method based on the small-signal model is used to analyze the variables and subsystems related to the oscillation modes. Using the eigenvalue method, it has been found that PMSGs connected to the power system could reduce the system stability under small disturbances [

10]. In [11], based on the participation factor (PF), it is found that PMSG does not contribute to the low-frequency oscillation (LFO) of the system, which contradicts the conclusion of [10].

2) The IMA method is a frequency domain method that explains the interaction mechanism between subsystems based on their external impedance characteristics. The effect of the PMSGs on the sub-synchronous torsional interactions (SSTI) of thermal power units is investigated by IMA, taking into account the field winding and damper winding of synchronous machines. It is shown that PMSGs can lead to insufficient damping of SSTI [

12].

3) The open-loop-mode resonance analysis method has investigated the mechanism of strong interaction by studying the open-loop-mode coupling. It is found that the simultaneous involvement of the PMSG and synchronous generator (SG) in oscillation modes is due to the proximity of the converter oscillation mode (COM) and the electromechanical oscillation mode (EOM) [

13]. In [14], [15], modal resonance occurs when the open-loop mode of the phase-locked loop (PLL) and the open-loop mode of the power system approach on the complex plane, which degrades the damping of the oscillation modes.

4) In the DTA method, the studied system is equated to be second-order due to the similarity between the system and the rotor motion equation of the SG. The damping torque coefficient is used to study the vibration characteristics of the system. In [

16], based on the rotor motion equation, the voltage source converter (VSC) model is presented to describe its internal voltage dynamics in the time scale of the direct current (DC)-link voltage control.

However, the eigenvalue analysis and open-loop mode resonance analysis methods suffer from the “dimensional disaster” problem and are unable to analyze the transmission path of frequency disturbances. IMA can explain the vibration mechanism at the physical level and can be used to model the “black (gray) box” system. However, it is difficult to uncover the dynamic interactions within the system from the external impedance characteristics of the device alone. The DTA method can only study the damping characteristics of the system itself and cannot separate the dynamics of the interactions between the various links in the system. The differential algebraic equation (DAE) is used to describe the system dynamics in time-domain simulation methods, but it cannot reveal the oscillation coupling mechanism. Time-domain simulation method can be used to verify the validity of the theoretical analysis. Given the limitations of the above methods, it is crucial to explore the dynamic interaction mechanism from a new perspective.

The DTA [

17]-[19], IMA [20], and open-loop-mode resonance [21], [22] methods have been used for the analysis of the DC capacitance-dominated oscillation modes in power systems. In [17]-[19], the dynamic DC capacitance is expressed as a second-order differential equation, and the DTA method is used to analyze the stability of the DC capacitance. The definitions of equivalent inertia, synchronization power, and damping power are also discussed. It is found that the insufficient positive synchronization and damping power can lead to the instability. In [23]-[25], small-signal models of the PMSG and the doubly-fed induction generator (DFIG) are established to study the stability of the DC-link voltage. However, the dynamic interaction mechanism between the subsystems has not been revealed, and the effect of various interactions on the damping of oscillation modes cannot be quantified. To address this issue, the stability of the DC capacitance oscillation mode and the PLL oscillation mode have been investigated using the transfer-function block diagram and the damping separation method [26]-[30]. It has been found that the DC capacitance-dominated sub-synchronous oscillation (SSO) is more prone to instability when the PMSG is connected to a weak alternating current (AC) grid or a line-commutated converter-based high-voltage direct current (LCC-HVDC) system [26]-[28]. However, the external AC grid is equivalent to a voltage source, and the dynamics of the SG are not involved [26]-[30]. It is essential to develop the SG model to analyze the damping characteristics of wind-thermal-bundled systems.

In this paper, the path analysis method (PAM) is used to analyze the stability of the DC capacitance oscillation mode by combining the transfer-function block diagram and DTA. Compared with other methods, PAM can elucidate the path of disturbance transmission and the coupling relationships between the subsystems. The effect of the interaction between the subsystems on the damping of the oscillation modes is quantified by the damping reconstruction. The contributions of this paper include the following aspects.

1) The linear models (LMs) of the PMSG, SG, and AC grid are established using the transfer function equations, and the closed-loop transfer-function block diagram of the wind-thermal-bundled system is derived.

2) The closed-loop transfer-function block diagram shows the damping path and the dynamic interaction process of the subsystem. The block diagram is reconstructed to achieve damping separation based on the superposition principle. The DTA method is used to quantify the effect of dynamic interaction between subsystems on the DC capacitance oscillation mode.

3) The effect of system parameters on the damping of the DC capacitance oscillation mode is analyzed using the damping separation method and the eigenvalue method, and verified by time-domain simulation.

The remainder of this paper is organized as follows. In Section II, the LM for the wind-thermal-bundled system is established, and the transfer-function block diagram is obtained. Sections III and IV conduct damping characteristic analyses based on PAM and eigenvalue analysis method, respectively. Section III shows the coupling relationships between the subsystems based on the block diagram, which is constructed to achieve damping separation. In Section IV, the state-space model of the wind-thermal-bundled system is established, and the oscillation mode of the system is analyzed with varying parameters. In Section V, the effect of the system parameters on the damping of the DC capacitance oscillation mode is confirmed by simulation results. Conclusions are given in Section VI.

II. LM for WIND-THERMAL-BUNDLED SYSTEM

In this section, the transfer function equations are used to establish the LMs of the PMSG, SG, and AC grid. Then, the transfer-function block diagram of the wind-thermal-bundled system is derived, and its validity is verified by the step response characteristics of the electromagnetic transient model (ETM). The diagram of the wind-thermal-bundled system is shown in Fig. 1.

Fig. 1 Diagram of wind-thermal-bundled system.

In the following, the voltage u_r is used as an example, and $u_{r} = [u_{r x}; u_{r y}]$ , $u_{r d q} = [u_{r d}; u_{r q}]$ , $u_{r d 2 q 2} = [u_{r d 2}; u_{r q 2}]$ .

A. LM of PMSG

The PMSG subsystem consists of the DC capacitance, the GSC, the PLL, the filter inductance, and the grid-connected transmission lines. The control diagram of the GSC is shown in Fig. 2.

Fig. 2 Control diagram of GSC.

In Fig. 2, $P I_{d c} (s)$ and $P I_{c c} (s)$ are the DC voltage outer loop control and the current inner loop control, respectively.

The dynamics of the constant DC voltage outer loop control for the GSC are given as:

\{\begin{array}{l} Δ i_{g d r e f} = P I_{d c} (s) Δ U_{d c} \\ Δ i_{g q r e f} = 0 \end{array}

(1)

where $P I_{d c} (s) = K_{p d c} + K_{i d c} / s$ .

The bandwidth of the DC voltage outer loop is designed to be one-tenth of the bandwidth of the current inner loop. Therefore, the grid-side current can follow its reference value based on the current inner loop. It can be expressed as $Δ i_{g d} = Δ i_{g d r e f}$ and $Δ i_{g q} = Δ i_{g q r e f}$ . As the power factor of the GSC is 1, $Δ i_{g q r e f}$ is set to be 0. Then, both $Δ i_{g q}$ and i_gq are equal to 0. So, the output active power of the GSC is:

Δ P_{e} = 1.5 (u_{t d 0} Δ i_{g d} + i_{g d 0} Δ u_{t d}) = G_{1} [\begin{matrix} Δ i_{g d} \\ Δ u_{t d} \end{matrix}]

(2)

where G₁ is the transfer function matrix from $Δ i_{g d}$ and $Δ u_{t d}$ to $Δ P_{e}$ . The expression for G₁ is given in (A1) in Appendix A.

The dynamic equation for the DC capacitance is:

\{\begin{array}{l} Δ U_{d c} = G_{d c} (s) (Δ P_{i n} - Δ P_{e}) \\ G_{d c} (s) = \frac{1}{s C_{d c} U_{d c 0}} \end{array}

(3)

The dynamic equation for the PLL is:

Δ θ_{P L L} = \frac{k_{i P L L} + s k_{p P L L}}{s^{2}} Δ u_{g q} = H_{P L L} (s) Δ u_{g q}

(4)

The filter line dynamics are:

[\begin{matrix} Δ u_{t d} \\ Δ u_{t q} \end{matrix}] = [\begin{matrix} Δ u_{g d} \\ Δ u_{g q} \end{matrix}] + [\begin{matrix} R_{g} + s L_{g} & - ω_{g} L_{g} \\ ω_{g} L_{g} & R_{g} + s L_{g} \end{matrix}] [\begin{matrix} Δ i_{g d} \\ Δ i_{g q} \end{matrix}]

(5)

In (5), $Δ u_{t d}$ can be written as:

Δ u_{t d} = G_{2} (s) [\begin{matrix} Δ i_{g d} \\ Δ u_{g d} \end{matrix}]

(6)

where $G_{2} (s)$ is the transfer function matrix from $Δ i_{g d}$ and $Δ u_{g d}$ to $Δ u_{t d}$ . The expression for $G_{2} (s)$ is given in (A2) in Appendix A.

The dynamic equations for the capacitance and impedance of the lines connected with the network are given as:

[\begin{matrix} Δ u_{g x} \\ Δ u_{g y} \end{matrix}] = G_{3} (s) [\begin{matrix} Δ i_{g x} \\ Δ i_{g y} \end{matrix}] + G_{4} (s) [\begin{matrix} Δ i_{1 x} \\ Δ i_{1 y} \end{matrix}]

(7)

[\begin{matrix} Δ i_{1 x} \\ Δ i_{1 y} \end{matrix}] = G_{5} (s) [\begin{matrix} Δ u_{g x} \\ Δ u_{g y} \end{matrix}] + G_{6} (s) [\begin{matrix} Δ u_{r x} \\ Δ u_{r y} \end{matrix}]

(8)

where $G_{3} (s)$ and $G_{4} (s)$ are the transfer function matrices from $Δ i_{g}$ and $Δ i_{1}$ to $Δ u_{g}$ , respectively; $G_{5} (s)$ and $G_{6} (s)$ are the transfer function matrices from $Δ u_{g}$ and $Δ u_{r}$ to $Δ i_{1}$ , respectively; and the expressions for $G_{3} (s)$ - $G_{6} (s)$ are given in (A3) and (A4) in Appendix A.

For u_g and i_g, the transformation equations between the x-y frame and the d-q frame are:

[\begin{matrix} Δ u_{g d} \\ Δ u_{g q} \end{matrix}] = T_{u g} [\begin{matrix} Δ u_{g x} \\ Δ u_{g y} \\ Δ θ_{P L L} \end{matrix}]

(9)

[\begin{matrix} Δ i_{g x} \\ Δ i_{g y} \end{matrix}] = T_{i g} [\begin{matrix} Δ i_{g d} \\ Δ i_{g q} \\ Δ θ_{P L L} \end{matrix}]

(10)

where T_ug is the transfer function matrix from $Δ u_{g}$ and $Δ θ_{P L L}$ to $Δ u_{g d q}$ ; and T_ig is the transfer function matrix from $Δ i_{g d q}$ and $Δ θ_{P L L}$ to $Δ i_{g}$ . The detailed expressions of T_ug and T_ig are given in (A5) and (A6) in Appendix A.

Combining (7), (8), and (10) yields $Δ i_{1}$ :

[\begin{matrix} Δ i_{1 x} \\ Δ i_{1 y} \end{matrix}] = G_{7} (s) [\begin{matrix} Δ i_{g d} \\ Δ θ_{P L L} \\ Δ u_{r x} \\ Δ u_{r y} \end{matrix}]

(11)

where $G_{7} (s)$ is the transfer function matrix from $Δ i_{g d}$ , $Δ θ_{P L L}$ , $Δ u_{r}$ to $Δ i_{1}$ . As $G_{7} (s)$ is complicated, it will not be listed in this paper.

Combining (7), (9), and (10), we can obtain $Δ u_{g d q}$ as:

[\begin{matrix} Δ u_{g d} \\ Δ u_{g q} \end{matrix}] = G_{8} (s) [\begin{matrix} Δ i_{g d} \\ Δ θ_{P L L} \\ Δ i_{1 x} \\ Δ i_{1 y} \end{matrix}]

(12)

where $G_{8} (s)$ is the transfer function matrix from $Δ i_{g d}$ , $Δ θ_{P L L}$ , $Δ i_{1}$ to $Δ u_{g d q}$ .

Based on (1)-(4), (6), (11), and (12), the LM of the PMSG is shown in Fig. 3. The input variables are $Δ u_{r x}$ , $Δ u_{r y}$ , and $Δ P_{i n}$ , while the output variables are $Δ i_{l x}$ and $Δ i_{l y}$ .

Fig. 3 LM of PMSG.

B. LM of SG

The voltage equation for SG is given as:

\{\begin{array}{l} Δ u_{o d_{2}} = - R Δ i_{o d_{2}} - X_{q}^{'} Δ i_{o q_{2}} \\ Δ u_{o q_{2}} = Δ E_{q}^{'} - R Δ i_{o q_{2}} + X_{d}^{'} Δ i_{o d_{2}} \end{array}

(13)

The third-order dynamic model of SG is given as:

\{\begin{array}{l} s Δ δ = ω_{0} Δ ω \\ M s Δ ω = Δ P_{m} - Δ P_{t} - D Δ ω \\ T_{d 0}^{'} s Δ E_{q}^{'} = Δ E_{f d} - Δ E_{q}^{'} + (X_{d} - X_{d}^{'}) Δ i_{o d_{2}} \end{array}

(14)

Considering the governor and turbine of SG, the relationship between the $Δ ω$ and $Δ P_{m}$ is:

Δ P_{m} = - \frac{F_{H P} K_{p} K_{δ}}{(1 + T_{1} s) (1 + T_{C H} s)} Δ ω = - G_{ω - P_{m}} (s) Δ ω

(15)

In the d₂-q₂ frame, the dynamic equation of the SG grid-connected transmission line is:

[\begin{matrix} Δ u_{o d_{2}} \\ Δ u_{o q_{2}} \end{matrix}] = [\begin{matrix} Δ u_{r d_{2}} \\ Δ u_{r q_{2}} \end{matrix}] + [\begin{matrix} R_{o} + s L_{o} & - ω_{g} L_{o} \\ ω_{g} L_{o} & R_{o} + s L_{o} \end{matrix}] [\begin{matrix} Δ i_{o d_{2}} \\ Δ i_{o q_{2}} \end{matrix}]

(16)

Combining (13) and (16), we can obtain:

[\begin{matrix} Δ i_{o d_{2}} \\ Δ i_{o q_{2}} \end{matrix}] = {[\begin{matrix} - R - R_{o} - s L_{o} & - X_{q}^{'} + ω_{g} L_{o} \\ X_{d}^{'} - ω_{g} L_{o} & - R - R_{o} - s L_{o} \end{matrix}]}^{- 1} [\begin{matrix} Δ u_{r d_{2}} \\ Δ u_{r q_{2}} - Δ E_{q}^{'} \end{matrix}]

(17)

Substituting (13) and (17) into the third formula of (14), we can obtain:

Δ E_{q}^{'} = G_{9} (s) [\begin{matrix} Δ u_{r d_{2}} \\ Δ u_{r q_{2}} \end{matrix}]

(18)

where $G_{9} (s)$ is the transfer function matrix from $Δ u_{r d_{2} q_{2}}$ to $Δ E_{q}^{'}$ , and the detailed expression of $G_{9} (s)$ is given in (A7) in Appendix A.

By substituting (18) into (16), $Δ i_{o d_{2} q_{2}}$ can be written as:

[\begin{matrix} Δ i_{o d_{2}} \\ Δ i_{o q_{2}} \end{matrix}] = G_{10} (s) [\begin{matrix} Δ u_{r d_{2}} \\ Δ u_{r q_{2}} \end{matrix}]

(19)

where $G_{10} (s)$ is the transfer function from $Δ u_{r d_{2} q_{2}}$ to $Δ i_{o d_{2} q_{2}}$ , and the detailed expression of $G_{10} (s)$ is given in (A8) in Appendix A.

The active power of SG is:

Δ P_{t} = u_{o d_{2} 0} Δ i_{o d_{2}} + i_{o d_{2} 0} Δ u_{o d_{2}} + u_{o q_{2} 0} Δ i_{o q_{2}} + i_{o q_{2} 0} Δ u_{o q_{2}}

(20)

By substituting (16) and (19) into (20), $Δ P_{t}$ is obtained as:

Δ P_{t} = G_{11} (s) [\begin{matrix} Δ u_{r d_{2}} \\ Δ u_{r q_{2}} \end{matrix}]

(21)

where $G_{11} (s)$ is the transfer function matrix from $Δ u_{r d_{2} q_{2}}$ to $Δ P_{t}$ .

For $Δ u_{r}$ and $Δ i_{o}$ , the conversion relation between the d₂-q₂ frame and the x-y frame is given as:

[\begin{matrix} Δ u_{r d_{2}} \\ Δ u_{r q_{2}} \end{matrix}] = G_{12} [\begin{matrix} Δ u_{r x} \\ Δ u_{r y} \\ Δ δ \end{matrix}]

(22)

[\begin{matrix} Δ i_{o x} \\ Δ i_{o y} \end{matrix}] = G_{13} [\begin{matrix} Δ i_{o d_{2}} \\ Δ i_{o q_{2}} \\ Δ δ \end{matrix}]

(23)

where G₁₂ is the transfer function matrices from $Δ u_{r}$ and $Δ δ$ to $Δ u_{r d_{2} q_{2}}$ ; and G₁₃ is the transfer function matrices from $Δ i_{o d_{2} q_{2}}$ and $Δ δ$ to $Δ i_{o}$ . The detailed expressions of G₁₂ and G₁₃ are given in (A9) and (A10) in Appendix A, respectively.

Based on (14), (19), (21)-(23), the LM of SG is shown in Fig. 4. The input variables are $Δ u_{r x}$ and $Δ u_{r y}$ , while the output variables are $Δ i_{o x}$ and $Δ i_{o y}$ .

Fig. 4 LM of SG.

C. LM of AC Grid

The dynamics of the AC grid are given as:

[\begin{matrix} Δ i_{2 x} \\ Δ i_{2 y} \end{matrix}] = G_{14} (s) [\begin{matrix} Δ u_{r x} \\ Δ u_{r y} \end{matrix}] + G_{15} (s) [\begin{matrix} Δ u_{2 x} \\ Δ u_{2 y} \end{matrix}]

(24)

where $G_{14} (s)$ and $G_{15} (s)$ are the transfer function matrices from $Δ u_{r}$ and $Δ u_{2}$ to $Δ i_{2}$ , respectively. The expressions of $G_{14} (s)$ and $G_{15} (s)$ are given in (A11) in Appendix A.

Omitting the change of $Δ u_{2}$ , (24) can be rewritten as:

[\begin{matrix} Δ i_{2 x} \\ Δ i_{2 y} \end{matrix}] = G_{14} (s) [\begin{matrix} Δ u_{r x} \\ Δ u_{r y} \end{matrix}]

(25)

According to Fig. 1, (26) is obtained as:

[\begin{matrix} Δ i_{2 x} \\ Δ i_{2 y} \end{matrix}] = [\begin{matrix} Δ i_{1 x} \\ Δ i_{1 y} \end{matrix}] + [\begin{matrix} Δ i_{o x} \\ Δ i_{o y} \end{matrix}]

(26)

The combination of (25) and (26) gives:

[\begin{matrix} Δ u_{r x} \\ Δ u_{r y} \end{matrix}] = G_{16} (s) ([\begin{matrix} Δ i_{1 x} \\ Δ i_{1 y} \end{matrix}] + [\begin{matrix} Δ i_{o x} \\ Δ i_{o y} \end{matrix}])

(27)

where $G_{16} (s)$ is the transfer function matrix from $Δ i_{1}$ and $Δ i_{o}$ to $Δ u_{r}$ . The detailed expression of $G_{16} (s)$ is given in (A12) in Appendix A.

Based on (27), the aggregated LM of the AC grid is shown in Fig. 5. The input variables are $Δ i_{o}$ and $Δ i_{1}$ , while the output variables are $Δ u_{r}$ .

Fig. 5 Aggregated LM of AC grid.

D. Closed-loop Transfer Function Model

By combining (2), (3), (5), (11), and (12) to eliminate the intermediate variables $Δ i_{1}$ , $Δ θ_{P L L}$ , $Δ i_{g}$ , and $Δ u_{t}$ , the output power of the PMSG is calculated as:

Δ P_{e} = G_{p 1} (s) Δ U_{d c} + G_{p 2} (s) [\begin{matrix} Δ u_{r x} \\ Δ u_{r y} \end{matrix}]

(28)

where $G_{p 1} (s)$ is the transfer function from $Δ U_{d c}$ to $Δ P_{e}$ ; and $G_{p 2} (s)$ is the transfer function matrix from $Δ u_{r}$ to $Δ P_{e}$ .

By combining (1), (4), (11), and (12) to eliminate the intermediate variables $Δ θ_{P L L}$ and $Δ i_{g}$ , $Δ i_{1}$ can be rewritten as:

[\begin{matrix} Δ i_{1 x} \\ Δ i_{1 y} \end{matrix}] = G_{i 1} (s) Δ U_{d c} + G_{i 2} (s) [\begin{matrix} Δ u_{r x} \\ Δ u_{r y} \end{matrix}]

(29)

where $G_{i 1} (s)$ and $G_{i 2} (s)$ are the transfer function matrices from $Δ U_{d c}$ and $Δ u_{r}$ to $Δ i_{1}$ , respectively.

Based on (3), (28), and (29), the aggregated LM of the PMSG is shown in Fig. 6. The input variables are $Δ u_{r}$ and $Δ P_{i n}$ , while the output variables are $Δ i_{1}$ .

Fig. 6 Aggregated LM of PMSG.

By substituting (22) into (21), the active power of SG is obtained as:

Δ P_{t} = G_{p 3} (s) Δ δ + G_{p 4} (s) [\begin{matrix} Δ u_{r x} \\ Δ u_{r y} \end{matrix}]

(30)

where $G_{p 3} (s)$ is the transfer function from $Δ δ$ to $Δ P_{t}$ ; and $G_{p 4} (s)$ is the transfer function matrix from $Δ u_{r}$ to $Δ P_{t}$ . The detailed expressions of $G_{p 3} (s)$ and $G_{p 4} (s)$ are given in (A13) in Appendix A.

Combining (19), (22), and (23), we can obtain:

[\begin{matrix} Δ i_{o x} \\ Δ i_{o y} \end{matrix}] = G_{i 3} (s) Δ δ + G_{i 4} (s) [\begin{matrix} Δ u_{r x} \\ Δ u_{r y} \end{matrix}]

(31)

where $G_{i 3} (s)$ and $G_{i 4} (s)$ are the transfer function matrices from $Δ δ$ and $Δ u_{r}$ to $Δ i_{o}$ , respectively. The expressions of $G_{i 3} (s)$ and $G_{i 4} (s)$ are given in (A14) and (A15) in Appendix A, respectively.

Substituting (15) into (14), we obtain the transfer function from $Δ P_{t}$ to $Δ δ$ :

G_{P - δ} (s) = \frac{1}{M s^{2} + D s + s G_{ω - P_{m}} (s)}

(32)

Based on (14) and (30)-(32), the aggregated LM of SG is shown in Fig. 7. The input variables are $Δ u_{r}$ , while the output variables are $Δ i_{o}$ .

Fig. 7 Aggregated LM of SG.

According to (3), (14), and (27)-(32), the transfer-function block diagram of the wind-thermal-bundled system is shown in Fig. 8.

Fig. 8 Transfer-function block diagram of wind-thermal-bundled system.

From the above derivation, it is clear that the transfer-function block diagram of the system before and after the polymerization corresponds to each other, and the dynamic interaction analysis can be performed based on the transfer-function block diagram.

E. Validation of LM

Before analyzing the dynamic interaction mechanism, the accuracy of the LM in MATLAB should be validated using the ETM in DIgSILENT/PowerFactory. The reference DC voltage U_dcref has a step change from 1.1 p.u. to 1.05 p.u. at $t = 1$ s, and the response curves of the system under LM and ETM are shown in Fig. 9. In Fig. 9, it is shown that the LM response is consistent with the ETM response, which verifies the correctness of the LM.

Fig. 9 Response curves following a change of U_dcref from 1.1 p.u. to 1.05 p.u.. (a) P_t of SG. (b) I_o of SG. (c) P_e of GSC. (d) U_dc of PMSG.

III. DAMPING CHARACTERISTIC ANALYSIS BASED ON PAM

In this section, the analysis of the damping path is presented based on the transfer-function block diagram. The block diagram is reconstructed to isolate the interaction damping. The factors influencing the damping characteristics are studied based on the damping reconstruction.

A. Damping Path Analysis

The block diagram of the damping path analysis, as shown in Fig. 10, clearly illustrates the relationship between each state variable and the corresponding link. This is beneficial for separating the disturbance transfer paths within the system. In the transfer function block diagram, the closed loop passing through the transfer function $G_{d c} (s)$ is defined as the SSO damping path. In Fig. 10, there are two damping paths which are represented by red and purple solid ovals, respectively, and the arrows indicate the direction of the disturbance transfer.

Fig. 10 Block diagram of damping path analysis.

Damping path 1 can be described as the path through which the disturbance is transmitted from $Δ U_{d c}$ to $Δ P_{e}$ , assuming that $Δ P_{i n}$ remains constant.

Damping path 1 consists of the transfer functions $G_{d c} (s)$ and $G_{p 1} (s)$ . This indicates that path 1 is related to the DC capacitance of the PMSG and the outer loop control of the GSC. Damping path 1 is referred to as the internal oscillation transfer path of the PMSG.

Damping path 2 can be described as follows. When $Δ P_{e}$ changes, the disturbance is transmitted from $Δ U_{d c}$ to $Δ i_{l}$ , then from $Δ i_{l}$ to $Δ u_{r}$ , and finally from $Δ u_{r}$ to $Δ P_{e}$ . Damping path 2 passes through three subsystems: PMSG, SG, and AC grid, and includes two closed-loop paths denoted as a and b, which are indicated by oval dashed lines. The closed-loop path a passes through $Δ i_{l}$ and $Δ u_{r}$ , and consists of the transfer function matrices $G_{i 2} (s)$ and $G_{16} (s)$ , which represent the interaction between the PMSG and the AC grid. The closed loop path b passes through $Δ i_{o}$ and $Δ u_{r}$ and consists of the transfer function matrices $G_{i 4} (s)$ and $G_{16} (s)$ , reflecting the interaction between the AC system and the SG. The closed loops a and b interact at the PCC voltage $Δ u_{r}$ to form a closed crossed loop. When the voltage $Δ u_{r}$ is disturbed, the currents $Δ i_{l}$ and $Δ i_{o}$ change due to the existence of closed loops a and b, which in turn affect the dynamics of $Δ u_{r}$ . Thus, damping path 2 is referred to as the oscillation coupling path among the PMSG, SG, and AC grid.

Furthermore, the interaction processes can be divided into two groups: the controller interaction in the PMSG, and the interaction among the PMSG, SG, and AC grid. The controller interaction in the PMSG can be understood as the dynamic process of controller interaction in the PMSG system, which is caused by disturbances through a closed loop (damping path 1). The interaction between the PMSG, the SG, and the AC grid can be understood as the dynamic interplay between the subsystems triggered by voltage and current disturbances at the PCC driven by the closed loop (damping path 2).

B. Damping Separation Method

According to the superposition principle of linear systems, the transfer function block diagram is reconstructed, and the damping provided by the dynamic interaction is quantified. The damping separation procedure is as follows.

Step 1: based on (14) and (30), we can obtain (33) and Fig. 11(a).

Fig. 11 Damping reconstruction process. (a) Step 1. (b) Step 2. (c) Step 3. (d) Step 4.

Δ δ = G_{m} (s) [\begin{matrix} Δ u_{r x} \\ Δ u_{r y} \end{matrix}]

(33)

where $G_{m} (s)$ is the transfer function matrix from $Δ u_{r}$ to $Δ δ$ . The expression of $G_{m} (s)$ is given in (A16) in Appendix A.

Step 2: combining (28) and (21), we obtain (34) and Fig. 11(b).

[\begin{array}{l} Δ i_{o x} \\ Δ i_{o y} \end{array}] = G_{n} (s) [\begin{matrix} Δ u_{r x} \\ Δ u_{r y} \end{matrix}]

(34)

where $G_{n} (s)$ is the transfer function matrix from $Δ u_{r}$ to $Δ i_{o}$ . The expression of $G_{n} (s)$ is given in (A17) in Appendix A.

Step 3: combining (27), (29), and (34), we obtain (35) and Fig. 11(c).

[\begin{matrix} Δ u_{r x} \\ Δ u_{r y} \end{matrix}] = G_{o} (s) Δ U_{d c}

(35)

where $G_{o} (s)$ is the transfer function matrix from $Δ U_{d c}$ to $Δ u_{r}$ .

The open-loop transfer function matrix $G_{o} (s)$ represents the interaction between the PMSG and the SG and the interaction between the PMSG and the AC system. If $Δ i_{o} = 0$ , i.e., the SG is not connected to the system and the interaction between the PMSG and the SG is not considered, the block diagram is simplified. The corresponding open loop transfer function matrix $G_{o 1} (s)$ is obtained by substituting (29) into (27). $G_{o 1} (s)$ represents the interaction between the PMSG and the AC grid. According to the superposition principle of linear systems, $G_{o} (s)$ is subtracted from $G_{o 1} (s)$ to obtain $G_{o 2} (s)$ , where $G_{o 2} (s)$ represents the interaction between the PMSG and the SG. The expressions for $G_{o} (s)$ , $G_{o 1} (s)$ , and $G_{o 2} (s)$ are given in (36), and the transfer function block diagram is shown in Fig. 11(c).

\{\begin{array}{l} G_{o} (s) = {(I - G_{16} (s) (G_{i 2} (s) + G_{n} (s)))}^{- 1} G_{16} (s) G_{i 1} (s) \\ G_{o 1} (s) = {(I - G_{16} (s) G_{i 2} (s))}^{- 1} G_{16} (s) G_{i 1} (s) \\ G_{o 2} (s) = G_{o} (s) - G_{o 1} (s) \end{array}

(36)

Step 4: when $Δ P_{i n}$ is selected as the input variable and $Δ$ U_dc is selected as the output variable, three transfer functions from $Δ$ U_dc to the variables $Δ$ P_e- $Δ$ P_in and $Δ$ P_e are obtained using the damping separation method, as shown in Fig. 11(d) and (37).

\{\begin{array}{l} G_{z 1} (s) = G_{p 1} (s) \\ G_{z 21} (s) = G_{p 2} (s) G_{o 1} (s) \\ G_{z 22} (s) = G_{p 2} (s) G_{o 2} (s) \end{array}

(37)

where $G_{z 1} (s)$ is the internal damping of the PMSG; $G_{z 21} (s)$ is the damping of the DC capacitance oscillation influenced by the interaction between the PMSG and the AC grid; and $G_{z 22} (s)$ is the damping influenced by the interaction between the PMSG and the SG system.

The corresponding damping coefficient can be calculated from:

\{\begin{array}{l} Z_{1} = I m (j ω_{d} G_{z 1} (j ω_{d})) / ω_{d} \\ Z_{21} = I m (j ω_{d} G_{z 21} (j ω_{d})) / ω_{d} \\ Z_{22} = I m (j ω_{d} G_{z 22} (j ω_{d})) / ω_{d} \\ Z_{2} = Z_{21} + Z_{22} \end{array}

(38)

where Z₁ is the PMSG self-damping coefficient; Z₂₁ is the PMSG-grid interaction damping coefficient; Z₂₂ is the PMSG-SG interaction damping coefficient; and Z₂ is the interaction damping coefficient. The total damping coefficient of the system is $Z = Z_{1} + Z_{2}$ .

C. Analysis of Influence Factors

1) Proportional Coefficient of DC Voltage Outer Loop of GSC

To study the influence of the proportional coefficient of DC voltage outer loop of GSC control K_pdc on the damping coefficient, Fig. 12(a) shows the frequency characteristic curves of damping coefficients Z₂₁ and Z₂₂ at different K_pdc. Figure 12(b) shows the frequency characteristic curves of self-damping coefficient Z₁, interaction damping coefficient Z₂, and total damping coefficient Z at different K_pdc. The values of K_pdc are 12, 18, and 25, respectively. The arrow indicates the change direction of the frequency characteristic curves with increasing K_pdc.

Fig. 12 Damping characteristic analysis with change of K_pdc. (a) Frequency characteristic curves of Z₂₁ and Z₂₂. (b) Frequency characteristic curves of Z₁, Z₂, and Z.

As shown in Fig.12(a), as K_pdc increases, the interaction damping between the PMSG and the SG increases, and the interaction damping between the PMSG and the AC grid decreases. As shown in Fig.12(b), as K_pdc increases, the frequency characteristic curves of Z₁, Z₂, and Z move upward, indicating that the increase in K_pdc causes the self-damping coefficient Z₁, the interaction damping coefficient Z₂, and the total damping coefficient Z to increase.

2) Integral Coefficient of DC Voltage Outer Loop of GSC

To study the influence of the integral coefficient of DC voltage outer loop of GSC control K_idc on the damping coefficient, Fig. 13(a) shows the frequency characteristic curves of damping coefficients Z₂₁ and Z₂₂ at different K_idc. Figure 13(b) shows the frequency characteristic curves of self-damping coefficient Z₁, interaction damping coefficient Z₂, and total damping coefficient Z at different K_idc. The values of K_id_c are 133, 233, and 333, respectively. The arrow indicates the change direction of the frequency characteristic curves as K_idc increases.

Fig. 13 Damping characteristic analysis with change of K_idc. (a) Frequency characteristic curves of Z₂₁and Z₂₂. (b) Frequency characteristic curves of Z₁, Z₂, and Z.

As shown in Fig. 13(a), as K_idcincreases, the interaction damping coefficient between the PMSG and the SG increases, and the interaction damping coefficient between the PMSG and the AC grid decreases. As shown in Fig. 13(b), as K_idc increases, the frequency characteristic curve of Z₁ moves upward, indicating that the positive damping effect provided by the PMSG increases. The frequency characteristic curve of Z₂ moves downward, indicating that the interaction damping decreases. The total damping coefficient Z decreases and the system stability is weakened.

3) Grid Strength

To study the influence of short-circuit ratio (SCR) on the damping coefficient, Fig. 14(a) shows the frequency characteristic curves of damping coefficients Z₂₁ and Z₂₂ under different SCRs. Figure 14(b) shows the frequency characteristic curves of self-damping coefficient Z₁, interaction damping coefficient Z₂, and total damping coefficient Z under different SCR. The values of SCR are 1, 2, and 3, respectively. The arrow indicates the change direction of the frequency characteristic curves as the SCR increases.

Fig. 14 Damping characteristic analysis with change of SCR. (a) Frequency characteristic curves of Z₂₁and Z₂₂. (b) Frequency characteristic curves of Z₁, Z₂, and Z.

As shown in Fig. 14(a), as the SCR increases, the interaction damping between the PMSG and the SG decreases, and the interaction damping between the PMSG and the AC power grid increases. As shown in Fig. 14(b), the frequency characteristic curve of Z₁ remains unchanged as the SCR increases, indicating that the positive damping effect provided by the PMSG is independent of the power system strength. The frequency characteristic curves of Z₂ and Z move upward, indicating that the interaction damping and the total damping increase.

4) Time Constant of Hydraulic Motor T₁

To study the influence of the time constant of the hydraulic motor T₁ on the damping coefficient, Fig. 15(a) shows the frequency characteristic curves of damping coefficients Z₂₁ and Z₂₂ under different T₁. Figure 15(b) shows the frequency characteristic curves of self-damping coefficient Z₁, interaction damping coefficient Z₂, and total damping coefficient Z under different T₁. The values of T₁ are 0.5, 2, and 5, respectively. The arrow indicates the change direction of the frequency characteristic curves as T₁ increases.

Fig. 15 Damping characteristic analysis with change of T₁. (a) Frequency characteristic curves of Z₂₁and Z₂₂. (b) Frequency characteristic curves of Z₁, Z₂, and Z.

As shown in Fig. 15(a), as T₁ increases, the interaction damping between the PMSG and the SG decreases, and the interaction damping between the PMSG and the AC grid remains unchanged. As shown in Fig. 15(b), the frequency characteristic curve of Z₁ remains unchanged as T₁ increases. The frequency characteristic curves of Z₂ and Z move downward, indicating that the interaction damping and the total damping decrease.

IV. DAMPING CHARACTERISTIC ANALYSIS BASED ON EIGENVALUE ANALYSIS METHOD

A. Eigenvalue Analysis

The small-signal models for PMSG and SG have been extensively studied, e.g., [

31]-[33], and will not be discussed here. The state space model for the system is given as:

\frac{d Δ X}{d t} = A Δ X + B Δ u

(39)

Based on (37) and the parameters in Appendix B Table BI, the primary oscillation modes of the wind-thermal-bundled system are shown in Table I.

TABLE I OSCILLATION MODES OF WIND-THERMAL-BUNDLED SYSTEM

Oscillation mode	Eigenvalue	Oscillation frequency (Hz)	Damping ratio
λ_1,2	$- 3.35 \pm j 15.870$	2.53	0.2063
λ_3,4	$20.77 \pm j 53.460$	8.51	-0.3621
λ_5,6	$- 10.50 \pm j 90.670$	14.43	0.1150
λ_7,8	$- 12.59 \pm j 321.130$	51.11	0.0392
λ_9,10	$- 22.53 \pm j 356.598$	56.75	0.0631
λ_11,12	$- 66.48 \pm j 1799.400$	286.39	0.0369

As shown in Table I, the system has six oscillation modes: one LFO mode $λ$ _1,2, two SSO modes $λ$ _3,4, $λ$ _5,6, and three medium-high frequency oscillation modes $λ$ _7,8, $λ$ _9,10, and $λ$ _11,12. The real parts of modes $λ$ _1,2, $λ$ _5,6, $λ$ _7,8, $λ$ _9,10, and $λ$ _11,12 are negative, i.e., these modes are stable oscillation modes. The real part of the mode $λ$ _3,4 is positive, i.e., mode 5 is an unstable SSO mode.

The normalized PFs of the oscillation modes have been calculated, and it has been shown that the oscillation mode $λ$ _3,4 is associated with the DC voltage control of the GSC. The effect of the system parameters on the mode $λ$ _3,4 is studied.

B. Influence of Parameters on Oscillation Mode

1) Proportional Coefficient of DC Voltage Outer Loop of GSC

Figure 16 shows the variation of the oscillation mode $λ_{3,4}$ and the damping ratio as K_pdc changes from 4 to 16. In Fig. 16, the arrows indicate the variation of $λ$ _3,4 and damping ratio as K_pdcincreases.

Fig. 16 Variation of oscillation mode λ_3,4 and damping ratio with different K_pdc. (a) Oscillation mode. (b) Damping ratio.

As shown in Fig. 16, the increase in K_pdc causes the oscillation mode $λ$ _3,4 to shift to the left, resulting in an increase in the damping ratio and indicating improved stability.

2) Integral Coefficient of DC Voltage Outer Loop of GSC

Figure 17 shows the variation of the oscillation mode $λ$ _3,4 and the damping ratio as K_idc changes from 40 to 120. In Fig. 17, the arrows indicate the variation of $λ$ _3,4 and damping ratio as K_idc increases.

Fig. 17 Variation of oscillation mode λ_3,4 and damping ratio with different K_idc. (a) Oscillation mode. (b) Damping ratio.

Figure 17 shows that as K_idc increases, $λ_{3,4}$ moves to the right and the damping ratio decreases, indicating a gradual destabilization of the system.

V. SIMULATION RESULTS

In this section, the wind-thermal-bundled system is constructed in DIgSILENT. At $t = 1$ s, K_idc is changed from 133 to 1000 to excite SSO. Other parameters are in accordance with Appendix B Table BI. The DC capacitance voltage curves with different parameters are shown to validate the above analysis. When the system is disturbed, the DC voltage curves with different K_pdc, K_idc, SCR, and T₁ are shown in Fig. 18.

Fig. 18 DC voltage response curves with different parameters. (a) K_pdc. (b) K_idc. (c) SCR. (d) T₁.

In Fig. 18, it can be observed that the DC voltage fluctuation decreases as K_pdc or SCRincreases. The DC voltage fluctuation increases as K_idc or T₁increases. The results of the damping characteristics analysis are validated by time-domain simulations.

VI. CONCLUSION

The damping characteristics of the wind-thermal-bundled system are analyzed using the PAM. The conclusions are given as follows.

1) The LMs for the PMSG, SG, and AC grid are established, respectively, and the LMs of the subsystems are connected to construct the transfer-function block diagram of the system.

2) The damping path analysis is presented based on the transfer-function block diagram. The interaction mechanism between the subsystems can be interpreted as a dynamic process driven by voltage disturbances and current disturbances at the PCC.

3) Based on the damping separation, the total damping of the DC capacitance-dominated oscillation mode can be divided into three parts: PMSG internal damping coefficient, PMSG-SG interaction damping coefficient, and PMSG-grid interaction damping coefficient.

4) The effect of the parameters on the damping characteristics is analyzed using the damping separation method and the eigenvalue analysis method, respectively. The analysis shows that K_pdc and SCR have a positive effect on the damping of the DC capacitance-dominated oscillation mode, while K_idc and T₁ have a negative effect. The analysis results are of great importance for practical engineering.

NOMENCLATURE

Symbol	——	Definition
A.	——	Variables
$ω_{g}$	——	Grid frequency
$ω_{0}$	——	Rated frequency
$δ, ω$	——	Power angle and frequency of synchronous generator (SG)
$Δ θ_{P L L}$	——	Phase locked loop (PLL) output angle
$Δ X$	——	State variable matrix
$Δ u$	——	Input variable matrix
A	——	Coefficient matrix
B	——	Input matrix
C₁	——	Line capacitance
C_dc	——	Direct current (DC) capacitance between machine-side converter (MSC) and grid-side converter (GSC)
E_fd	——	Forced no-load electromotive force of SG
$E_{q}^{'}$	——	q-aixs transient electromotive force of SG
F_HP	——	Power proportional coefficient of high-pressure cylinder
H_PLL(s)	——	Transfer function from $Δ u_{g q}$ to $Δ θ_{p l l}$
K_δ	——	Reciprocal of unequal rate
K_p	——	Proportional coefficient of proportional-integral-derivative (PID) link in governor system
K_pdc, K_idc	——	Proportional and integral coefficients of constant DC voltage control
k_pPLL, k_iPLL	——	Proportional coefficient and integral coefficient of PLL
M, D	——	Inertia and damping coefficients of SG
P_e	——	Input power of GSC
P_in	——	Output active power of MSC
P_m, P_t	——	Mechanical power and electromagnetic power of SG
R₁, L₁, i₁	——	Grid-side resistance, inductance, and current of permanent magnet synchronous generator (PMSG)
R₂, L₂	——	Equivalent resistance and inductance of alternating current (AC) grid
R_g, L_g	——	Filter resistance and inductance of PMSG
R_o, L_o	——	Resistance and inductance of grid-connected line
R, $X_{q}^{'}$	——	Resistance and q-axis transient reactance of SG
T₁	——	Time constant of hydraulic motor
T_CH	——	Time constant of steam chest
$T_{d 0}^{'}$	——	d-axis transient time constant of SG
U_dc	——	Voltage of DC capacitance
U_dcref	——	Voltage reference value of DC capacitance
u₂, i₂	——	Voltage and current of AC grid
u_g	——	Voltage of line capacitance
u_s, i_s	——	Stator voltage and current of wind turbine (WT)
u_t, i_g	——	Output voltage and current of GSC
u_r	——	Voltage at point of common coupling (PCC)
u_o, i_o	——	Output voltage and current of SG
X_d, $X_{d}^{'}$	——	d-axis reactance and d-axis transient reactance of SG
B.	——	Subscripts
d, q	——	d-axis and q-axis components of rotating reference frame signal
d₂, q₂	——	d₂-axis and q₂-axis components of rotating reference frame signal
ref, 0	——	Reference and steady-state values of variable
x, y	——	x-axis and y-axis components of rotating reference frame signal

Appendix

APPENDIX A

G₁ in (2) is given as:

G_{1} (s) = [\begin{matrix} 1.5 u_{t d 0} & 1.5 i_{g d 0} \end{matrix}]

(A1)

G₂ in (6) is given as:

G_{2} (s) = [\begin{matrix} s L_{g} & 1 \end{matrix}]

(A2)

$G_{3} (s)$ and $G_{4} (s)$ in (7) and (8) are given as:

\{\begin{array}{l} G_{3} (s) = - G_{4} (s) = [\begin{matrix} G_{C 1} (s) & G_{C 2} (s) \\ - G_{C 2} (s) & G_{C 1} (s) \end{matrix}] \\ G_{C 1} (s) = \frac{1}{C_{1} (s + ω_{0}^{2} / s)} \\ G_{C 2} (s) = \frac{ω_{0} G_{C 1} (s)}{s} \end{array}

(A3)

\{\begin{array}{l} G_{5} (s) = - G_{6} (s) = [\begin{matrix} G_{L 1} (s) & G_{L 2} (s) \\ - G_{L 2} (s) & G_{L 1} (s) \end{matrix}] \\ G_{L 1} (s) = \frac{R_{1} + s L_{1}}{L_{1}^{2} s^{2} + L_{1}^{2} ω_{0}^{2} + 2 s L_{1} R_{1} + R_{1}^{2}} \\ G_{L 2} (s) = \frac{ω_{0} L_{1}}{L_{1}^{2} s^{2} + L_{1}^{2} ω_{0}^{2} + 2 s L_{1} R_{1} + R_{1}^{2}} \end{array}

(A4)

T_ug and T_ig in (9) and (10) are given as:

\{\begin{array}{l} T_{u g} = [\begin{matrix} c o s θ_{P L L 0} & s i n θ_{P L L 0} & k_{1} \\ - s i n θ_{P L L 0} & c o s θ_{P L L 0} & k_{2} \end{matrix}] \\ T_{i g} = [\begin{matrix} c o s θ_{P L L 0} & - s i n θ_{P L L 0} & k_{3} \\ s i n θ_{P L L 0} & c o s θ_{P L L 0} & k_{4} \end{matrix}] \end{array}

(A5)

\{\begin{array}{l} k_{1} = - u_{g x 0} s i n θ_{P L L 0} + u_{g y 0} c o s θ_{P L L 0} \\ k_{2} = - u_{g x 0} c o s θ_{P L L 0} - u_{g y 0} s i n θ_{P L L 0} \\ k_{3} = - i_{g d 0} s i n θ_{P L L 0} - i_{g q 0} c o s θ_{P L L 0} \\ k_{4} = i_{g d 0} c o s θ_{P L L 0} - i_{g q 0} s i n θ_{P L L 0} \end{array}

(A6)

$G_{9} (s)$ in (17) is given as:

\{\begin{array}{l} G_{9} (s) = [\begin{matrix} a_{9_1} & a_{9_2} \end{matrix}] \\ a_{9_1} = {[T_{d 0}^{'} s + 1 + (X_{d} - X_{d}^{'}) a_{12}]}^{- 1} (X_{d} - X_{d}^{'}) a_{11} \\ a_{9_2} = {[T_{d 0}^{'} s + 1 + (X_{d} - X_{d}^{'}) a_{12}]}^{- 1} (X_{d} - X_{d}^{'}) a_{12} \\ [\begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}] = {[\begin{matrix} - R - R_{o} - s L_{o} & - X_{q}^{'} + ω_{g} L_{o} \\ X_{d}^{'} - ω_{g} L_{o} & - R - R_{o} - s L_{o} \end{matrix}]}^{- 1} \end{array}

(A7)

$G_{10} (s)$ in (18) is given as:

\{\begin{array}{l} G_{10} (s) = [\begin{matrix} a_{10_1} & a_{10_2} \\ a_{10_3} & a_{10_4} \end{matrix}] \\ a_{10_1} = a_{11} + a_{12} a_{9_1} \\ a_{10_2} = a_{12} + a_{12} a_{9_2} \\ a_{10_3} = a_{21} + a_{22} a_{9_1} \\ a_{10_4} = a_{22} + a_{22} a_{9_2} \end{array}

(A8)

G₁₂ and G₁₃ in (21) and (22) are given as:

\{\begin{array}{l} G_{12} = [\begin{matrix} c o s δ_{0} & s i n δ_{0} & k_{5} \\ - s i n δ_{0} & c o s δ_{0} & k_{6} \end{matrix}] \\ G_{13} = [\begin{matrix} c o s δ_{0} & - s i n δ_{0} & k_{7} \\ s i n δ_{0} & c o s δ_{0} & k_{8} \end{matrix}] \end{array}

(A9)

\{\begin{array}{l} k_{5} = - u_{r x 0} s i n δ_{0} + u_{r y 0} c o s δ_{0} \\ k_{6} = - u_{r x 0} c o s δ_{0} - u_{r y 0} s i n δ_{0} \\ k_{7} = - i_{o d 20} s i n δ_{0} - i_{o q 20} c o s δ_{0} \\ k_{8} = i_{o d 20} c o s δ_{0} - i_{o q 20} s i n δ_{0} \end{array}

(A10)

$G_{14} (s)$ and $G_{15} (s)$ in (23) are given as:

\{\begin{array}{l} G_{14} (s) = - G_{15} (s) = [\begin{matrix} G_{L 5} (s) & G_{L 6} (s) \\ - G_{L 6} (s) & G_{L 5} (s) \end{matrix}] \\ G_{L 5} (s) = \frac{R_{2} + s L_{2}}{L_{2}^{2} s^{2} + L_{2}^{2} ω_{0}^{2} + 2 s L_{2} R_{2} + R_{2}^{2}} \\ G_{L 6} (s) = \frac{ω_{0} L_{2}}{L_{2}^{2} s^{2} + L_{2}^{2} ω_{0}^{2} + 2 s L_{2} R_{2} + R_{2}^{2}} \end{array}

(A11)

$G_{16} (s)$ in (26) is given as:

G_{16} (s) = G_{14}^{- 1} (s)

(A12)

$G_{p 3} (s)$ and $G_{p 4} (s)$ in (29) are given as:

\{\begin{array}{l} G_{p 3} (s) = a_{11_1} k_{5} + a_{11_2} k_{6} \\ G_{p 4} (s) = [\begin{matrix} a_{p 4_1} & a_{p 4_2} \end{matrix}] \\ a_{p 4_1} = a_{11_1} c o s δ_{0} - a_{11_2} s i n δ_{0} \\ a_{p 4_2} = a_{11_1} s i n δ_{0} + a_{11_2} c o s δ_{0} \end{array}

(A13)

where $a_{11_1} = G_{11} (1,1)$ ; and $a_{11_2} = G_{11} (1,2)$ .

$G_{i 3} (s)$ and $G_{i 4} (s)$ in (30) are given as:

\{\begin{array}{l} G_{i 3} (s) = [\begin{matrix} a_{i 3_1} \\ a_{i 3_2} \end{matrix}] \\ G_{i 4} (s) = [\begin{matrix} a_{i 4_1} & a_{i 4_2} \\ a_{i 4_3} & a_{i 4_4} \end{matrix}] \end{array}

(A14)

\{\begin{array}{l} a_{i 3_1} = - s i n δ_{0} a_{10_2} k_{5} + c o s δ_{0} a_{10_1} k_{6} + k_{7} \\ a_{i 3_2} = c o s δ_{0} a_{10_2} k_{5} + s i n δ_{0} a_{10_1} k_{6} + k_{8} \\ a_{i 4_1} = - s i n δ_{0} c o s δ_{0} (a_{10_1} + a_{10_2}) \\ a_{i 4_2} = - {(s i n δ_{0})}^{2} a_{10_2} + {(c o s δ_{0})}^{2} a_{10_1} \\ a_{i 4_3} = {(c o s δ_{0})}^{2} a_{10_2} - {(s i n δ_{0})}^{2} a_{10_1} \\ a_{i 4_4} = s i n δ_{0} c o s δ_{0} (a_{10_1} + a_{10_2}) \end{array}

(A15)

$G_{m} (s)$ in (31) is given as:

G_{m} (s) = - {(1 + G_{p - δ} (s) G_{p 3} (s))}^{- 1} G_{p 4} (s)

(A16)

$G_{n} (s)$ in (32) is given as:

G_{n} (s) = G_{i 3} (s) G_{m} (s) + G_{i 4} (s)

(A17)

APPENDIX B

TABLE BI PARAMETERS OF WIND-THERMAL-BUNDLED SYSTEM

Module	Parameter	Value
PMSG	Rated power P_n (MW)	275
	DC capacitance C_dc (mF)	2
	Filter inductance L_g (H)	0.003
	Filter resistance R_g (Ω)	0.03
	Line resistance R₁ (Ω)	0.02
	Line inductance L₁(H)	0.002
	Line capacitance C₁ (μF)	5
GSC	Outer loop proportional coefficient K_pdc	18
	Outer loop integral coefficient K_idc	133
	Inner loop proportional coefficient K_p₁	45
	Inner loop integral coefficient K_i₁	2
PLL	Proportional coefficient k_pPLL	10
PLL	Integral coefficient k_iPLL	30
SG	Resistance R (p.u.)	0.1
	d-axis reactance X_d (p.u.)	1.59
	q-axis transient reactance $X_{q}^{'}$ (p.u.)	0.46
	d-axis transient reactance $X_{d}^{'}$ (p.u.)	0.25
	Line resistance R_o (Ω)	0.03
	Line inductance L_o (H)	0.001
	d-axis transient time constant $T_{d}^{'}$ (s)	0.68
	Inertia M	1.759
Turbine	Time constant of steam volume T_CH	0.5
	Time constant of reheater T_RH	8
	Time constant of cross-pipe T_CO	1
	Power proportional coefficients of high-pressure cylinder F_HP	0.3
	Power proportional coefficients of middle pressure cylinder F_IP	0.3
	Power proportional coefficients of low-pressure cylinder F_LP	0.4
Governor	Time constant of hydraulic motor T₁	0.5
	Amplification factor K_p	1
	Reciprocal of unequal rate K_δ	1
Exciter	Excitation regulation gain K_A	1
Exciter	Excitation regulation time constant T_A	0.2
AC grid	Equivalent resistance R₂ (Ω)	0.01
AC grid	Equivalent inductance L₂ (H)	0.1

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