Coordinated Frequency Control for Electric Vehicles and a Thermal Power Unit via an Improved Recurrent Neural Network

Abstract

With the advancement of intelligent power generation and consumption technologies, an increasing number of renewable energy sources (RESs), smart loads, and electric vehicles (EVs) are being integrated into smart grids. This paper proposes a coordinated frequency control strategy for hybrid power systems with RESs, smart loads, EVs, and a thermal power unit (TPU), in which EVs and the TPU participate in short-term frequency regulation (FR) jointly. All EVs provide FR auxiliary services as controllable loads; specifically, the EV aggregations operate in charging mode when participating in FR. The proposed coordinated frequency control strategy is implemented by an improved recurrent neural network (IRNN), which combines a recurrent neural network with a functional-link layer. The weights and biases of the IRNN are trained by an improved backpropagation through time (BPTT) algorithm, in which a chaotic competitive swarm optimizer (CCSO) is proposed to optimize the learning rates. Finally, the simulation results verify the superiority of the coordinated frequency control strategy.

1. Introduction

To achieve the goals of “carbon peak” and “carbon neutralization”, the share of RESs such as wind and solar power in power systems will continue to rise, and it is possible to completely replace traditional generators [1]. However, the volatility, uncertainty, intermittency, and limited inertia capability of RESs introduce inevitable concerns over the frequency stability of the power system [2,3] and hence impose challenges on frequency regulation. When a disturbance occurs, such as a sudden loss of generation or a rapid increase in load, the balance between supply and demand is disrupted, causing the grid frequency to deviate from its nominal value [4]. This frequency variation could have a series of adverse effects and potentially damage the stability and security of power grids if not effectively addressed in a timely manner [5]. Therefore, effective short-term frequency regulation is crucial for the reliable operation of power systems in the face of growing renewable energy penetration.

As an important subset of demand-side response, electric vehicles (EVs) are becoming one of the key policy choices to achieve decarbonization in the transport sector. Moreover, given their vehicle-to-grid (V2G) capability and rapid response characteristics, EVs are advised to engage in frequency regulation [6]. V2G technology employs an appropriate grid-connected converter control algorithm, which can fully or partially simulate the frequency control characteristics of a synchronous motor based on the distributed power supply of a grid-connected converter [7]. Several studies have investigated frequency regulation using V2G technology [8,9,10,11]. For instance, the authors in [8] developed a sectional droop charging control strategy for EV aggregators to maintain frequency stability within a microgrid with high penetration of RESs. In another study [9], the authors took into account the charging requirements of EV users and proposed a novel frequency control strategy that can provide virtual inertia, damping, and frequency support, thereby enhancing the frequency stability of the microgrid. Additionally, a hierarchical control strategy was suggested, consisting of a fuzzy rule-based controller, a two-degree-of-freedom fractional-order PI controller, and a proportional resonant controller, which was further analyzed for its impact on stability when EVs participate in frequency regulation [10]. Another study [11] presented an adaptive primary frequency support strategy for EV clusters considering the operational constraints defined by their charging behavior.

Recent studies have increasingly explored the coordinated control of EVs with various frequency regulation resources, including thermal power [12,13], hydropower [14,15], photovoltaics [16], and so on, yielding significant improvements in performance. Notably, building on the fast response capabilities and flexible charging–discharging patterns of EVs in frequency regulation, integrating EVs with traditional TPUs presents a promising approach to enhance grid stability and efficiency. While TPUs have traditionally been the backbone of power systems due to their reliable and consistent generation, their slower response to frequency fluctuations becomes a limitation, particularly as the grid incorporates more RESs that introduce variability and uncertainty. By coordinated frequency control between EVs and TPUs, the slower dynamics of TPUs are complemented, resulting in a more responsive and adaptive frequency regulation strategy. The coordinated frequency control proposed in [12] considered time-varying delays, introducing an improved robust stability criterion to estimate the asymptotic stability for a hybrid power system. In [13], EVs were utilized to assist power plants in swiftly damping oscillations in system frequency following load demand changes in two-area thermal and hydrothermal power systems.

The aforementioned coordinated frequency control approaches not only enhance the overall stability of the power system but also improve the efficiency of thermal power plants by reducing the need for frequent and rapid adjustments to their output. Unfortunately, they often rely heavily on accurate mathematical models of the system, which can be difficult to obtain, especially in complex or highly nonlinear environments. Moreover, these model-based approaches may struggle to adapt to dynamic changes or uncertainties within the system. These limitations highlight the need for more flexible and adaptive control strategies. Deep learning, with its ability to learn directly from data without requiring an explicit model, provides a powerful alternative that can handle complex, nonlinear systems and adapt to changing conditions in real time [17]. The approaches and concepts of deep learning in control are reviewed in [18,19]. Deep learning has been successfully applied across various fields, including chemical engineering [20], semiconductor manufacturing [21], renewable energy generation [22,23], networked systems [24], and nuclear reactor technology [25]. RNNs, in particular, have been recognized for their ability to generate optimal control signals through online learning processes, making RNN-based control schemes increasingly prominent in recent research [26]. Therefore, this paper adopts an improved RNN as a controller to manage the participation of EV aggregators in frequency regulation.

During the operation of power systems incorporating RESs, the system is subjected to non-Gaussian disturbances caused by wind speed and solar irradiance [27], which the objective functions employed in existing deep learning control strategies struggle to address effectively. Meanwhile, recent advancements in information theoretic learning techniques have introduced a variety of statistical indices, including entropy, correntropy, higher-order moments, and survival information potential (SIP), which have been applied to system identification, filtering, and control in non-Gaussian environments [28]. Among these, SIP, defined using the survival function, has been shown to surpass the commonly used minimum error entropy criterion in managing non-Gaussian disturbances [29]. Its ease of estimation and resistance to translation invariance make it particularly effective for complex and dynamic systems.

Furthermore, the selection of hyperparameters significantly affects the performance and effectiveness of neural networks. In recent years, swarm intelligence algorithms have gained popularity for optimizing network parameters due to their simplicity and extensive search capabilities [30]. For example, an improved whale optimization algorithm has been proposed to optimize bi-directional long short-term memory networks [31], and the sparrow search algorithm has been used to optimize the hyperparameters of fast learning networks [32]. The competitive swarm optimizer (CSO), first introduced by Cheng et al. in 2015 [33], has demonstrated notable strengths in exploration, exploitation, avoiding local optima, and enhancing convergence performance. However, there remains room for improvement in the quality of the initial population.

Inspired by the above research, this paper proposes an IRNN-based control method for EVs and thermal power units participating in frequency regulation jointly. The main contributions include the following:

(1)

To reduce computational complexity, which refers to the challenge of modelling thousands of EVs, an aggregate model of EVs is formulated using the averaging method. In this approach, all EVs operate exclusively in charging mode, which avoids additional stress on the EV batteries compared to V2G operations. The power controllability of all EV aggregators is further defined.

(2)

The coordinated controller is designed based on a functional-link recurrent neural network (FLRNN) to enhance frequency regulation by adaptively managing the power output of TPUs and the charging power of EVs. The weights and biases of the FLRNN are trained by an improved BPTT algorithm, with a chaotic competitive swarm optimizer (CCSO) employed to optimize the learning rates.

(3)

The non-Gaussian characteristics of renewable energy sources are taken into account. To this end, this paper introduces the concept of survival information potential (SIP) as a performance metric for training the weights and biases of FLRNN.

The rest of this paper is as follows: in Section 2, an EV aggregator charging control framework is proposed. Then, Section 3 introduces the model of EV aggregators and thermal power units. Next, the coordinated frequency control strategy for EV aggregators and TPU is displayed in Section 4. Section 5 presents the simulation results and analysis, which are implemented using MATLAB/Simulink 2024B. Finally, Section 6 summarizes this paper.

2. EV Aggregator Charging Control Framework

The EV-integrated hybrid power system studied is depicted in Figure 1, encompassing RESs such as PV arrays and wind turbines, thermal power units, EVs, loads, and various electrical components. Wind and solar energy would contribute to stochastic power generation. A substantial number of EVs are connected in the form of EV aggregators.

Considering the charging requirement of EVs participating in frequency regulation, this study adopts a charge management system of EV aggregators proposed in [8]. When EVs are required for frequency regulation, EV owners can opt to charge in either managed charging mode or unmanaged charging mode. In managed mode, the EV charging power can be adjusted across a theoretical range extending from zero up to the rated charging power. EVs in unmanaged mode are charged at the rated charging power until the expected SOC is reached.

To balance the charging power demand of EVs and satisfaction of the EV owners, EVs will charge at two different power levels:

(1)

Charging at the rated power: applies to EVs operating in unmanaged mode or EVs operating in managed mode but having insufficient remaining time.

(2)

Charging at power levels below the rated power: applies to EVs in managed mode with sufficient remaining time, enabling these normally managed EVs to engage in frequency regulation.

Under this strategy, the EV battery will always operate in the charging state without damaging the battery life. Therefore, encouraged by the charging price policy, EV owners are more inclined to choose the managed charging mode.

It is assumed that the SOC states of EVs do not change in a short time. Thus, the minimum charging power and maximum charging power of all aggregators can be obtained as follows [8]:

$$P_{min}={\displaystyle \sum }_{i=1}^N({\displaystyle \sum }_{r_i=1}^{R_i}{P}_{rate,r_i}+{\displaystyle \sum }_{m_i=1}^{M_i}{P}_{rate,m_i})$$

(1)

$$P_{max}={\displaystyle \sum }_{i=1}^N{\displaystyle \sum }_{q_i=1}^{Q_i}{P}_{rate,q_i}$$

(2)

where $R_i$ is the quantity of EVs operating in unmanaged mode in the $ith$ aggregator; $M_i$ stands for the quantity of EVs operating in managed mode but having insufficient remaining time in the $ith$aggregator; $Q_i$ represents the total of the charging EV in the $ith$ aggregator, with $i=1,2\dots N$; and $P_{rate}$ is the rated power of EVs.

3. Modelling of Hybrid Power System

3.1. Equivalent Model of EV Battery

The modelling of battery technology in EVs is fundamental and crucial for researching the involvement of electric vehicles in frequency regulation. General battery models encompass electrochemical, mathematical, and equivalent circuit models [9]. Among these, the equivalent circuit model is used in this paper due to its intuitive nature, allowing for the precise simulation of battery charging characteristics. Figure 2 depicts an equivalent battery model for EVs. This model captures the battery’s dynamic behavior and internal energy losses, enabling its effective use in simulations of EV systems.

In Figure 2, $U_b$ and $U_c$ are the terminal voltage and open-circuit voltage of the EV battery, respectively. $R,$ $R_1$, and $R_2$ represent the self-discharge resistance, over-voltage resistance, and inner-circuit resistance, respectively. $C_1$ denotes the over-voltage capacitance, which reflects the transient process of the battery, and $C$ stand for the equivalent capacitance of the EV battery. The dynamic equation of an EV battery is shown as follows:

$$\begin{array}{l}C{\displaystyle \frac{d{U}_c}{dt}}=I_b-{\displaystyle \frac{U_c}{R}}\\ C_1{\displaystyle \frac{d{U}_{c1}}{dt}}=I_b-{\displaystyle \frac{U_{c1}}{R_1}}\end{array}$$

(3)

To simplify the calculations, converter and transformer losses are ignored, allowing the battery’s output power to be calculated using the following:

$$P_{EV}=U_b{I}_b$$

(4)

This leads to the following equation:

$$\mathsf{\Delta }{P}_{EV}=\mathsf{\Delta }U{I}_{b0}+U_{b0}\mathsf{\Delta }I+\mathsf{\Delta }U\mathsf{\Delta }I$$

(5)

where $U_{b0}$ and $I_{b0}$ denote the initial DC voltage and current of the battery, respectively, and $∆U$ and $∆I$ represent the voltage and current variations. In the course of battery power regulation, the variation in current $∆I$ is minimal and can be effectively neglected. Thus, the battery’s output power simplifies to

$$\mathsf{\Delta }{P}_{EV}\approx \mathsf{\Delta }U{I}_{b0}$$

(6)

When the EV participates in the frequency regulation, the increase in DC voltage $∆U$ is controlled by the frequency deviation. The dynamic equation is then given by

$$\mathsf{\Delta }U=\mathsf{\Delta }f\cdot {\displaystyle \frac{k(SOC)}{1+T_{B}s}}$$

(7)

where $k$(SOC) is the proportionality coefficient related to the state of charge (SOC) and $T_B$ is the control signal delay time of the power electronics, which is minimal. Therefore,

$$\mathsf{\Delta }{P}_{EV}=I_{b0}\cdot \mathsf{\Delta }f\cdot {\displaystyle \frac{k(SOC)}{1+T_{B}s}}=\mathsf{\Delta }f\cdot {\displaystyle \frac{K(SOC)}{1+T_{B}s}}$$

(8)

with $K\left(SOC\right)=I_{b0}\cdot k\left(SOC\right)$, defining the participation factor for an EV in power grid frequency regulation.

3.2. Aggregate Model of EVs

Denote ${EV}_{ij}$ as the $jth$ normally managed EV in the $ith$ aggregator. When ${EV}_{ij}$receives a control signal to adjust its power consumption, it will either decrease or increase the charging power. The transfer function of ${EV}_{ij}$ is

$$G_{E{V}_{ij}}(s)={\displaystyle \frac{K_{ij}(SO{C}_{ij})}{1+s{T}_{ij}}}$$

(9)

where $K_{ij}\left(SO{C}_{ij}\right)$ is the charging coefficient of ${EV}_{ij}$ related to its SOC and $T_{ij}$ represents the time constant of ${EV}_{ij}$.

Denote the $ith$EV aggregator as $A_i.$ $K_i$, which stands for charging coefficient of $A_i$, can be calculated by [34]

$$K_i={{\displaystyle \int }}_0^1{K}_{ij}(SO{C}_{ij}){\varphi }_{so{c}_{ij}}d(SO{C}_{ij})$$

(10)

where ${\varphi }_{SO{C}_{ij}}$ is the probability distribution function of the real-time SOC of ${EV}_{ij}$with the mean value of $SO{C}_i$.

In this work, we assume that the time constant of ${EV}_{ij}$ in $A_i$ is identical for simplicity and then denote $T_{ij}=T_i$. The number of the normally controlled EVs $M_i$ in the $ith$ aggregator is regarded as its gain.

The transfer function of $A_i$ can be described by

$$G_{Ai}(s)={\displaystyle \frac{K_i\cdot M_i}{1+s{T}_i}}$$

(11)

As each EV involved in frequency regulation exclusively charges, EV aggregators reduce their charging power consumption when the frequency drops, virtually increasing the power supplied to the grid. Conversely, they increase their charging power consumption when the frequency rises. Denote $P_{cgrant}$ as the charging power granted by the smart grid to all EV aggregators when EVs are not participating in frequency regulation. Then, the power controllability of all EV aggregators ${∆P}_{EVA}$ can be expressed as

$$\mathsf{\Delta }{P}_{EVA}\in \left[\mathsf{\Delta }{P}_{EVA}^{-},\mathsf{\Delta }{P}_{EVA}^{+}\right]$$

(12)

where $∆P_{EVA}^{-}=P_{cgrant}-P_{max},∆P_{EVA}^{+}=P_{cgrant}-P_{min}$.

3.3. Model of TPU

A traditional TPU is mainly composed of a speed governor and reheat turbine, and the transfer function of a TPU is formulated as follows [35]:

$$G_T(s)={\displaystyle \frac{1+s{F}_{HP}{T}_{RH}}{(1+s{T}_G)(1+s{T}_{CH})(1+s{T}_{RH})}}$$

(13)

where $T_G$ is the action time constant of the governor, $T_{CH}$ is the time constant of the main steam inlet chamber, $T_{RH}$ is the time constant of the reheater, and $F_{HP}$ is the high-pressure turbine mechanical torque.

4. Coordinated Frequency Control Strategy for EV Aggregators and a TPU

In this section, a coordinated frequency control strategy for EV aggregators and a TPU is shown in Figure 3. To handle uncertain load changes and random power generations introduced by renewable energy, an improved RNN-based controller is developed to generate control signals ${∆P}_{CT}$ for the TPU and $∆P_{CEV1}$, $∆P_{CEV2}$, … $∆P_{CEVN}$ for the EV aggregators, respectively. $∆f$ stands for the system frequency deviation. ${∆P}_R$ and ${∆P}_L$ are the renewable power disturbance and load disturbance, respectively. $∆P_W$, $∆P_{PV}$, $∆P_T$, and ${∆P}_{EVA}$ are the wind power change, the PV power change, the thermal power change, and the output power variation of the EV aggregators, respectively. $H_{\mathsf{\Sigma }}$ and $D$ are the equivalent inertia constant and load damping coefficient, respectively.

4.1. FLRNN with SIP as the Performance Index

As shown in Figure 3, the FLRNN consists of one input layer, one functional expansion layer, two hidden layers, and one output layer. The input of the FLRNN $\mathit{u}\left(k\right)$ includes the frequency deviation $∆f\left(k\right)$ and the rate of change of frequency $df/dt\left(k\right)$ at instant k. The input layer utilizes functional-link expansion, chosen for its rapid convergence rate and reduced computational burden [36]. Each node in the input layer of the RNN can then be expanded by a trigonometric polynomial basis function. The two hidden layers shown in Figure 3 consist of a recurrent layer and a fully connected layer. The output $\mathit{y}\left(k\right)$of the FLRNN represents the control inputs to thermal power unit $∆P_{CT}$ and EV aggregators $∆P_{CEV1}$, $∆P_{CEV2}$, …, $∆P_{CEVN}$. These control signals enable more precise adjustments to system dynamics, providing enhanced adaptability in response to varying operating conditions compared to traditional methods. The forward pass of the FLRNN is as follows:

$$I(k)=\mathit{u}(k)$$

(14)

$$F(k)=[1,\sin (\pi u(k)),\cos (\pi u(k)),\sin (2\pi u(k)),\cos (2\pi u(k))]$$

(15)

$$\mathit{X}(k)=g(W_{xf}F(k)+W_{xx}\mathit{X}(k-1)+b_x)$$

(16)

$$C(k)=g(W_{cx}\mathit{X}(k)+b_c)$$

(17)

$$y(k)=W_{yc}C(k)+b_y$$

(18)

where $\mathit{I}\left(k\right)$ is the output of the input layer. $\mathit{F}\left(k\right)$ is the output of the functional expansion layer. $\mathit{C}\left(k\right)$ and $\mathit{F}\left(k\right)$ are the outputs of the hidden layers at instant k, respectively. ${\mathit{W}}_{xf},{\mathit{W}}_{xx},{\mathit{W}}_{cx}$, and ${\mathit{W}}_{yc}$ denote the corresponding weights between adjacent layers, while $b_x,b_c$, and $b_y$ are the corresponding biases. The activation function is a tanh function, $g\left(z\right)=(e^z-e^{-z})/(e^z+e^{-z})$.

In this context, the survival information potential (SIP) [29] is employed to handle non-Gaussian disturbances from wind and solar power. Here, $e\left(k\right)=\mathsf{\Delta }f\left(k\right)$, and the SIP of the frequency deviation serves as the cost function for training the FLRNN. It can be estimated using the Parzen window technique:

$$J(k)={\displaystyle \sum _{j=k-L+1}^k{\lambda }_j{e}^{\alpha }(j)}$$

(19)

where ${\lambda }_j={\left({\displaystyle \frac{k-j+1}{L}}\right)}^2-{\left({\displaystyle \frac{k-j}{L}}\right)}^2$ and the frequency deviation series [$e(k-L+1),e(k-L+2),...,e\left(k\right)]$ are collected within the sliding window, whose width is L. In this paper, $\alpha =2$.

4.2. BPTT Learning Rate Optimization Using CCSO

The FLRNN was trained to minimize the cost function of (19) by using an improved backpropagation through time (IBPTT) algorithm, which employs a chaotic competitive swarm optimizer (CCSO) by combing the theories of chaos maps and competitive swarm optimizer (CSO) to optimize the learning rates.

The competitive swarm optimizer (CSO) [33] is a swarm algorithm characterized by its pairwise competition mechanism. The general concept is illustrated in Figure 4. While the CSO has demonstrated notable capabilities in exploration, exploitation, local optima avoidance, and convergence performance, there remains significant potential for further enhancement. Meanwhile, chaotic maps exhibit traits such as unpredictability, ergodicity, randomness, aperiodicity, and sensitivity to initial conditions and parameters [37]. Utilizing chaotic maps to initialize the positions of a population can yield a more uniformly distributed population in space compared to random initialization. Therefore, a novel chaotic competitive swarm optimizer (CCSO) is proposed to solve the optimal optimization problem by combing the CSO and the theories of chaos maps. There are various forms of chaotic map equations, with typical examples including the Logistic map, Circle map, and Spatial Pyramid Matching (SPM) map, among others. This paper employs the SPM map for population initialization. The specifics of the CCSO are detailed below:

• Define the population size of the swarm $P\left(c\right)$ as $\overline{N}$, where $\overline{N}$ is an even number. The dimension of search paths is denoted as $\overline{D}$, and the maximum number of iterations is $maxN$. The position of the $l_{th}$ agent by ${\mathit{x}}_l=\left[x_l^1,\cdots ,x_l^d,\cdots ,x_l^{\overline{D}}\right],$ where $x_l^d$ is the value of the $d_{th}$ dimension of particle $l$.
• Initialize the particles of swarm $P\left(c\right)$ through the SPM map.

  $$x_{l+1}^d=\left\{\begin{array}{c}\mathrm{mod}\left({\displaystyle \frac{x_l^d}{p}}+\mu \sin (\pi x_l^d)+r,1\right),0\le x_l^d<p\\ \mathrm{mod}\left({\displaystyle \frac{x_l^d}{p(0.5-p)}}+\mu \sin (\pi x_l^d)+r,1\right),p\le x_l^d<0.5\\ \mathrm{mod}\left({\displaystyle \frac{1-x_l^d}{p(0.5-p)}}+\mu \sin (\pi (1-x_l^d))+r,1\right),0.5\le x_l^d<1-p\\ \mathrm{mod}\left({\displaystyle \frac{1-x_l^d}{p}}+\mu \sin (\pi (1-x_l^d))+r,1\right),1-p\le x_l^d<1\end{array}\right.$$

  (20)

  where $p\in (0,1)$, $\mu \in \left(0,1\right),x_1^d$, and $r$ are random numbers from 0 to 1, respectively.
• For each particle, a fitness value is evaluated using the integral of time-weighted absolute error (ITAE) as the fitness function, defined by the following equation:

  $$J_{ITAE}={\displaystyle \underset{\tau }{\overset{t}{\int }}(t-\tau )\left|\mathsf{\Delta }f\left(t\right)\right|dt}$$

  (21)
• Divide the swarm $P\left(c\right)$ into two components with equal size randomly. In each pair, a competition occurs between the two particles. The particle with smaller ITAE, termed as the winner, is selected to proceed directly to the next generation $P(c+1)$, while the loser particle updates its position and velocity by learning from the winner and is subsequently sent to $P(c+1)$.

  $$v_{l,\overline{k}}(c+1)=R_1(\overline{k},c)v_{l,\overline{k}}(c)+R_2(\overline{k},c)\left(x_{w,\overline{k}}(c)-x_{l,\overline{k}}(c)\right)+\phi R_3(\overline{k},c)\left({\overline{x}}_{\overline{k}}(c)-x_{l,\overline{k}}(c)\right)$$

  (22)

  $$x_{l,\overline{k}}(c+1)=x_{l,\overline{k}}(c)+v_{l,\overline{k}}(c+1)$$

  (23)

  where $\overline{k}=\overline{N}/2$. ${\mathit{R}}_1(\overline{k},c),{\mathit{R}}_2(\overline{k},c),{\mathit{R}}_3(\overline{k},c)\in [\mathrm{0,1}{]}^n$ represent three randomly generated vectors after the ${\overline{k}}_{th}$ round of competition and learning process in generation c, ${\overline{\mathit{x}}}_{\overline{k}}\left(c\right)$ stands for the mean position value of the relevant particles, and $\phi$ denotes a parameter that governs the influence of ${\overline{\mathit{x}}}_{\overline{k}}\left(c\right)$.
• Repeat step (4) until no particles remain in $P\left(c\right)$, then set $c=c+1$.
• Repeat steps (3) to (5) until the termination condition is satisfied.

The optimal learning rates for the BPTT are selected based on the solution with the lowest ITAE value. And a flow chart of the proposed CCSO is illustrated in Figure 5.

Finally, the procedure of training the FLRNN using improved BPTT with learning rate optimization by CCSO is summarized as shown in Algorithm 1.

Algorithm 1 Training the FLRNN using improved BPTT with learning rate optimization by CCSO

Initialization Initialize the weights and biases of the FLRNN. Forward $\begin{array}{l}I(k)=u(k)\\ F(k)=[1,\sin (\pi u(k)),\cos (\pi u(k)),\sin (2\pi u(k)),\cos (2\pi u(k))]\\ \widehat{\mathit{X}}(k)=W_{xf}F(k)+W_{xx}\mathit{X}(k-1)+b_x\\ \mathit{X}(k)=g(\widehat{\mathit{X}}(k))\\ \widehat{C}(k)=W_{cx}\mathit{X}(k)+b_c\\ C(k)=g(\widehat{C}(k))\\ y(k)=W_{yc}C(k)+b_y\\ J(k)={\displaystyle \sum _{j=k-L+1}^k{\lambda }_j{e}^2(j)}\end{array}$ Backward $\begin{array}{l}\delta y(k)=y(k)-y(k-1)\\ \delta C(k)={\displaystyle \raisebox{1ex}{$\partial J(k)$}\!\left/ \!\raisebox{-1ex}{$\partial C(k)$}\right.}=W_{yc}^T\delta y(k)\\ \delta \widehat{C}(k)={\displaystyle \raisebox{1ex}{$\partial J(k)$}\!\left/ \!\raisebox{-1ex}{$\partial X(k)$}\right.}=\delta C(k)\odot g^{\prime }(\widehat{C}(k))\\ \delta \mathit{X}(k)={\displaystyle \raisebox{1ex}{$\partial J(k)$}\!\left/ \!\raisebox{-1ex}{$\partial \mathit{X}(k)$}\right.}=W_{cx}^T\delta \widehat{C}(k)\\ \delta \widehat{\mathit{X}}(k)={\displaystyle \raisebox{1ex}{$\partial J(k)$}\!\left/ \!\raisebox{-1ex}{$\partial \widehat{\mathit{X}}(k)$}\right.}=\delta \mathit{X}(k)\odot g^{\prime }(\widehat{\mathit{X}}(k))\\ \delta F(k)={\displaystyle \raisebox{1ex}{$\partial J(k)$}\!\left/ \!\raisebox{-1ex}{$\partial F(k)$}\right.}=W_{xf}^T\delta \widehat{\mathit{X}}(k)+W_{xx}^T\delta \mathit{X}(k+1)\\ d{W}_{yc}\leftarrow d{W}_{yc}+\delta y(k)C(k)\\ d{b}_y\leftarrow d{b}_y+\delta y(k)\\ d{W}_{cx}\leftarrow d{W}_{cx}+\delta \widehat{C}(k)\mathit{X}(k)\\ d{b}_c\leftarrow d{b}_c+\delta \widehat{C}(k)\\ d{W}_{xf}\leftarrow d{W}_{xf}+\delta \widehat{\mathit{X}}(k)F(k)\\ d{W}_{xx}\leftarrow d{W}_{xx}+\delta \widehat{\mathit{X}}(k)\mathit{X}(k-1)\\ d{b}_x\leftarrow d{b}_x+\delta \widehat{\mathit{X}}(k)\end{array}$ Optimization Obtain the optimal learning rates $({\eta }_{xf},{\eta }_{xx},{\eta }_{cx},{\eta }_{yc},{\eta }_x,{\eta }_c,{\eta }_y)$ by CCSO Update $\begin{array}{l}{W}_m\leftarrow W_m+{\eta }_{m}d{W}_m;m\in \left\{xf,xx,cx,yc\right\}\\ b_n\leftarrow b_n+{\eta }_{n}d{b}_n;n\in \left\{x,c,y\right\}\end{array}$

5. Simulation Results and Discussion

The investigated power grid shown in Figure 1 is composed of a 600 MW reheat steam thermal power unit, a 150 MW PV plant with 300 × 0.5 MW PV arrays, a 150 MW wind power plant with 100 × 1.5 MW DFIG wind turbine generators, and two EV aggregators with 10,000 EVs and 20,000 EVs in managed mode, respectively. The EVs have a rated charging power of 3 kW. The value of the initial load is 603 MW. The parameters of the thermal power unit, EV aggregators, and optimal method are listed in

Table 1. Parameters of TPU, EV aggregators, CCSO, and FLRNN.

Parameter	Values	Parameter	Values
Governor time constant $T_G$	0.3 s	High-pressure turbine mechanical torque $F_{HP}$	0.4
Main steam inlet chamber constant $T_{CH}$	0.5 s	Equivalent inertia constant $H_{\mathsf{\Sigma }}$	6.5 s
Reheat time constant $T_{RH}$	14 s	Load damping coefficient $D$	0.012
Power variation lower limit for EV aggregators in frequency regulation ${∆P}_{EVA}^{-}$	0.06 p.u.	Power variation upper limit for EV aggregators in frequency regulation ${∆P}_{EVA}^{+}$	0.05 p.u.
Time constant of EV aggregator 1 $T_{A1}$	35 ms	Time constant of EV aggregator 2 $T_{A2}$	38 ms
Average SOC of EV aggregator 1 ${SOC}_1$	0.6 p.u.	Average SOC of EV aggregator 2 ${SOC}_2$	0.55 p.u.
Population size of the swarm $P\left(c\right)\overline{N}$	200	Dimension of search paths $\overline{D}$	20
Threshold constant of the SPM map $p$	0.4	Coefficient of the SPM map $\mu$	0.3
Number of neurons in the recurrent layer	12	Number of neurons in the connected layer	6

, and the distribution graph of the SPM map is shown in Figure 6.

5.1. Scenario 1

A 0.05 p.u. step load disturbance occurred at 5 s; consequently, the power system operated under rated frequency. The traditional droop control is compared with the proposed algorithm. The droop coefficients of the thermal plant and EV aggregators are 0.05 and 0.2, respectively. The system frequency response (SFR) curves are shown in Figure 7. It can be observed that the nadir of frequency in the SFR curves obtained from FLRNN + CCSO-based control, FLRNN-based control, and traditional droop control are 49.724 Hz, 49.656 Hz, and 49.613 Hz, respectively; the steady-state frequencies are reached at 49.961 Hz, 49.961 Hz, and 49.749 Hz, respectively. The frequency response using the FLRNN + CCSO-based controller exhibits smaller fluctuations. Correspondingly, the power changes of the thermal power unit and EV aggregators are shown in Figure 8 and Figure 9, respectively. The ITAE is used to evaluate the performance of three control strategies by measuring the cumulative time-weighted frequency deviation, with a lower ITAE value indicating better control system performance in maintaining the desired frequency over time. The ITAE values for the three control strategies are presented in

Table 2. ITAE values for three different control methods in Scenario 1.

Methods	FLRNN + CCSO	FLRNN	Droop
ITAE	70.17	73.43	317.98

, where the ITAE is 70.17, 73.43, and 317.98 for FLRNN + CCSO-based control, FLRNN-based control, and traditional droop control, respectively. While the ITAE values for the FLRNN + CCSO and FLRNN methods are similar, it is important to note that the nadir of the SFR curve differs between the two methods, with the frequencies being 49.724 Hz for FLRNN + CCSO and 49.656 Hz for FLRNN. The experimental results clearly demonstrate the superior performance of the proposed FLRNN control strategy and FLRNN + CCSO control strategy over conventional droop methods. Notably, the FLRNN + CCSO method exhibits exceptional capability in handling large-scale disturbances, effectively maintaining the system frequency within safe operational limits. Furthermore, the FLRNN method demonstrates remarkable frequency regulation performance under small disturbances while maintaining lower computational requirements.

The superiority of the proposed FLRNN + CCSO control strategy over the other two methods has been clearly demonstrated.

5.2. Scenario 2

In this scenario, random renewable energy sources are introduced, leading to power disturbances characterized by stochastic and intermittent patterns. Non-Gaussian wind power disturbances are shown in Figure 10, while non-Gaussian PV power disturbances are shown in Figure 11. The SFR curves under the non-Gaussian disturbances are shown in Figure 12. It is evident that even under complex and random disturbances, the proposed FLRNN + CCSO-based control strategy still achieves excellent control performance. In addition, it is obvious from Figure 13 that the probability density function (PDF) of the frequency deviation ${\gamma }_{∆f}$ with the FLRNN + CCSO-based control strategy can be quickly adjusted to a sharp and narrow shape near a specific value following a wind or PV change. The output power changes of the thermal power unit and EVs are depicted in Figure 14 and Figure 15, respectively. It can be seen that, under the FLRNN + CCSO-based control strategy, the output power of generators and especially EVs dynamically adjust their output to mitigate power mismatches, thereby minimizing frequency fluctuations and effectively utilizing EV flexibility. The simulation results, presented in

Table 3. ITAE values for three different control methods in Scenario 2.

Methods	FLRNN + CCSO	FLRNN	Droop
ITAE	30.24	34.62	92.98

, show ITAE values of 30.24, 34.62, and 92.98 for the three control strategies, respectively. Thus, it is evident that the proposed FLRNN + CCSO-based control method outperforms the other two control strategies.

6. Conclusions

This paper presents a coordinated control strategy for EVs participating in hybrid power system frequency regulation via a functional-link recurrent neural network. To address the non-Gaussian disturbances in power systems with RESs, the SIP is employed as a performance metric for FLRNN training. Furthermore, the weights and biases of the FLRNN are trained by an improved BPTT algorithm, with a CCSO employed to optimize the learning rates. The simulation results validate the efficacy of the proposed strategy in improving frequency regulation. Note that our study focuses on EVs participating in frequency regulation while in the charging state, thereby preventing any potential damage to battery life. Hence, future work will explore EVs participating in frequency regulation as responsive loads and power supplies and develop more effective control strategies.