A Survey on Load Frequency Control of Multi-Area Power Systems: Recent Challenges and Strategies

Abstract

Load frequency control (LFC) is well known for balancing the load demand and frequency for a multi-area power system. Studies have proven that LFC can improve the global performance of multi-area power systems. In recent years, the increasing proportion of renewable energy, integration of EVs, and cyber-attacks have become the main challenges in LFC power systems. Different strategies have been applied in the literature for LFC power systems and the possible impacts of renewable energy, EVs, and cyber-attacks. This survey paper is devoted to the research on directions in LFC multi-area power systems. The mathematical model of recent challenges in LFC multi-area power systems is summarized and the similarities and differences of these challenges are analyzed. The uncertainty of renewable energy is a frequently noted issue in LFC power systems; however, the uncertainty that exists in controller design is often ignored. In this survey, we analyze methods for treating the uncertainty of renewable energy and controller. This survey paper introduces the most recent research on LFC and acquaints anyone interested in its development, such that the most effective strategies can be developed by the researchers.

1. Introduction

Recently, social development and population growth have caused the demand for energy support to grow gradually. To guarantee the supply of power, the requirements for the power system are becoming higher and higher. Due to the steadily escalating size and intricacy of modern power systems [1], the problems and challenges that the power system may encounter can be expected to increase as well. Load frequency control (LFC) is widely used to judge whether the power system is in normal operation [2,3,4,5]. In practice, the valve opening of the governor is controlled by LFC, in which the control signal is made up of the frequency deviation. The volume of steam flow into the steam turbine changes when the real-time frequency deviates from the steady value. The frequency of the power generator is regulated to a steady value by controlling the rotor speed [6]. In a power system, LFC has been proven to effectively maintain the frequency and power flow of tie-lines between areas.

With increasing demand for electric power, the pressure on the power system is greatly increased. In ongoing research, scientists have realized that renewable energy is one of the best choices to lighten the power supply pressure of the traditional power system [7,8,9,10,11]. Environmental damage (e.g., climate change) caused by crude oil and fossil fuels is increasing the dependence on cleaner renewable energy. Normally, renewable energy includes solar, biomass, wind, hydro, and wave power. Among renewable energy resources, solar and wind power have been the most considered in relieving pressure on energy demand over the last few years. To maximize the utilization of renewable resources, more and more academics are placing their focus on these areas and intensifying research into them. In [12], a wind farm and photovoltaic (PV) model was integrated with an LFC power system. The LFC problem of multi-area wind power systems involving the communication delay was investigated in [13]. In [14], the LFC problem was investigated by considering the output wind power fluctuation for a significant wind power penetration time-delay power system. In [13], a fuzzy logic integral sliding mode controller was proposed for multi-area wind power systems. A new model was developed to explore the influence of the LFC power system frequency fluctuation on the PV output power in [15]. In [16], diesel generators, PVs, and wind turbines were considered as a multi-source power system, and LFC techniques were applied to balance the frequency and multi-source electricity. As the share of renewable energy in the power system increases, the system inertia is constantly reduced, which impacts system stability, leading to frequency deviations [17]. Based on a dual-structured fuzzy (DSF) model, a proportional and integral switched controller was proposed in [18] to improve the frequency regularization of the $\mu$-grid integrated with wind power. In [19], a fractional-order PID was developed for a multi-sources interconnected renewable power system. In [20], a non-conventional quasi-oppositional dragonfly algorithm was used to find the optimal parameters of the PID controller in the LFC renewable power system. Persistent large deviations in tie-line power exchange and system frequency in an LFC power system are harmful to appliances, can lead to instability of operation, reduce the lifetime of connected devices, and even result in a crash of the power system.

In addition to the above challenges, battery energy storage units are integrated into modern power systems. Due to the penetration of renewable energy into power grids, frequency fluctuations can be highly disturbed. Battery energy storage units can smooth these fluctuations, enhancing the transient response of the power system. This process is carried out through a continuous bidirectional power flow between the battery energy storage unit and the grid [21]. In [22], a battery energy storage unit was connected to a wind power system, reducing the adverse effects of frequency deviations. Today, batteries are widely applied in electric vehicles (EVs), and research attention has become focused on issues related to EVs. In [21], a modern power grid was constructed with generators, photovoltaic, solar thermal, wind, and energy storage for study of the LFC issue. EVs play an essential role in low-carbon development, and have significant implications for power grids. The adoption of EVs can effectively improve the power grid’s load factor and reduce the electricity supply cost [23]. To make the LFC problem more realistic, a hybrid power system (conventional thermal, electric vehicle (EV), and solar thermal) was considered in [16].

Open communication networks are used in LFC power system signals transmission [3]. Unfortunately, safety and other issues are gradually becoming more serious due to the openness of this network. Cyber-attacks on communications networks can impact the operation of power systems, potentially leading to serious disruptions [24]. A frequency fluctuation due to load change or cyber-attack in one area may spread to all connected neighboring areas and deal further damage to overall power system stability. The use of a communication network is vulnerable to different types of hostile attacks generated by adversaries. In LFC power systems, the most widely considered cyber-attacks are deception attacks and denial-of-service (DoS) attacks [24]. In [25], a handshake protocol was used to design event-triggered strategies and identify DoS attacks in an LFC power system. In [26], a switched system was constructed to analyze the stability of an LFC power system under DoS attacks and deception attacks. In [27], an event-triggered LFC mathematical model was constructed for a power system under deception attack. An event-based PI secure load-frequency controller was utilized for power systems suffering from deception attacks in [27]. To handle the significant wind power penetration and time-delay problem, a novel decentralized sliding-mode LFC strategy was designed for multi-area power systems in [14]. It can be seen from these studies that the challenges involved in integrating LFC with power systems are very important. Although renewable energy has many advantages and the network is convenient, the resulting stability issues for power systems cannot be ignored. To ensure the good performance of renewable power systems, various control strategies have been developed for the LFC issue. Figure 1 is shows the organization of the rest of this review paper.

In recent years, system control theory has been applied to solve the above-mentioned challenges in the power system [28]. In this survey, we mainly focus on the methods of traditional control, modern control, and intelligent control. Based on system control theory, many excellent control methods have been proposed for power systems to address various challenges.

Table 1. Comparison of recent studies on LFC in the literature.

Wind Power	Solar Energy	EVs	Deception Attacks	DoS Attacks	Feedback Control	PI/PID	SMC	Non-Fragile	Data-Driven
√	×	×	×	×	×	×	√	×	×	[13]
×	√	×	×	×	×	√	×	×	√	[29]
×	×	×	×	×	×	×	√	×	√	[30]
×	×	×	×	×	×	√	×	×	√	[31]
×	×	√	×	×	×	√	×	×	×	[32]
×	√	×	×	×	×	×	×	×	√	[33]
√	√	×	×	×	×	√	×	×	×	[16]
√	√	×	×	×	×	×	×	×	√	[28]
×	×	×	×	×	×	√	×	√	×	[34]
×	√	√	×	×	×	×	×	×	√	[21]
×	×	×	√	√	√	×	×	×	×	[35]
×	×	×	√	×	√	√	×	×	×	[27]
×	×	×	√	×	×	×	√	√	×	[36]
√	×	√	×	×	×	×	√	×	×	[9]
×	×	×	×	×	×	×	×	×	√	[37]
√	×	×	×	×	×	√	×	×	√	[38]

describes the contributions and limitations of recent research papers on these challenges.

In this survey, we review LFC power systems, recent challenges, and the strategies used to address them. The main contributions are summarised in the following aspects:

1.

This survey uses the flexible renewable power system model, in which the effect and uncertainty caused by renewable energy (wind and solar) are considered. Based on the detailed description of different mathematical models of renewable energy, the LFC problem can be solved with high productivity.

2.

Identifying types of cyber-attacks pertaining to LFC can help to formulate reasonable and effective strategies. We focus on the characteristics of these cyber-attacks and analyze their effects on LFC power systems.

3.

By optimizing or integrating control methods, excellent control strategies can be obtained and control performance can be improved. Intelligent control is more effective in solving the LFC issue for power systems with multiple renewable energy sources or those under cyber-attack due to its advantages in processing large amounts of data.

The remainder of this article is outlined as follows. The basis of LFC power systems in terms of the mathematic model and state space function is explained in Section 2. Section 3 introduces recent challenges related to LFC power systems. Section 4 describes defensive methods based on different control strategies. Finally, Section 5 concludes the survey.

2. Power System Model for LFC

To show the mathematical relationship and signal transmission of an LFC power system, a simplified model is provided here. Usually, the overall power system consists of M control areas connected by tie-lines [3,4,5]. Figure 2 presents a simplified i-th area system model.

The power system is a huge and very complex control system. Mostly, simplified linear models [3,4,5,6,7,8,9] have been proposed to show the operation of the actual dynamics for the multi-area power system. Based on device relationship and signal transmission, the mathematical model is expressed as

$$\left\{\begin{array}{cc}\hfill & \Delta {\dot{P}}_{mi}\left(t\right)=-\frac{\Delta P_{mi}\left(t\right)}{T_{chi}}+\frac{\Delta P_{vi}\left(t\right)}{T_{chi}},\hfill \\ \hfill & \Delta {\dot{P}}_{tie}^i\left(t\right)=2\pi {\displaystyle \sum _{j=1,i\ne j}^n}{T}_{ij}(\Delta f_i\left(t\right)-\Delta f_j\left(t\right)),\hfill \\ \hfill & \Delta {\dot{P}}_{vi}\left(t\right)=-\frac{\Delta f_i\left(t\right)}{R_i{T}_{gi}}-\frac{\Delta P_{vi}\left(t\right)}{T_{gi}}-\frac{\Delta P_{ei}\left(t\right)}{T_{gi}}+\frac{u_i\left(t\right)}{T_{gi}},\hfill \\ \hfill & \Delta {\dot{f}}_i\left(t\right)=\frac{\Delta P_{mi}\left(t\right)}{M_i}-\frac{D_i\Delta f_i\left(t\right)}{M_i}-\frac{\Delta P_{tie}^i\left(t\right)}{M_i}-\frac{\Delta P_{di}\left(t\right)}{M_i},\hfill \\ \hfill & AC{E}_i=\Delta P_{tie}+{\beta }_i\Delta f_i.\hfill \end{array}\right.$$

(1)

The nomenclature is summarised in

Table 2. Nomenclature of the power system.

Parameters	Mean	Parameters	Mean
$\Delta P_{mi}$	Generator power deviation	$\Delta P_{vi}$	Control valve position deviation
$\Delta P_{tie}^i$	Tie-line power exchange	$\Delta f_i$	Frequency deviation
$\Delta P_{ei}$	Integral of ACE	$\Delta P_{di}$	Load demand disturbance
${\beta }_i$	Frequency deviation factor	$D_i$	Load damping coefficient
$M_i$	Rotational inertia of the generator set	$\Delta P_{di}$	Load demand disturbancex
$T_{chi}$	Turbine time constant	$R_i$	Governor droop characteristic

.

We first define the following:

$x_i\left(t\right)={[\Delta P_{mi},\Delta P_{vi},\Delta P_{tie}^i,\Delta f_i]}^{\top }$, $y_i\left(t\right)=AC{E}_i$.

Furthermore,

$$A_i=\left[\begin{array}{cccc}-\frac{1}{T_{chi}}& \frac{1}{T_{chi}}& 0& 0\\ 0& -\frac{1}{T_{gi}}& 0& -\frac{1}{R_i{T}_{gi}}\\ 0& 0& 0& 2\pi {\displaystyle \sum _{j=1,j\ne i}^n}{T}_{ij}\\ \frac{1}{M_i}& 0& -\frac{1}{M_i}& -\frac{D_i}{M_i}\end{array}\right],\begin{array}{c}{B}_i={\left[0,\frac{1}{T_{gi}},0,0\right]}^{\top },\hfill \\ C_i={\left[0,0,1,{\beta }_i\right]}^{\top },\hfill \\ F_i={\left[0,0,0,-\frac{1}{M_i}\right]}^{\top }.\hfill \end{array}$$

(2)

Thus, the wind power system state vector space function is represented as

$$\left\{\begin{array}{c}{\dot{x}}_i\left(t\right)=A_i{x}_i\left(t\right)+B_i{u}_i\left(t\right)+F_i{w}_i\left(t\right),\hfill \\ y_i\left(t\right)=C_i{x}_i\left(t\right),\hfill \end{array}\right.$$

(3)

where $u_i\left(t\right)$ is defined as the control input and $w_i\left(t\right)=\Delta P_{di}$ is the system disturbance, which is bounded satisfying $\Delta P_{di}<ϵ$, where $ϵ$ is a positive constant.

3. Descriptions of Challenges

3.1. Renewable Energy

With the development of the power system, power demands have gradually increased. Therefore, the renewable energy power system is presented to lighten this pressure.

3.1.1. Wind Power

Wind power is a primary source of renewable energy. However, two main negative effects appear in the LFC wind power system. The ability to balance inertia is reduced. The complexity of variation of generation and additional uncertainty is raised.

(1) The wind power system integrates a capacity wind turbine generator (WTG) with a modern power system [7,8,9]. The dynamic model of the WTG can be written as

$$\Delta {\dot{P}}_{WTGi}=\frac{1}{T_{WTGi}}\Delta P_{w-i}-\frac{1}{T_{WTGi}}\Delta P_{WTGi}$$

(4)

where $P_{WTGi}$ is defined as the output power change of WTG, $P_{w-i}$ represent the wind power, and $T_{WTGi}$ is the time constant of WTG in the i-th area.

Therefore, the dynamic model of the i-th area wind power is represented as

$$\left\{\begin{array}{c}{\dot{x}}_{win}\left(t\right)=A_{win}{x}_{win}\left(t\right)+B_{win}{u}_i\left(t\right)+F_{win}{w}_{win}\left(t\right),\hfill \\ y_{win}\left(t\right)=C_{win}{x}_{win}\left(t\right),\hfill \end{array}\right.$$

(5)

where $x_{win}\left(t\right)={[x_i\left(t\right),\Delta P_{WTGi}]}^{\top }$ and $w_{win}\left(t\right)=[w_i\left(t\right),\Delta P_{w-i}]$.

The wind power in (5) is established by $P_{WTGi}$ and $P_{w-i}$. The output power change of WTG $P_{WTGi}$ is added to the power system state vector. Wind power is considered a kind of energy injected into the power system. However, the relationship between wind speed and wind power output cannot be explained. In other words, the fluctuation of renewable energy is ignored.

(2) Uncontrollable weather conditions usually lead to variable of WTG output power. Therefore, the wind power model should fully consider the impact of external interference. To realize the control of output power from the WTG, the pitch control technique has been developed [10,11,12,21]. Using the output feedback signal of the WTG, the angle of the blades can be controlled, meaning that the optimum angle of the blades tracks the variation in wind speed.

Figure 3 provides the mathematical model of a Nordex N43 turbine [10,11,12,21], where $\Delta P_{GW}$ is the wind power output deviation, $K_{p1}$, $K_{p2}$, and $K_{p3}$ represent the pitch control gain, hydraulic pitch actuator gain, and data fit pitch response gain, respectively, $T_{p1}$, $T_{p2}$, and $T_{p3}$ make up the time constant, $K_{pc}$ represents the blade characteristics, and $K_{fc}$ is the fluid coupling gain. The control signal $C_{1w}$ is obtained by the change in wind turbine mechanical output power. The inner controller $C_{2w}$ is provided by $C_{1w}$ and the inner process output $U_{1w}\left(s\right)$.

The relationship between the active power and input wind speed can be presented as

$$P_{GW}=\frac{\rho a^2{V}_w^3{C}_p}{2}(T_{SR},\beta )$$

(6)

where $\rho$ ($\mathrm{kg}/{\mathrm{m}}^3$) and a (${\mathrm{m}}^2$) is the area density and swept, respectively. $V_w$ ($\mathrm{m}/\mathrm{s}$) represents the wind speed, $\beta$ ($\mathrm{deg}$) represents the angle of the blade, $T_{SR}$ represents the tip speed ratio, and the rotor efficiency $C_p$ is provided as follows:

$$\begin{array}{cc}\hfill & C_p=\frac{T_{SR}-0.022{\beta }^2-5.6}{2}{e}^{-0.17T_{SR}}\hfill \\ \hfill & T_{SR}=\frac{r_{pm}\pi D}{60V}\hfill \end{array}$$

(7)

where the rotor speed is $r_{pm}$ and the diameter of blade rotor diameter is D$\left(m\right)$. It is worth noting that, as wind power is a renewable energy source, the power supply is always determined by environmental factors.

Considering the signal transmission shown in Figure 3, the transfer functions $G_{1w}$ and $G_{2w}$ [10,11,12,21] are obtained for the wind farm system:

$$\begin{array}{cc}\hfill & G_P\left(s\right)=\frac{K_{p1}(1+s{T}_{p1})}{(1+s)},G_H\left(s\right)=\frac{K_{p2}}{(1+s{T}_{p2})},\hfill \\ \hfill & G_D\left(s\right)=\frac{K_{p2}}{(1+s{T}_{p2})},G_I\left(s\right)=\frac{1}{(1+s{T}_w)}.\hfill \end{array}$$

(8)

where pitch control is defined as $G_P$, the hydraulic pitch actuator is defined as $G_H$, the data fit pitch response is defined as $G_D$, and the induction generator is defined as $G_I$.

By constructing the mathematical relationship of the wind and the generator, the output wind power $P_{WG}$ can be determined. The environment is considered based on the area density, swept area wind speed, and other factors. Differently, a few researchers have focused on the changes in the current and voltage of the WTG. According to the relationship between current and voltage, the WTGoutput can be easily obtained.

(3) Except for the turbine model in [10,11,12], a wind power generator can be modeled by considering changes in current and voltage [13,14,22,39,40,41,42,43]. In the wind power generation system, there exist different kinds of wind power generators, leading to different models.

A kind of squirrel-cage induction generator (SCIG) [14,22,39] has been widely considered for the output wind power. The SCIG is applied in wind power systems due to its high reliability and simple structure.

Figure 4 shows the basic configuration of the SCIG; $P_g$ is defined as the output power of SCIG, which can be calculated by the voltage. The detailed mathematical relationship is as follows:

$$P_g=\frac{-3V^{2}s(1+s)R_2}{{(R_2-s{R}_1)}^2+s^2{(X_1+X_2)}^2}$$

(9)

where the synchronous angular speed is ${\omega }_0$, $s=\frac{{\omega }_0-\omega }{{\omega }_0}$ is defined as the slip of generator, the stator resistance and reactance are defined as $R_1$ and $X_1$, respectively, and the rotor resistance and reactance are defined as $R_2$ and $X_2$, respectively. The wind power from SCIG is modeled for a multi-area power system in the case of $\omega \ge {\omega }_0$.

In wind power systems, the electric power is produced by the wind turbine, which is driven by wind power. The power coefficient is defined as $C_p\{\lambda ,\beta \}$, the pitch angle as $\beta$, and the area density as $\rho$, while $A_i$ is the cross-section of the rotor. With these, the output power can be determined. Based on [14,22,39], the output power is defined as

$$P_w=\frac{C_p(\lambda ,\beta )V_w^3\rho A_i}{2}$$

(10)

where $V_w$ is the wind speed. The mathematical relationship among the radius of the wind turbine R, the angular speed of the rotor $\omega$, and the wind speed $V_w$ is defined as the tip speed ratio $\lambda =\frac{R\omega }{V_w}$.

The following equation is used to obtain the angular speed of the rotor:

$${\omega }^2=\int \frac{2}{J}(P_w-P_g)dt$$

(11)

where the moment of inertia is defined as J. The pitch control system and hydraulic servo system have been studied based on the above definition. Compared with the explanation of a wind power system in [10,11,12,21], the relationship between the wind turbine and wind power is discussed in [14,22,39]. By constructing an SCIG mathematical model, it is possible to obtain find more information on the relationship between the turbine radius and the wind speed.

Another popular WTG for the wind power system is called a doubly-fed induction generator (DFIG). In [13,40,41,42,43], a DFIG system was discussed as a renewable resource. In previous survey research, it has been mentioned that most LFC wind power systems integrated with DFIG are trustworthy. To illustrate the difference between DFIG and [14,22,39,44,45], the mathematical structure of a simplified DFIG WTG is shown in Figure 5.

Based on the above mathematical relationships, we have the following equation for the rotor current of the q-axis component:

$${\dot{i}}_{qr}\left(t\right)=-\frac{i_{qr}\left(t\right)}{T_1}+\frac{X_2{V}_{qr}\left(t\right)}{T_1},$$

(12)

where $T_1=\frac{L_0}{w_s{R}_1}$, $L_0=L_r+L_m+\frac{L_m^2}{L_s+L_m}$, and $X_2=\frac{1}{R_2}$, in which the magnetizing inductance is $L_m$, the rotor leakage inductance is $L_r$, the synchronous speed is $w_s$, and $L_s$ represents the stator leakage inductance.

The rotational speed can be described by

$$\dot{w}\left(t\right)=-\frac{X_3{i}_{qr}\left(t\right)}{2H_t}+\frac{T_m\left(t\right)}{2H_t},$$

(13)

where $H_t$ is the given constant, $T_m$ represents the mechanical power, and $X_3=\frac{L_m}{L_s+L_m}$.

Therefore, the active power of WTG is obtained as follows:

$$P_e\left(t\right)=w\left(t\right)X_3{i}_{qr}\left(t\right),$$

(14)

The active power can be rewritten as

$$P_e\left(t\right)=w_p{X}_3{i}_{qr}\left(t\right),$$

(15)

where $w_p$ is the rotational speed at the operating point and the electromagnetic torque is described by

$$T_e\left(t\right)=i_{qs}\left(t\right)=-\frac{L_m}{L_s+L_m}{i}_{qr}\left(t\right).$$

(16)

Defining $x_w\left(t\right)={[\Delta X_{gi}\left(t\right),\Delta P_{gi}\left(t\right),\Delta f_i,\Delta P_{tie}^i\left(t\right),\Delta i_{qr,i}\left(t\right),\Delta w_i\left(t\right),\Delta E_i\left(t\right)]}^{\top }$, the WTG power system state equation is obtained as follows:

$${\dot{x}}_w\left(t\right)=A_w{x}_w\left(t\right)+B_w{u}_w\left(t\right)+F_w\Delta P_{di}\left(t\right).$$

(17)

In this subsection, we have discussed the different models of wind power. First, we introduced wind power as directly considered in the power system, with renewable resource energy supported by a capacity wind induction generator. Then, we discussed how to control the WTG through the pitch control method. Lastly, we considered how the different kinds of wind power generators influence the construction of power system models integrated with WTG. As a result of differences in the internal structure of various generators, their mathematical models are different. Focusing on this, the working principles of the SCIG WTG and DFIG WTG are described in detail using mathematical relationships.

3.1.2. Solar Energy

Another important and common renewable energy source is solar energy. To capture solar energy and make full use of it, a proper device must be used. There are two main methods of using solar energy [21].

(1) A large diode PV panel is subjected to sunshine. Using PV panels, solar energy is converted into a current that can be transmitted through conducting wires. In theory [46], an ideal PV solar energy can be modeled as the integration of a photocurrent source and diode. The variation in ambient temperature and sunlight irradiance is proven to influence the power supplied by PV panels [12]. To study the solar energy applied in power systems, a simplified PV panel model can be constructed considering the relationship of various electronic components [12,21,33].

The single-diode PV module equivalent circuit model is shown in Figure 6.

To analyze the voltage and current of the PV panel, the photocurrent $I_P$ is considered. The PV current is described by the following equation:

$$I=I_p-I_{sd}-I_r$$

(18)

where $I_p$, $I_{sd}$, and $I_r$ are the photocurrent, diode current, and current of the parallel resistance, respectively.

When subjected to sunshine at a certain temperature, a PV produces a photocurrent $I_p$, which can be described as follows:

$$I_p=I_S\left(\frac{S}{1000}\right)+J_t(T-T_{re})$$

(19)

where S represents the solar irradiance, $I_s$ represents the short-circuit current, $J_t$ represents the temperature coefficient, T represents the PV temperature, and $T_{re}$ represents the reference temperature.

The diode current $I_{sd}$ can be expressed as

$$I_{sd}=I_o\left[\exp \left(\frac{V+I{R}_s}{nmT}\right)-1\right]$$

(20)

in which the terminal voltage V, current I, series resistance $R_s$, diode ideality factor n, and Boltzmann’s constant m are considered. The saturation current $I_o$ can be obtained by

$$I_o=I_d{\left(\frac{T}{T_{re}}\right)}^3\exp \left[\frac{q{E}_g}{nm}\left(\frac{T-T_{re}}{T_{re}T}\right)\right]$$

(21)

where $I_d$ is the diode reverse current, $E_g$ is the bandgap energy of the cell semiconductor, and q is the charge of the electron.

Finally, $I_r$ can be obtained as follows:

$$I_r=\frac{V+I{R}_s}{R_h}$$

(22)

where $R_h$ is the parallel resistance.

In several studies [16,21,47,48], a first-order transfer function has been constructed to described the PV system. The transfer function of the PV–grid interface is

$$\Delta P_v=\frac{K_{pv}}{1+s{T}_{pv}}\Delta P_{solar}$$

(23)

where $K_{pv}$ is the gain of the PV and can be chosen as one, while $T_{pv}$ is the time constant. The dynamic behavior of the PV power conversion systems is neglected.

It is worth noting that a double-diode PV model can be designed by using two diodes [33]. A model with diodes D1 and D2 is shown in Figure 7. The mathematical relationships of a two-diode PV system are mostly the same as in one-diode PV systems.

(2) Solar thermal technologies are another technique to extract power from sunshine. Solar energy is transferred to a thermodynamic cycle to drive a mechanical device, creating electricity. It is worth noting that more governors and turbines are used in the solar thermal power system. Unlike wind power plants and PV power plants, an autonomous controller is needed for LFC solar thermal power systems [21]. The solar thermal power system model can be described using the solar energy, control signal, governor, and turbine, as shown in Figure 8.

Similarly, solar thermal technology can be represented as a first-order lag [49]. Based on signal transmission, we have

$$\Delta P_{solar}=\frac{1}{s{T}_t+1}\Delta P_G$$

(24)

where $K_S$ is the solar energy gain, $T_s$ is the time constat of collar energy, and $T_g$ and $T_t$ are the time constant of governor and turbine, respectively. The $ACE$ signal is taken as the control input in the LFC strategy. Figure 8 shows the steam produced in the heat exchangers to drive the turbine. Further, the deviation of $\Delta P_{solar}$ is considered in the power system and influences the balance of load and frequency.

Uncertainty or disturbance due to renewable energy sources cannot be avoided. The intermittent nature of renewable energy sources, system uncertainties with respect to the control and parameters, and linearization error all increase the difficulty of the LFC problem. Furthermore, there exist parameter uncertainties in any practical power system due to the changes and constantly varying load at the operation point. An uncertainty model with bounded uncertainties [50,51] can be defined as follows:

$$g_i\left(t\right)=\Delta A_i{x}_i\left(t\right)+\Delta B_i{u}_i\left(t\right)+(F_i+\Delta F_i)(\Delta P_{di}+P_{gi})$$

(25)

where $\Delta A_i$, $\Delta B_i$, $\Delta F_i$ are mismatched parametric uncertainties, $\Delta P_{di}$ is the system disturbance, and $P_{gi}$ represents the uncertainty in the renewable energy source.

It can be understood that renewable energy is important to our life and industry. Reasonable and efficient use of these energy sources can undoubtedly reduce the pressures caused by obtaining electric power in traditional ways. To estimate the disturbance due to load demand, system uncertainty, and WTG, in [52] the authors proposed an extended state observer. In another study, an LFC method was applied to ensure the performance of multi-area wind power systems [53]. Generally, wind power fluctuation, load disturbances, and system parameter uncertainties lead to larger voltage deviation [54]. Due to the complexity of the actual situation, different modeling methods can provide different solutions for the efficient use of renewable energy. Model description can help to handle the LFC issues in today’s modern power systems, and as such is an important research area closely connected with renewable energy.

3.2. Electric Vehicles

To achieve low-carbon targets, EVs represent an excellent advance. EVs can help to improve the power grid’s load factor while minimizing fluctuations. In addition, the integration of electric vehicles into the power system can balance power demand and reduce the prime cost of electricity supply [55]. Positive externalities for renewable energy integration from the side of power supply can be served by EVs [23]. The ratio of coverage of EVs helps to determine the power system’s operation. This means that reasonable use of EVs offers a chance to accelerate the transformation of the power system.

In EVs, the battery is the unique power supply. The battery has to face the problems of reliability, safety, and lifetime, which are key factors in realizing widespread coverage of EVs. The battery’s state of charge (SOC) reveals the relationship between stored energy and its capacity, which is an important feature of an EV. Considering an electric vehicle connected to the power system, a simplified model is shown in Figure 9.

The power output deviation of the EVs can be described as

$$\Delta {\dot{P}}_{EV}=\frac{K_{ev}}{T_{ev}}{u}_i\left(t\right)-\frac{1}{T_{ev}}\Delta P_{EV}$$

(26)

where $K_{ev}$ is the EV gains and $T_{ev}$ represents the constant parameters. Defining ${\widehat{x}}_i\left(t\right)=[x_i\left(t\right),\Delta P_{EV}]$, a power system model with EVs can be constructed easily. To clearly analyze the LFC problem, we need to know the inner signal transmission.

Furthermore, it is necessary to explore the EV model in a way that can explain the impact of EVs on the LFC. A straightforward EV model is proposed in Figure 10.

The electric vehicle model consists of an open circuit voltage source connected in series, with a series resistance $R_s$ and a parallel branch $R_t$, $C_t$ [56]. The terminal electric vehicle voltage can be obtained according to the the open-circuit voltage and the voltage drop across the series resistance and the R, C branch. Based on the mathematical relationship in Figure 10, the relationship between the electric vehicle open circuit voltage $V\left(SOC\right)$ and nominal voltage $V_n$ is as follows:

$$V\left(SOC\right)=V_n+\frac{SRT}{F}ln\left(\frac{SOC}{C_n-SOC}\right)$$

(27)

where F, T, R, are constant parameters and S is the sensitivity parameter between SOC and open circuit voltage.

Integration of power systems with EVs has become a primary research area, with a particular focus on EVs and power system operation. Optimized charging for reduced cost, electricity market participation, and scheduling and control techniques are among the most considered directions. Importantly, it is necessary to avoid causing network overloading or compromising system stability due to EV charging.

3.3. Cyber-Attacks

The timescale of seconds on which LFC signals are generated means that complex data validation algorithms cannot be used for estimation and validation of measurement data. This has contributed to issue of attackers being able to rewrite transmitted data by exploiting simple mathematics [57]. This is a major reason for power systems being easily damaged by cyber-attacks. Weaknesses or characteristics that exist in the power system can be used by attackers to design possible attack strategies. It is possible to mitigate or eliminate the impact of attacks on the power system by designing detection or control methods. To construct data integrity attacks, a common assumption about attackers having prior model knowledge of physical systems was relaxed in [58]. In the view of [24], DoS attacks and data integrity attacks are the most common cyber-attacks targeting LFC signals. Normally, it is necessary to assume a particular attack type. Therefore, we focus here on the impact of the most typical DoS and deception attacks.

3.3.1. Dos Attacks

When a DoS attack occurs, communication resources become occupied and transmission signals are delayed, causing devices to operate under incorrect conditions. DoS attacks take two forms, namely, stochastic and queueing.

(1) A Bernoulli process or Markov process is used to build a DoS attacks model. In such models, there is a possibility of a DoS attack happening in the communication channel at each sampling moment.

A power system model under DoS attack with a Bernoulli process [25,26,59] is provided as follows:

$$\left\{\begin{array}{c}{\dot{x}}_i\left(t\right)=A_i{x}_i\left(t\right)+(1-{\alpha }_i)B_i{u}_i\left(t_k\right)+{\alpha }_i{B}_i{u}_i\left(t_{k-1}\right)+F_i{w}_i\left(t\right),\hfill \\ y_i\left(t\right)=C_i{x}_i\left(t\right),\hfill \end{array}\right.$$

(28)

in which ${\alpha }_i\in 0,1$ is the Bernoulli distribution related to occurrences of DoS attack, and is described as

$$E\left({\alpha }_i\right)=\alpha ,E\left({({\alpha }_i-\alpha )}^2\right)=\alpha (1-\alpha ),$$

(29)

where $\alpha$ is the probability of a DoS attack.

A power system model under DoS attack with a Markov process [60] can be described as follows:

$$\left\{\begin{array}{c}{\dot{x}}_i\left(t\right)=A_i{x}_i\left(t\right)+{\beta }_i{B}_i{u}_i\left(t\right)+F_i{w}_i\left(t\right),\hfill \\ y_i\left(t\right)=C_i{x}_i\left(t\right),\hfill \end{array}\right.$$

(30)

where ${\beta }_i\in \{0,1\}$ represents the Markov modulated attack sequence that prevents the signal from being transmitted to the next device.

(2) In the queueing model, the DoS on/off action $h_n$ is considered, assuming the time instants at which the action of DoS on/off is changed (when it is possible to interrupt communication). The following time interval describes the sequence of DoS attacks:

$$H_n=h_n\cup [h_n,h_n+{\tau }_n]$$

(31)

where ${\tau }_n$ is the attack length during which communication is interrupted. When ${\tau }_n=0$, a single pulse at time $h_n$ can be seen as the sequence of DoS attacks. The most recent data received from the controller during the DoS attack are taken as the actuator signal, while $\Gamma \left(t\right)$ is the time interval.

$$\Gamma \left(t\right)=[\tau ,t]\backslash \{H_n\cap [\tau ,t]\}$$

(32)

3.3.2. Deception Attacks

False data injection (FDI) attacks and malicious attacks, and are defined as deception attacks on the LFC. Deception attacks seek to modify the transmitted information at certain points in an LFC power system [61]. Due to the different possible attack modes, there are many types of deception attacks.

First, we consider deception attacks to inject false data into the power system. These deception attacks corrupt the real-time transmission signal in the LFC power system. To realize this, deception attacks target the control or measurement channels. The attack vectors are designed based on attack templates or data corruption strategies. The state space function under deception attack is as follows:

$$\left\{\begin{array}{c}{\dot{x}}_i\left(t\right)=A_i{x}_i\left(t\right)+B_i(u_i\left(t\right)+{\alpha }_i{D}_{F1})+F_i{w}_i\left(t\right),\hfill \\ y_i\left(t\right)=C_i{x}_i\left(t\right)+{\alpha }_i{D}_{F2},\hfill \end{array}\right.$$

(33)

in which ${\alpha }_i$ is the Bernoulli distribution related to occurrences of deception attacks, $D_{F1}$ are the deception attacks on the actuator, and $D_{F2}$ are deception attacks on the system output.

In addition, deception attacks can be implemented by replaying recorded data. This kind of deception attack is called a replay attack. First, a disclosure attack is carried out to gather data sequences from the compromised resources. Then, the gathered data sequences are replayed. Replay attacks exploit valid data via delayed transmission or repeat the same signal fraudulently [62]. For example, attackers may iterate past data from the attacked actuators or sensors for a definite time in a Byzantine replay attack. A sensor channel under replay attack can be described as follows:

$$D_a=-C{x}_i\left(t\right)+y_i(t-{\tau }_i\left(t\right))$$

(34)

where ${\tau }_i\left(t\right)\in (0,t)$, $y_i(t-\tau )$ represents the sensor data. Replay attacks can be described as time-varying delays ${\tau }_i\left(t\right)$, in which the rate of change and upper bound are unknown. Replay attacks have the characteristic of a distinct monitoring phase and replay phase.

In addition to the types of deception attacks mentioned above, other deception attacks may involve covert attacks [58], zero-dynamics attacks [63], and more.

4. Strategies

In this section, we describe the various control strategies, including classical control, modern control, and intelligent control. New challenges such as renewable energy, electric vehicles, and cyber-attacks raise the degree of difficulty; therefore, modern control methods have been developed and applied to guarantee stable systems [64]. With the continuous development of artificial intelligence [28], many excellent new data-driven control methods have been applied to solving LFC problems.

4.1. Classical Control Method

Of the classical control methods, the most considered are PI and PID. In LFC, PI or PID control methods use the tie-line power deviation and frequency deviation to design the controller. As shown in Figure 2, the area control error $AC{E}_i$ is summarised based on the signal transmission $\Delta P_{tie}$ and $\Delta f_i$. The PI controller is given as

$$u_i=K_{p}AC{E}_i+K_i\int AC{E}_i$$

(35)

where $K_p$, $K_i$ represent the controller gain. By defining $y_i\left(t\right)=[AC{E}_i,\int AC{E}_i]$, the controller can be expressed as

$$u_i=K_{pi}{y}_i\left(t\right)$$

(36)

where $K_{pi}=[K_p,K_i]$, $K_{pi}$ will be designed.

The PID controller [19,65,66] can be designed as follows:

$$K_i\left(s\right)=K_p+\frac{K_i}{s}+s{K}_d$$

(37)

and the control input is

$$u_i=K_i\left(s\right)y_i\left(t\right)$$

(38)

Based on the frequency-domain approach, a fractional order PID controller cascaded with a first-order filter for the delayed LFC system was discussed in [67]. By using an improved version of particle swarm optimization [16], a PID-based secondary controller was proposed to stabilize the system. Based on the gain-scheduling decentralized (GSD) scheme, the authors in [68] compared a GSD-PID controller and a GSD-PI controller. To overcome uncertainties and reduce the computational cost, the authors of [69] described a novel numerical method to obtain stabilizing sets of PID or PI controllers. A tuning-free PID control strategy was developed in [70].

To address recent challenges, PI and PID controllers for LFC power systems have been widely researched. In [71], a hybridized fuzzy-PI-linear active disturbance rejection controller was designed for a power system with renewable energy and EVs. In [27], the $H-\infty$ performance was guaranteed by an event-based PI LFC scheme for power systems suffering from deception attacks. In [72], a novel cascade fractional order controller comprising a three-degrees-of-freedom PID and tilt-integral controllers was proposed for dealing with the LFC issue in a WTG power system. A PID controller was applied as the supplementary control task in a renewable power system in [20]. A PID and fuzzy controller were integrated into a fractional order environment for stabilizing power system frequency in [73].

4.2. Modern Control Method

Many control strategies have been developed for the LFC issue, including adaptive control, variable structure, intelligent techniques, digital control, and robust control. In variable structure control methods, SMC is a typical example. Aside from its simple implementation, SMC is insensitive to disturbance and parameter variations [74]. It is important that the useable sliding mode surface satisfies $s\left(t\right)=0$ and $\dot{s}\left(t\right)=0$. Researchers have made many attempts to design a proper SMC.

In [13,75,76], a kind of PI-type SMC was applied to an LFC power system

$$s\left(t\right)=G_i{x}_i\left(t\right)-{\int }_0^t{G}_i(A_i-B_i{K}_i)$$

(39)

where $G_i$ and $K_i$ are a constant matrix and $G_i$ is chosen to ensure that $G_i{B}_i$ is non-singular.

Mostly, the system state is applied to the SMC design. However, in many cases only the system output is available, such as in the case of system states that cannot be completely measured or may be overly costly to measure. Therefore, two kinds of methods are provided to design the output sliding mode controller [77]; the first is to estimate the states of the system using a state observer, while the second is to design the controller using the system output directly. Based on the system state and output, an output sliding mode surface [78] is designed as

$$s\left(t\right)=G_i{y}_i\left(t\right)-{\int }_0^t{C}_i{x}_i\left(\vartheta \right)d\vartheta$$

(40)

where $y_i\left(t\right)$ is the system output and $G_i$ is selected to ensure that $G_i{C}_i{B}_i$ is non-singular. The problems of system states that cannot be completely used are well solved by the output feedback SMC. In addition, considering the uncertainty or disturbance that exists in renewable power systems, a full-order observer can be applied to estimate the states of the system [79]. An output-dependant SMC was designed as follows:

$$s\left(t\right)=G_i(y_i\left(t\right)-y_i\left(0\right))-{\int }_0^t{L}_i\left(q\right)dq$$

(41)

with observed data $L_i\left(t\right)$ as follows:

$$L_i\left(t\right)=-K_i(x_i\left(t\right)-e_i\left(t\right))$$

(42)

where $e_i\left(t\right)$ is the state error.

On this basis, many excellent designs of sliding mode surfaces have been proposed. An adaptive fast terminal sliding mode controller was designed in [80] in which the bound of the unknown disturbances was estimated using an adaptive law. The authors of [52] designed a fractional-order integral SMC that avoided chattering of the frequency deviation. The authors of [53] explored the design of a derivative and integral terminal SMC strategy to handle the LFC problem. By taking advantage of the adaptive SMC and sliding mode observer, a novel control strategy can be proposed to help stabilize power systems [54]. Most control problems are solved by the design of the controller, and as such the controller needs to be seen as completely believable. However, many neglected factors, such as the round-off error in numerical computation, the aging/failure of system components, and the finite precision of measuring equipment can make the related assumptions may be infeasible [81].

To theoretically guarantee system stability, the controller should be designed to suppress the entire uncertainty. In Section 3.1, we show how uncertainties or disturbances from renewable energy can be considered together using an aggregated mismatched uncertainty $g_i\left(t\right)$ model. Similarly, it is possible that there may exist uncertainty in the controller. To describe such situations, the non-fragile control problem is proposed. A non-fragile controller is described as

$${\tilde{K}}_i=K_i+\Delta K_i$$

(43)

where the controller gain perturbation $\Delta K_i$ is defined as

$$\Delta K_i=M_i{\xi }_i{N}_i$$

(44)

with the uncertainty matrix satisfying $\parallel {\xi }_i\parallel <I$, $M_i$, and $N_i$ being the constant matrix.

To obtain a more practical controller, the non-fragile control problem can be used. In [81], an extended dissipative fuzzy non-fragile PID controller was designed with controller parameter perturbations. The authors in [82] proposed a non-fragile control approach such that the controller parameter variations can be ignored without loss of system performance. Considering the existence of control uncertainty, a non-fragile PI-control scheme [34] was discussed in the context of the LFC approach for nonlinear power systems. For a linear control system under DoS attack, a non-fragile feedback controller was applied in [83]. Considering uncertainties and external disturbances EVs power system, a full-order observer-based finite-time non-fragile LFC was designed in [84].

4.3. Intelligent Control

Intelligent control can be applied to improve the performance of LFC power systems. With intelligent control, relevant information about the power system is processed based on algorithms or logic. With the increasing size scale of power systems and data, it may be doubted whether data-driven control schemes represent a promising solution to balancing system frequency. In [85], a fuzzy-PSO-PIDLFC was obtained based on particle swarm optimization (PSO). Based on multi-agent deep reinforcement learning with continuous action domain, a data-driven cooperative LFC method for a power system was discussed in [37]. To realize the LFC strategy, area control errors need to be minimized. The objective Q is defined as unscheduled tie-line power flow interchange and the squared sum of frequency deviations [37], which is provided as

$$Q^{\mu }(s,\alpha )=-\sum _{t=1}^T[\Delta t\sum _{i=1}^n[{({\beta }_i\Delta f_i)}^2+{(\Delta P_{tie})}^2]]$$

(45)

where s is the state information, $\Delta t$ is the step size, and T represents the simulated total time. After exploration, the system frequency under the data-driven strategy is suitable for all scenarios. During centralized learning, a multi-agent deep deterministic policy gradient is applied to determine the controller parameters, which is optimized based on the coordinates in order to improve the load frequency control performance. An agent parameters initialization process is applied to accelerate the process of multi-agent deep reinforcement learning.

In addition, the data-driven method can be applied to improve the performance of traditional or modern control methods [86,87]. This means that data-driven control has great development potential in control theory. Many strategies have been proposed to realize data-driven LFC. An improved version of particle swarm optimization was developed to accelerate the optimization of the nonlinear LFC problem in [68]. Based on hybrid bacteria foraging-oriented particle swarm optimization, a linear quadratic regulator (LQR) controller was designed for AGC in [88]. PSO was applied to help design a fractional-order type-2 fuzzy logic controller for a frequency control problem of a renewable power system in [38]. To realize LFC, a generalized compression neural network (GRNN) was integrated for the design of a controller in [89]. In [71], optimal controller parameters were obtained using a new quasi-opposition-based artificial electric field algorithm. The selection of controller parameters is always a major challenge; the imperialist competitive algorithm was fruitfully exploited for solving this problem in [90]. Based on deep reinforcement learning in the continuous action domain, a model-free method for data-driven LFC was developed for renewable power systems with uncertainties in [91]. In [92], an event-triggered data-driven model-free adaptive LFC method was proposed for power systems. To solve the randomness from the power load demand and renewable generations, an adaptive learning strategy was applied in [93] to design the SMC for an LFC power system. An evolutionary imitation curriculum multi-agent deep deterministic policy gradient algorithm was developed in [28] to realize coordinated control.

4.4. Others

In addition to the control methods mentioned above, there are many control methods with excellent performance in solving LFC problems. In [94], the authors attempted to realize the integration of renewables into modern power systems using the concept of a grid-forming (GFM) converter. A whale optimization algorithm was used in [95] to enhance the virtual inertia control loop by optimizing the parameters of a virtual inertia controller while considering the uncertainties of system inertia with renewable energy sources. Virtual inertia control was applied to the parameter regulator of a WTG based on direct heuristic dynamic programming in [96]. In [97], a new application of a robust virtual inertia control-based coefficient diagram method controller was proposed for an islanded MG considering high-level RES penetration for enhancement of system validity and robustness in the face of disturbances and parametric uncertainties. In [98], dynamic deloading was proposed considering the influence of the safe limit of a WTG, the wind power prediction error, spinning reserve of the SG, and system operating costs. Based on a deep learning algorithm, an adaptive nonlinear deloading strategy for a wind turbine generator integrated with an interconnected LFC power system was proposed in [99]. The salp swarm algorithm (SSA) ws applied to promote the LFC performance of a multi-area hybrid renewable nonlinear power system in [100]. An optimal LFC was designed through combined state and control gain estimation for noisy measurements in [101]. A decentralized resilient $H-\infty$ LFC was proposed in [102] for multi-area power systems under DoS attacks.

5. Conclusions and Future Directions

5.1. Conclusions

This survey has explored the recent challenges in LFC power systems. Focusing on the mathematical modeling and state space function, the effect of certain challenges (wind, solar, EVs, and cyber-attacks) facing power systems has been investigated in detail. For all of these different challenges, we have considered the problem from the perspective of the relevant models and power system state function. For each challenge, we have analyzed the impact on the power system from the perspective of different models. This survey has shown that identifying the most common types of cyber-attacks targeting the LFC helps in formulating reasonable and effective control strategies. Aiming at the uncertainty problems that many researchers ignore, this paper studies renewable energy uncertainty and controller uncertainty. To handle such challenges in the LFC power systems, different control methodologies have been studied for traditional control, modern control, and intelligent control. According to publications on recent challenges and control methods, the deficiencies of existing research have been identified, which can help to inspire future researchers to seek more reliable and efficient control methods.

5.2. Future Directions

At present, the stability problem and communication security problem with respect to renewable energy interconnected power systems is well worth noting. Integration and optimization of multiple renewable energy sources represents a major challenge facing future research. Furthermore, the development of innovative technologies such as cloud computing, big data, and artificial intelligence shows great potential for solving LFC-related problems. The current challenges and future research directions are discussed below.

(1)

Renewable energy. With the deepening of research on renewable energy, certain sources of renewable energy that could not be used before have been gradually developed for power systems. The diverse power output of renewable energy sources connected to the power system increases the complexity variability in terms of power generation and adds uncertainty. This presents challenges for the stabilization of LFC power systems integrated with renewable energy sources. In addition, the integration of multiple renewable energy sources means that more electronic equipment needs to be used. The resulting need to ensure matching between devices, efficient information transmission, and other issues are unavoidable challenges facing LFC power systems.

(2)

Cyber security. Cyber-security problems are directly related to the normal operation of power systems. Compared with a single type of cyber-attack, hybrid cyber-attacks can have greater impact. Furthermore, new types of cyber-attacks on LFC power systems deserve attention. According to our investigation, most of the cyber-attacks considered in the context of LFC power systems can be modeled using the queueing model or stochastic model. However, the types of cyber-attack are increasing. One possible development direction of cyber-attacks involves smart design. For example, attackers may use devices to obtain information about the power system and determine vulnerable nodes in the network according to the obtained information. Cyber-attacks are expected to remain an important problem in LFC power systems in the future.

(3)

Control capability. In recent years, a large number of intelligent algorithm-based control methods have been researched for LFC power systems. Multiple issues around renewable energy, cyber-attacks, and other issues are focusing greater development space and attention on intelligent control methods. How to improve control capability is becoming an important issue. To achieve this goal, researchers need to choose their research direction; possibilities include combining traditional control methods with modern control methods, using intelligent control methods, and developing new intelligent algorithms.

(4)

Flexibility. A highly flexible and secure interconnected power system is a necessity due to the high penetration and networking requirements of renewable energy sources. The transition of the current energy production mix can be aided by hosting a larger proportion of renewable energy sources. In this way, multi-area power system can become more flexible and reliable, resulting in the development of novel intelligent algorithms along with control, storage, and market approaches.