Delay-Dependent Stability Analysis of Load Frequency Control for Power System With EV Aggregator

In this paper, the stability of load frequency control (LFC) for delayed power systems with an electric vehicle (EV) aggregator is studied based on Lyapunov theory and linear matrix inequalities (LMIs). Through mechanism analysis, the LFC of power systems with an EV aggregator based on a proportional–integral–differential (PID) controller is modeled. By constructing a delay interval information correlation functional and estimating its derivative using Wirtinger inequality and extended reciprocally convex matrix inequality, a new stability analysis criterion is proposed. Finally, in order to verify its advantage, the proposed method is used to discuss the influence of EV aggregator gains and PID controller gains on the delay margins for LFC of power systems with EV aggregator participation in frequency regulation.


INTRODUCTION
Under the guidance of sustainable development concept, the generation of renewable energy sources (RESs) such as wind power, hydropower, and photovoltaic power has developed rapidly in recent years, and part of traditional thermal power generation will be gradually replaced (Zhou et al., 2018). However, the grid connection of these RESs also brings some problems, especially the wind power generation with great intermittency and volatility Shi et al., 2021). These problems aggravate the imbalance between generation and load consumption in the power systems, resulting in obvious frequency fluctuation. Therefore, load frequency control (LFC) is widely used in power systems (Jin et al., 2019;Shangguan et al., 2021b). The frequency deviation caused by an intermittent energy grid connection is difficult to be eliminated by traditional generator sets. With the grid connection of controllable loads such as electric vehicles (EVs) and the rapid response characteristics of batteries, some studies paid attention to vehicle-to-grid technology, which provides frequency regulation services with a large number of converging EVs (Peng et al., 2017;Jia et al., 2018;Pinto et al., 2021;Teng et al., 2021).
In traditional power systems, the time delay phenomenon of the LFC system is not obvious. However, modern power systems tend to use flexible and open communication networks for information exchange (ShangGuan et al., 2021). For power systems with EVs and intermittent wind power connected, the EV aggregator needs to transmit the control command to the EVs through open communication networks (Ko and Sung, 2019;Li et al., 2019). The use of such networks will inevitably bring unreliable factors, such as time delay, packet loss, and potential failure, which may lead to instability of LFC for power systems Shangguan et al., 2021a). Therefore, it is very important to analyze the influence of time delays on the LFC of power systems with an EV aggregator. In addition, in order to ensure the stability of power system LFC, it is necessary to calculate the delay margins and determine all parameters of the proportional-integral-differential (PID) controller (Naveed et al., 2019b;Tek et al., 2020).
In recent years, EVs have been widely used in power systems, and there are also some studies on the influence of time delays and EV aggregator on LFC stability. The Rekasius substitution method is used to determine the stability delay margins of LFC with constant communication delays for an EV aggregator (Naveed et al., 2019a). Then, Naveed et al. presented a graphical method to describe the trajectory of the stable boundary and studied the influence of EV aggregators with communication delays on the stability regions and stability delay margins of the LFC system (Naveed et al., 2021). Based on Lyapunov theory and linear matrix inequalities (LMIs), stability criteria for timevarying delays using the Wirtinger-based improved integral inequality are proposed to calculate the delay margins for LFC with EVs, and the relationship between the gains and the delay margins of the PI controller is given in detail (Ko and Sung, 2018). Two stability criteria are derived, respectively, using Bessel-Legendre inequality and model reconstruction technique, and the interregional delay interaction and the effect of EV gain on the delay margins are discussed (Zhou et al., 2020). Khalil et al. proposed a microgrid model of photovoltaic power generation and EVs considering communication delay, and the maximum allowable delay bound for the stable operation of microgrids is calculated by solving the LMIs (Khalil et al., 2017). Dong et al. characterized the asymptotic stability of EV aggregation delays by using the delay distortion matrix structure of infinite operator dimension reduction and proved that convergence delay affects frequency stability in the form of low-frequency oscillation through three unstable modes (Dong et al., 2020). Although there have been some studies on the stability of delayed LFC systems with an EV aggregator, there are few studies on LFC of renewable energy power systems with an EV aggregator. Also, how to obtain more accurate delay margins remains a challenge.
In this paper, the stability of LFC for power systems with EV aggregator participation in frequency regulation is considered, and the influence of EV aggregator and controller gains on the delay margins is studied. Firstly, based on the PID controller, the LFC of power systems with an EV aggregator is modeled. Then, a new delay stability criterion using Wirtinger inequality and extended reciprocally convex matrix inequalities is proposed. Finally, according to the proposed stability criterion, the delay margins of LFC for power systems with an EV aggregator are obtained, and case studies are performed to show the advantage of the proposed method.

MODEL OF LFC FOR POWER SYSTEM WITH EV AGGREGATOR
The block diagram of the LFC for power systems with an EV aggregator is given in Figure 1, and the controller is the PID controller. e −sτ and e −sd denote the time delay of the frequency regulation circuit involved in the EV aggregator and the secondary frequency regulation circuit, respectively; K EV is the gain of the EV aggregator; Δf, ΔP EV , ΔP WTG , ΔP m , ΔP v , and ΔP d are the deviation of frequency, EV aggregator power output, wind turbine generator (WTG) power output, mechanical output of the generator, valve position, and load disturbance, respectively. Definitions of other related symbols in the figure are shown in Table 1. Select the following state variables, output variables, disturbance, and control input: Then, the following system state space model can be obtained: The controller is designed as and D 0 0 CF T .
In order to simplify the analysis, it is assumed that the delay τ(t) of the frequency regulation circuit involved in the EV aggregator is consistent with the delay d(t) of the secondary frequency regulation circuit. Then, the closed-loop state space equation of LFC for the delayed power system with an EV aggregator can be obtained as follows: where

DELAY-DEPENDENT STABILITY ANALYSIS
When discussing the internal stability of the power system, the influence of external disturbance can be ignored. The model of LFC for the delayed power system with an EV aggregator is obtained as follows: where h 1 ≤ d(t) ≤ h 2 and ∀t > t 0 .

Based on Wirtinger inequality, we have
h 12 where ε 1 (t) Using extended reciprocally convex matrix inequality to estimate Eq. 15 yields Then, we can get Applying Eq. 14 and Eq. 17 to Eq. 13, the following holds: By using the Schur complement, Eqs 10, 11 are equal to the following inequalities: which implies Thus, it follows from Eq. 21 that _ V(t) + αV(t) ≤ 0, which further leads to Noting that V(t) ≥ ρ x(t) 2 , V(t 0 ) ≤ β ϕ 2 , ρ > 0, and β > 0, we have which implies the system (Eq. 9) is exponentially stable (Yang et al., 2020). According to the above, system (Eq. 9) is exponentially stable if Eqs 10, 11 hold. The proof is completed.
Remark 1. The method proposed in this section establishes the constraint relation between the delay information and the exponential stability of the LFC for power systems with an EV aggregator, which can be used to analyze the influence of delays on the stability of the system and calculate the delay margins. The margins represent the time delay tolerance range of the system to ensure exponential stability, which is composed of the delay lower bound h 1 and delay upper bound h 2 .  Remark 2. In Theorem 1, the LFC for power systems with an EV aggregator is exponentially stable if Eqs 10, 11 are satisfied. The calculation steps of the delay margins for the stability of the system can be briefly summarized as follows: 1) Construct the LFC closed-loop model for power systems with an EV aggregator and a PID controller. 2) Choose the values of α, the EV aggregator gain K EV , and the allowable lower bound h 1 . 3) Calculate the delay margin h 2 of the power system by using the binary search technique (Zhang et al., 2013) and MATLAB/ LMI toolbox to solve the LMIs in Theorem 1.

CASE STUDIES
Case studies of LFC for power systems with an EV aggregator are presented to verify the advantage of the proposed method and study the influence of PID controller and EV aggregator gains on the delay margins. The related parameters of the system are shown in Table 1.

Comparison With the Existing Research
The method proposed by Jiang et al. (2012) is used to verify the advantage of the proposed method. Set h 1 0, α 0, K d 0, and k EV 1, and the system can be considered asymptotically stable if the conditions in Theorem 1 are true. Then, the delay margins of the method proposed in this paper are compared with the delay margins of time-varying delay (μ 0.9) in the study of Jiang et al. (2012). It is clear from Table 2 that the results of the proposed method are less conservative.

Effect of PID Controller and EV Aggregator Gains
The gains of the PID controller and EV aggregator have an important effect on the delay margins of the LFC for power systems with an EV aggregator. Firstly, let h 1 0, α 0.01, k EV  1, and PID controller parameters K be different; the delay margins of the system are obtained, and the related results are shown in Tables 3-5. As shown in Table 3, when K d 0 (PI controllers), for fixed K p , the delay margins decrease gradually with the increase of K i . For fixed K i , with the increase of K p , the delay margins decrease gradually. As can be seen from Tables 3-5, when K d is not 0 (PID controllers), the delay margins gradually become smaller with the gradual increase of K d . For fixed K p , the delay margins increase gradually with the increase of K i ; for fixed K i , the delay margins decrease as K p increases. To sum up, the delay margins under PI controllers are larger than that under the PID controller. The larger K p or K d is, the smaller delay margins are.
Then, the frequency deviations of LFC for power systems with an EV aggregator under the delay of 5.15s and different PID controller gains are simulated. It is assumed that the power deviations of load and WTG fluctuate randomly in the range of 0.19-0.21 p. u. and 0.49-0.51 p. u., respectively. As shown in Figure 2, when K [0.2 0.2 0], the system is stable. When K p and K i decrease (K [0.1 0.1 0]), the frequency deviation also tends to zero. But when K p and K i are increased (K [0.4 0.4 0]), or K d is increased (K [0.2 0.2 0.5]), it is clear that the frequency deviations do not converge in these cases. Therefore, Figure 2 validates the analysis in Tables 3-5, and appropriate selection of PID controller gains K is very important for the stability of LFC for power systems with an EV aggregator.
Finally, the gain of the EV aggregator K EV is also an important factor affecting the delay margins of LFC for power systems with an EV aggregator. As shown in Table 6, regardless of how the gains of the PID controller change, the delay margins of the delayed LFC system with an EV aggregator and intermittent wind energy decrease with the increase of K EV .

CONCLUSION
In this paper, the LFC stability of delayed power systems with an EV aggregator was studied. The LFC of the power system was modeled as a delayed linear system with an EV aggregator. Based on Lyapunov stability theory and the linear matrix inequality approach, a new stability criterion was proposed by using Wirtinger inequality and improved inverse convex matrix inequality. Finally, the influence of EV aggregator gains and PID controller gains on the delay margins was studied, and some case studies have shown the advantage of the results. The research of this paper can solve the delay margins more accurately and guide the design of PID controllers of LFC for power systems with an EV aggregator effectively.

DATA AVAILABILITY STATEMENT
The original contributions presented in the study are included in the article/Supplementary Material, and further inquiries can be directed to the corresponding author.