Model Predictive Hybrid PID Control and Energy-Saving Performance Analysis of Supercritical Unit

Abstract

In response to the escalating challenges of rapid load fluctuations and intricate operating environments, supercritical power units demand enhanced control efficiency and adaptability. To this end, this study introduces a novel model predictive hybrid PID control strategy that integrates PID with model predictive control (MPC), leveraging the operational characteristics of multi-loop systems. The proposed strategy adeptly marries the swift response of PID controllers with the foresight and optimization capabilities of MPC. A dynamic model of a supercritical unit is constructed using the subspace identification method. The model’s high precision is confirmed by its alignment with field data. Load change simulations demonstrate that the PID–MPC hybrid controller shows faster response times and more precise tracking capabilities compared to the feedforward-PID strategy. It achieves substantial improvements in the IAE index for three loops, with increases of 29.2%, 54.1%, and 57.3% over the feedforward-PID controller. An energy-saving performance analysis indicates that the proactive control actions of both the PID–MPC and MPC strategies lead to dynamic exergy efficiency and coal consumption rates with a broader range of dynamic process changes. The disturbance scenario simulation regarding the proposed controller achieves faster settling times and minimizes control deviation compared to the traditional controller.

1. Introduction

In the age of swift renewable energy expansion, while celebrated for their cleanliness and low-carbon profiles [1], the inherent unpredictability and intermittency of these sources present substantial challenges to the stability and dependability of electrical grids. Consequently, coal-fired power plants are evolving from their traditional single-energy supply paradigm towards a more agile and efficient energy supply model [2]. Supercritical coal-fired power plants (SCFPPs), recognized for their high efficiency in power generation, are assuming an increasingly stable role within the power production sector [3,4]. Nevertheless, with escalating and fluctuating power demands, particularly the prevalence of rapid load shifts and complex operating scenarios, the control efficiency and flexibility of supercritical units are subjected to more stringent demands.

The performance of a power unit’s control system is directly linked to the safety, economy, and stability of the unit [5]. Among the various control challenges in SCFPPs, the coordinated control of boilers and turbines stands out as one of the core issues that urgently requires innovative solutions. For a long time, simple and easy-to-implement PID controllers have dominated the coordinated control systems of boiler–turbine units [6,7]. However, when faced with complex, nonlinear, and multivariable coupled control systems, the limitations of PID controllers become increasingly apparent, making it difficult to achieve the desired control effects. Consequently, academia and industry have been continuously exploring new control strategies, such as gain-scheduling [8], sliding mode [9], and adaptive control [10]. Among the many advanced control strategies, model predictive control (MPC) stands out due to its unique predictive mechanism and ability to handle multivariable coupling and complex constraints [11,12,13]. It has demonstrated significant advantages in improving dynamic response and operational efficiency over a wide range of load conditions for power units.

Many scholars have explored the application of MPC in the control systems of power plants. Wu et al. [14] presented a distributed MPC approach for coal-fired power plants that incorporates carbon capture technology, which led to a significant improvement of 33.1% in the load tracking performance. In another study, He and Lima [15] designed an enhanced MPC algorithm based on modified sequential quadratic programming (SQP) and successfully implemented it in an SCFPP. Zhu et al. [16] developed a three-term MPC strategy and successfully applied it to a 1030 MW supercritical coal power plant. Compared with the existing PID controller, the new strategy significantly improves the performance of load and steam pressure control.

However, while MPC theoretically demonstrates significant potential for managing complex control systems and optimizing performance, it encounters substantial challenges in practical implementation. These include high computational complexity and the difficulty in meeting real-time requirements. Particularly in the operation of supercritical units, which are subject to rapid load changes and complex working conditions, the question of how to effectively reduce computational costs and enhance system response speed without compromising control precision has become a critical issue that needs to be addressed.

To tackle the aforementioned challenges, an innovative model predictive hybrid PID control strategy (PID–MPC) was developed. Some scholars have studied the feasibility of PID–MPC hybrid control. Sanama, Xia, and Nguepnang [17] implemented PID–MPC to enhance the performance of a chiller-fan coil unit. Sen, Singh, and Ramachndran [18] designed a hybrid MPC–PID controller. Compared with the pure PID controller, the results show that the performance of the control loop is improved under the hybrid control system. Existing studies have demonstrated the application potential of a PID–MPC hybrid strategy.

In this study, the application of PID–MPC hybrid control in power plant units is explored. This approach aims to seamlessly integrate the immediate feedback adjustments of PID control with the predictive optimization capabilities of MPC. The strategy leverages the complementary strengths of these control techniques through multi-loop decoupling, particularly under complex operating conditions. The paper commences by constructing a high-precision dynamic simulation model of a supercritical unit using the subspace identification method. Subsequently, the PID–MPC hybrid control strategy is meticulously designed based on this model. To assess the control performance, a set of control and energy-saving performance indicators are proposed. Ultimately, the effectiveness and robustness of the proposed strategy are comprehensively evaluated through extensive multi-scenario simulation tests.

2. Supercritical Unit Coordination System

2.1. System Description

The schematic diagram of the supercritical unit in this study is shown in Figure 1. Coal is burned in the furnace as fuel to release heat energy. The feed water enters the water wall after being preheated by an economizer and becomes a two-phase flow after being heated at a high temperature in the furnace. The steam is output by the separator and then heated into high-temperature and high-pressure steam through the multistage heating surface. Steam turbines and motors convert thermal energy from product steam into electrical energy for power generation. The entire energy conversion process within a power plant is a complex orchestration involving the synchronized operation of numerous components.

2.2. Coordinated Control System

To maintain the stability and efficiency of supercritical units, effective boiler–turbine coordination is crucial. However, the multivariable coupling nature of these systems demands sophisticated control strategies. Traditionally, a combination of feedforward functions and PID control has been used, as depicted in Figure 2a. While this approach is straightforward and adaptable, it falls short as grid flexibility demands and unit capacities increase. Frequent load changes exacerbate internal variable coupling, heightening sensitivity to disturbances and uncertainties. Moreover, the need for rapid power plant load adjustments necessitates a control system that is capable of swift, accurate responses and flexible adaptation to dynamic changes, presenting a significant challenge to the conventional control strategy.

This study presents a novel PID–MPC hybrid control strategy to tackle the challenges faced by supercritical units, as shown in Figure 2b. The implementation of a PID–MPC hybrid control strategy in industrial applications faces challenges such as system nonlinearity, external disturbances, and the technical limitations of sensors and actuators. Our strategy effectively addresses these challenges by integrating the rapid response of PID with the predictive and optimization capabilities of MPC. The predictive and optimization features of MPC enable the direct consideration of operational constraints, a capability not present in traditional PID control, which is particularly crucial when dealing with industrial systems that have hard constraints. The PID controller swiftly adjusts the main steam valve in response to load changes, while MPC precisely controls coal and water feed to optimize pressure and temperature.

3. Methods

3.1. Subspace Identification Modeling

Subspace identification (SID) has become extensively used in the realm of system identification [19]. Assuming that the n-th order continuous linear state-space equation of a CFPP can be expressed in the following form:

$$\left\{\begin{array}{c}\dot{x}(t)=A_{c}x(t)+B_{c}u(t)+K_{c}e(t)\\ y(t)=C_{c}x(t)+D_{c}u(t)+e_c(t)\end{array}\right.$$

(1)

where A_c, B_c, C_c, D_c, and K_c are the continuous state-space matrices. u(t) are the inputs, and y(t) are the outputs. e(t) are the disturbances, and x(t) are the vectors of n-th states. For dynamic systems, D is fixed to zero by default, meaning that the system has no feedthrough [19].

The parameters in system model (1) are determined by SID method, and the Hankel matrix composed of input and output data blocks is as follows:

$${\mathit{U}}_{\left(0\left|2i-1\right.\right)}\underset{=}{def}\left[{\displaystyle \frac{\begin{array}{cccc}u(0)& u(1)& \cdots & u(j-1)\\ u(1)& u(2)& \cdots & u(j)\\ ⋮& ⋮& \ddots & ⋮\\ u(i-1)& u(i)& \cdots & u(i+j-2)\end{array}}{\begin{array}{cccc}u(i)& u(i+1)& \cdots & u(i+j-1)\\ u(i+1)& u(i+2)& \cdots & u(i+j)\\ ⋮& ⋮& \ddots & ⋮\\ u(2i-1)& u(2i)& \cdots & u(2i+j-2)\end{array}}}\right]=\left[{\displaystyle \frac{\mathit{U}(0|i-1)}{\mathit{U}(i|2i-1)}}\right]=\left[{\displaystyle \frac{{\mathit{U}}_p}{{\mathit{U}}_f}}\right]$$

(2)

$${\mathit{Y}}_{\left(0\left|2i-1\right.\right)}\underset{=}{def}\left[{\displaystyle \frac{\begin{array}{cccc}y(0)& y(1)& \cdots & y(j-1)\\ y(1)& y(2)& \cdots & y(j)\\ ⋮& ⋮& \ddots & ⋮\\ y(i-1)& y(i)& \cdots & y(i+j-2)\end{array}}{\begin{array}{cccc}y(i)& y(i+1)& \cdots & y(i+j-1)\\ y(i+1)& y(i+2)& \cdots & y(i+j)\\ ⋮& ⋮& \ddots & ⋮\\ y(2i-1)& y(2i)& \cdots & y(2i+j-2)\end{array}}}\right]=\left[{\displaystyle \frac{\mathit{Y}(0|i-1)}{\mathit{Y}(i|2i-1)}}\right]=\left[{\displaystyle \frac{{\mathit{Y}}_p}{{\mathit{Y}}_f}}\right]$$

(3)

where i and j are user-defined parameters for adjusting identification accuracy. Y_p and Y_f represent past and future outputs, respectively. U_p and U_f represent past and future inputs, respectively.

Similarly, a corresponding definition of the state matrix is formulated to derive the generalized input–output matrix equation as follows:

$${\mathit{X}}_{\mathit{f}}{=\mathit{A}}^{\mathit{i}}{\mathit{X}}_{\mathit{p}}{+\mathsf{\Delta }}_{\mathit{i}}^{}{\mathit{U}}_{\mathit{p}}$$

(4)

$${\mathit{Y}}_{\mathit{p}}{=\mathsf{\Gamma }}_{\mathit{i}}{\mathit{X}}_{\mathit{p}}{+\mathit{H}}_{\mathit{i}}^{}{\mathit{U}}_{\mathit{p}}$$

(5)

$${\mathit{Y}}_{\mathit{f}}{=\mathsf{\Gamma }}_{\mathit{i}}{\mathit{X}}_{\mathit{f}}{+\mathit{H}}_{\mathit{i}}^{}{\mathit{U}}_{\mathit{f}}$$

(6)

where X_p and X_f represent past and future state variables, respectively. ${\mathsf{\Gamma }}_{\mathit{i}}$ is the generalized observable matrix, and H_i is the low-dimensional partitioned triangular Toeplitz matrix.

By combining Formulars (4)–(6),

$$\begin{array}{l}{\widehat{\mathit{Y}}}_{\mathit{f}}{=\mathit{L}}_{\mathit{w}}{\mathit{W}}_{\mathit{p}}{+\mathit{L}}_{\mathit{u}}^{}{\mathit{U}}_{\mathit{f}}\\ {\mathit{L}}_{\mathit{w}}=\left[\begin{array}{cc}{\mathit{A}}^{\mathit{i}}{\mathsf{\Gamma }}_{\mathit{i}}^{†}& {\mathsf{\Delta }}_{\mathit{i}}^{}-{\mathit{A}}^{\mathit{i}}{\mathsf{\Gamma }}_{\mathit{i}}^{†}{\mathit{H}}_{\mathit{i}}\end{array}\right]\\ {\mathit{W}}_{\mathit{p}}=\left[\begin{array}{c}{\mathit{Y}}_{\mathit{p}}\\ {\mathit{U}}_{\mathit{p}}\end{array}\right]\\ {\mathit{L}}_{\mathit{u}}^{}={\mathit{H}}_{\mathit{i}}^{}\end{array}$$

(7)

The current state can be characterized by historical input and output data, with L_w representing the state subspace matrix and L_u denoting the deterministic input subspace matrix.

The subspace matrices L_w and L_u can be identified from the Hankel matrix employing the least squares method. Subsequently, leveraging a linear observer grounded in (7), the system matrix [A, B, C, D] is solved by least squares method as follows:

$$\left[\mathit{A},\mathit{B},\mathit{C},\mathit{D}\right]=\arg \underset{A,B,C,D}{\min }{‖\left(\begin{array}{l}{\tilde{\mathit{X}}}_{i+1}\\ {\mathit{Y}}_i\end{array}\right)-\left(\begin{array}{cc}\mathit{A}& \mathit{B}\\ \mathit{C}& \mathit{D}\end{array}\right)\left(\begin{array}{l}{\tilde{\mathit{X}}}_{i+1}\\ {\mathit{U}}_{i|i}\end{array}\right)‖}_F^2$$

(8)

To reflect the dynamic operation characteristics of the unit as the coordinated control object, this study is based on the historical data of a cooperative power plant in China as the identification data. In order to optimize the control design, the input variables were selected as coal feed, water feed, and throttle valve opening, and the output variables include power, main steam pressure and water wall outlet temperature. The SID identification method in MATLAB R2022 B system identification toolbox is used to identify the system state space model.

3.2. PID–MPC Hybrid Control Strategy

3.2.1. PID Controller for Output Power

In the scenario of power generation control, the PID controller accurately regulates the opening of the main steam valve through real-time power feedback signals collected so as to achieve efficient and stable control of the power generation of the supercritical unit. The expression is as follows:

$$u(t)=K_P\cdot e(t)+K_I\cdot {\displaystyle \int e(t)\mathrm{d}t}+K_D{\displaystyle \frac{\mathrm{d}e(t)}{\mathrm{d}t}}$$

(9)

where K_P is the proportional coefficient, K_I is the integral coefficient, and K_D is the integral coefficient. After controller parameters are adjusted, the value of K_P is 58, K_I is 0.1, and K_D is 0.3.

3.2.2. Design of MPC

In view of the nonlinear, time-varying, and strong coupling characteristics of supercritical unit coordination system, MPC control strategy is proposed to regulate the main steam pressure and the outlet temperature of water wall in the coordination system.

Utilizing the general form (1) of the identified state-space model, discretization and difference processing are implemented to derive an augmented model as follows:

$$\left\{\begin{array}{l}\underset{x(k+1)}{\underbrace{\left[\begin{array}{c}\Delta x(k+1)\\ y(k+1)\end{array}\right]}}=\underset{A}{\underbrace{\left[\begin{array}{cc}{A}_m& 0_m^{{\rm T}}\\ C_m{A}_m& 1\end{array}\right]}}\underset{x\left(k\right)}{\underbrace{\left[\begin{array}{c}\Delta x(k)\\ y(k)\end{array}\right]}}+\underset{B}{\underbrace{\left[\begin{array}{c}{B}_m\\ 0\end{array}\right]}}\Delta u(k)\\ y(k)=\underset{C}{\underbrace{\left[\begin{array}{cc}{0}_m& 1\end{array}\right]}}\left[\begin{array}{c}\Delta x(k)\\ y(k)\end{array}\right]\begin{array}{ccc}\begin{array}{cc}\begin{array}{ccc}& & \end{array}& \end{array}& & \end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}\end{array}\right.$$

(10)

where $0_m^{}\in {ℝ}^{n_x}$ represents the 0 matrix with dimension n_x. $△u\left(k\right)$ is the increment of control input at time k.

Define the vectors of the predicted output Y and the control input ΔU as

$$\begin{array}{c}Y={\left[y{\left(k+1\left|k\right.\right)}^{{\rm T}}\begin{array}{c}\end{array}y{\left(k+2\left|k\right.\right)}^{{\rm T}}\cdots \begin{array}{c}\end{array}y{\left(k+N_P\left|k\right.\right)}^{{\rm T}}\right]}^{{\rm T}}\\ \mathsf{\Delta }U=\left[\mathsf{\Delta }u{\left(k\right)}^{{\rm T}}\begin{array}{c}\end{array}\mathsf{\Delta }u{\left(k+1\right)}^{{\rm T}}\begin{array}{c}\cdots \end{array}\begin{array}{c}\end{array}\mathsf{\Delta }u{\left(k+N_C-1\right)}^{{\rm T}}\right]\end{array}$$

(11)

Based on model (10), recursively in the prediction horizon N_P, the set of prediction output Y is obtained as follows:

$$\begin{array}{c}\begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}Y\left(k\right)=F_{}\widehat{\overline{x}}\left(k\right)+\Phi \mathsf{\Delta }U\left(k\right)\\ F=\left[\begin{array}{c}CA\\ C{A}^2\\ ⋮\\ C{A}^{N_P}\end{array}\right],\Phi =\left[\begin{array}{cccc}CB& 0& \cdots & 0\\ CAB& CB& \cdots & 0\\ ⋮& & \ddots & ⋮\\ C{A}_{}^{N_P-1}B& C{A}_{}^{N_P-2}B& \cdots & C{A}_{}^{N_P-N_C}B\end{array}\right]\end{array}$$

(12)

where F and Φ represent the constructed coefficient matrix, respectively.

For the coordinated control problem of supercritical power plant in this study, the goal of controller design is to minimize the absolute difference between the control output and the set point in the future prediction horizon N_P. Define the cost function of the quadratic form reflecting the control objective as follows:

$$\begin{array}{c}{}_{}^{}J={\left(R-Y\right)}^{{\rm T}}S\left(R-Y\right)+\mathsf{\Delta }{U}^{{\rm T}}\overline{R}\mathsf{\Delta }U\\ {{{\displaystyle }}^{}}_{}^{}{{{\displaystyle }}^{}}_{}^{}{R}^{{\rm T}}=\underset{N_P}{\underbrace{\left[1\begin{array}{c}\end{array}1\begin{array}{c}\end{array}\begin{array}{c}\cdots \end{array}\begin{array}{c}\end{array}1\right]}}{r}_i\left(k\right)\\ {{{\displaystyle }}^{}}_{}^{}{{{\displaystyle }}^{}}_{}^{}\overline{R}=r_w{I}_{N_C\times N_C}\end{array}$$

(13)

where J is the cost function of energy-aware MPC. $r_i\left(k\right)$ and $y_i\left(k\right)$ are the i-th reference and measure outputs at time k, respectively. R is the reference value matrix. S and $\overline{R}$ are error and control weight matrix, respectively. r_w is used as a tuning parameter for the desired response speed.

Necessitating the substitution of (12) with (13),

$$J={\left(R-F_{}\widehat{\overline{x}}\left(k\right)\right)}^{{\rm T}}{S}_q\left(R-F_{}\widehat{\overline{x}}\left(k\right)\right)-2\mathsf{\Delta }{U}^{{\rm T}}{\Phi }^{{\rm T}}{S}_q\left(R-F_{}\widehat{\overline{x}}\left(k\right)\right)+\mathsf{\Delta }{U}^{{\rm T}}\left({\Phi }^{{\rm T}}{S}_q\Phi +S_r\right)\mathsf{\Delta }U$$

(14)

To facilitate optimal solution, (14) is transformed into quadratic programming (QP) form as follows:

$$\left\{\begin{array}{l}\underset{\mathsf{\Delta }U}{\min }J=\underset{\mathsf{\Delta }U}{\min }\left(\mathsf{\Delta }{U}^{{\rm T}}H\mathsf{\Delta }U-2f^{{\rm T}}\mathsf{\Delta }U\right)\\ H={\Phi }^{{\rm T}}S\Phi +\overline{R}\\ f={\left(R-F_{}\widehat{\overline{x}}\left(k\right)\right)}^{{\rm T}}S\Phi \end{array}\right.$$

(15)

where H is the symmetric matrix of the quadratic term and f is the matrix of the linear term.

In the realm of coal-fired power plant operations, safety regulations and equipment specifications necessitate the imposition of upper and lower bounds on control variables. For regulating variables, coal feed rate, water feed rate, and valve opening, the constraints are set as follows:

$$U^{\min }\left(k\right)\le U\left(k\right)\le U^{\max }\left(k\right)$$

(16)

$$\Delta U^{\min }\left(k\right)\le \Delta U\left(k\right)\le \Delta U^{\max }\left(k\right)$$

(17)

The MPC optimization problem is composed of objective function (15) and constraint conditions (16) and (17). The optimal control sequence is obtained by solving the optimization problem at k time. According to the MPC principle, the MPC rolling optimization control is realized when only the first item in the prediction sequence is applied to the system.

After many tests in the simulation environment, the key parameters of MPC applied to the control of the main steam pressure of the coordination stage and the outlet temperature of the water wall are set in

Table 1. Parameter settings of MPC.

Parameter	Value
Sampling time	4
Predictive horizon	25
Control horizon	10
Error weight of main steam pressure	1
Error weight of water wall outlet temperature	0.9

.

3.3. Perfermance Index

3.3.1. Control Performance

Considering the long-term operational performance of the control system and the comprehensive impact of error amplitude and persistence, the Integrated Absolute Error (IAE) index is employed as follows:

$$IA{E}_i={\displaystyle \sum _{k=1}^N\left|r_i\left(k\right)-y_i\left(k\right)\right|}\Delta t\underset{}{\overset{}{{\displaystyle }}}{{\displaystyle }}_{}^{} i=1,2,3$$

(18)

The maximum error (ME) is a comprehensive measure of the maximum difference between the actual output and the expected output. The formula is as follows:

$$M{E}_i=\underset{1\le k\le N}{\max }\left[r_i\left(k\right)-y_i\left(k\right)\right]{{\displaystyle }}_{}^{}\underset{}{\overset{}{{\displaystyle }}} i=1,2,3$$

(19)

3.3.2. Energy-Saving Performance

The exergy analysis method based on the second law of thermodynamics is a crucial tool for studying and optimizing the thermodynamic performance of systems [20,21]. The coal supply serves as the energy input source for an SCFPP. The electrical power is the product energy of SCFPP. Exergy represents the maximum potential of a system to perform work. The equations are as follows:

$$\dot{E}{x}_{\mathrm{coal}}={\dot{m}}_{\mathrm{coal}}\cdot e{x}_{\mathrm{coal}}$$

(20)

$$\dot{E}{x}^W=\dot{P}$$

(21)

where $\dot{E}{x}_{\mathrm{coal}}$ is the dynamic exergy rate of coal feed, kW; ${\dot{m}}_{\mathrm{coal}}$ is the mass flow rate of coal feed, kg/s; $e{x}_{\mathrm{coal}}$ is the specific exergy of coal, kJ/kg; $\dot{P}$ is the electric power, kW.

The dynamic exergy efficiency of the overall cycle of an SCFPP is precisely defined as follows:

$${\eta }_{ex}={\displaystyle \frac{\dot{P}}{\dot{E}{x}_{\mathrm{coal}}}}$$

(22)

where ${\eta }_{ex}$ represents dynamic exergy efficiency, %.

Coal consumption is a vital consideration in the operation and optimization of CFPPs. As a key indicator of energy saving and carbon reduction, the coal consumption rate (CCR) accurately reflects the amount of coal consumed per kilowatt-hour of electricity generated

$$\mathrm{CCR}={\displaystyle \frac{M_{\mathrm{coal}}}{\dot{P}\times t_d}}$$

(23)

where CCR is the coal consumption rate, kg/MW·h; M_coal is the total coal consumption during operation period, kg; t_d is the operation period, h.

4. Results and Discussion

4.1. Model Validation

In this study, a dataset comprising 4500 samples, each taken at 4 s intervals, is extracted from a 350 MW supercritical unit at a cooperative power plant in China for the purpose of system identification modeling. To assess the model’s validity, 9000 additional unidentified data are employed for testing. Figure 3 shows the model identification and testing results. Figure 3 demonstrates that the model’s output closely aligns with the actual field data, and the trend of the three output groups shown in Figure 3b is consistent with the power plant’s data, thereby validating the model’s high accuracy and reliability.

4.2. Wide-Range Load Change

To assess the variable load control performance of various control strategies, a comparative analysis of the PID–MPC hybrid controller, the classical MPC controller, and the feedforward-PID controller is conducted, as depicted in Figure 4. Figure 4 reveals that, while the feedforward-PID controller exhibits minimal overshoot during the response phase, it suffers from a significant delay in response time, underscoring its lack of responsiveness. Conversely, both the PID–MPC and MPC controllers display superior rapid response capabilities, effectively and swiftly addressing the load change signals. The performance index is detailed in

Table 2. Performance indexes of different strategies.

Controller	IAE1	IAE2	IAE3
PID–MPC	2320.4	452.7	1669.8
MPC	1998.2	430.1	2067.4
Feedforward-PID	3278.3	985.6	3909.9

. According to the table, the PID–MPC hybrid controller achieves a control performance nearly equivalent to that of MPC, with substantial improvements in the IAE index for the three loops, showing increases of 29.2%, 54.1%, and 57.3% over the feedforward-PID controller. Furthermore, the energy-saving performance depicted in Figure 4c indicates that the more proactive control actions of PID–MPC and MPC result in dynamic exergy efficiency and CCRs with a broader range of dynamic process changes.

4.3. Uncertainty Disturbances

To assess the performance of various control strategies in the presence of uncertain disturbances, step disturbances are introduced at 500 s, 4000 s, and 7500 s for three critical control variables: coal feed rate, water feed rate, and throttle valve opening. Figure 5 compares the responses of the PID–MPC hybrid controller, the MPC controller, and the feedforward-PID control to these disturbances. Figure 5a illustrates that, across the three disturbance scenarios, the feedforward-PID controller experiences the most significant overshoot. Its adjustment process is characterized by substantial fluctuations and a slow convergence to the set reference value. In contrast, the PID–MPC hybrid control strategy and the MPC controller demonstrate superior anti-interference capabilities. They can swiftly identify and respond to disturbances, markedly reducing settling time and effectively minimizing control deviation. The energy-saving performance following the disturbances, as influenced by the controllers, is depicted in Figure 5c. This figure indicates that the feedforward-PID exhibits more gradual changes, while PID–MPC and MPC show larger response peaks.

Table 3. Performance index of different strategies.

Case	Controller	IAE1	IAE2	IAE3	ME1 (MW)	ME2 (MPa)	ME3 (℃)	${\mathit{\eta }}_{\mathit{e}\mathit{x},\mathbf{ave}}$(%)	CCRave (kg/MW·h)
Case 1: Throttle valve opening disturbance (7500 s)	PID–MPC	34.2	20.5	114.7	0.10	0.02	−0.14	30.47	613.7
	MPC	13.1	12.7	99.5	−0.03	0.01	−0.12	30.47	613.7
	Feedforward-PID	327.4	103.7	201.9	−0.32	−0.09	−0.33	30.45	614.1
Case 2: Coal feed rate disturbance (500 s)	PID–MPC	21.2	4.7	69.7	0.08	0.02	0.23	30.48	613.6
	MPC	1.2	4.1	46.7	0.01	0.01	0.16	30.48	613.6
	Feedforward-PID	73.5	20.8	550.9	0.09	−0.03	1.32	30.49	613.3
Case 3: Water feed rate disturbance (4000 s)	PID–MPC	56.2	3.8	25.4	−0.10	−0.01	−0.09	30.48	613.4
	MPC	5.1	2.4	20.5	−0.16	−0.01	−0.06	30.49	613.5
	Feedforward-PID	116.4	3.2	48.6	−0.20	0.01	−0.09	30.49	613.4

provides a quantitative summary of the performance indicators, highlighting that, under three distinct interference conditions, the PID–MPC hybrid controller and the MPC controller significantly outperform the traditional feedforward-PID control method in terms of the ME and IAE indicators. Notably, as indicated in bold in Table 3, the MPC controller achieves the best performance in the indicators, while the PID–MPC hybrid control strategy closely matches MPC’s performance but with a substantially reduced computational complexity. This reduction enhances the real-time performance and decreases resource consumption in practical applications, underscoring its significant potential and practical value in the optimization of control strategies.

5. Conclusions

This paper introduces a novel PID–MPC hybrid control strategy that synergistically merges the immediate feedback adjustment capabilities of PID control with the predictive optimization features of MPC. The main conclusions are as follows:

• A dynamic model of a supercritical unit is constructed using the subspace identification method. The model’s high precision is confirmed by its alignment with field data, providing a solid foundation for the design of subsequent control strategies.
• In variable load simulation tests, the PID–MPC hybrid control strategy shows faster response times and more precise tracking capabilities compared to the traditional feedforward-PID strategy. It achieves substantial improvements in the IAE index for three loops, with increases of 29.2%, 54.1%, and 57.3% over the feedforward-PID controller, closely matching the performance of a classic MPC controller and effectively enhancing the system’s dynamic response performance.
• The energy-saving performance analysis indicates that the proactive control actions of both the PID–MPC and MPC strategies lead to dynamic exergy efficiency and CCRs with a broader range of dynamic process changes.
• Faced with a variety of control disturbance scenarios, the PID–MPC hybrid controller proves its robust anti-interference capabilities. This strategy consistently shows less overshoot, achieves faster settling times, and minimizes control deviation when compared to the conventional feedforward-PID controller.

Future research will place greater emphasis on directly considering the operational constraints in the energy efficiency and process optimization of MPC. Additionally, there will be an exploration of tighter integration between MPC and PID, along with more extensive application testing, particularly in industrial applications.