Time Delay Stability Analysis and Control Strategy of Wind Farm for Active Grid Frequency Support

Abstract

With the rapid development of wind turbines and rising penetration levels, grid codes now require wind farms to provide active frequency support. However, time delays in fast power response reduce the stability of system frequency modulation. This study focuses on integrated inertia control and droop control of wind turbines with explicit consideration of time delays. First, the frequency modulation process is analyzed, and the main sources of time delay are identified. A system frequency response model is then developed, incorporating the time delay link into the state-space equations. Based on this model, frequency-domain and linear matrix inequality methods are applied to analyze delay-independent stability and time delay margins of wind turbines. A PI controller for the synchronous unit is designed, and compensation parameters for wind turbine delay are calculated to enhance system stability. Simulation results show that with a wind penetration level of 50%, the system becomes unstable when the delay reaches 0.32 s. By applying the proposed strategy, stability can be maintained even with a delay of 0.5 s. These results confirm the effectiveness of the proposed strategy and highlight its potential for improving frequency regulation in high-renewable power systems.

1. Introduction

Wind and solar power lack inertia and active support, which weakens system dynamics during disturbances [1,2,3]. Power converters block the transfer of turbine inertia to the grid. Also, they operate in Maximum Power Point Tracking (MPPT) mode, which fails to effectively suppress the maximum rate of frequency change and maximum frequency deviation during disturbances [4].

The doubly-fed induction generator (DFIG) is the dominant wind power technology. Its frequency modulation (FM) methods fall into power reserve control [5,6] (including overspeed [7] and pitch angle control [8]) and rotor kinetic energy control (including virtual synchronous machine technology, virtual inertia control [9,10], droop control [11], and comprehensive inertia control). Reference [12] provides an emergency FM scheme for isolated DFIG-dominated systems using nonlinear regulators; Reference [13] describes an optimal reserve allocation to maximize kinetic energy storage for droop-based primary frequency regulation (PFR); Reference [14] provides a fuzzy-adaptive virtual inertia strategy for DFIGs and energy storage to address reliability–frequency deviation trade-offs in PFR; and Reference [15] proposes a system of adaptive droop feedback control which can provide dual frequency regulation capability.

In practice, rotor kinetic energy control is a control process based on the deviation of the frequency state, so there must be a time delay in the response. While extensive research has been conducted on power systems with time delays [16,17], few studies have focused on time delays in wind power generation processes. Reference [18] discusses the response delay of the virtual inertia of grid-following devices and analyzes its influence on transient indicators. Reference [19] analyzes the influence of virtual inertia delay based on simulation results and concludes that a larger delay is actually beneficial to the frequency modulation effect. Reference [20] reveals the influence of the delay link in wind turbine (WT) frequency modulation at a theoretical calculation level. Reference [21] approximates the time delay using the first-order Taylor and Pade methods and, on this basis, derives a frequency-modulation aggregation model considering the time delay. Reference [22] briefly discusses the impact of time delay on the minimum frequency and the output power of wind turbines and thermal power units, providing theoretical guidance for the design of converter control strategies. However, most existing studies simplify the delay link as a first-order inertia element.

When droop control is applied, the system can be represented as a retarded-type delay system, in which the derivative of the state depends only on delayed states. For such systems, the frequency-domain method is appropriate, since it allows direct derivation of delay-independent stability conditions and explicit computation of delay margins, which cannot be conveniently obtained by classical Nyquist or Routh-Hurwitz criteria. In contrast, when integrated inertia control is included, the model becomes a neutral-type delay system, where derivatives of delayed states appear. Neutral systems pose additional challenges, as their stability is affected by high-frequency root distributions. The methods described above are not well-suited for such systems, whereas Lyapunov functional approaches can be applied but often yield conservative results. Therefore, the linear matrix inequality (LMI) method is employed, since it provides a rigorous and computationally efficient framework for establishing stability criteria, calculating delay margins, and synthesizing controllers. A summary comparison of different methods for retarded and neutral systems is presented in

Table 1. Comparison of stability analysis methods for droop control and integrated inertia control.

Method	Retarded Delay System (Droop Control)	Neutral Delay System (Integrated Inertia Control)	Limitation
Frequency-domain method	√	×	derivation complex for high-order models
Routh–Hurwitz stability criterion	√	×	approximation introduces errors no exact delay margins
Nyquist stability criterion	√	×	need approximation cannot directly compute delay margins
Lyapunov functionals	×	√	conservative depends on functional construction not convenient for controller design
LMI	×	√	conservative relies on numerical solvers, may be heavy for large-scale systems
Approximation + Root locus	√	×	accuracy depends on order of approximation may miss high-frequency dynamics

√ indicates that the row corresponding method is applicable to the column corresponding control method; × indicates that it is not applicable.

to further clarify this choice.

This paper investigates comprehensive inertia control and droop control for wind turbines. It first analyzes delay components in FM and derives a state-space equation for the system frequency response (SFR) with time delay. Based on this model, frequency-domain and LMI methods are applied to analyze delay-independent stability and time delay margins. A PI controller for synchronous units and compensation parameters for wind turbines are then designed using the LMI method. Simulations validate the proposed strategy. In addition,

Table 2. All key symbols and abbreviations with definitions.

Symbol	Definition
f	System frequency
Δf	Frequency deviation
dΔf/dt	Rate of change of frequency
Pad	Additional power output of WT
PMPPT	Power output under maximum power point tracking
Pe	Electrical output power of WT
PREF	Reference active power of WT
ΔPW, ΔPW-i	Additional power injected by wind farm/WT i
ΔPG	Change in synchronous generator output power
ΔPL	Load disturbance power
Kp, Kp-i	Virtual inertia control parameters of wind farm/WT i
Kd, Kd-i	Droop control parameters of wind farm/WT i
τ	Time delay
τmax	Maximum permissible time delay
D	System damping coefficient
R	Primary frequency modulation droop coefficient
M	System inertia time constant
TR	Reheat time constant
FH	High-pressure cylinder power proportion
TG	Turbine governor time constant
TCH	High-pressure cylinder steam chamber volume time constant
α, αi	Wind power penetration level
n	Number of WTs
ΔPv	No practical significance, only convenient for calculation
ΔPm	No practical significance, only convenient for calculation

lists all key symbols and abbreviations with their definitions, allowing readers to easily reference technical notations.

2. SFR Model Considering Time Delay

In this section, the causes of time delays in the FM process of wind turbines (WTs) are first analyzed. Considering the above-mentioned causes of time delays, a SFR model of a wind farm is established.

2.1. The Causes of Time Delay

In practice, there will be response time delays for WTs applying FM, including:

• Frequency measurement: When WTs apply FM through rotor kinetic energy control, they take the frequency signal as input to change their output electric power. The frequency measurement requires 5–10 power frequency cycles to ensure accuracy.
• Artificial time delay: When WTs apply FM through virtual inertia control or comprehensive inertia control, the input signals include the differential of the frequency. Since the ideal differential signal cannot be realized, a low-pass filter is generally introduced to prevent the sensitive false triggering of FM due to noise or incorrect command generation. The low-pass filter produces a certain time delay.
• Generation of command signal: After the substation controller collects the frequency deviation signal and the differential signal, it takes a certain amount of time to generate the power command signal according to the established algorithm.
• Communication link: It takes a certain amount of time for the frequency signal measured at the point of connection of the wind farm to the grid to be transmitted to the substation controller and for the control signal to be sent to the rotor-side converter of the WT.
• Control implementation: After the WT receives the control signal, it takes a certain amount of time to respond to the given input control signal.

According to the data of a certain wind farm, the frequency measurement requires 5 power-frequency cycles. The time delay to generate the command signal is between 30–40 ms, and the total time delay in the control implementation and communication links is approximately 100 ms. Therefore, when WTs applies FM, the time delay is in the order of hundreds of milliseconds, reaching 300–500 ms. The factors of the time delay in the FM process of WTs are shown in Figure 1.

Similar measurements from other operating plants indicate that the overall delay in frequency modulation typically falls within 300–500 ms. Reference [19] reports 350–500 ms total delay for virtual inertia control response in commercial DFIG wind farms, while reference [20] observed delays in the range of 300–480 ms in a provincial grid-connected wind power base. Reference [23] points out that the measurement delay is assumed to be 100 ms, and the power order delay is assumed to be 0.5 s. Reference [24] points out that high delays in power commands can cause wind farm oscillations, emphasizing that communication segments can reach hundreds of milliseconds and are sensitive to stability. Reference [25] points out that to measure, compute, and transmit the frequency signal to the wind turbine plant controller, a response time on the scale of hundreds of milliseconds is required.

2.2. System Frequency Control Framework

Under normal circumstances, WTs operate in MPPT mode. When a frequency deviation event occurs, WTs apply FM based on current frequency deviation signal Δf and frequency differential dΔf/dt. The additional power signal Pad is obtained through the K_p and K_d gains, respectively, and then added to power P_MPPT, calculated by MPPT, to get given power signal P_REF of the converter for control. After the WT receives the control signal, it outputs FM power P_e after the required response time. To simplify the model calculation, it is assumed that P_e is equal to P_REF. The calculation formula of P_e is shown in (1) for cases where the WT applies FM through comprehensive inertia control.

$$\left\{\begin{array}{l}{P}_e=P_{REF}=P_{MPPT}+P_{ad}\\ P_{ad}=-K_p\Delta f-K_d\cdot d\Delta f/dt\end{array}\right.$$

(1)

According to the analysis in Section 2.1, there are time delay factors in the process of WTs applying FM which lead to a delay in the conversion from frequency deviation signal Δf to FM power P_e output by the WT, thus affecting the dynamic performance of the SFR. Therefore, compensated time t − τ is used to replace the t related to the frequency signal, and the calculation formula of P_e, considering the time delay factors, is obtained, as shown in (2), where τ is the total time delay in the FM process of the WT.

$$\left\{\begin{array}{l}{P}_e=P_{REF}=P_{MPPT}+P_{ad}\\ P_{ad}=-K_p\Delta f(t-\tau )-K_d\cdot d\Delta f(t-\tau )/dt\end{array}\right.$$

(2)

Thus, the system frequency control framework for WTs participating in grid FM is obtained as shown in Figure 2. It should be emphasized that the system frequency dynamics in this study are represented by the Center of Inertia (COI) model, which reflects the weighted average frequency of all synchronous machines. This modeling choice facilitates tractable analyses of time-delay stability issues. However, the COl model is a simplification and does not account for generator heterogeneity, spatial variations in frequency, or network topology effects. In practice, frequency measurements are location-dependent, and local responses may deviate from the COl average. Therefore, the results should be interpreted as providing general system-level insights rather than detailed locational predictions. Future work will incorporate more detailed network-based models to address these limitations.

The parameters of the traditional synchronous generator sets are all defined according to the typical values, as described in reference [26]: system damping coefficient D = 2; primary frequency modulation droop coefficient R = 0.05; system inertia time constant M = 8.0 s; reheat time constant T_R = 7.0 s; high-pressure cylinder power proportion F_H = 0.3; turbine governor time constant T_G = 0.2 s; and high-pressure cylinder steam chamber volume time constant T_CH = 0.3 s. The synchronous unit model uses a reheat steam turbine unit, whose block diagram is shown in Figure 3.

Based on the above framework, the system frequency control equation under droop control can be obtained, as shown in (3). The system frequency control equation under comprehensive inertia control is presented in (4).

$$\left\{\begin{array}{l}\Delta P_{W-i}={\alpha }_i{K}_{P-i}\Delta f(t-\tau )\\ \Delta P_W={\displaystyle \sum _{j=1}^n\Delta P_{W-j}}\\ \Delta P_G={\displaystyle \frac{(1-\alpha )}{R}}\cdot {\displaystyle \frac{1}{1+T_{G}s}}{\displaystyle \frac{1}{1+T_{CH}s}}{\displaystyle \frac{(1+F_H{T}_{R}s)}{1+T_{R}s}}\cdot \Delta f\\ \Delta f={\displaystyle \frac{1}{(1-\alpha )(Ms+D)}}\cdot (\Delta P_L-\Delta P_W-\Delta P_G)\end{array}\right.$$

(3)

$$\left\{\begin{array}{l}\Delta P_{W-i}={\alpha }_i(K_{P-i}\cdot +K_{d-i}s)\Delta f(t-\tau )\\ \Delta P_W={\displaystyle \sum _{j=1}^n\Delta P_{W-j}}\\ \Delta P_G={\displaystyle \frac{(1-\alpha )}{R}}\cdot {\displaystyle \frac{1}{1+T_{G}s}}{\displaystyle \frac{1}{1+T_{CH}s}}{\displaystyle \frac{(1+F_H{T}_{R}s)}{1+T_{R}s}}\cdot \Delta f\\ \Delta f={\displaystyle \frac{1}{(1-\alpha )(Ms+D)}}\cdot (\Delta P_L-\Delta P_W-\Delta P_G)\end{array}\right.$$

(4)

where α_i represents the output ratio of WT i in active power distribution, n is the total number of wind turbines, and the sum of the output ratios of each WT is α. PW denotes the total change in wind farm output, and K_P-i and K_d-i are the comprehensive inertia control parameters of WT i. ΔP_G represents the aggregated output of traditional synchronous generators. On this basis, P_v and P_m are introduced and defined, as shown in (5) and (6):

$$s\Delta P_v={\displaystyle \frac{1-\alpha }{R{T}_G}}\Delta f-{\displaystyle \frac{1}{T_G}}\Delta P_v$$

(5)

$$s\Delta P_m={\displaystyle \frac{1}{T_{CH}}}\Delta P_v-{\displaystyle \frac{1}{T_{CH}}}\Delta P_m$$

(6)

Considering wind farm aggregation and expressing ΔPw as Δf(t − τ), the matrix-form state space equation for system frequency control under droop control is obtained, as shown in (7), where x_d denotes x(t − τ).

$$\left\{\begin{array}{l}\dot{x}=Ax+A_1{x}_d+Bu\\ x={\left[\Delta f,\Delta P_V,\Delta P_M,\Delta P_G\right]}^T,u=\Delta P_L\\ A=\left[\begin{array}{cccc}-{\displaystyle \frac{D}{M}}& 0& 0& -{\displaystyle \frac{1}{(1-\alpha )M}}\\ {\displaystyle \frac{1-\alpha }{R{T}_G}}& -{\displaystyle \frac{1}{T_G}}& 0& 0\\ 0& {\displaystyle \frac{1}{T_{CH}}}& -{\displaystyle \frac{1}{T_{CH}}}& 0\\ 0& {\displaystyle \frac{F_H}{T_{CH}}}& {\displaystyle \frac{T_{CH}-T_R{F}_H}{T_{CH}{T}_R}}& -{\displaystyle \frac{1}{T_R}}\end{array}\right]\\ A_1=\left[\begin{array}{c}-{\displaystyle \frac{\alpha K_p}{(1-\alpha )M}}\end{array}\right]e_1{e}_1^T\\ B={\displaystyle \frac{1}{(1-\alpha )M}}{e}_1\end{array}\right.$$

(7)

The matrix-form state space equation for system frequency control under comprehensive inertia control is presented in (8).

$$\left\{\begin{array}{l}\dot{x}-C{\dot{x}}_d=Ax+A_1{x}_d+Bu\\ C={\displaystyle \frac{\alpha K_d}{(1-\alpha )M}}{e}_1{e}_1^T\end{array}\right.$$

(8)

3. Stability Criteria of WT Frequency Regulation with Time Delay

The presence of time delay can deteriorate the system transient performance, and in severe cases, lead to system instability. Under the current background of increasing wind power penetration, it is necessary to determine the maximum time delay that a wind farm can tolerate during its application of FM.

Currently, methods for judging stability margins of time delay systems include frequency-domain and time-domain approaches [27]. In this section, for WT FM under droop control, since the state space equation has a low dimension, the frequency-domain method is adopted to analyze the time delay margin quickly and effectively without considering conservatism. However, due to the existence of differential terms of past values of state variables in the system frequency control state space equation under comprehensive inertia control, which form neutral functional differential equations, the frequency-domain method is no longer applicable. Therefore, the LMI approach in the time-domain method is used.

3.1. Time Delay Stability of Droop Control

The stability of the time delay equation can be determined by the distribution of its characteristic roots. For the time delay (7), a characteristic equation is presented (9).

$$\det (sI-A-A_1{e}^{-\tau s})=s^4+a_3{s}^3+a_2{s}^2+a_{1}s+a_0+(b_3{s}^3+b_2{s}^2+b_{1}s+b_0)e^{-\tau s}$$

(9)

The coefficients in the equation are defined in (10) and (11).

$$\left\{\begin{array}{l}{a}_3={\displaystyle \frac{D}{M}}+{\displaystyle \frac{1}{T_G}}+{\displaystyle \frac{1}{T_{CH}}}+{\displaystyle \frac{1}{T_R}}\\ a_2={\displaystyle \frac{D}{M{T}_G}}+{\displaystyle \frac{D}{M{T}_{CH}}}+{\displaystyle \frac{D}{M{T}_R}}+{\displaystyle \frac{1}{T_G{T}_{CH}}}+{\displaystyle \frac{1}{T_G{T}_R}}+{\displaystyle \frac{1}{T_{CH}{T}_R}}\\ a_1={\displaystyle \frac{1}{T_G{T}_{CH}{T}_R}}+{\displaystyle \frac{D}{M{T}_G{T}_{CH}}}+{\displaystyle \frac{D}{M{T}_{CH}{T}_R}}\\ +{\displaystyle \frac{D}{M{T}_G{T}_R}}+{\displaystyle \frac{F_H}{RM{T}_G{T}_{CH}}}\\ a_0={\displaystyle \frac{D}{M{T}_G{T}_{CH}{T}_A}}+{\displaystyle \frac{1}{RM{T}_G{T}_{CH}{T}_R}}\end{array}\right.$$

(10)

$$\left\{\begin{array}{l}{b}_3={\displaystyle \frac{\alpha k_p}{(1-\alpha )M}}\\ b_2={\displaystyle \frac{\alpha K_P}{(1-\alpha )M}}({\displaystyle \frac{1}{T_G}}+{\displaystyle \frac{1}{T_{CH}}}+{\displaystyle \frac{1}{T_R}})\\ b_1={\displaystyle \frac{\alpha K_P}{(1-\alpha )M}}({\displaystyle \frac{1}{T_G{T}_{CH}}}+{\displaystyle \frac{1}{T_G{T}_R}}+{\displaystyle \frac{1}{T_{CH}{T}_R}})\\ b_0={\displaystyle \frac{\alpha K_P}{(1-\alpha )M}}\cdot {\displaystyle \frac{1}{T_G{T}_{CH}{T}_R}}\end{array}\right.$$

(11)

When (9) has no characteristic roots in the right half of the complex plane or on the imaginary axis, the system is stable. If the system remains stable for all τ, it indicates delay-independent stability. It is worth noting that a only appears in the time delay term, representing a strong correlation between α and the time delay margin.

Due to the presence of time delay τ, exponential terms are introduced into the system characteristic equation. Through Taylor expansion, it can be observed that the system has infinitely many roots. Therefore, the magnitude-phase matching method is considered to avoid directly solving the characteristic equation.

When τ = τ_max, the system reaches critical stability. Substituting s = ±jω into (8) to make its value zero results in (12).

$$\left\{\begin{array}{l}{w}^4-j{a}_3{w}^3-a_2{w}^2+j{a}_{1}w+a_0+(-j{b}_3{w}^3-b_2{w}^2+j{b}_{1}w+b_0)e^{-jw\tau }=0\\ w^4+j{a}_3{w}^3-a_2{w}^2-j{a}_{1}w+a_0+(j{b}_3{w}^3-b_2{w}^2-j{b}_{1}w+b_0)e^{jw\tau }=0\end{array}\right.$$

(12)

which may be defined as follows:

$$b=(-j{b}_3{w}^3-b_2{w}^2+j{b}_{1}w+b_0)e^{-jw\tau }$$

(13)

Thus, τ only exists in the phase of b, which indicates that it is possible to determine the amplitude relationship of b and then derive the phase of b to solve for τ. Starting from (12), the amplitude relationship is obtained, as shown in (14).

$$\left\{\begin{array}{l}{R}_e(b)=a_2{w}^2-w^4-a_0\\ I_m(b)=a_3{w}^3-a_{1}w\\ Ab{s}^2(b)=R_e{(b)}^2+I_m{(b)}^2\end{array}\right.$$

(14)

Considering the amplitude relationship of b from (13), the result is (15).

$$Ab{s}^2(b)={(b_0-b_2{w}^2)}^2+{(b_{1}w-b_3{w}^3)}^2$$

(15)

By solving (14) and (15) simultaneously, an eighth-order equation for ω is derived, from which the sequence {ω*} can be solved. This sequence contains possible solutions and also provides potential imaginary axis crossing points of the characteristic root distribution of (9) as t varies.

When there are no solutions in the set {ω*}, it indicates that the distribution curve of the system characteristic roots with varying time delay does not cross the imaginary axis, meaning the system is delay-independent stable. It should be noted that α only appears in $b_3^2$, $b_2^2$, $b_1^2$, and $b_0^2$, while ${\overline{b}}_3$, ${\overline{b}}_2$, ${\overline{b}}_1$, and ${\overline{b}}_0$ are defined as shown in (16).

Thus, the α quadratic equation in terms of α/(1 − α) is obtained as shown in (17). The solution of this equation gives the delay-independent stability boundary for WTs applying FM.

$$\left\{\begin{array}{l}{\overline{b}}_3={\displaystyle \frac{k_p}{M}}\\ {\overline{b}}_2={\displaystyle \frac{K_P}{M}}({\displaystyle \frac{1}{T_G}}+{\displaystyle \frac{1}{T_{CH}}}+{\displaystyle \frac{1}{T_R}})\\ {\overline{b}}_1={\displaystyle \frac{K_P}{M}}({\displaystyle \frac{1}{T_G{T}_{CH}}}+{\displaystyle \frac{1}{T_G{T}_R}}+{\displaystyle \frac{1}{T_{CH}{T}_R}})\\ {\overline{b}}_0={\displaystyle \frac{K_P}{M}}\cdot {\displaystyle \frac{1}{T_G{T}_{CH}{T}_R}}\end{array}\right.$$

(16)

$${\displaystyle \frac{{\alpha }^2}{{(1-\alpha )}^2}}=\underset{w>0}{\min }{\displaystyle \frac{{(a_2{w}^2-w^4-a_0)}^2+{(a_3{w}^3-a_{1}w)}^2}{{({\overline{b}}_0-{\overline{b}}_2{w}^2)}^2+{({\overline{b}}_{1}w-{\overline{b}}_3{w}^3)}^2}}$$

(17)

When there are solutions in the set {ω*}, starting from (12), the phase relationship is obtained, as shown in (18).

$$arg(b)=arctan({\displaystyle \frac{a_3{w}^3-a_{1}w}{a_2{w}^2-w^4-a_0}})+{\displaystyle \frac{1}{2}}({\displaystyle \frac{a_2{w}^2-w^4-a_0}{1-|a_2{w}^2-w^4-a_0|}})\pi$$

(18)

Considering the amplitude relationship of b from (15), the result is (19).

$$arg(b)=\arctan ({\displaystyle \frac{b_3{w}^3-b_{1}w}{b_0-b_2{w}^2}})+{\displaystyle \frac{1}{2}}(1-{\displaystyle \frac{b_0-b_2{w}^2}{|b_0-b_2{w}^2|}})\pi +w\tau$$

(19)

From this, the sequence of t, {τ*} = {${\tau }_1^{*}$ + 2kπ/ω, ${\tau }_2^{*}$ + 2kπ/ω, ${\tau }_3^{*}$ + 2kπ/ω, ${\tau }_4^{*}$ + 2kπ/ω}, k ∈ Z can be solved. The smallest positive value in the sequence is the stability margin of the system at this time.

3.2. Time Delay Stability of Comprehensive Inertia Control

Under comprehensive inertia control, the state-space equation for system frequency control has a differential link of the past values of state variables, which makes it a neutral functional differential equation. The Lyapunov functional method has yielded numerous results when applied to neutral functional differential equations [28,29].

In this section, the stability criterion for neutral time delay linear continuous systems based on the LMI method is used to obtain the calculation method of the system time delay margin [30].

Theorem 1.

For the neutral time delay system as in (8), given a time delay τ > 0, if there are symmetric positive definite matrices P, Q1, Q2, Q3, Q4, R, Z, and X and matrices N1, N2, N3, N4, N5, and N6 of appropriate dimensions such that (20) holds, the system is asymptotically stable.

$$\Phi <0,\mathsf{\Psi }>0$$

(20)

where Φ and Ψ are symmetric matrices. The elements of Φ and Ψ are defined in (21):

$$\begin{array}{l}{\Phi }_{11}=sym\left(PA+N_1\right)+Q_1+\tau X_{11}\\ {\Phi }_{12}=N_2^T+\tau X_{12},{\Phi }_{13}=P{A}_1-N_1+N_3^T+\tau X_{13}\\ {\Phi }_{14}=N_4^T+\tau X_{14},{\Phi }_{15}=N_5^T+\tau X_{15}\\ {\Phi }_{16}=PC+N_6^T+\tau X_{16},{\Phi }_{17}=A^T(R+\tau Z)\\ {\Phi }_{22}=-Q_1+Q_2+\tau X_{22},{\Phi }_{23}=-N_2+\tau X_{23}\\ {\Phi }_{24}=\tau X_{24},{\Phi }_{25}=\tau X_{25},{\Phi }_{26}=\tau X_{26}\\ {\Phi }_{33}=-Q_2+Q_3-sym\left(N_3\right)+\tau X_{33}\\ {\Phi }_{34}=-N_4^T+\tau X_{34},{\Phi }_{35}=-N_5^T+\tau X_{35}\\ {\Phi }_{36}=-N_6^T+\tau X_{36},{\Phi }_{37}=A_1^T(R+\tau Z)\\ {\Phi }_{44}=-Q_3+Q_4+\tau X_{44},{\Phi }_{45}=\tau X_{45},{\Phi }_{46}=\tau X_{46}\\ {\Phi }_{55}=-Q_4+\tau X_{55},{\Phi }_{56}=\tau X_{56},{\Phi }_{66}=-R+\tau X_{66},\\ {\Phi }_{17}=C^T(R+\tau Z), {\Phi }_{77}= R+\tau Z\\ {\mathsf{\Psi }}_{11}=X, {\mathsf{\Psi }}_{12}={\left[N_1,N_2,N_3,N_4,N_5,N_6\right]}^T, {\mathsf{\Psi }}_{22}=Z\end{array}$$

(21)

By verifying the conditions, the time delay margin of the system can be obtained. It is worth noting that when K_d = 0, the stability under droop control mode is determined. However, due to the inherent conservatism of the Lyapunov functional, the calculated time delay margin is smaller than that obtained via the frequency-domain method.

4. Analysis of Time Delay Stability and Control Strategies

Different classes of compensation controllers addressing time delay have been studied in power systems. A PID controller with derivative filtering was shown to improve transient response, but it was more sensitive to measurement noise and communication delays. Adaptive and robust controllers provide better performance under parameter uncertainties, but their tuning is more complex and requires additional system identification. Predictive control methods can explicitly handle multi-variable constraints and future system trajectories, yet they demand high computational resources and accurate models. This paper employed a PI controller due to its simplicity and practical prevalence.

4.1. PI Controller Design for Dynamic Performance Enhancement

To enhance the dynamic performance of frequency modulation, a PI controller for traditional synchronous machine aggregation is introduced, as shown in Figure 4.

Consequently, the state space equation is obtained as presented in (22), where $K_p^s$ and $K_i^s$ denote the parameters of the introduced PI controller.

$$\left\{\begin{array}{l}\dot{\overline{x}}-\overline{C}{\dot{\overline{x}}}_d=(\overline{A}+K)\overline{x}+{\overline{A}}_1{\overline{x}}_d+\overline{B}u\\ \overline{x}={\left[\begin{array}{cc}{x}^T& {\displaystyle \int \Delta fdt}\end{array}\right]}^T\\ \overline{A}=[\begin{array}{cc}A& 0\\ 1& 0\end{array}],\overline{C}=[\begin{array}{cc}C& 0\\ 0& 0\end{array}]\\ {\overline{A}}_1=[\begin{array}{cc}{A}_1& 0\\ 0& 0\end{array}],\overline{B}={[\begin{array}{cc}{B}^T& 0\end{array}]}^T\\ K=K_1{K}_2{K}_3,K_1={\displaystyle \frac{1-\alpha }{T_G}}{e}_2\\ K_2=[\begin{array}{cc}{K}_p^s& K_i^s\end{array}],K_3={[\begin{array}{cc}{e}_1& e_5\end{array}]}^T\end{array}\right.$$

(22)

To design the PI controller, based on the Theorem 1, a congruence transformation is applied by multiplying both sides of Φ′ with diag[P⁻¹, P⁻¹, P⁻¹, P⁻¹, P⁻¹, P⁻¹, P⁻¹, I, I] to eliminate the nonlinearity in the LMI approach. R + τZ = λP, $\overline{P}$ = P⁻¹, $\overline{Q}i$ = P⁻¹QiP⁻¹ (i = 1, …, 6), $\overline{R}$ = P⁻¹RP⁻¹, $\overline{Z}$ = P⁻¹ZP⁻¹, $\overline{X}$ = P⁻¹XP⁻¹, $\overline{N}i$ = P⁻¹NiP⁻¹ (i = 1, …, 6) are introduced. Next, we define U = K₂K₃P⁻¹ to get Theorem 2.

Theorem 2.

For the neutral time delay system as described in (22), given a time delay τ > 0, if there are symmetric positive definite matrices $\overline{P}$, $\overline{Q}i$ (i = 1, …, 6), $\overline{ R}$, $\overline{Z}$, and $\overline{X}$ and matrices $\overline{N}i$ (i = 1, …, 6) such that (23) holds, then the system is asymptotically stable.

$${\Phi }^{\prime }<0,\mathsf{\Psi }>0$$

(23)

The elements that need to be revised are as shown in (24), and the definitions of the remaining elements are the same as those in (21):

$$\begin{array}{l}{{\Phi }^{\prime }}_{11}=sym\left(A\overline{P}+K_{1}U+{\overline{N}}_1\right)+{\overline{Q}}_1+\tau {\overline{X}}_{11}\\ {{\Phi }^{\prime }}_{13}=A_1\overline{P}-{\overline{N}}_1+{\overline{N}}_3^T+\tau {\overline{X}}_{13}\\ {{\Phi }^{\prime }}_{16}=C\overline{P}+{\overline{N}}_6^T+\tau {\overline{X}}_{16}\\ {{\Phi }^{\prime }}_{17}=\lambda \overline{P}{A}^T,{{\Phi }^{″}}_{37}=\lambda \overline{P}{A}_1^T\\ {{\Phi }^{\prime }}_{67}=\lambda \overline{P}{C}^T, {{\Phi }^{″}}_{77}=-\lambda \overline{P}\\ {{\Phi }^{\prime }}_{19}=\overline{P}{e}_1\end{array}$$

(24)

The parameters of the PI controller can be calculated by:

$$K_2=U{\overline{P}}^{-1}{K}_3^T{(K_3{K}_3^T)}^T$$

(25)

4.2. Additional Control for Time Delay Compensation of Wind Turbines

The PI controller for traditional synchronous machine aggregation can effectively help stabilize the system frequency modulation. However, at the initial stage of a disturbance event, the system frequency drops rapidly. The additional power of the wind turbine is relatively large and seriously lags behind the actual demand. This will cause the system to fall into severe oscillations in the initial stage of frequency modulation. Therefore, it is advisable to introduce additional control for the wind turbine’s frequency modulation to suppress the oscillation problem.

As shown in Figure 5, the red line represents a situation where there is a time delay throughout the entire frequency modulation cycle of the wind turbine, the blue line represents the ideal frequency modulation situation without time delay, and the yellow line represents the situation where the wind turbine lags behind the disturbance event only at the start-up moment. It is worth noting that the wind turbine does not actually output power before t₀. Here, t₀ is the start time of the wind turbine’s frequency modulation, t₁ is the intersection point of the blue and yellow lines, and t₂ is the point where the blue line approaches the red and yellow lines.

The transfer function of the system frequency modulation when the wind turbine is not started is shown in (26).

$$\begin{array}{l}R(s)={\displaystyle \frac{{b^{\prime }}_3{s}^3+{b^{\prime }}_2{s}^2+{b^{\prime }}_{1}s+{b^{\prime }}_0}{s^4+a_3{s}^3+a_2{s}^2+a_{1}s+a_0}}\\ {b^{\prime }}_i={\displaystyle \frac{1}{\alpha K_p}}{b}_i,i=1,2,3,4\end{array}$$

(26)

To calculate the frequency change after the wind turbine applies frequency modulation, the frequency after the time delay can be regarded as a new disturbance. Thus, the calculation framework is obtained as shown in Figure 6. Under this framework, the frequency response is the superposition of several response functions which are applied after the corresponding time delay. The response component after a time delay of nt is called the n-th order time delay component. In this calculation method, although each increase in the time delay leads to an increase in the order of the corresponding response function, the poles of the response functions are the same, so there is no difficulty in the calculation.

When taking into account the frequency-response function with only start-up time delay, it is only necessary to adjust D and M based on (27) and then find R1(s) using (26). Through calculating the difference in frequency responses between the scenario where a time delay exists and the one where only a start-up time delay exists, we can figure out the parameters of the compensation controller.

$$\begin{array}{l}{D}^{\prime }=D+{\displaystyle \frac{\alpha }{1-\alpha }}{K}_p\\ M^{\prime }=M+{\displaystyle \frac{\alpha }{1-\alpha }}{K}_d\end{array}$$

(27)

5. Case Study

By 2060, China’s wind power penetration is projected to reach 44%. Therefore, in the simulations presented in this section, the wind power penetration is set to 50% to validate the proposed strategy. To verify the stability impact of time delay on wind turbines applying FM and the improvement effect of the PI controller on the time delay margin, a two-area system simulation model including traditional synchronous units and wind turbines is built on the Simulink platform.

5.1. Time Delay Effects on System Frequency Stability in Droop Control

In this subsection, the delay-independent stability boundary and time delay margin of wind turbines under droop control are calculated using the frequency-domain method and validated through simulation.

The relationship between the time delay margin of wind turbines under droop control, wind power penetration, and K_p is shown in Figure 7, based on the calculations in Section 2.1. As wind power penetration increases, the overall system inertia decreases, causing the system frequency to change rapidly under disturbances. The buffering effect of synchronous units on the time delay impact of wind turbines applying FM weakens, leading to a reduction in the system’s FM time delay margin. As K_p increases, wind turbines will make more significant power adjustments during FM events, increasing the possibility of excessive frequency regulation. Under time delay effects, wind turbine power adjustments may occur at incorrect time nodes, causing system instability. Therefore, the system’s time delay margin decreases when K_p is large.

The relationship between the delay-independent stability boundary and K_p is shown in Figure 8. As K_p increases, the delay-independent stability boundary decreases. To examine the influence of time delay on the stability of system FM, simulations are carried out with time delays of 0.2 s, 0.5 s, and 1 s, aiming to analyze the dynamic impact on system frequency. The following simulation parameters are set: the initial wind speed of the wind farm is 12 m/s, the output power of the wind farm is 50 MW, the rated power of the wind farm is 100 MW, the value of K_p is 15, the rated power of the synchronous unit is 100 MW (which means α is 50%), and the initial active load of the system is 100 MW. A simulation of a sudden 10 MW load increase event in the system is conducted, and the results are presented in Figure 9.

When a disturbance occurs in the system, wind turbines participate in the system FM through droop control. However, due to the effect of time delay, there is a corresponding lag in the response time. In the early stage of the disturbance, only synchronous units support the frequency, causing the lowest point of the system frequency to decrease as the time delay increases. When the time delay of the wind turbines is less than the time delay margin (at this time, the time delay margin is 828.5 ms), the system frequency and the output power of the wind turbines show slight oscillations as the time delay increases. Conversely, when the time delay exceeds the time delay margin, both the system frequency and the output power of the wind turbines fall into severe oscillations. This clearly shows that time delay will deteriorate the stability of system FM.

5.2. Time Delay Effects on System Frequency Stability in Comprehensive Inertia Control

In this subsection, the LMI method is used to calculate the time delay margin of wind turbines under comprehensive inertia control and to verify the frequency response calculation considering time delay. The relationship between the time delay margin of wind turbines under comprehensive inertia control, wind power penetration, and control parameters K_d and K_p is shown in Figure 10 when α is 50%.

As shown in Figure 10, the system FM time delay margin decreases rapidly with increasing K_d, and its impact on the time delay margin is significantly greater than that of K_p. Therefore, virtual inertia control parameters must be carefully selected to avoid destabilizing system FM.

With K_d = 8 and K_p = 15, and other parameters consistent with Section 4.1, simulations are performed under time delays of 0.5 s.

According to the calculations in Section 3.2, the time delay components of each order under a 0.5 s time delay are shown in Figure 11a. The summation results match the frequency response simulation results without errors caused by Pade approximation or Routh approximation. As seen in Figure 11a, the peak values of higher-order time delay components increase rapidly. This is because the Laplace transform of higher-order components contains high-order terms whose time-domain functions include high-degree terms of τ. With fixed exponential decay rates and small denominator factorial terms, these components cannot suppress the steep peak growth induced by high-degree τ-terms. Figure 11b illustrates the variation of time delay components across different time delays. It can be observed that time delay components of the same order only experience time-domain translation as the delay changes. This explains why under small delays, adjacent-order components balance rapidly without oscillations. When the delay increases, certain time intervals are dominated by specific-order components, leading to oscillations, divergence, and eventual system instability.

5.3. Enhancement Strategies for System Frequency Time Delay Stability

Simulations are performed with K_d = 8, K_p = 15, α = 50%, and τ = 0.5 s to verify the effectiveness of the time delay margin improvement method. Under these conditions, the system’s time delay margin calculated via Theorem 1 of the LMI method is only 170.8 ms. The synchronous unit PI controller parameters derived from Theorem 2 are [−6.3172, 1.9045]. The simulation results after introducing the synchronous unit PI controller are shown in Figure 12.

Under these conditions, the system FM no longer exhibits divergent oscillations. It enters a stable recovery state after 6 s, but due to excessive early frequency fluctuations, the system frequency oscillates with a 0.5 s period before 6 s. This indicates that the primary cause of frequency oscillation is the inherent time delay in wind turbines, causing them to switch operating states every 0.5 s. To mitigate severe frequency changes and enable rapid stabilization, time delay compensation control is applied during the first two time delay cycles. By calculating the system frequency response function and performing regression analysis, the modified parameters for wind turbine comprehensive inertia control are derived. During the first time delay cycle when wind turbines start operating, [K_p, K_d] = [−14.42, −1.2074] reduces turbine output to align with frequency changes supported by synchronous units. During the second cycle, [K_p, K_d] = [−3.5634, 5.086] progressively increases output to avoid abrupt power oscillations in virtual inertia control after initial compensation. The simulation results with this time delay compensation controller are shown in Figure 13.

With the introduction of synchronous unit PI controllers and wind turbine time delay compensation, the system frequency recovery process becomes more stable, and the system time delay margin is significantly improved compared to those under conventional conditions.

6. Conclusions

This paper investigated both comprehensive inertia control and droop control for wind turbines. FM delay components were first analyzed, and a time delay system frequency response state-space equation was derived from the SFR model. Based on this, frequency-domain and LMI methods were used to analyze the delay-independent stability and time delay margin of wind turbines applying FM with time delays. Synchronous unit PI controllers and wind turbine time delay compensation parameters were designed via the LMI method to enhance system stability. Through theoretical analysis and simulation validation, the following conclusions were drawn:

(1)

Delay-independent stability boundaries exist for wind power penetration during system FM. These boundaries decrease with increasing wind turbine control parameters. Within the delay-dependent stability region, the system FM time delay margin decreases with larger control parameters, with virtual inertia control parameters exerting a more significant impact.

(2)

An accurate calculation method for system frequency responses considering time delays is proposed. This method ensures fixed system eigenvalues, avoiding infinite eigenvalues caused by exponential terms in time delays and errors from order-reduction approximations.

(3)

Synchronous unit PI controller parameters designed via the LMI method significantly improve the system time delay margin. Time delay compensation applied to wind turbines during early FM events effectively suppresses severe frequency fluctuations.

This research was based on the center of inertia assumption using aggregated models, focusing on comprehensive inertia control and droop control for wind turbine FM. Optimal control strategies under time delay effects require further investigation. It should be noted that the validation in this study was carried out on a two-area test system, which is a simplified model. Future work will extend the proposed analysis and control strategy to classical IEEE benchmark systems. In addition, verification using real measurement data will be pursued to further confirm the practical reliability of the approach.