1. Introduction
With the rapid development of today’s society and economy, there has been a significant increase in people’s demand for energy. However, the traditional reliance on fossil energy, primarily coal, not only leads to energy shortages but also exacerbates the global greenhouse effect. This puts immense pressure on the environment and human society. In the context of global carbon neutrality, there is an urgent need to establish a green, low-carbon, clean, and efficient renewable energy system [
1,
2,
3]. Among the various renewable energy sources, hydropower is the most widely utilized, accounting for over 70% of the total renewable energy [
4]. In comparison to wind and solar energy, hydropower is more flexible and reliable [
5]. Consequently, many countries have prioritized the development of hydropower energy, with China’s energy strategy giving it a crucial role [
6,
7,
8].
The hydraulic turbine governor system (HTGS) is a complex nonlinear dynamical system and a key component of the hydropower energy development process. It plays a central role in realizing hydropower energy production. Under normal operating conditions, the HTGS is in a steady state [
9]. However, during transitional operating conditions, the HTGS enters a non-steady state, and the dynamic response process occurs in each state variable. This implies that the system needs to rapidly adapt to different working conditions and load changes in order to ensure smooth operation. The governor is the core component of the control system. Based on the governor control strategy, HTGS can operate under specific requirements [
10,
11]. However, it should be noted that the regulating capability of the governor control strategy is limited. Currently, the most widely adopted control strategy is the proportional integral differential (PID) control strategy [
12,
13,
14,
15]. However, PID mostly provides good regulation quality only for linear systems. The nonlinear coupling characteristics of HTGS and the changing load make it difficult to design gain scheduling for the PID controller. Additionally, the PID controller has relatively poor adaptive ability and cannot handle larger perturbations, leading to deviations from the design operating point. As a result, it cannot adapt to the complex operating conditions of the hydraulic turbine and meet the control index requirements [
16]. Therefore, there are still some limitations to PID control in the hydraulic turbine regulation system.
In addition, the HTGS can exhibit chaotic behavior in certain cases. When the values of the PID parameters are not reasonable, the HTGS can become chaotic [
17]. Chaotic motion leads to instability and disorder in the HTGS, making it highly dangerous. Therefore, it is crucial to effectively regulate and control chaotic behavior in the HTGS. The PID control strategy alone is not sufficient to achieve this. It is necessary to introduce a suitable nonlinear control strategy [
18] in addition to the PID control strategy to effectively address and resolve the issue.
Among the nonlinear strategies, the sliding mode control strategy (SMC) is a commonly used control strategy for controlling chaotic behavior, which can effectively overcome the uncertainty of the system [
19,
20] and has strong robustness to unknown disturbances. In this paper, the sliding mode control problem of HTGS is studied, and the design of SMC is expected to realize the effective regulation of the chaotic behavior of HTGS. The authors of [
21] developed a fourth-order nonlinear mathematical model of HTGS with uncertain perturbations. They investigated the occurrence of chaos phenomena in HTGS under the influence of these perturbations. To address this issue, the authors employed the method of state-feedback linearization to establish the relationship between the governor output and the generator speed in the mathematical model of HTGS. They then designed a sliding-mode controller to achieve speed control of the chaotic HTGS, thereby ensuring its stability. However, it is important to note that while this method effectively controls the generator speed, it may not provide the same level of stability for the other state variables of the HTGS, potentially leading to a phenomenon of ‘false stability’. To address this limitation, several scholars have made improvements. The authors of [
22], introduce the concept of guide vane opening in the design of the sliding mode surface. This approach defines a new sliding mode surface and effectively mitigates the issue of ‘false stability’ that is commonly observed in traditional sliding mode control. However, to enhance the robustness of the process from the initial time instant [
23], some researchers have replaced the linear sliding mode surfaces with integral sliding mode surfaces. The authors of [
24,
25] have confirmed that the HTGS exhibits chaotic behavior when the PID parameter value is unreasonable. Building upon this, an integral sliding mode control law is designed for each state variable of the system as part of the design step of the integral sliding mode controller. The simulation results demonstrate that the designed controller effectively achieves quick and smooth control of the HTGS state variables, bringing them to equilibrium. This control approach successfully eliminates chaotic behavior and enhances the quality of regulation. However, a limitation of the above method is that it relies solely on the strong robustness of the SMC to overcome the uncertainty perturbations in the system. This approach is not optimal. To further enhance convergence speed and reduce the jitter caused by disturbances, it becomes necessary to estimate or observe the uncertainty disturbances and incorporate the estimated or observed values into the sliding mode controller.
In current research, various measures have been employed to predict or observe uncertain perturbations. These measures primarily involve two methods. The first method involves predicting the upper bound of uncertain perturbations through the use of adaptive laws in conjunction with adaptive control. The second method involves observing the tracking of uncertain perturbations through the design of perturbation observers [
26]. The authors of [
27] present a nonlinear mathematical model of HTGS, taking into account the uncertainty of energy loss as an internal disturbance. To enhance the efficiency and power stability of the HTGS, a non-singular fast terminal sliding mode controller (NFTSMC) based on the adaptive backstepping method is proposed. This controller designs the adaptive rate to anticipate the upper bound of the uncertainty disturbance. However, it should be noted that this method assumes linearity and time invariance against external disturbances, which may limit its applicability in the real operating environment of HTGS. In contrast, the authors of [
28] focus on investigating the backstepping sliding-mode fault-tolerant tracking control of HTGS in the presence of external disturbances. To mitigate the impact of unknown random disturbances, a nonlinear disturbance observer is designed to identify and estimate the disturbance term. This approach effectively improves the regulatory performance of the HTGS.
However, it should be noted that sliding mode control has some drawbacks. The traditional sliding mode convergence law includes a discontinuous
function, which leads to jittering of the controller. This results in lower control accuracy, high heat loss in the circuit, and wear and tear on moving parts [
29]. To address this issue, previous authors have made improvements by selecting a suitable convergence law that considers the dynamic quality of the convergence process and reduces system vibration. The authors of [
21,
30], used a continuous saturation function (
) instead of the
function in the convergence law, effectively reducing the jitter phenomenon of HTGS. Additionally, the authors of [
31] introduced a tangent function into the traditional power convergence law to develop a new composite sliding mode convergence law. This method is applied to the magnetic levitation system of synchronous motors and significantly eliminates the dynamic process of the system while attenuating the jitter vibration.
In summary, in order to stabilize the HTGS where chaotic behavior occurs, a novel integral sliding mode control method based on nonlinear disturbance observer compensation is investigated in this paper. The main contributions are as follows: (1) Modeling HTGS and conducting an in-depth analysis of its dynamic behavior under PID control revealed that the traditional PID controller has poor anti-interference ability. The Lyapunov exponential spectrum plot clearly indicates the occurrence of chaotic phenomena in HTGS when there are changes in PID parameters; (2) An ISMC is proposed for the HTGS where chaotic instability occurs. Compared to traditional SMC, the ISMC approach enhances robustness from the very beginning of the process. It achieves invariance from the initial time instant, which is not guaranteed in traditional SMC; (3) in addition, the uncertainty perturbations present in the HTGS are accurately observed using a nonlinear disturbance observer, and the observed values are compensated into an integral sliding mode controller to achieve the purpose of shortening the convergence speed of the system and improving the transient response performance; and (4) To address the limitations of the previously described SMC, we propose an improvement to the sliding-mode exponential convergence law. This improvement involves the introduction of a nonlinear power combination function, referred to as ‘’, instead of the conventional or ‘’ functions. By using this new function, we are able to further reduce system jitter while achieving fast and high-precision convergence of the system.
The organization structure for the rest of this article is as follows. The paper provides a detailed description of the HTGS model in
Section 2, followed by an analysis of its dynamic characteristics in
Section 3.
Section 4 proposes a controller design method for the HTGS, while
Section 5 presents a simulation analysis to verify the proposed control strategy. The paper concludes with a summary of the full text in
Section 6.
3. HTGS Dynamic Characteristics Analysis
Currently, the parallel PID control structure is commonly used as the main regulation strategy for HTGS control. Based on the literature [
18,
24] mentioned, the parallel PID control has the following form:
where
,
, and
are all dimensionless parameters in the PID control law.
It is now assumed that the HTGS is subject to uncertain nonlinear perturbations from itself and the outside world, and the effect of the nonlinear perturbations on its convergence state is observed. Introducing the uncertain perturbations
,
,
. Combining Equation (9), the model becomes as follows:
The dynamic characterization of HTGS under its PID control is conducted using MATLAB. The simulation time is set to 15 s, and the parameter list of the hydraulic turbine regulation system is provided in
Table 1. The initial states of the system are all set to 0.01. The governors
,
, and
are assigned values of 10, 0.5, and 1, respectively. The equilibrium point of the system is (0, 0, 0, 0). Initially, we observe the effect of PID in controlling the system without any perturbation (i.e.,
,
,
and
are all set to 0). The simulation timing diagram is depicted in
Figure 3. Immediately after this, we apply an uncertain perturbation to the system, which is defined as a delta function that varies with time t. The system is then subjected to an uncertain perturbation:
,
, and
. At this time, the PID control effect on the turbine regulation system timing and phase diagrams are shown in
Figure 4 and
Figure 5. Comprehensive
Figure 3,
Figure 4 and
Figure 5 demonstrate that the PID control can effectively stabilize the system’s state variables and bring them to equilibrium point 0 within a limited time. This control mechanism plays a significant role in controlling the system. However, when subjected to nonlinear disturbances, such as the relative deviation of the rotor angle
, the relative deviation of the generator rotational speed
, the relative deviation of the hydraulic turbine mechanical torque
, and the relative deviation of the guide vane opening
, there are noticeable nonlinear irregular oscillations near equilibrium point 0. This study demonstrates that while PID control can partially contribute to the control of highly nonlinear systems like HTGS, it exhibits limited anti-jamming capability and weak robustness when facing significant uncertain nonlinear perturbations. These limitations have a detrimental effect on the safe and stable operation of hydropower stations.
In addition, HTGS is affected by the PID parameters, and significant changes to them may cause the system to diverge and generate nonlinear oscillations. In this simulation, the governor parameters
,
, and
are set to 2, 1, and 7, respectively, while the uncertainty perturbation remains unchanged. The simulation time is set to 10 s. The system timing diagram and phase diagram are illustrated in
Figure 6 and
Figure 7, respectively.
Figure 6 indicates that the system tends to diverge instead of converging after the PID parameters are altered. Moreover, the two-dimensional and three-dimensional phase diagrams in
Figure 7 reveal the existence of strange attractors, which leads to the initial judgment that the system exhibits chaotic behavior. To verify the theory, a calculation of the Lyapunov index is necessary. This index is crucial in measuring the system’s dynamics and determining its chaotic characteristics. By observing the maximum Lyapunov index, we can determine if the system has chaotic motion. The value of
is changed within the range of [4, 7.5] and plotted in the Lyapunov exponent spectrum of the system, as depicted in
Figure 8. By observing
Figure 8, we can determine the presence of chaotic motion in the system based on its maximum Lyapunov exponent being positive. Furthermore, there are three positive Lyapunov exponents in the system, indicating the occurrence of hyperchaotic motion.
This section establishes an accurate mathematical model and a reasonable analysis method to accurately and comprehensively reflect the dynamic behavior of the HTGS. The purpose of this paper’s research is to solve the chaotic phenomenon occurring in the HTGS. In the following section, a stabilization approach is proposed by designing an integral sliding mode controller based on a disturbance observer and an improved convergence law. This approach aims to stabilize the system at the equilibrium point and achieve better dynamic performance compared to the HTGS controlled by the traditional PID.
4. Control Method Design
This section presents the design of a new nonlinear controller aimed at addressing the limitations of PID control. The proposed HTGS improved integral sliding mode control system, which incorporates a nonlinear disturbance observer, is illustrated in
Figure 9. The various functional components of the system work in unison to achieve closed-loop control.
4.1. Perturbation Observer and Integral Sliding Mode Control Method Design
Based on the fundamental concept of sliding mode control, an Integral Sliding Mode Controller (ISMC) can be formulated for the HTGS model. The HTGS model, which is to be designed, can be represented as follows:
The purpose of ISMC is to find the appropriate expression for the controller output
based on the HTGS model, with the objective of causing each state variable
of the system to converge to a desired value of 0. Equation (11) can be expressed as follows:
where
,
,
, and
.
is shown below:
let the equilibrium point
, and let the error
. Then we have the following equation:
The integral sliding surface
used in the literature [
24,
25] is chosen as follows:
where
and
are slip surface coefficients, and
, determine the quality of the final slip state.
, Equation (14) can be written as follows:
The traditional exponential convergence law is as follows:
where
. The role of
is to improve the dynamic quality of the system; adjusting the parameter can change the convergence speed of the system to the sliding mode surface; the gain parameter
of the sign function is the main parameter of the system to overcome uptake and external disturbances.
Take the first Lyapunov function as follows:
it is known that
. Deriving it gives:
based on the Lyapunov stability condition, the system approaches the sliding mode plane throughout the state space when the condition is satisfied by
nd arrives at the steady state asymptotically with the selected convergence law after entering the sliding mode. This condition is the basis for deriving the control law
.
The preliminary design of the control law
combined with the traditional exponential convergence law is as follows:
bringing
into
yields:
taking
greater than the upper bound of
, then
,
, and
,
are less than 0. That is,
. The control law
is reasonably designed.
When the value of ε exceeds the upper bound of
,
, thereby satisfying the Lyapunov finite-time stability condition. However, the existing literature [
24,
25] primarily relies on the robustness of ISMC itself to overcome uncertainty perturbations in the system, but this approach has certain limitations. When the value of
is relatively large, it results in a larger selection of
, which ultimately leads to significant sliding mode control jitter. The presence of large jitters not only decrease control accuracy but also causes frequent actuator startup and potential damage to the actuator. To address these shortcomings and further reduce sliding mode jitter, improve control accuracy, and enhance convergence speed, this paper proposes the use of a nonlinear disturbance observer to observe the integrated nonlinear disturbance
. The observed value
is then used to compensate for the nonlinear disturbance
, and this compensated value
is integrated into the ISMC.
Combining the disturbance observer and the integral sliding mode control, the control law
can be designed as follows:
Additionally, to ensure tightness, the perturbation needs to satisfy the following conditions [
34]:
Assumption 1. The perturbed signal is bounded, and there exists , satisfying , .
Assumption 2. The perturbed signal is slowly varying, so consider , .
Now design the nonlinear perturbation observer to estimate the uncertain perturbation
in the system. Let
, for the nonlinear system shown in Equation (12), design the nonlinear perturbation observer as follows:
where
is the internal state of the nonlinear perturbation observer to be designed, and the design
.
is the nonlinear perturbation observer gain.
is the internal state variable of the observer.
is defined as:
Define the observation error as follows:
where
.
According to Assumptions 1 and 2, the dynamic characteristics of its nonlinear perturbation observer can be described as follows:
take the second Lyapunov function
as follows:
this gives
. Deriving this gives:
From the above equation, if is a positive definite matrix, i.e., are greater than 0, then . It means that the nonlinear perturbed observer deviation will converge to 0 in finite time. This proves that the perturbed observer design is reasonable.
Based on the given Equation (28), if is a positive definite matrix with and greater than 0, then . This implies that the nonlinear perturbed observer deviation will converge to 0 within a finite time frame. Therefore, it can be concluded that the perturbed observer design is reasonable.
4.2. Improvement of the Convergence Law
The use of the sign function
in conventional sliding mode control can result in control discontinuity due to its switching action, leading to severe jitter in the system. To address this issue, the authors of [
5,
21,
30] proposed the use of a continuous saturation function
as an alternative to the
function. The saturation function is defined as follows:
where
is called the boundary layer.
is 0.1.
Using the conventional saturation function
, the control law
of the integral sliding mode controller (ISMC-NDO) based on nonlinear disturbance observer compensation is obtained as follows:
However, the approach used by the authors of [
5,
21,
30] is not optimal. In order to improve the convergence speed of the sliding mode surface and further reduce the jitter of the motion trajectory before the state variables converge to the origin, this paper proposes an improvement to the traditional exponential convergence law. The controller uses a nonlinear power combination function,
, to replace the
or
function, resulting in a smoother system. The expression for the
function is as follows:
where
are parameters to be designed, where
is the constant affecting the tracking speed, and
is the constant affecting the filtering effect [
35].
In summary, the NISMC-NDO control rate
can be obtained as follows:
substituting Equation (32) into (18) yields:
since and , , i.e., .
Similarly, when
,
since , , and , |, i.e., .
In conclusion, the system error will converge to 0 in finite time as . The nonlinear power-combined convergence law can guide the system to the sliding mode surface, and the amplitude of will gradually decay during the convergence of the system state trajectory to the sliding mode surface. The size of the system error will adaptively adjust the switching gain and quickly converge to stabilize the system at the equilibrium point. Therefore, the designed integral sliding mode controller with perturbation observer and improved convergence law is reasonably designed.
Equation (32) NISMC-NDO control law
is as follows:
This is the final control law for controlling the overall system. Now, its simulation results will be discussed.