Adaptive Modified Active Disturbance Rejection Control for the Superheated Steam Temperature System Under Wide Load Conditions

Abstract

The operation of the superheated steam temperature system significantly impacts the safety and economy of thermal power units. To ensure its stable operation under large-scale variable load conditions, a modified active disturbance rejection control strategy based on parameter adaptation is proposed. Firstly, a typical superheated steam temperature system model is introduced, and the cascade control structure is applied to the model. Then, on this basis, a modified active disturbance rejection control strategy based on parameter adaptation is proposed, and the parameter tuning method of the modified active disturbance rejection control is introduced. Finally, the control performance of the proposed control strategy under a wide range of variable loads is verified through comparative simulations under nominal working conditions and uncertain working conditions. To further illustrate the effectiveness of the proposed strategy, the method is applied to a certain 660 MW unit in the field. After implementing the method, the fluctuation range of superheated steam temperature on the A and B sides decreased to only 34.0% and 53.0% of the original, respectively, and the fluctuation variance on the A and B sides decreased to only 28.5% and 43.3% of the original, respectively. The above field application results fully demonstrate that the control strategy proposed does not merely remain at the theoretical simulation level, but is a key technical means that can be effectively implemented and effectively solve the problem of superheated steam temperature control in thermal power units.

1. Introduction

In recent years, more and more renewable energy is connected to the power grid. Due to its high sensitivity to environmental factors and inherent volatility, renewable energy causes large fluctuations in the power generation of the grid. To ensure stable grid operation, thermal power units in traditional power systems need to perform deep peak shaving and frequency regulation to meet the flexibility requirements from large-scale renewable integration. However, rapid load changes can affect the stable operation of thermal units. During fast load variations and under various disturbances, key operating parameters such as main steam pressure, superheated steam temperature, and reheat steam temperature fluctuate significantly, which leads to reduced control performance. The superheated steam temperature (SST) system is a critical component of a thermal power unit [1]. If the SST is too high or too low, it can cause irreversible damage to the unit. Excessively high temperature may damage equipment, while excessively low temperature reduces efficiency. Therefore, maintaining good control of SST is crucial for the stability and safety of thermal power units [2].

To ensure the safe operation of the unit, it is necessary to control the fluctuation range of the SST as much as possible. Only when the fluctuation range is small enough can the set value be increased, which improves the economic performance of the unit. However, the operation of the SST involves processes such as steam superheating, combustion, and heat transfer. Its temperature change shows large inertia and long delay characteristics [3], which makes the control of the SST difficult to carry out.

To suppress temperature fluctuations in the SST during rapid load changes and under various disturbances, many researchers have carried out related studies. A multi-model dynamic matrix control strategy based on enthalpy prediction of the intermediate point is proposed to improve traditional PID control [4]. An accurate feedforward mechanism is introduced into a cascade control system to achieve better performance than traditional cascade control [5]. Based on sliding mode variable structure control, a new sliding mode control method is designed to outperform conventional cascade PID control [6]. In addition, a nonlinear control method is developed using a high-order sliding mode observer to suppress multiple types of disturbances [7]. With the development of artificial intelligence, many researchers have also tried to apply AI technology to the SST. To improve the control of strongly coupled and long-time-delay processes, a deep neural network-based identification method is proposed [8], which has strong generalization ability and stability for the system. A prediction model for the SST is built using machine learning, with model parameters optimized by a random search algorithm [9]. lIn order to accurately predict the changing trend of superheated steam temperature, a superheated steam temperature prediction model based on an improved whale algorithm optimizing long short-term memory neural network is proposed [10]. A neural network-based prediction model is used for the superheated steam temperature system, and a feedforward-feedback control strategy is adopted to improve control performance [11].

On the one hand, although PID control boasts advantages such as a simple structure, easy implementation, fast dynamic response, and strong robustness, it still has drawbacks, including poor adaptability, limited anti-disturbance capability, and insufficient ability to handle large time-delay processes when applied to nonlinear systems with significant inertia [12]. Therefore, it often needs to be combined with other control strategies to achieve effective control under complex working conditions. On the other hand, artificial intelligence technologies such as neural networks and reinforcement learning have strong nonlinear modeling capabilities and offer major advantages in predictive modeling and dynamic identification. However, during the implementation process, they require a large amount of training data to support them, which takes a considerable amount of time. There are still considerable challenges in practical applications.

In summary, the controller for the SST should have strong anti-disturbance ability, high robustness, and should not rely on an accurate model. Active disturbance rejection control (ADRC) is proposed by researcher Han Jingqing based on PID control and modern control theory. It provides effective control for multi-input, multi-output and high-order systems, while also having a simple structure and strong robustness. ADRC combines the idea of eliminating errors based on errors in classical control theory and the idea of observing and compensating for errors through state observers in modern control theory. It can overcome the difficulties in solving nonlinear problems and the reliance on precise mathematical models. It regards various uncertainties in the controlled object, including internal and external disturbances, as total disturbances. It uses an extended state observer (ESO) to estimate and compensate for the total disturbances in real time, achieving active suppression and elimination of disturbances. In recent years, it has been applied in many fields. An ADRC is applied in a battery thermal management system, demonstrating excellent thermal stability [13]. An ADRC based on phase advance error is proposed and validated in an ultra-supercritical unit, showing excellent tracking and disturbance rejection performance [14]. To improve the performance of permanent magnet synchronous motor control systems, an ADRC strategy optimized by particle swarm optimization is introduced, which enhances system robustness and dynamic response speed [15]. For thermal power processes with large time-delay characteristics, Ref. [16] proposed a quantitative tuning rule for the time-delayed ADRC (TD-ADRC) structure based on the typical first-order plus time delay (FOPTD) model. Simulation and laboratory water tank experiments validated the tuning efficacy. Ref. [17] proposed a delay-inertia-compensation ADRC (DIC-ADRC) for time-delay dominant industrial systems, where these systems can be approximated as second-order plus time-delay (SOPTD) systems, and verified its robustness and control performance through on-site application in the main steam pressure process. An improved ADRC is proposed to accurately estimate the mover position and speed required by a permanent magnet synchronous linear motor system [18]. A modified ADRC based on gain scheduling is proposed and applied to the denitration system of a coal-fired unit, achieving better control performance [19]. In addition, an improved ADRC is presented for the SST along with its parameter tuning approach [20]. To improve the efficiency of wind energy conversion systems (WECS) equipped with permanent magnet synchronous generators (PMSG) under variable wind conditions, Ref. [21] enhanced the tracking performance and adaptive capability of the system by integrating ADRC with the perturb and observe (P&O) algorithm. Ref. [22] proposed an innovative solution combining an interleaved DC-DC boost converter with a nonlinear MPPT-ADRC strategy; the robust ADRC compensated for system disturbances and enhanced the stability of the hospital’s power grid system under fluctuating solar radiation. To address the challenges associated with weak active disturbances, substantial steady-state speed amplitude fluctuations, and difficulty in achieving a balance between overshoot and speed control in the sensorless PMSM control system, Ref. [23] proposed a sensorless vector ADRC method for permanent magnet synchronous motors based on the Luenberger observer and proved that this control method has the advantages of small steady-state error and strong self-disturbance resistance through simulation experiments. Considering the renewable energy sources and energy storage, a scheme of load frequency control is proposed based on the deep deterministic policy gradient and ADRC for multi-region interconnected power systems [24]. Ref. [25] theoretically analyzes multiple types of ADRC and applies them to the main steam pressure system, demonstrating their potential for practical use. In addition, ADRC has also been widely used in other fields such as intelligent ship system [26], robotic systems [27,28], aircraft control systems [29], rehabilitation exercise control systems [30], distillation process [31], and permanent magnet motor drive systems [32]. It can be seen that the ADRC methods mentioned in these references are widely applied in complex nonlinear systems due to their low dependence on accurate models and strong robustness.

However, significant limitations still exist when the conventional ADRC is directly applied to systems with large inertia characteristics, such as the SST system of thermal power plants. Specifically, the SST system faces two core challenges: first, due to the large inertia characteristic of the system, the output will be delayed, resulting in the input signal of ESO being out of sync and thus unable to achieve precise observation and compensation of the system. Second, the current operating conditions of thermal power plants are subject to changes, requiring large-scale load adjustments.

To address the large inertia of the SST system, Han [12] proposed two key perspectives: the control signal compensation and output prediction. Based on these two perspectives, time-delay ADRC [33] and smith-predictor ADRC [34] have been proposed. Nevertheless, both methods suffer from output oscillations and degraded control performance when there is a mismatch in time delay [35]. To solve this problem, a modified active disturbance rejection control (MADRC) strategy is proposed [35]. However, in the context of large-scale load changes, the fixed-parameter MADRC fails to fully guarantee control performance.

Therefore, this paper proposed a gain-scheduling-based parameter adaptive MADRC strategy. Specifically, for the system models corresponding to different load conditions, inertial compensation modules are constructed, respectively, and the controller parameters are tuned and optimized by adopting appropriate parameter tuning methods. This improvement effectively enhances the tracking accuracy and anti-disturbance capability of the SST system under three typical operating conditions. The main contributions of this paper are summarized as follows:

(1)

An adaptive modified active disturbance rejection control method is proposed for the superheated steam temperature system.

(2)

A parameter tuning method is presented for the adaptive modified active disturbance rejection control.

(3)

The effectiveness of the proposed method is verified through on-site applications. Notice that this method possesses the outstanding advantages of a simple principle, strong engineering practicability and high operational reliability and can significantly improve the control effect of the SST.

The paper is organized as follows: Section 2 introduces the structure and control model of the SST. Section 3 describes the control principle of ADRC and presents the proposed MADRC based on a parameter adaptation strategy. Section 4 compares the proposed MADRC with multiple controllers under different working conditions through simulations. Monte Carlo experiments are used to simulate scenarios with uncertain system parameters, verifying the control performance of the proposed method. The robustness and high-frequency noise suppression capability of the controller are also explored. The results demonstrate the effectiveness of the proposed control strategy. Section 5 applies the proposed method to a 660 MW unit for on-site application. Through the comparative analysis of the operation data before and after the method is put into use, the effectiveness of the proposed method is verified. Section 6 summarizes the paper and points out the broad application prospects of the proposed method.

2. Structure and Control Model of Superheated Steam Temperature System

The structure of the SST is shown in Figure 1. From Figure 1, it can be seen that the system is a two-stage process, which includes both low-temperature and high-temperature superheating stages. Steam exits the steam-water separator and enters the pipeline. It then goes through two superheating stages before entering the steam turbine. During this process, the temperature of the superheated steam is precisely controlled by adjusting the flow rate W of the attemperation water through the water injection valve. Throughout the process, the output main steam temperature T is mainly influenced by the attemperation water flow W, the inlet steam flow D, and the flue gas heat Q. Among these, the attemperation water flow W is the control variable of the system, while the steam flow D and the flue gas heat Q are the main disturbance variables.

Ref. [36] takes the high-temperature superheater of a supercritical 600 MW once-through boiler as the study object. It analyzes the dynamic characteristics and uses a linearized distributed parameter method to build a model of the superheated steam temperature system at typical load points. This model has been widely adopted. The specific mechanism model is as follows [36]:

$${\displaystyle \frac{T_{exit}\left(s\right)}{D_{sp}\left(s\right)}}=-K_{DT}·e^{-{\tau }_{0}s-{\displaystyle \frac{{\alpha }_D{T}_{m}s}{1+T_{m}s}}s}$$

(1)

where

$${\tau }_0=V·\overline{\rho }/D$$

(2)

$${\alpha }_D=({\alpha }_2·A)/(D·C_p)$$

(3)

$$T_m=(M·C)/({\alpha }_2·A)$$

(4)

$${\alpha }_2=B·{\displaystyle \frac{{\lambda }^{0.6}{C}_p^{0.4}}{{\mu }^{0.5}}}·D^{0.8}$$

(5)

$$K_{DT}=(I_1-I_{sp})/(D·C_{p1})$$

(6)

In Equations (1)–(6), $T_{exit}$ represents the outlet temperature of the superheater. D and $D_{sp}$ are, respectively, the main steam flow rate and the water spray flow rate. $C_p$ and $C_{p1}$ are, respectively, the average specific heat at constant pressure of the working medium in the process and the specific heat at constant pressure of the working medium at the process outlet. $I_1$ and $I_{sp}$ represent the enthalpy values of the steam at the water spray point and the enthalpy values of the desuperheating water, respectively; ${\tau }_0$ is the average time for steam to flow through the heat-receiving tube; ${\alpha }_D$ is the dynamic parameter in the flow process; V and $\overline{\rho }$ are, respectively, the volume of the section and the average density of the steam within the section; $T_m$ is the time constant for heat storage in metals; A and ${\alpha }_2$ are, respectively, the internal surface area and convective heat release coefficient of the link; M and C are, respectively, the mass and specific heat of the metal in the wall tube; $\lambda$ and $\mu$ are the thermal conductivity and dynamic viscosity of steam, respectively; B is a constant.

The model shown in Equation (1) can be regarded as a functional expression of the working condition parameters and can be represented by the high-order inertia link: $K/{(Ts+1)}^n$ [36].

Table 1. Multi-load SST system model.

Load/%	Leading Zone	Inert Zone
100	$-\frac{0.815}{{(18s+1)}^2}$	$\frac{1.276}{{(18.4s+1)}^6}$
75	$-\frac{1.657}{{(20s+1)}^2}$	$\frac{1.202}{{(27.1s+1)}^7}$
50	$-\frac{3.067}{{(25s+1)}^2}$	$\frac{1.119}{{(42.1s+1)}^7}$

shows the transfer functions of the lead zone and inertia zone of the SST at three typical load points: 100%, 75%, and 50% load. It can be seen from Table 1 that the model parameters change significantly with load variation. Especially, the time constant of the inertia zone changes notably. Therefore, conventional controllers cannot meet the control performance requirements of the system under multiple load conditions. It is necessary to design a control strategy that ensures a good control effect across various load conditions.

3. The Design of Modified Active Disturbance Rejection Control

First-order and second-order ADRCs are the most widely used types of active disturbance rejection controllers in engineering practice. Based on the theory of the ESO, which treats complex dynamics and uncertainties in the system as “total disturbance”, low-order ADRC can also control high-order systems. Moreover, it has been proven that low-order controllers can effectively control high-order systems in Ref. [37], and this approach has been applied in engineering practice. The following section discusses the first-order active disturbance rejection controller. Its structure is shown in Figure 2.

In general, the system can be described in the following form:

$$\dot{y}=\widehat{f}(y, \ddot{y}, \tau , \dots , d)+bu$$

(7)

where $\widehat{f}$ is a composite function that includes unknown dynamic terms such as time delay constants ($\tau$), high-order dynamics ($\ddot{y}$), and external disturbances (d). b is the uncertain input gain of the system. Equation (7) is modified to obtain the following equation:

$$\dot{y}=\widehat{f}+bu+b_{0}u-b_{0}u=f+b_{0}u$$

(8)

where $f=\widehat{f}+(b-b_0)u$ is called the total disturbance (including both internal and external disturbances). $b_0$ is the estimated value of b and is a known quantity. By defining y as the extended state $x_1$, and defining f as the extended state $x_2$, the state-space representation of Equation (8) is expressed as:

$$\left\{\begin{array}{c}\left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\end{array}\right]=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}{b}_0\\ 0\end{array}\right]u+\left[\begin{array}{c}0\\ 1\end{array}\right]\dot{f}\hfill \\ y=\left[\begin{array}{cc}1& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]\hfill \end{array}\right.$$

(9)

For the first-order system shown in Equation (9), the corresponding second-order extended state observer can be expressed as

$$\left[\begin{array}{c}{\dot{z}}_1\\ {\dot{z}}_2\end{array}\right]=\left[\begin{array}{cc}-{\beta }_1& 1\\ -{\beta }_2& 0\end{array}\right]\left[\begin{array}{c}{z}_1\\ z_2\end{array}\right]+\left[\begin{array}{cc}{b}_0& {\beta }_1\\ 0& {\beta }_2\end{array}\right]\left[\begin{array}{c}u\\ y\end{array}\right]$$

(10)

where ${\beta }_1,{\beta }_2$ are the gains of the observer. When ${\beta }_1,{\beta }_2$ are properly adjusted, $z_2$ can track the total disturbance f well.

The bandwidth parameterization method is used for tuning, as follows:

$$\left\{\begin{array}{c}{\beta }_1=2{\omega }_o\hfill \\ {\beta }_2={\omega }_o^2\hfill \end{array}\right.$$

(11)

where ${\omega }_o$ is the observer bandwidth.

The state feedback control law is designed as

$$u={\displaystyle \frac{u_0-z_2}{b_0}}$$

(12)

Substituting Equation (12) into Equation (8) gives the compensated system as

$$\begin{array}{cc}\hfill & \dot{y}=f+b_{0}u=f+b_0{\displaystyle \frac{u_0-z_2}{b_0}}\hfill \\ \hfill & \approx f+b_0{\displaystyle \frac{u_0-f}{b_0}}=u_0\hfill \end{array}$$

(13)

where $b_0$ is the estimated value of b and can be manually assigned, and f is the total disturbance (including both internal and external disturbances). It can be seen that the system becomes an integral series structure. From Figure 2, the control law is given as

$$u_0=k_p(r-z_1)$$

(14)

where r is the setpoint of the system, and $k_p$ is the controller bandwidth. Both have clear physical meanings. The parameter d in Figure 2 represents an external disturbance.

Conventional ADRC has been applied in many fields. However, when used for systems with large inertia and high-order characteristics, its control performance is often not ideal. Due to the high-order inertia in the system, the ESO cannot accurately estimate the state and disturbance of high-order systems, resulting in limited control effectiveness. A modified active disturbance rejection control (MADRC) strategy is proposed by designing a compensation link to approximately eliminate the high-order inertial link in Refs. [35,38], thereby enhancing the state estimation accuracy of ESO and improving the control performance for objects with large inertia and high-order characteristics. The MADRC proposed is based on first-order ADRC. It adds a compensation module to approximately cancel the large inertia links in high-order systems. For the compensation module, its function is to delay the control signal before it enters the ESO. Due to the large inertia and large delay characteristics of the SST, the system output has already been delayed because of the high-order inertia characteristics of the system itself. When using a conventional ADRC controller, the problem of the output signal being out of sync with the control signal will occur. This module will make the control signal and output signal entering the ESO more synchronous, thereby approximately offsetting the high-order inertial characteristics of the controlled object and improving the control accuracy of the system. The SST can usually be described as $G_p\left(s\right)=K/{(Ts+1)}^n$. The compensation module in the controller is selected as

$$G_{cp}\left(s\right)={\displaystyle \frac{1}{{\left(Ts+1\right)}^{n-1}}}$$

(15)

where T is the time constant of the compensation module. The design concept of MADRC is to compensate the controlled object into a system of order 1 through a compensation part before entering the ESO, as shown in Equation (16):

$$Y\left(s\right)={\displaystyle \frac{K}{{(Ts+1)}^n}}U\left(s\right)={\displaystyle \frac{K}{(Ts+1)}}{\displaystyle \frac{1}{{(Ts+1)}^{n-1}}}U\left(s\right)={\displaystyle \frac{K}{(Ts+1)}}{U}_f\left(s\right)$$

(16)

where $U_f\left(s\right)$ is the output of the compensation part. Then, one of the input signals of ESO changes from the control signal u of the system to the output $u_f$ of the compensation part. Equation (16) can be organized as

$$\dot{y}={\displaystyle \frac{K}{T}}{u}_f-{\displaystyle \frac{1}{T}}y=b_0{u}_f+\left({\displaystyle \frac{K}{T}}-b_0\right)u_f-{\displaystyle \frac{1}{T}}y=b_0{u}_f+f$$

(17)

where $b_0$ is the known input gain, and $f=(K/T-b_0)u_f-y/T$ is the total disturbance of the higher-order system after compensation. The structure of the MADRC can then be shown as in Figure 3.

Thus, compared to the regular design of ADRC described above, the ESO of MADRC can be implemented as follows:

$$\left[\begin{array}{c}{\dot{z}}_1\\ {\dot{z}}_2\end{array}\right]=\left[\begin{array}{cc}-{\beta }_1& 1\\ -{\beta }_2& 0\end{array}\right]\left[\begin{array}{c}{z}_1\\ z_2\end{array}\right]+\left[\begin{array}{cc}{b}_0& {\beta }_1\\ 0& {\beta }_2\end{array}\right]\left[\begin{array}{c}{u}_f\\ y\end{array}\right]$$

(18)

where $u_f$ is the output of the compensation part, and the state feedback control law is the same as Equation (12).

By adding the compensation module, the control signal entering the ESO is delayed. This allows the system output and the delayed control signal to synchronize, enabling the ESO to estimate the system state with higher accuracy. Since the stability proof of MADRC has been elaborated in detail in Ref. [35], this article does not provide further explanations but instead focuses more on the design of MADRC.

Based on this, the parameters of the MADRC are tuned. As mentioned above, the parameters to be tuned for MADRC are ${\omega }_o$, $b_0$, $k_p$, and T. Among these, T varies with the operating conditions and corresponds to three typical working points, so it does not require additional tuning. For the tuning of the three parameters ${\omega }_0$, $b_0$, and $k_p$, the following rules can be referred to:

1.

A larger $k_p$ and a smaller $b_0$ both make the system respond quickly. However, too large $k_p$ or too small $b_0$ can cause a large overshoot in the response and increase system fluctuations. When the value of $b_0$ falls within the interval [$K/2T$, ∞), the ESO has good stability and convergence.

2.

The disturbance compensation capability and observation ability increase with a larger ${\omega }_o$. However, this also increases the noise in the ESO. Therefore, it is necessary to gradually increase the value of ${\omega }_o$ to an appropriate level to ensure the control performance of the ESO. Based on extensive simulation experience, when the value of ${\omega }_o$ is in the range [0.3, 0.6], the system can maintain good control performance.

Based on the above tuning rules, a parameter tuning method for MADRC is developed through extensive simulation. The parameter tuning process is shown in Figure 4. The detailed steps are given below:

1.

First, $b_0$ should be fixed, and its value should fall within [$K/2T$, ∞).

2.

Choose a small $k_p$ and ${\omega }_o$, then gradually increase the value of ${\omega }_o$ until the output shows no significant change. Record the value of ${\omega }_o$ at this point.

3.

Gradually increase the value of $k_p$ until the system achieves satisfactory control performance. Record the value of $k_p$ at this point. If the control effect is not ideal, repeat the above steps until satisfactory performance is achieved.

Based on the above discussion of how parameters affect control performance, how to select the adaptive parameter is discussed. Using the single-variable method to explore how different ${\omega }_o$ values influence the system output. As can be seen from the following Figure 5, when ${\omega }_o$ reaches a certain size, the control effect on the system basically remains unchanged. Therefore, it is only necessary to ensure that ${\omega }_o$ is within a reasonable range. As for $b_0$, since this parameter is the denominator in the control rate, a change in $b_0$ may cause a jump in the control rate. Therefore, in actual operation, $b_0$ is not selected as the adaptive parameter. Based on the above analysis, $k_p$ is selected as the adaptive parameter for tuning.

To simplify the tuning process, $b_0$ is first fixed at 0.06, which meets the range requirement. Then, following the MADRC parameter tuning process shown in Figure 4, the tuned parameter ${\omega }_o=0.3$ is obtained through extensive cyclic testing. The values of the adaptive gain coefficient $k_p$ under 100%, 75%, and 50% load are $0.060$, $0.055$, and $0.050$, respectively.

It should be noted that it has already been shown through simulation that the choice of model order has little effect on the model in Ref. [39]. Moreover, Figure 6 shows the open-loop step response of the inert region transfer function model and its approximate seventh-order model under 100% full load as described in this paper. It can be observed from the figure that the waveforms of the two are almost the same, and the approximate result essentially has little impact on the closed loop. Therefore, this paper approximates the inertia zone transfer function at 100% full load as a seventh-order system, which is: $G_p\left(s\right)=1.276/{(16.0s+1)}^7$. This ensures consistent system order across the three typical load conditions and facilitates the design of MADRC.

4. Simulation Research

To verify the control effect of the proposed ADRC in the SST under variable load conditions, this experiment uses three controllers for comparison: the PID control based on integral gain adaptation from Ref. [40], the second-order ADRC based on the fruit fly algorithm from Ref. [41], and the nonlinear ADRC (NLADRC) from Ref. [12]. All the simulation experiments are conducted in the MATLAB 2023a version. In all control structures, the inner loop proportional control parameter is set to $k_{p1}=-2$. For the PID control based on integral gain adaptation, its parameters are $k_p=0.6171$ and $k_d=20.0000$. The values of the integral gain coefficient under 100%, 75%, and 50% load are $k_{i1}=0.0075$, $k_{i2}=0.0050$, and $k_{i3}=0.0030$, respectively. For the outer loop second-order ADRC, its parameters are $k_p={10}^{-4}$, $k_d=0.014$, $b_0=0.0073$, and ${\omega }_o=1.33$. For the NLADRC, its parameters are $b_0=0.1$, ${\beta }_1=0.1$, ${\beta }_2=0.006$, ${\beta }_3=0.0003$, ${\alpha }_1=0.5$, ${\delta }_1=0.1$, ${\alpha }_2=1.0$, ${\delta }_2=0.001$.

Step tests of setpoint and input disturbance are applied to the above controllers under both nominal and uncertain conditions across multiple load levels. Their control performance is observed and compared.

4.1. Control Effect Under Nominal Working Conditions

Under nominal conditions, the setpoint is changed as a step signal at 10 s. A step disturbance is added to the outer loop at 2000 s, and to the inner loop at 4000 s. The system output and control signal under 100%, 75%, and 50% load conditions are shown in Figure 7, Figure 8 and Figure 9.

From Figure 7, it can be observed that the proposed ADRC has the fastest tracking speed under 100% load. When the system is disturbed, it returns to the steady-state value in the shortest time. From Figure 8, it can be seen that under 75% load, the proposed ADRC still responds and converges faster than PID, ADRC and NLADRC. From Figure 9, under 50% load, both ADRC and NLADRC show significant overshoot, and the response speed of NLADRC is very slow, while the PID control still has noticeable fluctuations after tracking convergence. The proposed ADRC produces no overshoot and almost no fluctuation during the simulation, demonstrating the best control performance.

In addition, to test the control performance of the control strategy under different types of disturbances, a ramp disturbance with an amplitude of 2 and a change rate of 0.002, and a sine wave disturbance with an amplitude of 2 are selected for verification. The specific simulation process is as follows: At 10 s, the set value is stepped from 0 to 1. At 2000 s, an outer loop disturbance with a slope of 0.002 and an amplitude of 2 is added. At 5000 s, this disturbance is reduced to 0 at the same rate. At 10,500 s, a sinusoidal wave disturbance is given to the system. From Figure 10, it can be seen that the method proposed in this paper has the best anti-interference effect under different types of disturbances, thereby further verifying the effectiveness of the strategy proposed in this paper.

In summary, the proposed ADRC shows no overshoot at any of the three typical load points. When input disturbances occur, it tracks quickly without obvious fluctuations, showing high application potential.

4.2. Control Effect Under Uncertain Working Conditions

The SST is a typical nonlinear system with time-varying characteristics. It involves multi-physics coupling processes and has large time-delay elements, making it highly complex. Therefore, many simplifications are made during modeling, meaning the SST has strong uncertainty. It is necessary to examine whether the proposed control strategy can still achieve satisfactory performance under the above conditions. This paper uses Monte Carlo experiments to compare the control effects of various strategies when system uncertainty exists. The parameters of each controller are kept unchanged, while the time constants and gain coefficients in the transfer functions from Table 1 are randomly varied within ±10%. The simulation from Figure 7, Figure 8 and Figure 9 is repeated 200 times, and the results are shown in Figure 11, Figure 12 and Figure 13. As can be seen from Figure 11, the proposed ADRC achieves the best control performance at 100% load. The other three control strategies also keep the output within an acceptable range under all three typical load points, indicating that all three strategies have strong robustness.

4.3. Formatting of Mathematical Components

To better analyze the control capabilities of the three strategies quantitatively, the following performance metrics are calculated from the Monte Carlo experiments under the three typical load conditions: the integral absolute error (IAE), the overshoot, and the total variation (TV) of the control input. The formulas for IAE and TV are given below:

$$\mathrm{IAE}={\int }_0^{\infty }\left|r\left(t\right)-y\left(t\right)\right|\mathrm{d}t$$

(19)

$$\mathrm{T}\mathrm{V}=\sum _{i=0}^{n-1}|u(i+1)-u\left(i\right)|$$

(20)

It should be noted that a smaller IAE value indicates better control performance. This means the control strategy is more reliable under parameter changes and has stronger disturbance rejection ability. The TV index is defined as the sum of absolute differences in the control signal between adjacent time steps. A smaller TV value means the control signal is smoother and the actuator moves more gently. If the TV value is too large, it can cause mechanical wear on the actuator and harm stable system operation. From Figure 14, it can be observed that at 100% load, the proposed ADRC has the smallest IAE and TV values among the three control strategies. Although the proposed ADRC shows some large overshoot in a few simulations, the overall overshoot is kept within 1%, with a few cases within 4%, which is acceptable. From Figure 15, the proposed ADRC also has the smallest IAE value. Both its TV and overshoot are lower than those of PID and NLADRC, indicating that the proposed ADRC is more stable under parameter variations. From Figure 16, at 50% load, the IAE value of the proposed ADRC is similar to that of PID control, but its TV value is much smaller. The second-order ADRC shows overly scattered results and very high overshoot, which is no longer satisfactory. The NLADRC has the largest TV and IAE values, indicating that its performance is the worst among the three controllers. In summary, the proposed ADRC provides the strongest control performance against system uncertainty across the 50% to 100% load range. It also shows high robustness and great application value.

To conduct a more in-depth robustness analysis, $M_s$ constraints are adopted to measure the robustness of the controller. Since $M_s$ represents the frequency-domain index of a linear system, and NLADRC is essentially a nonlinear controller, it is impossible to analyze its $M_s$ index. Therefore, we only conducted analyses on PID, ADRC and MADRC. The maximum sensitivity function is often selected as the robustness index, and it is defined as

$$M_s=\underset{\omega }{max}\left|S\left(i\omega \right)\right|=\underset{\omega }{max}\left|{\displaystyle \frac{1}{1+G_c\left(i\omega \right)G_p\left(i\omega \right)}}\right|$$

(21)

The $M_s$ constraint can be described as the reciprocal of the shortest distance from a point to (−1, 0j) on the Nyquist curve of an open-loop system. The smaller the $M_s$, the better the robustness of the controller. The reasonable range of $M_s$ is 1.2 to 2.0. Figure 17 shows the Nyquist plots of three controllers under 100% load. It can be observed that the frequency characteristic trajectory of PID is the farthest from the center of the circle, that of ADRC is the closest, and MADRC is between the two and closer to PID. This indicates that both PID and MADRC have good robustness, while ADRC has poor robustness. Although the robustness of PID is better than that of MADRC, the robustness of MDADRC under 100% load is acceptable.

Table 2. The $M_s$ values of different controllers under three typical operating conditions.

	100% Load	75% Load	50% Load
PID	$1.3359$	$1.4856$	$1.5490$
ADRC	$2.3473$	$1.8381$	$1.7820$
MADRC	$1.4585$	$1.4928$	$1.4550$

presents the $M_s$ values of each controller under three typical operating conditions. As can be seen from Table 2, the $M_s$ value of MADRC under 75% load conditions is not much different from that of PID, while it outperforms PID under 50% load industrial control conditions. Overall, MADRC has excellent robustness and can perform well even when the system is operating at low load. Compared with other controllers, it has better robustness.

In addition, it is equally important to analyze the controller’s sensitivity to noise. Take MADRC as an example to analyze the derivation process of high-frequency noise. MADRC can be transformed into the TDOF equivalent form as shown in Figure 18 through Mason’s gain formula.

In Figure 18, d represents the external disturbance and w represents the external noise. Then, the high-frequency noise suppression capability of MADRC can be described by the transfer function from w to u,

$$G_{wu\_M}={\displaystyle \frac{-G_c\left(s\right)G_h\left(s\right)}{1+G_c\left(s\right)G_h\left(s\right)G_p\left(s\right)}}$$

(22)

where

$$G_c\left(s\right)={\displaystyle \frac{\frac{k_p}{b_0}(s^2+{\beta }_{1}s+{\beta }_2)}{s^2+{\beta }_{1}s+k_p{G}_{cp}\left(s\right)s+{\beta }_2[1-G_{cp}\left(s\right)]}}$$

(23)

$$G_h\left(s\right)={\displaystyle \frac{k_p({\beta }_{1}s+{\beta }_2)+{\beta }_{2}s}{k_p(s^2+{\beta }_{1}s+{\beta }_2)}}$$

(24)

$$G_p\left(s\right)={\displaystyle \frac{-2k_1{k}_2[{(T_{1}s+1)}^{n_1}-2k_1]}{{(T_{1}s+1)}^{2n_1}{(T_{2}s+1)}^{n_2}}}$$

(25)

Among them, $k_1$, $T_1$, and $n_1$ are parameters in the leading zone model, while $k_2$, $T_2$, and $n_2$ are parameters in the inert zone model. According to Equations (22)–(25) [25], we can obtain

$$G_{wu\_M}\underset{s\to \infty }{\sim }-{\displaystyle \frac{k_p{\beta }_1+{\beta }_2}{b_0}}{\displaystyle \frac{1}{s}}={\Lambda }_{wu\_M}{\displaystyle \frac{1}{s}}$$

(26)

The larger the value of ${\Lambda }_{wu\_M}$ is, the stronger the system’s ability to suppress high-frequency noise will be; conversely, the smaller the value is, the weaker the ability will be. Similarly, ADRC and PID can also be described in this way,

$$G_{wu\_A}\underset{s\to \infty }{\sim }-{\displaystyle \frac{{\beta }_3+k_d{\beta }_2+{\beta }_1{k}_p}{b_0}}{\displaystyle \frac{1}{s}}={\Lambda }_{wu\_A}{\displaystyle \frac{1}{s}}$$

(27)

$$G_{wu\_PID}\underset{s\to \infty }{\sim }-k_d{s}^2{\displaystyle \frac{1}{s}}={\Lambda }_{wu\_PID}{\displaystyle \frac{1}{s}}$$

(28)

Table 3. The ${\Lambda }_{wu}$ values of different controllers under three typical operating conditions.

	100% Load	75% Load	50% Load
MADRC	$-2.10$	$-2.05$	$-2.00$
ADRC	$-332.51$	$-332.51$	$-332.51$
PID	$-\infty$	$-\infty$	$-\infty$

shows the values of ${\Lambda }_{wu}$ for the three controllers under the three working conditions. From the data in Table 1, it can be seen that the values of MADRC are much larger than those of ADRC and PID under all three working conditions, indicating that MADRC has the best ability to suppress high-frequency noise.

To observe more intuitively the noise suppression capabilities of the three controllers, a noise with a power of 0.0005 is applied to the system. Figure 19, Figure 20 and Figure 21 show the control effects of each controller after adding noise under three typical loads. It can be seen from the output curves in Figure 19, Figure 20 and Figure 21 that the three controllers all have good high-frequency noise suppression capabilities under the three typical working conditions, and the output fluctuations are all within an acceptable range. By observing the control signal curve, it can be seen that the signal fluctuations of PID control and ADRC are both very strong and large in amplitude, indicating that PID and ADRC have undergone frequent and large-scale adjustments to counteract the noise. In contrast, the control signal of MADRC fluctuates the least, indicating the mildest adjustment. The greater the fluctuation amplitude of the control signal, the more frequently the actuator operates, and the higher the wear. To sum up, all three controllers have good high-frequency noise suppression capabilities. Among them, MADRC has the most stable control signal, the least actuator burden and the best performance.

In total, the proposed MADRC is based on the idea of synchronizing the system output y and the control input u. However, it has many improvements compared to the conventional ADRC. The main features are summarized as follows:

(1)

The proposed MADRC does not add any additional tuning parameters compared to the standard ADRC, and the method of parameter tuning is simple.

(2)

The proposed MADRC inherits the simple structure feature of the standard ADRC, making it easy to be implemented in the DCS system without additional cost investment.

(3)

While ensuring robustness, the proposed MADRC can improve the tracking performance and anti-interference ability of the system.

(4)

The proposed MADRC can enhance the estimation ability of ESO through synchronizing y and u, and also increases the upper limit of the observer bandwidth of ESO.

(5)

The proposed MADRC is applicable to a wide range of load conditions and various working scenarios, not limited to a specific working point.

In addition,

Table 4. Comparison of core characteristics among different ADRC variants.

Controller Type	Core Architecture	Design Goal	Added Module
ADRC	ESO + disturbance compensation, simplify system to $\dot{y}\approx u_0$	Handle system uncertainty and external disturbance	No extra module
Time-delay ADRC	Inherit conventional ADRC architecture	Solve ESO signal asynchrony caused by output delay	Pure time-delay module
Smith-predictive ADRC	Inherit conventional ADRC architecture	Solve ESO signal asynchrony caused by output delay	Output prediction module
MADRC	Inherit conventional ADRC architecture	Solve ESO signal asynchrony caused by output delay	Inertia link compensation
Adaptive MADRC	Based on MADRC, add a parameter adaptive method	Improve performance under wide load conditions, solve oscillation of time-delay ADRC	Parameter adaptive method

summarizes the comparison of the core characteristics of various modified ADRC controllers, including ADRC, time-delay ADRC, Smith-predictive ADRC, MADRC, and the MADRC proposed in this paper. From Table 4, the similarity among several modified ADRCs lies in that they are all based on the structure of conventional ADRC and incorporate different modules to enhance the control performance of systems with large inertia characteristics and time delays. The difference lies in the different methods adopted and the different effects achieved. Compared with time-delay ADRC, Smith-predictive ADRC and MADRC, the adaptive MADRC proposed can not only synchronize ESO input signals without causing output oscillations, but also improve the control performance of the system under wide loads and variable operating conditions, and has a very good application prospect.

5. Field Application

To further test the effectiveness of the method proposed in this paper, open-loop identification is used to build system models of a 660 MW supercritical unit under 100%, 75%, and 50% load. Following the approach presented, the parameters for the adaptive modified ADRC are obtained using the method shown in Figure 4. The modified ADRC is then implemented in the unit’s distributed control system through logic configuration. A linear parameter switching rate is designed.

$$k_p=\left\{\begin{array}{cc}{k}_{p1},\hfill & \hfill P=100\%\mathrm{Pe};\\ \frac{k_{p1}-k_{p2}}{175}(660-P)+k_{p1},\hfill & \hfill 75\%\mathrm{Pe}<P<100\%\mathrm{Pe};\\ k_{p2},\hfill & \hfill P=75\%\mathrm{Pe};\\ \frac{k_{p2}-k_{p3}}{175}(495-P)+k_{p2},\hfill & \hfill 50\%\mathrm{Pe}<P<75\%\mathrm{Pe};\\ k_{p3},\hfill & \hfill P=50\%\mathrm{Pe};\end{array}\right.$$

(29)

and

$$T=\left\{\begin{array}{cc}{T}_1,\hfill & \hfill P=100\%\mathrm{Pe};\\ \frac{T_1-T_2}{175}(660-P)+T_1,\hfill & \hfill 75\%\mathrm{Pe}<P<100\%\mathrm{Pe};\\ T_2,\hfill & \hfill P=75\%\mathrm{Pe};\\ \frac{T_2-T_3}{175}(495-P)+T_2,\hfill & \hfill 50\%\mathrm{Pe}<P<75\%\mathrm{Pe};\\ T_3,\hfill & \hfill P=50\%\mathrm{Pe};\end{array}\right.$$

(30)

Here, P represents the unit load, and $Pe$ is the rated load of the unit. In addition, some on-site details need to be explained. The sampling period of the DCS system used on-site is 250 ms, and it adopts K-type thermocouple sensors with a resolution of 0.1. Regarding the actuator restrictions, the controlled quantity is the valve opening degree, and its constraint is 0–100. For communication latency, on-site transmission is all carried out through hardwiring, so its communication latency is relatively small and can basically be ignored compared to the thermal inertia in the system. A comparison of performance before and after applying the proposed method is shown in Figure 22. The unit load varies between 50% and 100%. After the method is applied, the fluctuation in SST becomes significantly smaller.

Table 5. The values of the superheated steam temperature on sides A and B before and after investment.

	Before Investment (℃)			After Investment (℃)
	Max	Min	Difference	Max	Min	Difference
A-side	605.731	586.417	19.314	601.948	585.384	6.564
B-side	607.599	580.739	26.860	601.635	588.525	13.110

shows the statistics of the maximum and minimum values of the superheated steam temperature before and after the introduction of the method proposed in this paper. After calculation, the fluctuation range on sides A and B is reduced to only 34.0% and 53.0% of the original, respectively. The variance of the fluctuation on sides A and B is only 28.5% and 43.3% of the original, respectively. In summary, these results fully demonstrate the effectiveness of the proposed method.

It should be noted that for the set value strategy, the first thing to clarify is that the higher the set value, the better. However, if the superheated steam temperature is too high, it will cause irreversible damage to the superheated steam temperature pipeline, affecting the safety of the unit. If the temperature is too low, it will reduce the economic efficiency of the unit. In practical applications, as shown in Figure 22, there is a situation where the set value changes before the proposed method is put into use. This is because during actual operation, the temperature of the superheated steam needs to be constantly monitored. If it is too high, it will be lowered; if it is too low, it will be raised. This means that more intervention is required during operation, the burden is greater, and the control effect is not good. After the proposed method is put into use, the set value can remain unchanged for a period of time. This indirectly verifies the effectiveness of the proposed method.

It should also be noted that the actuator is a desuperheating water valve opening degree, with a constraint range of 0 to 100. Taking Figure 22 as an example, it drops to the lower limit of 0 at 500 s and is opened to the upper limit of 100 around 2300 s to adjust the temperature of the superheated steam.

Remark: For the division of working conditions, it is carried out according to the working conditions of 100%, 75% and 50% load in Equation (29). The more working condition divisions there are, the higher the precision of control will be, but at the same time, it will also increase the workload of parameter design and the implementation difficulty in the DCS system. To achieve a balance between higher precision and workload, the above division is chosen to avoid excessive complexity while ensuring accuracy.

6. Conclusions

This paper introduces a modified active disturbance rejection control strategy based on parameter adaptation. It aims to solve the control problems caused by high-order inertial parts in SST. The study first describes the SST and its models at three common load points: 100%, 75%, and 50% load. The basic control idea of active disturbance rejection control (ADRC) is also explained. A modified ADRC strategy with parameter adaptation is proposed. This strategy uses a cascade control structure. In this setup, the inner loop uses a PI controller, and the outer loop uses the modified ADRC controller. A method for setting the parameters of the modified ADRC is given. Parameter adjustment is also performed for system models under different load conditions. The control effect of the proposed method is tested by simulation. The tests include both normal and uncertain working conditions. Moreover, real operation data from a 660 MW unit show that the proposed control strategy works well. The fluctuation range on sides A and B is reduced to only 34.0% and 53.0% of the original, respectively. The variance of the fluctuation on sides A and B is only 28.5% and 43.3% of the original, respectively. The above field application results fully demonstrate that the control strategy proposed does not merely remain at the theoretical simulation level but is a key technical means that can be effectively implemented and effectively solve the problem of superheated steam temperature control in thermal power units. This strategy, through precise disturbance compensation and parameter adaptation, significantly reduces the amplitude and dispersion of steam temperature fluctuations, greatly enhancing the stability and safety of the unit operation, and providing a practical and feasible engineering solution for high-precision SST control. In future work, we will focus on the quantitative research of parameters and further delve into the study of working condition division intervals to enhance control effectiveness without increasing complexity. In addition, we will further derive the mathematical relationship between the load percentage and the controller bandwidth in subsequent research so as to better prove the system’s adaptability. Meanwhile, in the subsequent research, we will also explore the scheme of integrating MADRC with fault diagnosis technology in order to further enhance the overall reliability of the system.