Prescribed Performance Load Frequency Control for Regional Interconnected Power System Under Energy Storage System Output Constraints

Abstract

This study addresses the issue of frequency instability caused by an imbalance between load power and generation power in a power system. A state-space model of a two-area power system including a thermal power plant is first established, incorporating the output power limitations of the energy storage system, which is the actuator for frequency control. Under input saturation constraints, a frequency control strategy based on a prescribed performance control technique is proposed. This strategy not only ensures frequency stability but also achieves an optimal transient response curve. The proposed control strategy is theoretically validated and numerically simulated, demonstrating its effectiveness in suppressing frequency variations in power systems under constraints regarding the output power of the energy storage system.

1. Introduction

During the operation of power systems, a dynamic balance between load power and generation power is crucial for maintaining system stability. When this balance is disrupted, frequency deviations occur, posing significant threats to system security. In recent years, frequency-deviation-related grid failures have occurred frequently. For instance, in 2016, the South Australian grid experienced severe frequency variations due to imbalance, triggering emergency shutdowns of multiple generators and significantly disrupting power supply [1]. In 2019, in the United States, in-depth research into frequency regulation technologies was initiated to enhance the California power system’s ability to respond to sudden frequency variations [2]. In 2020, load-shedding measures were implemented in certain regions of the Indian grid to prevent system collapse caused by frequency variations, further highlighting the critical role of frequency control in grid operation [3]. Nowadays, power systems are a combination of renewable-energy power plants and conventional-energy power plants. The integration of multiple sources and frequently changing loads cause frequency variation, which is unexpected in a power system.

To address these issues, researchers [4,5,6] have investigated load frequency control (LFC), which focuses on maintaining the power balance among interconnected areas under various load conditions. Researchers [7,8,9] have also applied battery-based energy storage systems (BESSs) and redox flow batteries that can be rapidly activated during frequency anomalies to balance the mismatch between grid load and generation by injecting or absorbing power, thereby mitigating the impact of frequency variations. In addition to BESSs, compressed air energy storage (CAES) systems are also applied in LFC [10,11]. Compared with CAES, BESSs have advantages in efficiency and rapid response, for example, the latent heat of CAES usually accounts for more than 30% of the energy. The energy storage system serves as the vital link between generating and utilizing energy, playing a critical role in managing the variability of renewable energy sources and fortifying the stability of power grids. There are various kinds of energy storage systems, such as thermal energy storage, mechanical energy storage, chemical energy storage, electrochemical energy storage, and so on [12]. Among these, BESSs have been widely recognized by engineers for their excellent overall characteristics. With millisecond-level response times, BESSs are well suited to counteract rapid frequency changes and efficiently stabilize the grid frequency [13]. Hybrid mechanical and electrical systems, such as flywheel–battery hybrid energy storage systems or photovoltaic–battery energy storage systems, have also been applied to regulate frequency changes, such as in [14,15].

The increasing spatial imbalance between load demand and generation resources in modern power systems has intensified the need for cross-regional resource sharing and coordinated control. To tackle this challenge, multi-area power systems rely on tie-lines for inter-area resource allocation. However, regional load variations can still adversely affect global frequency or voltage stability [16]. Load frequency control, a core function of Automatic Generation Control, plays a pivotal role in maintaining the balance between generation power and load demand, ensuring system frequency stability [17,18].

Researchers have proposed various control strategies for LFC in multi-area interconnected power systems. The existing approaches include proportional–integral–derivative controllers [19], control strategies combining the multiverse optimizer algorithm with improved objective functions [20], methods integrating the improved particle optimization algorithm and proportional–integral–double derivative with fractional-order integration [21], and fractional-order PID controllers [22]. While these methods can theoretically drive frequency deviations to zero, they suffer from practical limitations such as excessive transient overshoot, prolonged convergence times, and significant fluctuations in controller output magnitudes, which hinder their applicability in ensuring secure and efficient power system operation.

Notably, when energy storage systems participate in frequency control, some physical constraints should be considered, for example, the State of Charge and maximum allowable current of the BESS [23], the limited network bandwidth of the communication network employed to transmit the control signal [24], the volume of the air storage system [10,11,25,26], and the mechanical limits of turbine/governor controllers [27]. These constraints can be modeled as saturation mathematically. Without such constraints, abrupt frequency changes could demand power outputs exceeding the design limits of the BESS. This may subject internal components (e.g., batteries, power converters, or assorted mechanical systems) to extreme stress, leading to overheating, swelling, explosions in batteries, or overload damage in converters [28]. Furthermore, unrestricted high-power demands accelerate the aging and performance degradation of energy storage systems, ultimately compromising their ability to support grid frequency stability [29]. Thus, researchers have been interested in the entire LFC process to ensure power system tolerance; as a result, the least mean square algorithm, model predictive control, and linear matrix inequality-based robust control have been proposed to enhance grid stability, improve power quality, and ensure reliability in grid applications [30,31,32].

It is worth mentioning that although the main objective of LFC is to maintain frequency stability, which can be viewed as a steady-state target, transient progress should also be given due attention. According to [33], transient performance in power system is a crucial evaluation index, especially in the case of disturbance and uncertainty [34]. In [35], an optimized integral sliding mode control scheme was applied to improve the transient progress, i.e., the overshoot and the steady-state error. In [36,37], artificial intelligence approaches such as neutral networks were applied for the parameter adjustment in the control progress. To address the limitations of existing methods, we propose a novel prescribed performance control method for LFC in a two-area interconnected power system. The aforementioned works on LFC [13,14,15,16,17,18] are mainly focused on the final stability of frequency, which is a steady-state target. However, during the process of achieving this steady-state goal, the system may experience unwanted transient responses. Our research is precisely inspired by this point, and the main contributions of our work are as follows:

1.

Unlike [25,26,27,28,29,30,31,32,33,34,35,36,37,38], we take the input saturation into consideration, which is a common constraint in real systems.

2.

Moreover, we focus on simultaneously realizing steady-state control objectives and maintaining transient performance, i.e., the final frequency error is limited, and the entire progress evolves within a predefined range.

The rest of this paper is organized in the following way: In Section 2, we propose the problem formulation for this study. A state-space model of the multi-area power system is established, and a prescribed performance controller is introduced. To enhance practical applicability, input saturation constraints are incorporated to ensure that the control signals remain within permissible ranges, preventing system instability, equipment damage caused by excessive control signals (encountered in [39]), and unsatisfactory performance in worst-case conditions (encountered in [40]). Then, in Section 3, we propose a strategy that generates the BESS input signals, enabling the rapid and stable compensation of power imbalances. A theoretical analysis is presented as well. This approach achieves dynamic load balancing across multiple regions, effectively maintaining overall system frequency stability. A numerical example is proposed to demonstrate the validity of the proposed controllers. Finally, Section 5 provides a summary.

2. LFC Model of Two-Area Interconnected Power System

As shown in Figure 1, the LFC model of a two-area interconnected power system consists of control area i and control area j, which operate independently but interdependently. Each control area includes units such as controllers, energy storage systems (ESSs), governors, reheat turbine units, and generator units. These components play distinct roles in dynamically regulating system frequency variations, achieving frequency stability and load balance through multi-level coordination.

In the control process, the controller first generates regulation signals based on the frequency variation within the area. To avoid overload or instability caused by excessive control signals, when an ESS participates in frequency regulation, the controller’s output signal, which is constrained by a saturation function, serves as the target power input for the BESS. The BESS then provides corresponding power compensation according to this input, rapidly responding to load frequency control demands to reduce the Area Control Error. Subsequently, the adjusted power signal is transmitted to the governor and the reheat turbine unit. By regulating the steam flow and rotational speed, these units further optimize output power to meet dynamic load balance requirements.

To enable inter-area interconnection and resource sharing, the output signals from each control area are transmitted to the other area via tie-lines, facilitating the coordinated regulation of frequency variations between the two areas. This interconnected architecture allows the dynamic adjustments in area i and area j to mutually influence each other, achieving global frequency coordination. Ultimately, through precise frequency deviation regulation and rapid power response, the collaborative operation of all units within each control area ensures dynamic frequency balance and enhanced stability in the two-area power system.

Through the rational division of responsibilities and cooperation among functional modules, this model guarantees both rapidity and stability in frequency regulation, effectively suppressing frequency variations caused by load fluctuations. It provides a reliable framework for designing LFC strategies in multi-area power systems.

The thermal power unit model includes the governor model and the reheat turbine model. Their transfer functions can be found in [24,35] and are as follows:

$$G_g={\displaystyle \frac{1}{T_{g}s+1}}$$

(1)

$$G_{t,r}={\displaystyle \frac{c{T}_{r}s+1}{\left(T_{t}s+1\right)\left(T_{r}s+1\right)}}$$

(2)

The transfer function of the generator set is

$$G_p={\displaystyle \frac{K_p}{T_{p}s+1}}$$

(3)

The physical meanings of the parameters are detailed in

Table 1. Parameters and their physical meanings in the regional interconnected power system.

Parameter	Physical Meaning	Parameter	Physical Meaning
$\Delta P_{\mathrm{g}}$	Governor output incremental change	$\Delta P_{\mathrm{d}}$	User load interference
$\Delta P_{\mathrm{r}}$	Incremental change in output thermal power of reheating unit	$\Delta f$	Incremental frequency variation in power system
$\Delta P_{\mathrm{m}}$	Governor-controlled increase in power	$\Delta P_{\mathrm{tie},i}$	Exchange power between contact lines
$T_{\mathrm{g}}$	Governor time constant	$c$	Reheat coefficient
$T_{\mathrm{r}}$	Reheat time constant	$K_{\mathrm{p}}$	Power system gain
$T_{\mathrm{t}}$	Time constant of steam turbine	$K_{\mathrm{AF}}$	Proportional negative feedback coefficient of frequency variation
$T_{\mathrm{p}}$	Time constant of power system	$T_{ij}$	Power synchronization coefficient of contact line between area i and area j
$R$	Speed control gain	$\beta$	Frequency variation setting

.

As shown in Figure 1, the state-space model of control region i can be expressed as

$$\left\{\begin{array}{l}{\dot{\mathit{x}}}_i={\mathit{A}}_i{\mathit{x}}_i+{\mathit{B}}_i{u}_i+{\mathit{F}}_i{\mathit{\omega }}_i\\ y_i={\mathit{C}}_i{\mathit{x}}_i\end{array}\right.$$

(4)

with

$${\mathit{x}}_i={\left[x_{i1},x_{i2},x_{i3},x_{i4},x_{i5},x_{i6}\right]}^T={\left[\Delta f_i,\Delta P_{\mathrm{m}i},\Delta P_{\mathrm{r}i},\Delta P_{\mathrm{g}i},\Delta P_{\mathrm{t}\mathrm{i}\mathrm{e},i},{\displaystyle \int A_{\mathrm{c}\mathrm{e},i}\mathrm{d}t}\right]}^{\mathrm{T}}$$

(5)

$$A_i=\left[\begin{array}{cccccc}{\displaystyle \frac{1}{T_{\mathrm{p}i}}}& {\displaystyle \frac{K_{\mathrm{p}i}}{T_{\mathrm{p}i}}}& 0& 0& -{\displaystyle \frac{K_{\mathrm{p}i}}{T_{\mathrm{p}i}}}& 0\\ 0& -{\displaystyle \frac{1}{T_{\mathrm{t}i}}}& {\displaystyle \frac{1}{T_{\mathrm{t}i}}}& 0& 0& 0\\ -{\displaystyle \frac{c_i}{T_{\mathrm{g}i}}}\left({\displaystyle \frac{1}{R_i}}+K_{\mathrm{A}\mathrm{F}i}\right)& 0& -{\displaystyle \frac{1}{T_{\mathrm{r}i}}}& {\displaystyle \frac{1}{T_{\mathrm{r}i}}}-{\displaystyle \frac{c_i}{T_{\mathrm{g}i}}}& 0& -{\displaystyle \frac{c_i}{T_{\mathrm{g}i}}}\\ -{\displaystyle \frac{1}{T_{\mathrm{g}i}}}\left({\displaystyle \frac{1}{R_i}}+K_{\mathrm{A}\mathrm{F}i}\right)& 0& 0& {\displaystyle \frac{1}{T_{\mathrm{g}i}}}& 0& -{\displaystyle \frac{1}{T_{\mathrm{g}i}}}\\ 2\pi \left({\displaystyle \sum _{i=1,i\ne j}^N{T}_{ij}}\right)& 0& 0& 0& 0& 0\\ K_i{\beta }_i& 0& 0& 0& K_i& 0\end{array}\right]$$

(6)

$${\mathit{B}}_i={\left[\begin{array}{cccccc}-\frac{K_{\mathrm{p}i}}{T_{\mathrm{p}i}}& 0& 0& 0& 0& 0\end{array}\right]}^T$$

(7)

$${\mathit{C}}_i=\left[\begin{array}{cccccc}{\beta }_i& 0& 0& 0& 1& 0\end{array}\right]$$

(8)

$${\mathit{F}}_i={\left[\begin{array}{cccccc}-\frac{K_{\mathrm{p}i}}{T_{p\mathrm{i}}}& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& -2\pi & 0\end{array}\right]}^{\mathrm{T}}$$

(9)

Remark 1.

As mentioned in [41,42,43], there may be other types of generating units, such as hydro-turbine units, wind turbine units and gas-engine in the two-area interconnected power system; moreover, the model can be nonlinear with time delays. In actual engineering applications, Automatic Generation Control has multiple techniques for LFC, and the frequency variation is very small around the equilibrium even under large disturbances. So, in this study, we formulate the system as a linear time invariant (LTI) system, as in (4). Uncertainties in real system, such as the transmission capacity of tie-lines, is viewed as bounded disturbance. Although the model is relatively simple, we consider a realistic constraint, i.e., input saturation. As stated in [44], saturation may lead to instability. As stated in [45], non-smooth saturation function approximation and auxiliary system are two typical techniques for handling the saturation constraint. Other artificial techniques such as fuzzy and data-driven control are also effective [46,47]. In this paper, we design an auxiliary system for saturation compensation.

3. Controller Design Considering Input Saturation Constraints

In order to ensure the safety and stability of the energy storage system during the charging and discharging process, a saturation function is introduced into the controller design to constrain the input signal, to prevent overload or loss of control of the energy storage device caused by large control signals. This not only effectively ensures the safety of the energy storage system but also extends the system’s service life and improves its reliability in long-term operation. Meanwhile, in order to improve the dynamic performance of the system during transient processes, the prescribed performance control (PPC) method is applied. Through setting upper and lower limits on the transient response, the output of the system can be kept within the allowable fluctuation range during its progress.

The operational flow of the controller is shown in Figure 2. First, the system is initialized and the output $y_i(t)$ is obtained. Next, error conversion is performed to obtain a new error signal ${\epsilon }_i=F_i(z_i(t))$ with $z_i=y_i\left(t\right)-y_{ref}\left(t\right)$. The purpose of this is to maintain the boundedness of ${\epsilon }_i$. Then, the control input signal $u_i$ is updated as an input to control the charge and discharge power of the energy storage system, and $\Delta u_i=sat(v_i)-v_i$ is transmitted to the auxiliary system.

During signal processing, control signals beyond the range are intercepted and set to the upper or lower allowed limit through the limiting effect of the saturation function $sat(v_i)$, so as to avoid the input signal exceeding the capacity limit of the energy storage system. Finally, the obtained control signal $u_i$ is updated until $z_i(t)$ reaches the range, and the process then ends.

The main feature of this study is that it takes into account the actual situation, i.e., considering not only the input saturation but also the transient performance. This design provides theoretical and technical support for the long-term operation and frequency stability control of energy storage systems.

Considering the characteristics of the actual system, the input amplitude is constrained by the saturation constraint as follows:

$$u_i=sat\left(v_i\right)=\left\{\begin{array}{ll}{\overline{u}}_m& \hfill v_i\ge {\overline{u}}_m\\ v_i & \hfill {\underset{¯}{u}}_m<v_i<{\overline{u}}_m\\ {\underset{¯}{u}}_m & \hfill v_i\le {\underset{¯}{u}}_m\end{array}\right.$$

(10)

For the sake of simplicity, we take $\left|{\overline{u}}_m\right|=\left|{\underset{¯}{u}}_m\right|$ in this paper.

In power systems, the control objective is to lower the Area Control Error as much as possible. Therefore, we set $y_i^{\mathrm{ref}}=A_{\mathrm{ce},i}^{\mathrm{ref}}=0$ and define the following error:

$$z_i=y_i\left(t\right)-y_{ref}\left(t\right)$$

(11)

Based on PPC, the analysis of $z_i$ is converted into a boundedness analysis of the conversion error ${\epsilon }_i$, which is defined as

$${\epsilon }_i={F_i}^{-1}\left({\displaystyle \frac{z_i}{w_i}}\right)={\displaystyle \frac{1}{2}}\ln \left({\displaystyle \frac{\frac{z_i}{w_i}+1}{1-\frac{z_i}{w_i}}}\right)$$

(12)

with $w_i\left(t\right)=\left(w_{i0}-w_{i\infty }\right)e^{-at}+w_{i\infty }$. Taking the derivative of ${\epsilon }_i$ yields

$${\dot{\epsilon }}_i=\left({\displaystyle \frac{w_i}{{w_i}^2-{z_i}^2}}\right){\dot{z}}_i-\left({\displaystyle \frac{z_i}{{w_i}^2-{z_i}^2}}\right){\dot{w}}_i=r_i{\dot{z}}_i+h_i$$

(13)

with $r_i={\displaystyle \frac{w_i}{{w_i}^2-{z_i}^2}}$ and $h_i=-{\displaystyle \frac{z_i}{{w_i}^2-{z_i}^2}}{\dot{w}}_i$. Define

$${\overline{\epsilon }}_i={\epsilon }_i-{\lambda }_i$$

(14)

with

$${\stackrel{.}{\lambda }}_i=-C_1{\lambda }_i+r_{i}CB\Delta u$$

(15)

and

$$\Delta u=sat\left(v_i\right)-v_i=u_i-v_i$$

(16)

The following can then be obtained:

$$\stackrel{.}{{\overline{\epsilon }}_i}=r_i{\stackrel{.}{z}}_i+h_i+C_1{\lambda }_i-\Delta u=r_{i}CA{x}_i+r_{i}CB{v}_i+r_{i}CF{w}_i+h_i+C_1{\lambda }_i$$

(17)

Define the Lyapunov function as

$$V={\displaystyle \frac{1}{2}}{\overline{\epsilon }}_i^2$$

(18)

Its derivative is

$$\dot{V}={\overline{\epsilon }}_i\cdot {\dot{\overline{\epsilon }}}_i=r_{i}A{x}_i{\overline{\epsilon }}_i+r_{i}F{w}_i{\overline{\epsilon }}_i+h_i{\overline{\epsilon }}_i-C_1\lambda {\overline{\epsilon }}_i+r_i{c}_{i}B{v}_i{\overline{\epsilon }}_i$$

(19)

Take

$$v_i={\displaystyle \frac{1}{r_i{c}_{i}B}}\left(\begin{array}{l}-r_i{c}_{i}A{x}_i-r_i{c}_{i}F{w}_i\\ -h_i-C_1{\lambda }_i-k_1{\overline{\epsilon }}_i\end{array}\right)$$

(20)

Then, we have

$$\dot{V}=-k_1{{\overline{\epsilon }}_i}^2\le 0$$

(21)

From (15), it can be seen that ${\overline{\epsilon }}_i$ belongs to a bounded-input, bounded-output (BIBO) system; therefore, ${\lambda }_i$ and ${\epsilon }_i$ are bounded. According to PPC conclusions, $z_i\left(t\right)$ is constrained within the preset range.

Remark 2.

The purpose of this study is to improve the transient performance of the control process. It can be seen that in the PPC boundary function $w_i\left(t\right)=\left(w_{i0}-w_{i\infty }\right)e^{-at}+w_{i\infty }$, the constants $w_{i0},w_{i\infty }$, and $a$ are preset by the designer, which provides a range for the indices during transient progress. Therefore, the overshoot, rise time, and setting time are regulated by properly setting the parameters in the PPC boundary functions.

4. Simulation Results

In this section, we propose a numerical example using the MATLAB/Simulink R2022a platform to verify the effectiveness of the controller. The $i$-th control region in the LFC model of the two-region interconnected power system shown in Figure 1 is modeled and simulated. In a real system, the parameters in

Table 2. Variables and values for the interconnected power system in region $i$.

Variable	Numerical Value	Variable	Numerical Value
$T_{gi}$	0.08 s	$C_i$	0.35
$T_{ri}$	4.2 s	$K_{pi}$	120
$T_{ti}$	0.3 s	$K_{AFi}$	1.1
$T_{pi}$	20 s	$T_{ij}$	0.0707 s
$R_i$	2.4	${\beta }_i$	0.425

have specific physical meanings, which can be obtained according to the product manual such as GE Speedtronic™ Mark VI Gas Turbine Governor, USA and Siemens SST-400 Reheat Steam Turbine, Germany [43,48].

The simulation results are shown in Figure 3, Figure 4, Figure 5 and Figure 6. Figure 3 shows the curve of the error $z_i\left(t\right)$. It can be seen that $z_i\left(t\right)$ evolves within the region encompassed by the PPC function. Thus, the control objective is realized. Figure 4 shows the states of $x_{i1}$ and $x_{i5}$, indicating that $\Delta P_{tie}$ and $\Delta f$ converge to the target value. This shows that the control strategy can effectively balance the power in each region and stabilize the system frequency to ensure the system’s safety. In order to highlight the advantages of the method proposed in this study, we present comparison results in Figure 5. For the same model and simulation settings, Figure 5a shows the curves of $\Delta P_{tie}$ and $\Delta f$ with controller (10), and Figure 5b shows the curves of $\Delta P_{tie}$ and $\Delta f$ with a traditional PID controller. Both controllers are effective in stabilizing the system in the presence of disturbance and uncertainty, but the transient progress of Figure 5a is significantly superior to that of Figure 5b, i.e., our proposed controller has a significant advantage in improving the transition process. Figure 6 shows the trajectories of $u_i$ and $v_i$. It can be seen that the input signal is constrained by the saturation before $t_{sat}=0.12s$. This also makes sense, as the comparatively large initial value needs a large control input to regulate it, which is limited by the tolerance of the storage system. Thus, the energy storage system operates within a safe area, which not only extends the system’s service life but also improves its stability. Figure 7 shows ${\lambda }_i$ for the auxiliary system; as mentioned above, it is bounded. Thus, the stability and transient performance of the regional interconnected power system are verified by the simulation results.

5. Conclusions

In this study, we designed a controller with input saturation constraints for regional interconnected power systems to ensure these systems’ stability and transient performance. With the application of this control strategy, the error evolved within the preset region, and the actual input was limited, meeting the requirements of practical application. Finally, a simulation was performed on the MATLAB/Simulink platform to provide an example, i.e., load frequency control of a regional interconnected power system. The effectiveness and reliability of the designed control strategy in practical applications were thus verified. As stated in [41,49], time delay and actuator failure are common in real-life systems; therefore, our future work will focus on solving the LFC problem in the presence of time delay and actuator failures. In addition, we will conduct a theoretical analysis of the influences of parameter values on system performance, thus providing valuable suggestions for the choice of parameters.