Control Method for Ultra-Low Frequency Oscillation and Frequency Control Performance in Hydro–Wind Power Sending System

Abstract

In a hydropower-dominated power grid, the primary frequency regulation (PFR) capability of hydropower units is typically compromised to suppress ultra-low frequency oscillations (ULFOs). However, as renewable wind power is further integrated, a practicable solution to damp ULFOs has emerged, which is to adjust the frequency control parameters of wind turbine (WT) units. Driven by the goals of overall damping enhancement and ULFO suppression, this paper first establishes an extended unified frequency model (EUFM) of a hydro–wind power sending system. Based on EUFM, the damping torque of the hydro–wind power sending system is derived, and the specific impact of WT control parameters on ULFOs and PFR characteristics is investigated. Then, a novel optimization objective function considering damping in the ultra-low frequency band and PFR is formulated and solved using an intelligence algorithm. By optimizing the parameters of the WT to suppress ULFOs, the PFR capability of hydropower units can be released. Finally, simulation results verify that the optimized WT parameters can simultaneously address the ULFO problem and guarantee PFR performance, thereby enhancing the frequency dynamic stability of the sending system.

1. Introduction

In the past decade, in the hydropower-dominated power grid, which is asynchronous interconnecting through a back-to-back high-voltage DC (HVDC) transmission system, ultra-low frequency oscillations (ULFOs) have occurred multiple times in China [1]. Unlike traditional low-frequency oscillations (LFOs) in power systems, ULFOs manifest as the frequency of the whole units oscillating in the same mode, which means there are no relative oscillations between generator rotors [2]. There is a common sense that ULFOs are within the scope of frequency stability issues, characterized by long oscillation periods and extremely low oscillation frequencies (less than 0.1 Hz), posing a significant threat to the safe and stable operation of power systems [3].

The current research mainly focuses on asynchronous interconnected or islanded hydropower grids. From a mechanistic perspective, studies have indicated that the main reason for the occurrence of ULFOs is the inability of hydropower units to provide sufficient damping torque within the ultra-low frequency band (0~0.1 Hz) [4]. To find out the specific factors, reference [5] investigates the governing system’s parameters, and in [6], the PID controller of the governing system and its parameters are given in detail. The above studies indicate that the damping level of the system is determined by the proportional and integral coefficients of the PID controller as well as the water starting time. In addition to damping level, thermal power ratio and dead zone setting are crucial to affect the oscillation mode of ULFOs [7,8].

Currently, methods to suppress ULFOs mainly include optimizing the parameters of the hydro turbine (HT) governor and additional damping control. Reasonably increasing the ratio of governor proportional and integral gain can enhance damping to suppress ULFOs [9]. Based on this, in [10], a parameter design method for HT governors is proposed using the structural singular value theory. In [11], deep learning algorithms are applied to ensure the effectiveness of the optimized PID parameters for various extreme operating conditions. Additionally, a parameter optimization strategy is suggested in [12], which considers both damping characteristics and PFR of hydropower units, aiming to suppress ULFOs by optimizing the parameters of hydropower unit governor systems. Based on these findings, reference [13] proposed a CFS_GPSS strategy to compensate phase and increase system damping, which enhances PFR performance while suppressing ULFO. However, the above methods have weakened the PFR ability of hydropower units, resulting in a weakened ability of the power grid to resist active power shocks.

In the realm of additional damping control, power system stabilizers (PSS) are commonly utilized for mitigating LFOs [14]. However, traditional PSS are unable to suppress ULFOs. Therefore, reference [15] suppresses ULFOs through PR-PSS control. Reference [16] demonstrates through small signal analysis that GPSS control has a suppressive effect on ULFOs and optimizes appropriate parameters to suppress ULFOs. Furthermore, in [17], the crossbreed operation in the genetic algorithm is introduced to the particle swarm optimization (PSO) algorithm to form a hybrid PSO so as to optimize PSS4B parameters; the optimized PSS4B can provide effective damping in different frequency bands. Additionally, the ability of DC transmission to provide frequency support is explored [18]. DC additional control has also been applied for the suppression of ULFOs, such as a frequency limiting controller (FLC) [19] and a damping controller [20].

In recent years, with the significant integration of wind power into the power system, wind power has played an increasingly crucial role in ensuring the safe and stable operation of the power system [21]. Therefore, in many countries, wind farms are required to be equipped with PFR controllers. The participation of wind turbines in the primary frequency regulation of the system requires the use of auxiliary frequency regulation control links, which can be divided into two methods: power shedding control [22] and rotor kinetic energy control [23]. Additional frequency controllers are designed to meet strict frequency regulation requirements to ensure sufficient frequency and power support against load disturbances. A previous study [24] optimizes the parameters of WT with the PFR performance of WT participating in PFR as the goal. However, the design process did not consider the damping of ULFOs, potentially leading WT to provide negative damping in the ultra-low frequency band, exacerbating the threat of ULFOs in certain circumstances. Hence, in power grids with a high proportion of hydropower at the sending end, it is necessary to readjust the frequency control parameters of WT to reduce the risk of ULFOs. Reference [25] reveals the dynamic behavior of WT and its interference mechanisms with HT, explaining the system’s ULFO modes. It also analyzes the impact of different parameters of WT in various operating modes on ULFOs, indicating that different parameters have varying degrees of suppression on ULFOs. Furthermore, reference [26] investigates the issue of ULFOs in wind-hydro hybrid systems, demonstrating that WT can alleviate ULFOs but has not optimized the best frequency control parameters.

To fill the gap on additional control of ULFO based on renewable energy, this paper aims to explore the considerable potentiality in terms of both ULFO suppression and PFR performance of WT units. The contributions include the following three parts:

(1)

For ULFO mechanism analysis, an EUFM considering hydropower and wind units is established. The damping torque coefficients of the hydro–wind power system are derived.

(2)

Another effort is to reveal the detailed influence of WT frequency control parameters on frequency characteristics. In this aspect, the damping level and key performance indicators of PFR are investigated.

(3)

From the perspective of ULFO suppression by fulfilling the damping control potential of wind units. An optimization model of WT control parameters is established. The objective function balances both ULFO suppression and PFR performance, which is solved by the PSO algorithm. Then, the superior PFR performance of hydro units is profitable.

The organization of this paper is as follows: Section 2 establishes the EUFM of the hydro–wind sending system. Section 3 investigates the influence of WT frequency control parameters on damping characteristics and frequency characteristics. Section 4 proposes an optimal control method of WT for damping enhancement and PFR improvement comprehensively. Section 5 simulates and validates the aforementioned analysis. Finally, the main conclusions are drawn in Section 6.

2. Extended Unified Frequency Model of Hydro–Wind System

The damping coefficients of the hydropower unit and DFIG in the ultra-low frequency range are derived using the damping torque method. Moreover, an extended unified frequency model of hydropower and WT units is established to analyze the overall damping coefficient of the system in this section.

2.1. The Model of Hydropower Unit

It is convincingly proved that hydropower units would provide negative damping in the ultra-low frequency band. This is primarily attributed to the unique water hammer effect of HT and the FPR parameters of the governor. In that case, to simplify the analysis of hydropower units’ impact on ULFOs, we focus solely on the open-loop model of the governor and HT, as illustrated in Figure 1. The open-loop transfer function G_m(s) is derived as the multiplication of the transfer functions of the governor and HT, denoted as G_gov(s) and G_tur(s), which are defined by Equations (1) and (2), respectively.

$$G_{gov}(s)={\displaystyle \frac{(K_P{T}_d+K_D)s^2+(K_P+K_I{T}_{\mathrm{d}})s+K_I}{(b_p{K}_P{T}_d+b_p{K}_D+T_d)s^2+(b_p{K}_P+b_p{K}_I{T}_d+1)s+b_p{K}_I}}$$

(1)

where K_P, K_I, and K_D are the proportional, integral, and differential coefficients of the PID governor, respectively; b_p is the permanent slip coefficient.

$$G_{tur}(s)={\displaystyle \frac{1-T_{w}s}{1+0.5T_{w}s}}$$

(2)

where T_w is the water starting time.

The complex torque coefficient (CTC) method is a common approach for ULFO analysis [27]. By determining the damping characteristics, the nature of ULFO modes can be easily characterized, and the adverse effects of the governor can be quantitatively assessed. Based on the CTC method, the transfer function G_m(s) can be decomposed in Δδ-Δω coordinate system, damping torque K_Dm, and synchronous torque K_Sm, as shown in Equation (3). Moreover, K_Dm is related to the ability to suppress oscillations and is mainly determined by the parameters of HT governor and water starting time, with the variation trend illustrated in Figure 2. Currently, in order to enhance the damping coefficient in the ultra-low frequency band of hydropower units, there is a general increase in the governor coefficient K_P and a decrease in K_I, leading to a decrease in the PFR capability of hydropower units. In the hydro–wind system, the damping torque of the WT can be increased to enhance the overall damping of the system, thereby releasing some of the PFR capability of the hydro turbine. Therefore, in Section 2.3, we established the EUFM to analyze the damping torque of the hydro–wind system.

$$G_{\mathrm{m}}(s)={\displaystyle \frac{\Delta T_{\mathrm{m}}}{\Delta \omega }}=G_{\mathrm{gov}}(s)G_{tur}(s)=K_{\mathrm{Sm}}\Delta \delta +K_{\mathrm{Dm}}\Delta \omega$$

(3)

2.2. The Model of Doubly Fed Induction Generator

The WT discussed in this study are doubly fed induction generators (DFIGs), which are the most widely used wind turbines, consisting of a wound-type asynchronous generator with stator winding directly connected to a fixed-frequency three-phase power grid and a bidirectional back-to-back IGBT voltage source converter installed on the rotor winding. For DFIG with traditional power control methods, it is not feasible to respond to changes in system frequency. Thus, to make it possible for DFIG to participate in system frequency regulation, comprehensive inertia control is popular, including virtual inertia control (VIC) and droop control, by which the external characteristics of DFIG are closest to synchronous generators, as shown in Figure 3. It should be noted that K_d and K_f are the VIC coefficient and droop control coefficient, respectively. DFIG responds rapidly to frequency rate changes through the VIC while also adjusting the electromagnetic power of the WT through the droop control in response to system frequency deviations. In addition to frequency control, it also includes pitch angle control and speed control. When the wind speed changes, the tip speed ratio is adjusted by changing the WT speed to achieve maximum power tracking (MPPT). When the wind speed is higher than the set value, the maximum power is maintained through pitch angle control. These three parts constitute the main DFIG control system.

Similarly, based on the CTC method, the damping torque of the DFIG can be expressed as follows [26]:

$$D_W={\displaystyle \frac{-K_d{\omega }_d^4{a}_i}{{(-{\omega }_d^2+b_i)}^2+{({\omega }_d{a}_i)}^2}}+{\displaystyle \frac{-K_{\mathrm{f}}{\omega }_d^2(-{\omega }_d^2+b_i)}{{(-{\omega }_d^2+b_i)}^2+{({\omega }_d{a}_i)}^2}}$$

(4)

Under varying wind speed conditions, the operational mode of the WT varies, with a_i and b_i representing the coefficients in different modes.

2.3. Extended Unified Frequency Model

The ULFO features longer oscillation periods and lower oscillation frequencies. Thus, it can be assumed that faster dynamic response processes such as rotor angle and AC voltage have decayed. Subsequently, the system frequency deviation is rather small [28]. In that case, based on the power characteristics of hydropower and wind units, respectively. We deduce an EUFM of hydro–wind sending system for ULFO analysis, which is illustrated in Figure 4. And ω is the system average frequency; P_L is the system active load; P_G and P_W are the output active power of HT and WT units, respectively.

Based on EUFM, it can be derived as:

$$-{\displaystyle \frac{\Delta P_L}{\Delta \omega }}=(M_G+M_W+M_T)s+(D_G+D_W+D_T)$$

(5)

where M_G and D_G are the equivalent inertia coefficient and damping coefficient of N hydropower units, respectively; M_T and D_T are the synchronous torque and damping torque of N HT units, respectively; M_W and D_W are the synchronous torque and damping torque of M WT units, respectively. In Equation (5), the summation of D_G, D_W, and D_T represents the total damping torque of the system.

Based on the mechanism analysis, the occurrence of ULFOs is more likely when the damping torque value is negative. Most solutions focus on improving the damping torque of the hydro units by adjusting the governor parameters. However, it is disappointing that the PFR capability of the hydro units is simultaneously reduced, posing potential risks to frequency stability.

3. The Influence of DFIG on ULFO and PFR Performance

To enhance the overall damping torque in hydro–wind systems, increasing the WT’s damping torque D_W has been identified as an effective strategy to mitigate ULFOs. At the same time, it is necessary to consider the frequency regulation of DFIG. This section mainly analyzes the influence of DFIG parameters on damping and PFR performance.

3.1. ULFO Damping Characteristics of DFIG

Based on the aforementioned analysis, the damping torque of DFIG is primarily related to VIC and droop control. At rated wind speed, the impact of K_d and K_f on the damping level in the ultra-low frequency band is illustrated in Figure 5.

It is obvious that in the ultra-low frequency band of 0–0.1 Hz, the VIC with larger K_d would provide a more severe negative damping effect. In contrast, the influence of droop control is delightful in the ultra-low frequency band; specifically, the larger K_f leads to a greater positive damping torque coefficient. Accordingly, to strengthen the damping in the ultra-low frequency band, a novel solution is to minimize K_d and increase K_f of DFIG.

However, regarding PFR performance, excessively small K_d may cause significant frequency rate changes and maximum frequency deviations. Conversely, an excessively large K_f may promote WT to release much rotor kinetic energy when participating in frequency regulation, resulting in severe secondary frequency drops during PFR, which we will discuss in Section 3.2. Therefore, it is essential to tune the parameters appropriately to balance the damping in the ultra-low frequency band and the performance of PFR.

3.2. PFR Performance of DFIG

The control methods for DFIG to participate in PFR include power load shedding control and rotor kinetic energy control. The former diminishes the economic viability of wind farms and burdens long-term economic losses. In this paper, we employ the later control for DFIG. To characterize the PFR performance of DFIG, the typical frequency response curve is illustrated in Figure 6; some important indicators, including the rate of change in frequency (RoCoF), maximum frequency deviation Δf_max1, and maximum secondary frequency drop deviation Δf_max2 are investigated. It should be noted that when the power grid oscillates at t₀, DFIG conducts PFR from t₀ and exits at t_off, causing a secondary frequency drop while restoring rotor speed.

3.2.1. The Rate of Change in Frequency

The magnitude of RoCoF is dependent on system inertia and load increment, which can be expressed as:

$$\mathrm{RoCoF}={\displaystyle \frac{\Delta P_L{f}_0}{2H_{sys}{S}_b}}$$

(6)

where ΔP_L is the system load increment, f₀ is the initial frequency, S_b is the system capacity, and H_sys is the system equivalent inertia.

Moreover, H_sys can be expressed as:

$$H_{sys}=\frac{{\displaystyle \sum _{i=1}^N{H}_i{P}_i}+{\displaystyle \sum _{j=1}^M{H}_j{P}_j}}{S_b}$$

(7)

where N and M are the numbers of HT and WT, respectively; H_i and H_j are the inertia time constants of the i-th and j-th HT and WT, respectively; P_i and P_j are the capacities of the i-th and j-th HT and WT, respectively.

Generally, the virtual inertia time constant of DFIG is determined by the virtual inertia coefficient K_d. When the inertia time constant of HT remains constant, RoCoF decreases with an increase in K_d.

3.2.2. The Maximum Frequency Deviation

The maximum frequency deviation Δf_max1 of the system can be derived from the rotor motion equation [24], as shown in the specific formula Equations (8)–(14).

$$\Delta f_{\max 1}=1-M{e}^{{\displaystyle \frac{\alpha (\pi /2-\theta -\gamma )}{\omega }}}\cos (\pi /2-\gamma )-C$$

(8)

$$\alpha =-{\displaystyle \frac{T_J+K_d+K_f{T}_1}{2T_1(T_J+K_f)}}$$

(9)

$$\omega =\sqrt{{\displaystyle \frac{K_1+K_f}{T_1(T_J+K_d)}}}-{\alpha }^2$$

(10)

$$M={\displaystyle \frac{\Delta P_L}{\omega (T_J+K_d)}}\sqrt{{\displaystyle \frac{K_1}{K_1+K_f}}}$$

(11)

$$\theta =\arccos [{\displaystyle \frac{\omega (T_J+K_d)}{\sqrt{K_1(K_1+K_f)}}}]$$

(12)

$$C=1-{\displaystyle \frac{\Delta P_L}{K_1+K_f}}$$

(13)

$$\gamma =\arctan (\omega /\alpha )$$

(14)

where T_J is the inertia constant of the hydropower unit; T₁ and K₁ are the coefficients of the identified low-order model of the WT governor.

The maximum deviation of system frequency with changes in DFIG parameters K_d and K_f is shown in Figure 7. It indicates that K_d and K_f can both reduce the maximum frequency deviation.

3.2.3. Maximum Secondary Frequency Drop Deviation

Besides RoCoF and maximum frequency deviation, it is worth mentioning that the risk of secondary frequency drop is also of great concern. Under rotor kinetic energy control, the DFIG can release rotor kinetic energy to provide active power support when the system frequency drops. Specifically, when K_f and K_d are larger, the output active power of DFIG during PFR will be greater, which means more rotor kinetic energy or PFR ability will be released. However, when the DFIG exits PFR, more active power will be absorbed from the power grid to restore the speed, resulting in a larger secondary frequency drop [29].

The secondary frequency drop Δf_max2 is determined as follows:

$$\Delta f_{\max 2}=1-M_1{e}^{{\displaystyle \frac{\alpha (\pi /2-\theta -\gamma )}{\omega }}}\cos (\pi /2-\gamma )-f_{off}+\frac{\Delta P_u}{K_1}$$

(15)

$$M_1={\displaystyle \frac{\Delta P_u}{\omega (T_J+K_d)}}\sqrt{{\displaystyle \frac{K_1}{K_1+K_f}}}$$

(16)

The system frequency that DFIG exits PFR is given by:

$$f_{off}=M{e}^{\alpha (t_{off}-t_0)}\cos [\omega (t-t_0)+\theta ]+C$$

(17)

where ΔP_u is the unbalanced power when the DFIG exits PFR.

When the PFR exiting time t_off of DFIG is constant, the relationship between the secondary frequency drop with K_d and K_f is investigated in Figure 8. The deviation of the secondary frequency drop becomes severe when K_d increases from 0 to 20 and 40 to 60, while the impact is not significant between 20 and 40. The frequency drop initially decreases when K_f is 0~8, and it will increase with the larger K_f.

4. Comprehensive Control Optimization of DFIG for ULFO Suppression and PFR

DFIG provides a significant solution to support frequency control. This section presents a comprehensive parameter optimization method for DFIG engaged in balancing the damping of ULFOs and PFR performance.

4.1. Objective for ULFO Damping Control of DFIG

For ULFO suppression, the damping level is an important metric indicator. We propose the average value D_f of the damping torque coefficient within a certain ultra-low oscillation frequency band, which is represented as:

$$D_f={\displaystyle \sum _{i=1}^n\frac{D_W(f_i)}{n}}$$

(18)

where n is the total number of frequency sampling points selected within the ultra-low oscillation frequency band; f_i is the system frequency value of the i-th sampling point.

The objective function for ULFO damping control of DFIG can be concluded as follows:

$$\{\begin{array}{c}{A}_1(K)=D_f\\ K=[K_d,K_f]\end{array}$$

(19)

4.2. Objective for PFR Performance of DFIG

In order to enhance the PFR performance, we put forward a PFR performance indicator A₂(K), which comprehensively considers RoCoF, maximum frequency deviation, and maximum secondary frequency drop deviation, which is represented by Equation (20).

$$A_2(K)=k_1\mathrm{RoCoF}+k_2\Delta f_{\max }+\Delta$$

(20)

where k₁ and k₂ are the weights for RoCoF and total maximum frequency deviation, respectively; Δ is the penalty function; Δf_max is the larger value between the maximum frequency deviation and the maximum secondary frequency drop deviation, i.e., It is expressed in Equation (21).

$$\Delta f_{\max }=\max [\Delta f_{\max 1},\Delta f_{\max 2}]$$

(21)

According to the Technical Requirements and Test Procedures for the PFR of WT in China [30], the PFR dynamic performance of WTs should meet the following requirements.

(1)

The lag time of PFR should not exceed 2 s;

(2)

The rise time of PFR should not exceed 9 s;

(3)

The adjustment time of PFR should not exceed 15 s;

To comply with the technical requirements and test procedures for PFR, when the WT parameters are designed unreasonably and do not meet the regulatory requirements, Δ is set to a large value to discard the set of parameters.

4.3. Comprehensive Control Optimization Model and Solution

The comprehensive, objective function for optimizing main control parameters, including K_d and K_f, considering the damping level and PFR performance. Thus, the comprehensive control optimization model is defined as follows:

$$\{\begin{array}{c}\min A(K)=k_3{A}_1(K)+k_4{A}_2(K)\\ T_L\le 2s\\ T_R\le 9s\\ T_A\le 15s\end{array}$$

(22)

where k₃ and k₄ are the weights for the damping torque coefficient index and PFR performance index. T_L, T_R, and T_A are the lag, rise, and adjustment times of PFR, respectively. To ensure fairness in parameter comparison, each parameter must undergo normalization and dimensionless processing.

To solve the proposed comprehensive control model, we adopt the PSO algorithm to obtain optimized control parameters of DFIG. PSO is an evolutionary computation technique that originated from the study of bird flock foraging behavior. It finds wide application in the field of electrical engineering [31]. The PSO simulates the sharing of information among individuals in a bird flock; each particle represents a solution, and it continuously adjusts its velocity and position to search for the optimal solution.

In a population with m particles, each particle having n dimensional variables, the position and flight velocity of the i-th particle in the k-th iteration process are denoted as X_i^k = [x_i,1^k,x_i,2^k, … ,x_i,n^k] and V_i^k = [v_i,1^k,v_i,2^k, … ,v_i,n^k], respectively. The individual best position P_i^k = [p_i,1^k,p_i,2^k, … ,p_i,n^k] and the global best position G_i^k = [g_i,1^k,g_i,2^k, … ,g_i,n^k] of the population are determined by calculating the fitness value of the objective function for each particle, and the velocity and position of each particle in the next iteration are computed as follows:

$$\{\begin{array}{c}{v}_{i,j}^{k+1}=\sigma v_{i,j}^k+c_1{r}_1(p_{i,j}^k-x_{i,j}^k)\\ \begin{array}{cc}+c_2{r}_2(g_{i,j}^k-x_{i,j}^k)& j=1,2,\cdots ,n\end{array}\\ x_{i,j}^{k+1}=x_{i,j}^k+v_{i,j}^{k+1}\end{array}$$

(23)

where r₁ and r₂ are random numbers uniformly distributed on the interval [0, 1]; c₁ and c₂ are learning factors that determine the positional information characteristics of the population, with larger learning factors leading to shorter local optimization times; σ is the inertia weight, which is calculated as follows:

$$\sigma ={\sigma }_{\max }-{\displaystyle \frac{{\sigma }_{\max }-{\sigma }_{\min }}{K_{\max }}}k$$

(24)

where K_max is the maximum number of iterations, σ_max is set to 0.9, and σ_min is set to 0.4.

The optimization scheme based on the PSO algorithm is illustrated in Figure 9.

5. Simulations and Verification

To validate the effectiveness of the proposed control method, two simulation systems are constructed using PSCAD v50/EMTDC software. Case 1 involves a hydro–wind two-machine islanding system, while case 2 represents an asynchronous hydro–wind system transmitted by HVDC in Southwest China.

5.1. Case 1: Hydro–Wind Two-Machine Islanding System

In case 1, a hydro–wind two-machine islanding system is presented. Detailed parameters are shown in

Table 1. Simulation parameters of the two-machine islanding system.

Symbol of Islanding System	Value	Symbol of Islanding System	Value
KP	3.6	Tw	2.4 s
KI	1.9	Vspeed	11 m/s
KD	0.5	Pwind	30.5 MW
TJ	10 s	Phy	69.5 MW
Bp	0.04	Pload	100 MW

.

Based on the analysis above, we select four typical control parameters to simulate the effect of DFIG, which are given below.

Par 1:

No additional control, K_d = 0, K_f = 0.

Par 2:

Only considering damping optimization, K_d = 0, K_f = 50.

Par 3:

Focusing on PFR performance only (Method of Reference [24]), K_d = 35.2, K_f = 25.4.

Par 4:

Comprehensive optimization considering both damping and frequency response, K_d = 15.3, K_f = 30.4.

At 10 s, there is a sudden increase of 10 MW load, and then the system experiences the risk of ULFOs. At 20 s, the DFIG exits frequency regulation. The frequency response curves of the system under different DFIG parameters are shown in Figure 10. To identify the oscillation modes, including damping ratio and oscillation frequency, the TLS-ESPRIT method is applied. When DFIG participates in frequency regulation under different parameters, the corresponding oscillation modes are shown in

Table 2. Case 1: Frequency deviation and damping with different DFIG parameters.

DFIG Parameters	Δfmax (Hz)	Damping Ratio 1	Damping Ratio 2
Par 1	0.092	0.201	0.201
Par 2	0.064	0.434	0.211
Par 3	0.054	0.287	0.220
Par 4	0.056	0.357	0.253

. In addition, damping ratios 1 and 2 represent the damping ratio before and after the DFIG exits PFR, respectively.

With Par 1, the DFIG does not participate in system frequency regulation, and it exhibits significant oscillation amplitudes that are difficult to suppress. Par 2 results in higher damping during the DFIG’s involvement in PFR, but the excessive release of rotor kinetic energy after exiting from PFR causes a significant secondary frequency drop, which makes subsequent oscillations challenging to mitigate. Par 3 yields optimal RoCoF and maximum frequency deviation, but with relatively low damping, oscillations persist and are slow to dampen. Par 4 significantly enhances damping compared to Par 3, with damping ratios 1 and 2 increasing by 24.4% and 15.0%, respectively. Meanwhile, it minimizes the impact on PFR performance, leading to a better suppression effect of ULFOs.

5.2. Case 2: Asynchronous Hydro–Wind System Transmitted by HVDC

In case 2, an asynchronous hydro–wind system transmitted by HVDC is presented as shown in Figure 11. The asynchronous interconnection operation situation is simulated, which is more rational to the occurrence scenario of ULFOs. Detailed parameters are shown in

Table 3. Simulation parameters of asynchronous hydro–wind system transmitted by HVDC.

Symbol of Sending System	Value	Symbol of Sending System	Value
KP	4	Tw	2.5 s
KI	2	Vspeed	11 m/s
KD	0.5	Pdc	3000 MW
TJ	8 s	Pwind	935 MW
Bp	0.04	Phy	2885 MW

.

The small disturbance is a sudden increase of 600 MW load at 10 s, and then ULFOs are excited. Figure 12 illustrates the frequency response curves under different DFIG parameters, while

Table 4. Case 2: Frequency deviation and damping with different DFIG parameters.

DFIG Parameters	Δfmax (Hz)	Damping Ratio 1	Damping Ratio 2
Par 1	0.231	0.175	0.175
Par 2	0.129	0.382	0.177
Par 3	0.110	0.257	0.189
Par 4	0.111	0.321	0.218

presents the corresponding damping ratios when DFIG participates in frequency regulation.

In Case 2, four selected parameters are as follows:

Par 1:

No additional control, K_d = 0, K_f = 0.

Par 2:

Consideration of damping optimization only, K_d = 0, K_f = 70.

Par 3:

Focus on PFR performance only (Method of Reference [24]), K_d = 43.2, K_f = 30.6.

Par 4:

Comprehensive optimization of damping and PFR performance, K_d = 20.5, K_f = 36.8.

It is obvious that when the DFIG is not involved in system frequency regulation, the system exhibits large oscillation amplitudes and prolonged oscillation durations. When DFIG participates in system frequency regulation with Par 2, it results in a significant secondary frequency drop. Moreover, Par 3 achieves optimal RoCof and maximum frequency deviation but exhibits relatively low oscillation damping, making it difficult to quickly suppress oscillations. Finally, compared to Par3, optimized Par 4 based on the proposed control method shows that the maximum frequency deviation increases by only 0.001 Hz, and the oscillation damping increases by 24.9% and 15.3%, respectively.

Based on the simulations of both Case 1 and Case 2, it can be further confirmed that the comprehensively optimized control method proposed in this paper significantly enhances the ability to suppress ULFOs while guaranteeing PFR performance. Meanwhile, it retains the frequency regulation of hydropower units.

6. Conclusions

This paper establishes an EUFM of hydro–wind power sending system and investigates the specific impact of WT control parameters on ULFOs and PFR characteristics. Then, a novel optimization objective function considering damping in the ultra-low frequency band and PFR is formulated and solved using the particle swarm optimization (PSO) algorithm. Simulation results verify that the optimized WT parameters can simultaneously address the ULFO problem and guarantee PFR performance. The conclusions can be summarized as follows:

(1)

The negative damping of hydropower units is the fundamental cause of ULFOs, and unlike the LFO issue, ULFO is within the scope of frequency stability.

(2)

In the ultra-low frequency band of 0–0.1 Hz, the VIC provides a negative damping torque coefficient. In contrast, the droop control provides a positive damping torque coefficient.

(3)

Regarding the influence of DFIG parameters on PFR performance, RoCoF decreases with an increase in K_d; larger K_d and K_f can both reduce the maximum frequency deviation, but the values of K_d and K_f should not be set too large to prevent excessive secondary frequency drop.

(4)

The integration of wind power provides a viable option for ULFO suppression. The objective function is formulated to balance both ULFO suppression and PFR performance. Simulation results verify that the optimized WT parameters improve more than 15% damping ratio while guaranteeing PFR performance.

(5)

In recent years, the control scheme of energy storage systems has been improved [32]. Future research will focus on further investigating wind energy storage systems, considering the strategy of coordinating wind turbines and energy storage to suppress ULFOs.