Frequency Stability Analysis and Control Parameter Optimization in High-Voltage Direct Current-Asynchronous Power Systems with Automatic Generation Control

Abstract

Asynchronous interconnected systems connected by High-Voltage Direct Current (HVDC) transmission lines struggle with frequency support between interconnected areas, increasing the risk of frequency instability and resulting in low efficiency in frequency resource utilization. This study establishes a frequency dynamic analysis model for asynchronous interconnected power grids and develops an HVDC frequency controller for frequency control in these systems. It analyzes the control effects of HVDC frequency under two scenarios: without Automatic Generation Control (AGC) and with AGC. The research conducts an in-depth study on the system stability and frequency control parameter optimization for HVDC-asynchronous interconnected systems, significantly improving the system’s response speed and accuracy under various conditions through parameter optimization. Furthermore, the introduction of AGC demonstrates good adaptability, and a comparative analysis of HVDC frequency control effects under different AGC control modes is conducted. Finally, case studies validate the effectiveness and robustness of the proposed optimization scheme for frequency stability control in asynchronous interconnected systems under various fault scenarios.

1. Introduction

1.1. Motivation

In recent years, High-Voltage Direct Current (HVDC) transmission technology has been considered an efficient solution for power system interconnection due to its lower construction costs, smaller transmission losses, and minimal environmental impact [1,2]. Asynchronous grid interconnection, a key application field of HVDC technology, has achieved non-synchronous operation between regional power grids through back-to-back HVDC projects such as LuXi and YuE [3]. However, the isolating characteristics of HVDC transmission to some extent weaken the mutual support power capabilities between grids [4,5]. In asynchronous power grid systems connected solely by HVDC links, the frequency of the two grids is independent, presenting new challenges to the stability of the frequency and power support of the grid [6,7,8]. Therefore, in the context of asynchronous interconnection, optimizing dispatch strategies and reasonably allocating and coordinating frequency regulation resources within the grid to maintain frequency stability and ensure system reliability and security has become a critical challenge that needs to be addressed in the current field of power system research.

1.2. Literature Review

HVDC power transmission can respond quickly to changes in the frequency of the grid, provide the necessary power support, and improve the stability of the grid’s frequency.

Currently, research on HVDC frequency control strategies mainly focuses on the application of single-side grids in HVDC transmission lines. The frequency instability issue is likely to occur between the sending-end grid and receiving-end grid connected by HVDC transmission lines. As a result, many researchers have focused on the frequency stability of both the sending-end and receiving-end grids. Reference [9] addresses the frequency over-limit issue in the sending-end grid of an HVDC transmission system and proposes a coordinated frequency control strategy for wind power and hybrid ultra-high-voltage direct current (UHVDC) transmission systems in the power sending-end system. Reference [10] presents a coordinated control strategy based on fuzzy logic control for HVDC and wind farm collaborative participation in frequency regulation for the sending-end system, effectively improving the frequency characteristics of the HVDC sending-end system. Reference [11] proposes a frequency control scheme for the LCC-HVDC sending-end system based on rapid power compensation, which effectively enhances the frequency performance of the sending-end system. Reference [12] introduces a coordinated frequency control strategy for UHVDC participation in system frequency regulation, effectively increasing the equivalent inertia of the sending-end system and suppressing Rate of Change of Frequency and frequency deviations. Reference [13] investigates the frequency issues in the sending-end system under HVDC faults, proposing and verifying that HVDC-based fuzzy logic control can effectively suppress transient frequency deviations. Reference [14] presents a frequency coordination control strategy based on situational awareness for the sending-end system frequency stability, significantly reducing frequency deviations and frequency recovery time. Reference [15] addresses the voltage and frequency stability of the receiving-end system, establishing an optimization model considering transient voltage and frequency stability constraints, and proposes a coordinated emergency control strategy. Reference [16] proposes a real-time coordinated control strategy based on an online-updated frequency response model for a transient frequency stability issue in the receiving-end system, achieving real-time optimization of emergency control and effectively solving the transient frequency instability problem. Reference [17] introduces an additional frequency control method for the receiving-end system, effectively reducing overvoltage and frequency oscillations. Additionally, some researchers have conducted more in-depth analysis and studies on single-side grids in multi-area HVDC interconnections or on grids with low inertia under HVDC interconnections [18,19,20,21,22,23]. However, all of the above studies focus only on frequency control strategies for single-side power system of the HVDC interconnected system, without considering the impact of frequency changes on both sides of the HVDC system and how the control strategies affect the frequency characteristics of both sides.

In addition to research on frequency control strategies, many scholars have also paid significant attention to the optimization of HVDC frequency control parameters. The authors of reference [24] address the frequency stability issue of the receiving-end grid and the rational allocation of frequency regulation resources, proposing a method for determining virtual inertia and droop parameters, which effectively enhances frequency support capability. Reference [25] focuses on the frequency stability issue of weakly interconnected grids, proposing an HVDC voltage coordination control strategy based on the Virtual Synchronous Generator (VSG) method, and provides a detailed analysis of the tuning methods for control parameters. However, like the aforementioned studies, these also only analyze the frequency stability characteristics and control parameter optimization for single-side grids in asynchronous interconnection, without considering the overall frequency dynamic characteristics of both sides of an HVDC-interconnected system. Reference [26] designs a droop frequency limit controller (FLC) strategy for the frequency stability issue of asynchronously interconnected grids and proposes a multi-objective optimization method for the droop FLC parameters. The authors of reference [27] designed a droop control tuning scheme for converter control parameters, minimizing the lowest frequency deviation and stabilization time in the primary frequency response. Reference [28] introduces a new bilateral inertia and damping emulation (BIDE) control scheme, and analyzes and optimizes control parameters, effectively improving the frequency stability of both sides of the HVDC system. Reference [29] proposes a new inertia emulation and rapid frequency control strategy, and optimizes the key parameters of the controller to enhance overall frequency stability. Reference [30] establishes a more accurate linearized model of the high-voltage direct current (HVDC) transmission system for the asynchronous interconnected HVDC systems on both sides, utilizing additional frequency control (AFC) to achieve frequency modulation reserve sharing in the asynchronous interconnected system to participate in system frequency regulation, and analyzes the design of AFC parameters. Beyond the optimization of control parameters, some scholars have also conducted in-depth studies on other aspects of HVDC-asynchronous interconnected grids [31,32,33]. However, the above studies only consider the primary frequency regulation and HVDC participation in frequency control characteristics in HVDC-asynchronous interconnected grids without taking into account the impact of secondary frequency regulation on the interconnected system after HVDC participation in frequency control.

Reference [34] addresses the frequency stability issue of two asynchronous interconnected areas by proposing a structure for a frequency synchronous operation controller of the partitioned asynchronous power grid and optimizing the determination of key control parameters for operation, but lacks more in-depth theoretical analysis, such as stability analysis and a more precise optimization of parameter settings.

In the current context, the frequency stability issue of HVDC-asynchronous interconnected power systems needs to be given high priority. More attention and research are urgently needed with regard to HVDC frequency control in HVDC-asynchronous interconnected two-area systems. Additionally, research into the theoretical analysis, optimization of key control parameters, and adaptability of AGC control under HVDC participation in frequency control is becoming increasingly pressing.

1.3. Contributions

To address this issue, this paper conducts a system stability analysis of asynchronously interconnected power systems under the influence of HVDC frequency controllers and the optimization of control parameter tuning, and an adaptability analysis of AGC control. Its specific contributions are as follows:

• Building a frequency dynamic analysis model of an asynchronous interconnected two-area system with AGC in frequency synchronous control, conducting an adaptive analysis of HVDC frequency control and AGC control, analyzing system stability under changes in HVDC frequency control parameters, and analyzing system frequency characteristics and HVDC power under different AGC control modes.
• Developing a parameter optimization method for an asynchronous interconnected power grid on both sides to synchronize frequency, optimizing the proportional coefficient $k_p$ and integral coefficient $k_i$ in the HVDC frequency controller to achieve the best frequency control effect.

The comparison between the proposed method and the existing method is shown in

Table 1. Comparison of frequency control research in HVDC-asynchronous interconnected power systems.

Reference	Connection	Considered Grids	Control Method	AGC Adaptability Analysis
[25]	VSC-HVDC	Weak grid	VSG + HVDC voltage control	No
[12]	UHVDC	Sending end	Droop + virtual inertia control	No
[16]	HVDC	Receiving end	Real-time control	No
[30]	LCC-HVDC	Both grids	AFC (PI controller)	No
This paper	VSC-HVDC	Both grids	Frequency controller (PI controller)	Yes

. The rest of this paper is organized as follows. Section 2 builds a frequency dynamic analysis model of the asynchronous interconnected two-area system with HVDC under the same frequency control without AGC and with AGC. Section 3 provides a stability analysis of the system with HVDC participating in the asynchronous interconnected two-area frequency control, control parameter optimization, and the adaptability analysis of the HVDC frequency control and AGC control. The case analysis is described in Section 4. Finally, Section 5 summarizes the research results and conclusions of this paper.

2. Frequency Dynamic Analysis Model of the Asynchronous Interconnected Two-Area System with HVDC

2.1. Implementation and Challenges of HVDC-Asynchronous Interconnected Power Grids

Asynchronous interconnection means that power grids in different areas are non-synchronously connected through power electronic devices. As shown in Figure 1, the current situation of the asynchronously interconnected power grid is that the Yunnan power grid forms an asynchronous interconnection structure with the main Southern power grid through HVDC transmission lines such as Chusui, Puqiao, Luxi, and so on. The asynchronous interconnection power grid, through the high controllability of the HVDC transmission system, achieves precise control and rapid adjustment of the transmitted power, enhancing the safety and flexibility of the power grid.

The Luxi back-to-back converter station is the first back-to-back HVDC project constructed and operated by the China Southern Power Grid Corporation, as shown in Figure 2. The Luxi back-to-back project adopts a combination of high-capacity flexible HVDC and conventional HVDC, which can enhance the power grid’s flexible control over power, reduce transmission losses, and decrease the occurrence of transmission faults, ensuring the safe and stable operation of the power grid.

HVDC-asynchronous interconnection enables the exchange of electrical energy between AC power systems of different frequencies, which can solve the stability issues of cross-regional power grids, but at the same time, it also loses the frequency support capability between regions.

2.2. Frequency Dynamics Analysis Model of HVDC-Asynchronous Interconnected Two-Area Systems

This article aims to enhance the frequency support capability between two areas interconnected through an HVDC link by using a PI controller as the HVDC frequency controller to realize the synchronous control of the frequency in the asynchronously interconnected areas. Taking the frequency deviation of both sides as the input, the PI controller dynamically adjusts the output power of the HVDC system to regulate the frequency stability of both areas. For example, when an area is affected by a power disturbance and the frequency drops, the HVDC frequency controller detects a frequency deviation between the two sides, and by using the PI control of the HVDC system, it outputs more power to the low-frequency area, thereby achieving frequency support from the high-frequency area to the low-frequency area.

The model of the two areas interconnected asynchronously through an HVDC link is shown in Figure 3, in which $H_1$ and $H_2$ are the equivalent inertias of the two areas’ systems, $D_1$ and $D_2$ are the equivalent damping coefficients, $G_H\left(s\right)$ is the prime mover model of the hydraulic turbine in the hydropower unit, $R_H$ is the regulation coefficient of the hydropower unit, $K_H$ is the proportion of power generation of the hydropower unit, $G_T\left(s\right)$ is the steam turbine prime mover model, $R_T$ is the regulation coefficient of the thermal power unit, $K_T$ is the proportion of power generation of the thermal power unit, $\Delta P_t$ is the transmission power of the HVDC tie line, and $G_{FC}\left(s\right)$ is the HVDC frequency controller, $B_1$ and $B_2$ represent the area frequency deviation coefficients (MW/0.1 Hz) of the two sides of the power grid, $k_{i\_agc1}$ and $k_{i\_agc2}$ are the integral coefficients of the AGC for both sides of the grid, $\Delta P_{t1}$ and $\Delta P_{t2}$ are the HVDC output powers of the two sides of the grid, $T_{CH}$ is the time constant of the steam volume, and $T_W$ is the water hammer effect time constant.

Figure 4 depicts the model of the HVDC frequency controller, which employs a PI control strategy with the frequency difference as the input. Here, $K_p$ denotes the proportional gain, and $K_i$ represents the integral gain.

The HVDC frequency controller utilizes the power regulation characteristics of the HVDC system to link the frequencies of both grids. When a power disturbance occurs in one grid, causing a frequency deviation, the HVDC frequency controller detects the frequency difference and generates control instructions based on a proportional–integral (PI) control algorithm to adjust the HVDC transmission power by increasing or decreasing the power transfer from one grid to the other, allowing their frequencies to gradually converge. Due to the fast response of the HVDC system, it can suppress frequency deviations and restore grid stability in a short time, thereby achieving frequency support and synchronization between asynchronously interconnected grids.

3. Stability Analysis and Control Parameter Determination for Asynchronously Interconnected Two-Area Frequency Control Involving HVDC Participation

3.1. Stability Analysis of Asynchronously Interconnected Two Areas with HVDC Frequency Control

The values of the control parameters $k_p$ and $k_i$ of the HVDC frequency controller not only affect the stability of the HVDC interconnection system but are also related to the control effect of the HVDC frequency controller on the frequency of both sides of the power grid. Therefore, it is necessary to first conduct a theoretical analysis of the system’s stability to determine the impact characteristics of the HVDC frequency controller parameters on the frequency stability of both sides of the interconnection system.

In this subsection, we will analyze the relationship between the HVDC frequency control and primary frequency control of the system, without considering AGC control. In Figure 3, $u_1$ and $u_2$ represent the power disturbances of Area 1 and Area 2, respectively. The output y is the difference in frequency between the two areas. As shown in the figure, the HVDC frequency controller takes the frequency difference between two areas as input and the HVDC power as output.

In control system analysis, the closed-loop transfer function describes the mathematical relationship between the output response and the input signal within a closed-loop control system. Correctly analyzing the closed-loop transfer function can ensure that the system is stable and has the dynamic and steady-state characteristics required for specific applications.

By analyzing the system structure diagram, taking the power disturbances of Area 1 and Area 2 as inputs, and the frequency deviation $\Delta f_1-\Delta f_2$ of the two areas as the output, the corresponding closed-loop system transfer functions $G_1\left(s\right)$ and $G_2\left(s\right)$ are analyzed, as shown in Equations (1) and (2).

$$G_1\left(s\right)={\displaystyle \frac{Y\left(s\right)}{U_1\left(s\right)}}={\displaystyle \frac{a_1{s}^4+a_2{s}^3+a_3{s}^2+a_4{s}^1+a_5}{f_1{s}^5+f_2{s}^4+f_3{s}^3+f_4{s}^2+f_5{s}^1+f_6}}$$

(1)

$$G_2\left(s\right)={\displaystyle \frac{Y\left(s\right)}{U_2\left(s\right)}}={\displaystyle \frac{b_1{s}^4+b_2{s}^3+b_3{s}^2+b_4{s}^1+b_5}{f_1{s}^5+f_2{s}^4+f_3{s}^3+f_4{s}^2+f_5{s}^1+f_6}}$$

(2)

Here, Y(s) represents the Laplace transform of the frequency difference y(t) between the two areas, while $U_1\left(s\right)$ is the Laplace transform of the power disturbance $u_1\left(t\right)$ in Area 1, and $U_2\left(s\right)$ is the Laplace transform of the power disturbance $u_2\left(t\right)$ in Area 2.

The closed-loop system transfer function of the asynchronously interconnected two-area model has a denominator polynomial that is a 5th-order polynomial in ‘s’. Setting this polynomial to zero allows the construction of the system’s characteristic equation. The characteristic equation plays a crucial role in the stability analysis of the control system because it is directly related to the poles of the closed-loop system. The stability of the closed-loop system is entirely determined by the roots of the closed-loop characteristic equation (closed-loop poles). The denominator polynomial $D\left(s\right)$ is shown in Equation (3).

$$D\left(s\right)=f_1{s}^5+f_2{s}^4+f_3{s}^3+f_4{s}^2+f_5{s}^1+f_6$$

(3)

In the transfer functions $G_1\left(s\right)$ and $G_2\left(s\right)$ and the polynomial denominator $D\left(s\right)$, the coefficients $a_i$, $b_i$, and $f_i$ are related to the parameters of the system of two asynchronously interconnected areas, and the detailed value of the coefficients is shown in Appendix A.

The Routh criterion is a mathematical method used to determine the stability of a system. It analyzes the coefficients of the polynomial in the denominator of the closed-loop transfer function. When all poles of the closed-loop transfer function are located in the left half of the complex plane, the system is considered stable. If there are any poles in the right half, the system is unstable. By constructing a Routh table, the need to solve high-order characteristic equations is eliminated, which simplifies the process of stability analysis.

According to the analysis using the Routh criterion, it is known that a necessary and sufficient condition for the system to be stable is that all coefficients of the characteristic equation are positive, and all elements of the first column of the Routh table are positive; therefore, the necessary and sufficient condition for the stability of the asynchronous interconnection system is as shown in Equation (4).

$$\left\{\begin{array}{c}\left(1\right)f_1>0,f_2>0,f_3>0,f_4>0,f_5>0,f_6>0\\ \left(2\right)f_1>0,f_2>0,f_7>0,f_9>0,f_{10}>0,f_6>0\end{array}\right.$$

(4)

The expressions for $f_7$, $f_8$, $f_9$, and $f_{10}$ are shown as follows: $f_7=(f_2{f}_3-f_1{f}_4)/\left(f_2\right)$, $f_8=(f_2{f}_5-f_1{f}_6)/\left(f_2\right)$, $f_9=(f_7{f}_4-f_2{f}_8)/\left(f_7\right)$, and $f_{10}=(f_9{f}_8-f_7{f}_{10})/\left(f_9\right)$

By constructing the Routh table for the denominator polynomial of the closed-loop transfer function of the interconnected system, and calculating and analyzing the controller parameters under the necessary and sufficient conditions for system stability, the corresponding relationship between system stability and control parameters $k_p$ and $k_i$ is obtained, thereby determining the stability region of the system concerning $k_p$ and $k_i$.

3.2. Optimization of Control Parameters for HVDC Frequency Controllers

When a power disturbance occurs, the frequency of a single-sided power grid changes, and the frequency difference between the two sides increases. The HVDC frequency controller begins to respond, adjusting the magnitude of the HVDC output power to reduce the frequency difference until the frequencies of both power grids return to consistency.

The parameters $k_p$ and $k_i$ of the HVDC frequency controller have a certain impact on frequency control performance. From a control theory perspective, the proportional coefficient $k_p$ affects the system’s feedback strength. When $k_p$ is small, the feedback signal is insufficient, resulting in a slow response to deviations and increased frequency deviation, which may lead to oscillations near the stability region boundary. The integral coefficient $k_i$ is mainly used to eliminate steady-state error. A small $k_i$ causes the system to maintain a frequency deviation for a longer time, requiring more time to reach stability. As $k_i$ increases, the system gradually enhances its correction of accumulated error, reducing the frequency deviation and shortening the stabilization time. While both $k_p$ and $k_i$ improve system performance as they increase, there is a threshold beyond which further increases in these control parameters have diminishing effects on frequency control and may even lead to adverse effects. Choosing appropriate values for $k_p$ and $k_i$ is essential to optimize system performance, ensuring minimal frequency deviation and stabilization time while avoiding oscillations and overcorrection.

The settling time is used to describe the system’s ability to enter and remain near the final steady-state value after experiencing a step input, with a shorter settling time indicating that the system can reach a stable state more quickly. In the analysis above, when the $k_p$ and $k_i$ values are within an appropriate range, the maximum frequency deviation control effect is relatively good and close. The settling time required for the frequency difference between the two areas to return to the reference value of 0 is usually the frequency stability index that is more valued in engineering. This paper will comprehensively consider the time needed for the frequency deviation to recover consistency when both sides of the power grid are disturbed, that is, the comprehensive settling time $T_s$. The expression for $T_s$ is shown in Equation (5), where the stability range is set to $\pm 2\%$ of the maximum difference in frequency between the two sides after the disturbance occurs, that is, $\pm max\left(\right|\Delta f_1-\Delta f_2\left|\right)\ast 2\%$. The smaller the settling time, the better the frequency synchronization control effect of the HVDC frequency controller.

$$T_s=0.5\ast T_{s_{Area1}}+0.5\ast T_{s_{Area2}}$$

(5)

Here, $T_s$ represents the comprehensive settling time, $T_{s_{Area1}}$ refers to the step settling time of the closed-loop transfer function $G_1\left(s\right)$ when Area 1 experiences a power step disturbance, and $T_{s_{Area2}}$ refers to the step settling time of the closed-loop transfer function $G_2\left(s\right)$ when Area 2 experiences a power step disturbance.

3.3. Analysis of the Adaptability of HVDC Frequency Control with AGC

Primary frequency control involves a differential adjustment and cannot restore the grid frequency to the standard value, while secondary frequency control allows for zero-differential adjustment, enabling more precise maintenance of system frequency stability. Secondary frequency control primarily relies on AGC, which automatically manages the load of each power plant unit through computer control, achieving full automation of the frequency control process. The dynamic analysis model of the frequency in a two-area system with AGC is shown in Figure 3.

The control modes of AGC mainly include three types: Flat Frequency Control(FFC), Flat Tie-line Control(FTC), and Tie-line Bias Control(TBC). The FFC mode refers to the control target of the control area, which is to maintain the system frequency of the area at a set value, that is, to keep the frequency constant. The FTC mode’s control target is to maintain the tie-line power exchange constant. The purpose of the TBC mode is to control the tie-line power between various areas in a multi-area interconnected power system to maintain the system frequency and tie-line power within the predetermined range. In practical applications, power systems often adopt two control modes: FFC and TBC. Therefore, this paper will analyze the characteristics of HVDC frequency control under the FFC and TBC control modes.

Due to the different control objectives of various control modes, the calculation method for the area control error (ACE) also varies, thus affecting the power grid frequency differently. The calculation formulas for the ACE under the FFC and TBC modes are shown in Equations (6) and (7), respectively.

$$AC{E}_{FFC}=-10B\Delta f$$

(6)

$$AC{E}_{TBC}=-10B\Delta f+\Delta P_t$$

(7)

Here, B represents the area frequency deviation coefficient, and $\Delta P_t$ represents the tie-line power of the interconnected system.

The TBC mode exhibits stronger responsiveness and faster recovery in terms of frequency restoration and HVDC power balance, while the FFC mode tends to provide more stable frequency regulation. Therefore, by reasonably combining the TBC and FFC control modes, leveraging the fast recovery capability of the TBC mode and the stable regulation of the FFC mode, system performance can be optimized in different scenarios. This approach ensures rapid frequency recovery while avoiding excessive initial frequency deviation or power fluctuations.

4. Case Study

4.1. Case Study of the HVDC Asynchronously Interconnected Two-Area Model Without AGC

The parameters of the HVDC asynchronous interconnection system model are listed in

Table 2. Parameters of HVDC asynchronous interconnection system.

Area	Parameters	Value
Area 1	Equivalent inertia $H_1$/s	6
	Equivalent damping coefficient $D_1$	1
	Regulation coefficient $R_H$	0.04
	The proportion of power generation of the hydropower unit $K_H$	0.712
	Water hammer effect time constant $T_W$/s	0.5
	HVDC power coefficient $K_1$	0.0174
Area 2	Equivalent inertia $H_2$/s	6
	Equivalent damping coefficient $D_2$	1
	Regulation coefficient $R_T$	0.05
	The proportion of power generation of the thermal power unit $K_T$	0.5385
	Steam volume time constant $T_{CH}$/s	0.3
	HVDC power coefficient $K_2$	0.0072

. By substituting these parameters into the denominator polynomial of the closed-loop transfer function as shown in Equation (3) and applying the Routh stability criterion, the range of values for $k_p$ and $k_i$ that ensure the stability of the system can be derived. The range of values for the control parameters of the HVDC frequency controller under the condition of system stability is indicated in the shaded area of Figure 5.

When the control parameters $k_p$ and $k_i$ are appropriately set, the variation in the frequencies in the two areas under HVDC frequency control, as well as the frequency change in the power grid on the disturbance side without HVDC frequency control, are shown in Figure 6.

As shown in Figure 6, without HVDC frequency control, the maximum frequency deviation of Area 1 after a disturbance reaches −0.036 Hz. With HVDC frequency control, the maximum frequency deviation is −0.018 Hz. It can be seen that with appropriate values of the control parameters $k_p$ and $k_i$, the frequency deviation can be significantly reduced, and the time for the frequency to stabilize will also be greatly shortened, achieving timely and effective frequency support for asynchronous interconnected two areas. The HVDC frequency controller takes the frequency deviation between the two areas as the input, with a deviation of 0 as the reference value. After a disturbance in Area 1, the frequency drops sharply, and the frequency of Area 2 follows synchronously under the action of the controller. With appropriate values of $k_p$ and $k_i$, the frequencies of the two areas can be consistent within a few seconds.

Besides system stability, the values of the control parameters $k_p$ and $k_i$ have a significant impact on the minimum frequency of both areas and the stabilization time. When a disturbance occurs in Area 1, the frequency changes in Area 1 under different $k_p$ and $k_i$ parameters are shown in Figure 7 and Figure 8. The frequency changes in Area 2 are not displayed in the figures; refer to Figure 6 for the frequency synchronization effect.

As shown in Figure 7, when $k_i$ is set to an appropriate value, a smaller $k_p$ results in a larger frequency deviation, and oscillations occur due to $k_p$ being close to the critical value of the stability domain, leading to a longer stabilization time; as $k_p$ increases, the frequency deviation decreases and the stabilization time is reduced; when $k_p$ exceeds a certain level, the impact of further increases in $k_p$ on the frequency stability indicators diminishes, the maximum frequency deviation slightly increases in the reverse direction, and the frequency stabilization time remains almost unchanged.

As shown in Figure 8, when $k_p$ is set to an appropriate value, a smaller $k_i$ results in a larger frequency deviation and a longer stabilization time; as $k_i$ increases, the frequency deviation gradually decreases, and the stabilization time decreases; when $k_i$ exceeds a certain level, further increases in $k_i$ have little effect on the frequency stabilization time, and a reverse increase in the maximum frequency deviation occurs.

As shown in Figure 9, only points with a comprehensive stabilization time of less than 25 s are retained for display. When the control parameters are near the boundary of the stability domain, the stabilization time increases significantly, and the frequency change exhibits oscillatory behavior. As the parameters $k_p$ and $k_i$ gradually increase, the stabilization time decreases accordingly. When $k_i$ increases to a certain extent, if $k_i$ continues to increase, the comprehensive stabilization time $T_s$ will decrease at an extremely slow rate, meaning the benefit of further increasing $k_i$ is almost zero. When $k_p$ increases to a certain extent, if $k_p$ continues to increase, the comprehensive stabilization time will show a slightly increasing trend. In summary, the best benefit in terms of comprehensive stabilization time $T_s$ is achieved when it is reduced to within 3 s and the control parameters $k_p$ and $k_i$ are minimized. Through optimization analysis using MATLAB mathematical tools, it is found that the optimal HVDC frequency control effect is achieved when $k_p=1609$ and $k_i=6136$.

Through the above research, the control parameters $k_p$ and $k_i$ of the HVDC frequency controller were optimized and their adaptability to AGC was analyzed. To verify the adaptability and adjustment characteristics of the optimized HVDC frequency control under different working conditions, an HVDC asynchronous interconnected two areas model and an HVDC asynchronously interconnected two-area model with AGC were built in Matlab/Simulink. The system frequency variation characteristics and the variation characteristics of the HVDC frequency regulation power were studied under three conditions: power disturbance in Area 1, power disturbance in Area 2, and HVDC blocking.

In the case study analysis of this paper, the grid capacity of Area 1 is 115,049 MW, and the grid capacity of Area 2 is 277,906 MW. Three scenarios were simulated by applying a 1000 MW power disturbance to Area 1 and Area 2 at the 5 s mark, as well as simulating a 1000 MW HVDC blocking fault, to verify the frequency control effect of the HVDC frequency controller through simulation analysis.

As shown in Figure 10, we conducted a simulation analysis on the two-area model without the AGC. Figure 10a–c illustrate the frequency variations in Area 1 under a 1000 MW power disturbance, the frequency difference between the two areas, and the HVDC frequency control power changes under a 1000 MW HVDC lockout fault, respectively. The control parameters for the HVDC frequency controller were set as $k_p=1609$ and $k_i=6136$.

From Figure 10a, we can see that under the action of the HVDC frequency controller, the maximum frequency deviation of the grid is 0.0099 Hz, which is reduced to about one-fourth of the maximum frequency deviation in Area 1 without HVDC control. Additionally, after 2.71 s, the frequencies of both areas tend to converge, showing a significant improvement in stability time compared to the uncontrolled scenario. (In this section, the stability range is defined as 2% of the maximum frequency difference after the disturbance occurs, i.e., $\pm max\left(\right|\Delta f_1-\Delta f_2\left|\right)\ast 2\%$.)

After the 1000 MW disturbance, the HVDC frequency control output power decreases, and during this process, part of the disturbance is shared with the grid in Area 2 through the HVDC transmission line. The frequency control resources in Area 2 are utilized for rapid support to prevent further frequency deviation. With the combined effects of the generator units and HVDC frequency control, a new power load balance state is reached, stabilizing the HVDC frequency control output power at approximately 600 MW lower. Figure 10b,c can be analyzed similarly, where the maximum frequency deviations under HVDC frequency control are effectively managed, and the frequencies of both areas become consistent after 3.60s and 2.62s, respectively. Detailed results are shown in

Table 3. The result of HVDC frequency control under different working conditions without AGC.

	Disturbance in Area 1	Disturbance in Area 2	HVDC Blocking Fault
The maximum frequency deviation of Area 1/Hz	−0.0099	−0.0096	0.0070
The maximum frequency deviation of Area 1 (without control)/Hz	−0.0368	/	0.0368
The maximum frequency deviation of Area 2/Hz	−0.0096	−0.0100	−0.0020
The maximum frequency deviation of Area 2 (without control)/Hz	/	−0.0153	−0.0153
Ts/s	2.71	3.60	2.62

.

In summary, the HVDC frequency controller, with appropriate control parameters, can effectively respond to frequency variations under different disturbance faults and provide rapid response. It can quickly align the frequencies of both grids, achieving mutual support between the two sides of the HVDC asynchronous interconnection and efficiently utilizing the frequency control resources of both grids.

4.2. Case Study of the HVDC Asynchronously Interconnected Two-Area Model with AGC

After considering secondary frequency modulation, the frequencies of the two interconnected asynchronous two-area systems can be restored to the standard frequency values, and the HVDC frequency controller still effectively controls the frequencies on both sides, as detailed in Figure 11.

In the asynchronous interconnection system model shown in Figure 3, $k_{i\_agc1}$ and $k_{i\_agc2}$ are the integral coefficients of AGC. When the absolute value of these coefficients is too large, it can lead to system instability, and they are typically set to around −0.3. By setting the integral coefficients of AGC on both sides of the power grid to −0.3, the impact of different AGC control mode combinations on the frequency changes of both power grids is analyzed. The relationship between the frequency changes of the interconnected system’s power grids and the AGC control mode combinations is shown in Figure 12. The relationship between the HVDC transmission power involved in frequency modulation and the AGC control mode is shown in Figure 13.

As shown in Figure 12, the solid line represents the frequency deviation in Area 1, and the dashed line represents the frequency deviation in Area 2. When Area 1 and Area 2 use the TBC+FFC and TBC+TBC modes, the maximum frequency deviation is slightly greater than that in the FFC+FFC and FFC+TBC modes; and in the TBC+TBC mode, the time required for the frequency of both sides of the power grid to return to the standard frequency value is the shortest.

As shown in Figure 13, the disturbance occurs in Area 1, and the negative value of the HVDC power involved in frequency modulation indicates that power needs to be transmitted from Area 2 to Area 1 to achieve frequency support. When Area 1 and Area 2 adopt the TBC+FFC and TBC+TBC modes, the maximum output of HVDC power is greater than that in the FFC+FFC and FFC+TBC modes. However, under the FFC+FFC mode, the HVDC power for frequency regulation cannot return to zero; the restoration of HVDC power to zero requires that one side of the power grid adopts the TBC control mode. Meanwhile, in the TBC+TBC mode, the time required for power to recover to zero is the shortest.

In summary, the four combinations of AGC control modes have quite similar control effects on the maximum frequency deviation and the maximum HVDC power involved in frequency modulation. However, different combinations have different impacts on the recovery time of frequency and HVDC power recovery, with the TBC+TBC mode showing the best control effect for both the recovery of frequency and the recovery of HVDC power.

As shown in Figure 14, we conducted a simulation analysis on the two-area model with AGC. Figure 14a–c illustrate the frequency variations in Area 1 under a 1000 MW power disturbance, the frequency difference between the two areas, and the HVDC frequency control power changes under a 1000 MW HVDC lockout fault, respectively. The control parameters for the HVDC frequency controller were set as $k_p=1609$ and $k_i=6136$, while the AGC control mode adopted a TBC+TBC combination with integral coefficients $k_{i\_agc1}$ and $k_{i\_agc2}$ set to −0.3.

From the figure, we can see that under the combined action of AGC in the three scenarios, the maximum absolute values of frequency deviation in the asynchronous interconnection using the HVDC frequency controller are 0.0103 Hz, 0.0087 Hz, and 0.0074 Hz, all of which are lower than the maximum absolute frequency deviation without HVDC frequency control. Additionally, due to the rapid action of the HVDC frequency control power, the rate of frequency change in the disturbed grid is slightly reduced.

Comparing the frequency changes under HVDC frequency control without AGC, it is evident that adding AGC still results in excellent control performance. The frequency stabilization times under the HVDC frequency controller are 5.64 s, 11.37 s, and 8.11 s, allowing the frequencies of the two interconnected grids to quickly align, achieving mutual support between them. The HVDC frequency control power under different operating conditions is shown in Figure 14. In TBC mode, the calculation of ACE simultaneously considers the grid frequency deviation and HVDC frequency control power, allowing the HVDC frequency control power to gradually return to zero under the influence of AGC. Detailed results are shown in

Table 4. The result of HVDC frequency control under different working conditions with AGC.

	Disturbance in Area 1	Disturbance in Area 2	HVDC Blocking Fault
The maximum frequency deviation of Area 1/Hz	−0.0103	−0.0087	0.0074
The maximum frequency deviation of Area 1 (without control)/Hz	−0.0385	/	0.0385
The maximum frequency deviation of Area 2/Hz	−0.0093	−0.0092	−0.0020
The maximum frequency deviation of Area 2 (without control)/Hz	/	−0.0126	−0.0126
Ts/s	5.64	11.37	8.11

.

5. Conclusions

The inefficiency in sharing and coordinating frequency regulation resources between asynchronous regions further exacerbates the system’s stability issues. At the same time, although the introduction of AGC into asynchronous interconnected grids can achieve a certain degree of automation and optimization in frequency regulation, the adaptability of AGC in the context of asynchronous interconnection still requires further in-depth research.

This study establishes a dynamic frequency analysis model for asynchronous interconnected power grids with AGC, and studies the system stability analysis of asynchronously interconnected power systems under the influence of HVDC frequency controllers, optimization of control parameter tuning, and adaptability analysis of AGC control. The HVDC frequency controller parameters $k_p$ and $k_i$ significantly impact system stability. When these parameters are too small, frequency control performance is poor due to proximity to the stability region boundary; if they are too large, the effectiveness does not improve and may even slightly decrease. Optimized parameters enhance the system’s response speed and steady-state accuracy across different operating conditions. Additionally, when combined with AGC, the HVDC controller demonstrates strong adaptability, especially under TBC mode, restoring frequency modulation power. This coordinated control improves frequency support and resource sharing in asynchronous interconnected grids, addressing frequency stability challenges.

This study also has some limitations, such as the influence of model assumptions and system complexity on the results. Future research could further explore how to optimize frequency control strategies in more complex system structures to achieve higher operational efficiency and stability.