Load Frequency Model Predictive Control of a Large-Scale Multi-Source Power System

Abstract

With increased interests in affordable energy resources, a cleaner environment, and sustainability, more objectives and operational obligations have been introduced to recent power plant control systems. This paper presents a verified load frequency model predictive control (MPC) that aims to satisfy the load demand of three practical generation technologies, which are wind energy systems, clean coal supercritical (SC) power plants, and dual-fuel gas turbines (GTs). Simplified state-space models for the two thermal units were constructed by concepts of subspace identification, whereas the individual wind turbine integration was implicated by the Hammerstein–Wiener (HW) model and then augmented from the output to simulate the effect of a wind farm, assuming similar power harvesting from all turbines in the farm. A practical strategy of control was then suggested, which was as follows: with a changing load demand, the available harvested wind energy must be fully admitted to the network to cover part of the load demand with the free energy, and the resultant load signal will then be instructed to the MPCs designed for the coal and gas units for the coordination of generation. The load signal, after being penetrated by wind, has more transients and faster changes, and needs a more sophisticated control in order to follow the load demand of the flexible coal and gas units. Furthermore, as the level of wind penetration increases, the power system frequency excursions are higher. The simulation results show an acceptable performance for linear MPCs embedded to the GT and coal units, with around a 90 MW share of wind without exceeding the safe restrictions of the plants and allowable reasonable frequency excursions. The complete simulation framework can be used to facilitate wind energy penetration in such power systems and train the operators and future engineers with subsequent power system frequency simulation studies.

1. Introduction

1.1. Aims

The adequacy, security, and stability of power systems are major issues in energy utilities and power engineering research, and have recently received much interest for enhancing power system operations after the integration of considerable levels of renewable energy (RE). The dominant RE technologies are well known to be wind and solar. Despite the fact that solar and wind power capacities are the fastest growing RE technologies, with rates of around 40% and 20%, respectively [1], solar energy is recognized to be easier to model and predict than wind. On the other hand, with the research efforts conducted on clean fossil technologies with the aims of improving energy efficiency and reducing CO₂ emissions, the power system has been integrated with hybrid power generation devices and auxiliaries that have different operational characteristics and, thus, have introduced more challenging objectives to load frequency control (LFC) systems. The usual way to deal with the load frequency control of a two-area power system or multiple generation systems is shown in Figure 1, which is a standard figure that is commonly found in many textbooks for basic control actions carried out for the aim of generation control [2,3,4].

However, it must be emphasized that more sophisticated models and control are needed when the issue is further complicated by the presence of mixed technologies of wind turbines, modern clean coal technologies, and dual-fuel gas turbines. The system is made more realistic using the system identification of state-space models, which can then be embedded with an appropriate MPC strategy in order to modernize the conventional load frequency control.

The one that we suggest in this paper is based on verified state-space models of a real 600 MW supercritical (SC) cleaner power plant, a 250 MW Siemens Dual-Fuel gas turbine GT, and a Hammerstein–Wiener model for a wind farm that is assumed to be composed of 190 wind turbines (TWT-1.65). The data set for every energy resource mentioned was gathered from previous research work of the corresponding author [5,6,7]. Furthermore, every model has an individual MISO, structured to enhance the accuracy and produce more realistic results. The proposed upgraded system in this paper is shown in Figure 2. The new proposed system with the aforementioned resources would offer better environmental insight while keeping the system secure and stable through a promising MPC strategy. The Hammerstein–Wiener model in the figure was used to predict the harvested wind power to satisfy part of the load demand signal, and the remaining load instructed the MPCs that control the clean coal unit and gas unit to cover the remaining load with minimum fuel. With this insight, the significance of the paper can be stated as follows:

• Facilitating a cleaner and environmentally friendly power system from verified models and simulations of a hybrid power generation system.
• Realizing the importance of mixed generation technology in terms of system strength and environmental effect.
• Investigating the effect of wind penetration on traditional power systems fed by coal power plant and coal/gas stations.
• Determination of the most practically feasible level of wind energy penetration without causing extensive negative effects on the frequency, which is necessary for all stability classes (dynamic and steady-state stabilities).

1.2. Literature Review and Paper Contributions

The review of literature has focused on modelling and control of multi-source power systems, which are naturally hybrid from the generation side. The review should then cover different features and methodologies of modelling and power systems, with emphasis on system dynamics and control. A wide range of publications were studied in order to have a clear picture of the practical importance of the research area.

Mohamed et al. [8] have reviewed new trends for supercritical and ultra-supercritical power plants with two main categories of models: physical models, and empirical models of SC and USC power plants, which have been simulated using APROS^®, APD^®, FORTRAN, SimuEngine, GSE, MATLAB^®, and Thermolib. Because of the distinct advantages of each method, both methods are unquestionably eligible for modelling SC and USC units. The control system strategies have been also reviewed and the dominant control approach was MPC.

Rehman [9] has presented a study that establishes a reliable coordination between renewable resources in Hybrid Power Systems (HPSs). The most popular scheme has been found to be the Wind-PV systems with and without storage. Wind-PV total capacity has been varied from 1.6 to 120.0 kW, with an average size of 21.0 kW and an average COE of 0.458 US$/kWh. Simulation has been carried out by HOMER and HOMER Pro software for the design and optimization of HPS, in addition to HOGA and MATLAB. The favoured targets of hybrid power systems have been compiled during the survey. However, the system is relatively small and may not be suitable for large-scale loads due to the low energy density of renewables.

Li et al. [10] has implemented a model of a hydrogen storage wind and gas complementary power generation system for energy management. A dynamic mathematical model of a micro gas turbine has been combined with an aerodynamic model of a wind turbine. The alkaline electrolyser model has developed the economic and environmental cost of the system as an objective function, and establishes the capacity optimization model of the wind–gas complementary power generation system. The results have been compared with variable loads in a specific area during the northern winter and it has been concluded that the wind–gas complementary system can boost energy use while lowering wind curtailment.

Tsoutsanis [11] has implemented a generic model from MATLAB/SIMULINK for hybrid power plants, consisting of 14 wind turbines and a gas turbine. The author has investigated the performance and challenges of hybrid power plants, has studied different scenarios, and has compared the dynamic response of a hybrid power plant and a double gas turbine power plant for 10 h, which have shown improvements in terms of emissions and fuel consumption.

Barelli et al. [12] have presented an MPC strategy and applied it to a hybrid system that is structured by a solid oxide fuel cell (SOFC) and a recuperative GT for following a typical daily load demand in microgrids. The simulation has been carried out in MATLAB^® and a Simulink environment, and has shown a promising system performance with high average efficiency (approximately 54.5%) and load changes with quick response .

Di Gaeta et al. [13] have shown a dynamic model of a 100 kW commercial micro gas turbine (MGT) fed with a mixture of standard and alternative fuels and the MGT emulated dynamics have been based on first order differential equations with different operating conditions. Customized genetic algorithms (GAs) and non-linear east squares (NLS) direct search methods have been used. The simulations have been carried out via the Dormand–Prince integration solver with a time step of 0.25 s on a MATLAB/SIMULINK environment.

Bizon [14] has proposed proton exchange membrane fuel dell (FC) HPSs to study optimization, switching function, and balance of DC power flow. The model has been built and simulated using MATLAB/SIMULINK. The study has shown the classification of seven FC HPSs with the batteries operated in charge-sustaining mode in order to notice the decrease in size, maintenance, and lifetime. The best design in each category has been chosen to focus on in the experiment, obtaining a 99.56% conversion of hydrogen.

Bensaber et al. [15] have proposed HPSs using PV, wind turbines, and batteries to meet load demand and control the voltage and frequency. The challenges of the proposed HPS in terms of controlling and managing the power flow have been discussed.

Avagianos et al. [16] have published a review of the solid-fuel flexible thermal power plants, discussing the effect of the penetration of non-dispatchable power plants upon wind and solar, for example, and analysing flexible operation requirements that influence their economic viability: fast start-up process, low minimum load, and ramping.

Zhang et al. [17] have proposed a fuzzy model predictive controller (MPC) for wide-range load tracking. The researchers have assembled an extended state observer for the plant behaviour and the MPC with unknown disturbances. The research has been carried out to simulate the extended state observer-based fuzzy model predictive control (ESOFMC) on a 1000 MW ultra-supercritical (USC) boiler–turbine unit. The final results represent the performance of load following over a wide range. However, despite the detailed consideration of the USC controller, no renewable penetrations have been practically considered.

Annaluru [18] has studied the flexibility of the Indian power grid and its challenges in terms of Renewable Energy (RE) with different variable generation (VG) penetration, such as solar PV and wind. The author has studied the Indian grid’s net load curve, with emphasis on three potential penetration levels for RE scenarios. Improving the plant flexibility has been presented with three alternative solutions.

Qatamin et al. [7] have proposed System Identification (SI) techniques for prediction of wind turbine output power using a generalized state-space model. The method uses subspace methods mainly based on a numerical algorithm for subspace state space system sdentification (N4SID) and prediction error method (PEM), with multi input single output (MISO) to investigate the differences in computational capabilities. It has been noticed that there is a considerable relation between model order and simulation accuracy, which motivates the study of optimal order that achieves the maximum possible accuracy. A comparison between PEM and the N4SID has been carried out in terms of the accuracy, complexity, and speed of simulation. However, the research lacks extension of LFC studies.

Ersdal et al. [19] have proposed a model predictive controller for automatic generator control (AGC) of the Nordic power system, based on a simplified system model. The study considers the limitations of different parts of the power system, such as tie-line power flow, capacity, and rate of change in generation, which lead to more flexibility and coordination between multiple inputs. The Kalman filter has been integrated with the MPC algorithm for load frequency control with state estimation. The research finally has given a comparison between MPC and the conventional (LFC/AGC) with proportional-integral (PI) controllers.

Shakibjoo et al. [20] have proposed a new load frequency control (LFC)/type-2 fuzzy control (T2FLC) with multi-areas, thermal units, wind farms, demand response (DR), and a battery energy storage system (BESS). The proposed linear model has been implemented using MATLAB software with four simulation scenarios of the 10-machine New England 39-bus test system (NETS-39b) to obtain an approximately 20% improvement.

From the literature analysis, it has been deduced that the load frequency control still needs further study that should be preferably closer to the practical sense, which can be applicable—when normalized—for conventional centralized power systems or embedded generation. The paper contributions can be then clarified as follows:

• A cleaner power system framework has been presented from verified simulators of three different generating units, which has been built with the aim of satisfying system stability and environmental requirements. These units are a clean coal supercritical unit, a dual-fuel gas turbine unit, and a wind farm.
• A Hammerstein–Wiener Model has been built to predict the wind penetration level according to weather data inputs. The typical example has given many levels of approximately 90 MW penetration peak. On the other hand, the other two flexible units are emulated with identified state-space models. The parameters of all models have been identified and verified with real sets of data, unlike the previously published literature of load frequency control studies that usually adopt standard transfer functions and form a block diagram without enough attention to parameter calibration to fit a real set of data.
• A simplified and practically feasible operation and control strategy has been suggested for heavily loaded power systems: the wind farm output should be admitted to the network from economic and environmental view-points, the remaining load signal, heavily affected by the randomness of the wind, is instructed to the MPC controllers of the flexible generators (coal and gas), and the wind farm is relieved from duty of load frequency control because this obligation is completely carried out by the flexible units and their associated controls, which in turn, reduces the storage technology requirements of the wind turbines where that storage would be needed only in case of curtailed or low load.

The paper has been organized as follows: Section 2 presents an overview on state-space and HW identification algorithms; Section 3 shows the general mechanism of MPC through its mathematical description with its application to the coal and gas units; Section 4 depicts the simulations of the verified models and of the control performance analysis; and Section 5 concludes the research work and states the future suggestions.

2. An Overview of State-Space and Hammerstein-Wiener Systems’ Identification

2.1. Theory and Basic Mechanism for State-Space Subspace Identification

In system identification, an informative set of data must be available in advance to setup the optimized parameters of a new model or calibration of existing model parameters. Once the identification phase has been finished, the identified model should be checked via a different data set in order to ensure the model validity and integrity [21]. In this paper, every set of data has been split to have 50% of the samples for identification and the other 50% of data samples for verification. The approach of identification that has been adopted in this paper is the subspace N4SID algorithm. The general following explanation is applicable to any subspace algorithm, which have been published in many references [22,23,24]. Using the following linear, discrete time-invariant state-space model (SSM):

$$x_{k+1}=A{x}_k+B{u}_k+w_k$$

(1)

$$y_k=C{x}_k+D{u}_k+v_k$$

(2)

The assumed pair $\left(C, A\right)$ is observable and the pair $\left(A, B\right)$ is controllable. Where $x_k\in R^n$ is the state vector, $u_k\in R^r$ is the input vector, $y_k\in R^m$ is the output vector, and $w_k, v_k$ are zero mean white Gaussian noise. The most used tool to be highlighted is the singular value decomposition (SVD), which is used to produce the required state-space matrices. However, some concepts and preliminaries must be first explained, as in the following subsections.

2.1.1. Inputs and Outputs

Arranged inputs and outputs data as a Hankel matrix, shown in extended data vectors and extended data matrices:

The extended data vectors, given a number $L$ refers to known output vectors and a number $L+g$ refers to known input vectors, are given as:

$${\mathrm{y}}_{k|L}\stackrel{\mathrm{def}}{=}\left[\begin{array}{c}{y}_k\\ y_{k+1}\\ ⋮\\ y_{k+L-1}\end{array}\right]\in R^{Lm}$$

(3)

$$u_{\mathrm{k}|\mathrm{L}+\mathrm{g}}\stackrel{\mathrm{def}}{=}\left[\begin{array}{c}{u}_k\\ u_{k+1}\\ ⋮\\ u_{k+L+g-2}\\ u_{k+L+g-1}\end{array}\right]\in R^{\left(L+g\right)m}$$

(4)

The extended data matrices define the following output data matrix with $L$ block rows and $K$ columns:

$$Y_{k|L}\stackrel{\mathrm{def}}{=}\left[\begin{array}{ccccc}{y}_k& y_{k+1}& y_{k+2}& \dots & y_{k+K-1}\\ y_{k+1}& y_{k+2}& y_{k+3}& \dots & y_{k+K}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ y_{k+L-1}& y_{k+L}& y_{k+L+1}& \dots & y_{k+L+K-2}\end{array}\right]\in R^{Lm\times K}$$

(5)

This is the known data matrix of output variables, and the matrix:

$$U_{k|L+g}\stackrel{\mathrm{def}}{=}\left[\begin{array}{ccccc}{u}_k& u_{k+1}& u_{k+2}& \cdots & u_{k+K-1}\\ u_{k+1}& u_{k+2}& u_{k+3}& \cdots & u_{k+K}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ u_{k+L+g-2}& u_{k+L+g-1}& u_{k+L+g}& \cdots & u_{k+L+K+g-3}\\ u_{k+L+g-1}& u_{k+L+g}& u_{k+L+g+1}& \cdots & u_{k+L+K+g-2}\end{array}\right]\in R^{\left(L+g\right)r\times K}$$

(6)

is the known data matrix of input variables.

The reversed extended observability matrix ($O_i$) are defined as the following definitions according to SSM, Equations (1) and (2). And the number of block rows is denoted by subscript $i$.

$${\mathrm{O}}_i\stackrel{\mathrm{def}}{=}\left[\begin{array}{c}C\\ CA\\ ⋮\\ C{A}^{i-1}\end{array}\right]\in R^{im\times n}$$

(7)

and a reversed extended controllability matrix ($C_i^d$) for pair $\left(\mathrm{A}, \mathrm{B}\right)$:

$$C_i^d\stackrel{\mathrm{def}}{=}\left[\begin{array}{cccc}{A}^{i-1}B& A^{i-2}B& \dots & B\end{array}\right]\in R^{n\times ir}$$

(8)

It is important to mention that there are many applications for observability and controllability matrices, such as monitoring lithium-ion battery [25]. However, the application here is somehow different as it focuses on building the matrices necessary for off-line identification of linearized discrete time models of the flexible generating units, and the decomposition of those matrices in the final stage of identification.

The triangular Toeplitz matrix ($H_i^d$): The columns result from multiplying between the extended observability matrix and the transposed extended controllability matrix.

This matrix contains the four matrices (C, A, B, D).

The triangular Toeplitz matrix ($H_i^d$).

$$H_i^d\stackrel{\mathrm{def}}{=}\left[\begin{array}{cccc}D& 0& \cdots & 0\\ CB& D& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ C{A}^{i-2}B& C{A}^{i-3}B& \cdots & D\end{array}\right] \in {ℝ}^{im\times \left(i+g-1\right)}$$

(9)

The number of block rows is denoted by subscript $i$. The block columns are denoted by $i+g-1$.

The lower block stochastic triangular Toeplitz matrix ${\mathcal{H}}_{\mathrm{i}}^{\mathrm{s}}$ for the pair (C, A).

Define the Toeplitz matrix ($H_i^s$)

$$H_i^s\stackrel{\mathrm{def}}{=}\left[\begin{array}{cccc}0& 0& \cdots & 0\\ C& 0& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ C{A}^{i-2}& C{A}^{i-3}& \cdots & 0\end{array}\right]\in {ℝ}^{im\times il}$$

(10)

Finally, $X_i$ is the state vector is denoted as:

$$X_i=\left[\begin{array}{cccc}{x}_i& x_{i+1}& \dots & x_{i+j-1}\end{array}\right]$$

(11)

The main target here is to recover the hidden state $x_k$ and if this hidden state is recovered, by using singular value decomposition (SVD), the $X_i$ and the system order $\mathrm{n}$ can be recovered and known.

The basic realization theory of subspace identification as a part of the deterministic case is explained briefly in the next subsection.

2.1.2. Realization Theory

Realization theory is discussed by defining the impulse response model: A linear state-space model $x_{x+1}=A{x}_k+B{u}_k$ and $y_k=C{x}_k+D{u}_k$ with the initial state $x_0$. The impulse response model can be given as:

$$y_k=C{A}^k{x}_0+{\displaystyle \sum }_{i=1}^{k}C{A}^{k-i}B{u}_{i-1}+D{u}_k$$

(12)

So the matrix at time instant $k-i+1$ is as follows:

$$H_{k-i+1}=C{A}^{k-i}B \in {ℝ}^{ny\times nu},$$

(13)

The impulse response matrix at time is $k-i+1$. Then, the output, $y_k$, at time $k$ is defined in terms of impulse response matrices as following: $H_1=CB$, $H_2=CAB$, and so on …., $H_k=C{A}^{k-1}B$.

The impulse responses $H_{i }\forall i=1,\dots .,L$+J., can be used to develop the Hankel matrices that contain necessary information of the system matrices.

The Hankel matrices

$$H=\left[\begin{array}{ccccc}{H}_1& H_2& H_3& \dots & H_J\\ H_2& H_3& H_4& \dots & H_{J+1}\\ H_3& H_4& H_5& \ddots & H_{J+2}\\ ⋮& ⋮& \ddots & \ddots & ⋮\\ H_{L+1}& H_{L+2}& \dots & \dots & H_{L+J}\end{array}\right] \in R^{ny\left(L+1\right)\times nu.J}$$

(14)

the submatrices $H_A$, $H_B$,$H_C$ can be then extracted [22].

$$H_{1|L}=H_n=\left[\begin{array}{ccccc}{H}_1& H_2& H_3& \dots & H_J\\ H_2& H_3& H_4& \dots & H_{J+1}\\ H_3& H_4& H_5& \ddots & H_{J+2}\\ ⋮& ⋮& \ddots & \ddots & ⋮\\ H_{\mathrm{L}+1}& H_{\mathrm{L}+2}& \dots & \dots & H_{\mathrm{L}+\mathrm{J}-1}\end{array}\right] \in R^{ny.\mathrm{L} \times \mathrm{nu}.\mathrm{J}}$$

(15)

$$H_{2|L}=H_A=\left[\begin{array}{ccccc}{H}_2& H_3& H_3& \dots & H_{J+1}\\ H_3& H_4& H_4& \dots & H_{J+2}\\ H_4& H_5& H_5& \ddots & H_{J+3}\\ ⋮& ⋮& \ddots & \ddots & ⋮\\ H_{\mathrm{L}+1}& H_{\mathrm{L}+2}& \dots & \dots & H_{\mathrm{L}+\mathrm{J}}\end{array}\right] \in R^{ny.\mathrm{L} \times \mathrm{nu}.\mathrm{J}}$$

(16)

$$H_B=\left[\begin{array}{c}{H}_1\\ H_2\\ H_3\\ ⋮\\ H_L\end{array}\right] \in R^{ny.\mathrm{L}\times \mathrm{nu}}$$

(17)

$$H_C=\left[\begin{array}{ccccc}{H}_1& H_2& H_3& \dots & H_J\end{array}\right] \in R^{ny\times \mathrm{nu}.\mathrm{J}}$$

(18)

These matrices are closely related to the observability and the controllability matrices $(O_L$, $C_J)$, respectively, which are given to facilitate of realization theory, where $O_L$ is the observability matrix defined by:

$$O_L=\left[\begin{array}{c}C\\ CA\\ C{A}^2\\ ⋮\\ C{A}^{L-1}\end{array}\right] \in R^{ny.\mathrm{L} \times \mathrm{nx}}$$

(19)

and $C_J$ is the controllability matrix defined by:

$$C_J=[\begin{array}{cccc}B& AB& \dots & A^{J-1}B] \in R^{nx\times \mathrm{nu}.\mathrm{J}}\end{array}$$

(20)

a suitable factorization of the Hankel matrix, $H_n$ = $H_{1|L}$, may be used to determine $O_L, C_J$. Then the system matrices to satisfy the following three matrix relations are chosen.

$$H_{1|L}=H_A=O_{L}A{C}_J, H_B=O_{L}B, H_C=C{C}_J$$

(21)

Conclude the system matrices as follows [22]:

$$A={\left(O_L^T{O}_L\right)}^{-1}{O}_L^T{H}_A{C}_J^T{\left(C_J{C}_J^T\right)}^{-1}$$

(22)

$$B={\left(O_L^T{O}_L\right)}^{-1}{O}_L^T{H}_B$$

(23)

$$C=H_C{C}_J^T{\left(C_J{C}_J^T\right)}^{-1}$$

(24)

The realization of the parameter of matrices is obtained from SVD.

In linear algebra, the singular value decomposition (SVD) is defined as follows [26], in an analysis of real or complex matrices. Now we can apply SVD of the finite block to Hankel matrix $H_n$.

$$H_{1|L}=H_n=O_L{C}_J=US{V}^T=U{S}_1{S}_2{V}^T$$

(25)

Results are directly obtained from the SVD when applying factorization of the Hankel matrix into the product of the observability and controllability matrices. For example, the internally balanced:

$$O_L=U{S}_1, C_J=S_2{V}^T$$

(26)

Then the system matrices are estimated as follows and defined as internally balanced:

$$\begin{gathered} A=S_1^{-T}{U}^T{H}_{A}V{S}_2^{-T} \\ B=S_1^{-T}{U}^T{H}_B \\ C=H_{C}V{S}_2^{-T} \end{gathered}$$

(27)

The different subspace identification algorithms are discussed originally in [23]. Since we have used the subspace method as tool identification, any subspace algorithm can be used to fulfil the modeling aims in the paper. N4SID has been selected to identify the models of SCPP and GT. The next section discusses the basic theory of MPC.

2.2. The Hammerstein-Wiener Model

The Hammerstein–Wiener (HW) model is introduced here as nonlinear component in multisource load frequency control schemes. The HW model in this paper is used to predict the harvested wind power from the individual wind turbine given. The inputs of temperature, air pressure, wind speed, pitch angle, and humidity, are admitted to an input nonlinearity block, to a linear transfer function, then to the output nonlinearity block to form the well-known Hammerstein–Wiener model. The rule of this component is to accurately predict the wind share, although there is no reason to prevent it from being used for control system design. However, the wind subsystem is left as open loop in the proposed model for the aforementioned reasons in the paper contributions. Therefore, several benefits of the Hammerstein–Wiener models are mentioned, such as achieving higher accuracy by simplifying the control process by allowing for the checking of each parameter separately. As stated earlier in the paper, the Hammerstein–Wiener model has been built to ensure the integrity of the input parameters of the wind model, but this shall not affect the linear MPC performance of the thermal units.

The Hammerstein–Wiener model is mostly used when the system output depends nonlinearly on its inputs. The input–output relationship, in this case, could be decomposed into two interconnected elements. The linear transfer function is used to represent the system’s dynamics, and the nonlinear functions are used to capture nonlinearities for the inputs and outputs of the linear system [27]. The standard block diagram of the Hammerstein–Wiener model component is shown in Figure 3.

Where:

• $\omega \left(t\right)=f\left(u\left(t\right)\right)$ is a nonlinear function for transforming input data $u\left(t\right)$.
• $x\left(t\right)=T\left(.\right).w\left(t\right)$ and $y\left(t\right)$ have the same dimensions, $x\left(t\right)$ has a linear relation with $w\left(t\right)$; where $w\left(t\right)$ and $x\left(t\right)$ are the input and output of the linear block, respectively, and are defined as internal variables.
• $y\left(t\right)=h\left(x\left(t\right)\right)$ is a nonlinear function that maps the output of the linear block $x\left(t\right)$.

Various scalar nonlinearity estimators are provided by MATLAB^® through the system identification toolbox. Hammerstein–Wiener models have nonlinearity estimators for both input nonlinearity $f$ and output nonlinearity $h$. The nonlinearity estimators could be configured as a saturation, dead zone, unit gain, wavelet network, sigmoid network, piecewise linear function, or other options. In this paper, the piecewise functions for input and output nonlinearities are configured. Finally, an iterative search algorithm has been used with appropriate settings to estimate the HW model of the wind turbine. The individual turbine output is then augmented through a linear gain to formulate the effect of a huge wind farm of 190 turbines.

3. The General Mechanism of Model Predictive Control and Its Application to the Flexible Units

The model predictive control (MPC) is considered as one of the well-recognized control technologies for industrial process applications, especially in thermal power generation plants, used to calculate the predicted plant output after simulation and to control the output of SCPP. There are two reasons to utilize these advanced control techniques. Firstly, there are several variety control topologies in the MPC model. Secondly, there is the ability to use wide operational constraints in the MPC model to ensure the safety and reliability of the power plant [28].

The action of the MPC can be summarized as follows:

• Predict the output trends.
• Perform optimization to reduce the errors.
• Calculate the control shifts or changes in inputs, then execute only the first sample of control inputs.
• Repeat these steps every sampling time.

In the section “Control Strategy Testing with MPC”, a validated state-space mathematical model was used for the 600-MW cleaner coal power plant and 250-MW gas turbine unit, which represents the production of a supercritical coal-fired power plant and gas power plant. The linear state-space model was identified as the control strategy. The discrete-time state-space model could be used for prediction as follows [29]:

3.1. Prediction of State and Output Variables

The future control trajectory (predicted niput) is represented by:

Assume the initial condition $u$ equal zero.

$$\mathrm{\Delta }u\left(k_i\right), \mathrm{\Delta }u\left(k_i+1\right),\dots , \mathrm{\Delta }u\left(k_i+N_C-1\right).$$

(28)

The future state variables are represented by:

$$x\left(k_i+1| k_i\right),x\left(k_i+2| k_i\right),\dots , x\left(k_i+m| k_i\right),\dots ,x\left(k_i+N_p| k_i\right)$$

(29)

where $k_i>0$ is the sampling instant, $N_C$ is the control horizon, and $N_p$ is the prediction horizon. Also referring to the length of the optimization, $N_C$ should be less than $N_p$. Prediction horizon is the length of time for the predicted future output, whereas the control horizon represents the length of time for the optimized trajectory of the manipulated inputs.

The future state variables (means future output), based on the state-space model $\left(A, B, C\right),$ are represented as:

$$\begin{gathered} x\left(k_i+1| k_i\right)=Ax\left(k_i\right)+B\mathrm{\Delta }u\left(k_i\right) \\ x\left(k_i+2| k_i\right)=Ax\left(k_i+1| k_i\right)+B\mathrm{\Delta }u\left(k_i+1\right)=A^{2}x\left(k_i\right)+AB\mathrm{\Delta }u\left(k_i\right)+B\mathrm{\Delta }u\left(k_i+1\right) \\ ⋮ \\ x\left(k_i+N_p| k_i\right)=A^{N_p}x\left(k_i\right)+A^{N_p-1}B\mathrm{\Delta }u\left(k_i\right)+A^{N_p-2}B\mathrm{\Delta }u\left(k_i+1\right) \\ +\dots +A^{N_p-N_C}B\mathrm{\Delta }u\left(k_i+N_C-1\right). \end{gathered}$$

Predict output variables (power) from predicted state variables directly after assume $D=0$:

$$\begin{gathered} y\left(k_i+1| k_i\right)=CAx\left(k_i\right)+CB\mathrm{\Delta }u\left(k_i\right) \\ y\left(k_i+2| k_i\right)=C{A}^{2}x\left(k_i\right)+CAB\mathrm{\Delta }u\left(k_i\right)+CB\mathrm{\Delta }u\left(k_i+1\right) \\ ⋮ \\ y\left(k_i+N_p| k_i\right)=C{A}^{N_p}x\left(k_i\right)+C{A}^{N_p-1}B\mathrm{\Delta }u\left(k_i\right)+C{A}^{N_p-2}B\mathrm{\Delta }u\left(k_i+1\right) \\ +\dots +C{A}^{N_p-N_C}B\mathrm{\Delta }u\left(k_i+N_C-1\right). \end{gathered}$$

Then arrange input and output in vectors:

$$Y={\left[y\left(k_i+1| k_i\right) y\left(k_i+2| k_i\right)y\left(k_i+3| k_i\right)\dots y\left(k_i+N_p| k_i\right)\right]}^T$$

(30)

$$U={\left[\mathrm{\Delta }u\left(k_i\right) \mathrm{\Delta }u\left(k_i+1\right) \mathrm{\Delta }u\left(k_i+2\right) \dots \mathrm{\Delta }u\left(k_i+N_C-1\right)\right]}^T$$

(31)

Collect the two vectors $Y$ and $U$ together in a compact matrix form as:

$$Y=Fx\left(k_i\right)+\phi \mathrm{\Delta }U$$

(32)

where:

$$\mathrm{F}=\left[\begin{array}{c}CA\\ C{A}^2\\ C{A}^3\\ ⋮\\ C{A}^{N_p}\end{array}\right]; \phi =\left(\begin{array}{ccccc}CB& 0& 0& \cdots & 0\\ CAB& CB& 0& \cdots & 0\\ C{A}^2& CAB& CB& \cdots & 0\\ ⋮& ⋮& ⋮& \cdots & ⋮\\ C{A}^{N_{p-1}}B& C{A}^{N_{p-2}}B& C{A}^{N_{p-3}}B& \cdots & C{A}^{N_p-N_c}B\end{array}\right)$$

(33)

3.2. Optimization of Control Signals

The very basic case is to find the first derivative of the cost function with respect to $\mathrm{\Delta }U$ and obtain the vector sequence $\mathrm{\Delta }U$ as the future control law.

• ∙ Set data vector that contains the set-point:

$$R_s^T=\left[1 1 \dots 1\right]r\left(k_i\right)$$

(34)

Define cost function $J$ as following:

$$J={\left(R_s-Y\right)}^T\left(R_s-Y\right)+\mathrm{\Delta }{U}^T\overline{R}\mathrm{\Delta }U$$

(35)

where $\overline{R}$ is a diagonal matrix and equal $r_w$ $I_{N_{c\times }{N}_c}$; $r_w$ is used as a tuning parameter for the MPC, and they mainly affect the performance of the controller and computation time demands, the aim would be solely to make the error as small as possible.

• Find the optimal $\mathrm{\Delta }U$ that will minimize $J$, write $J$ as following:

  $$J={\left(R_s-Fx\left(k_i\right)\right)}^T\left(R_s-Fx\left(k_i\right)\right)-2\mathrm{\Delta }{U}^T{\phi }^T\left(R_s-Fx\left(k_i\right)\right)+\mathrm{\Delta }{U}^T\left({\phi }^T\phi +\overline{R}\right)\mathrm{\Delta }U$$
• Take the first derivative of the $J$ and to obtain the minimum $J$ put condition $\frac{\partial J}{\partial \mathrm{\Delta }U}$ equal to zero:

  $$\frac{\partial J}{\partial \mathrm{\Delta }U}=-2{\phi }^T\left(R_s-Fx\left(k_i\right)\right)+2\left({\phi }^T\phi +\overline{R}\right)\mathrm{\Delta }U$$
• The optimal solution for the control signal:

$$\mathrm{\Delta }U={\left({\phi }^T\phi +\overline{R}\right)}^{-1}{\phi }^T\left(R_s-Fx\left(k_i\right)\right)$$

(36)

Applying the aforementioned concepts to the hybrid system, the advancement in control strategy with the MPC leads the system for the two generating units to be as shown in Figure 4.

The inputs and outputs are adopted with regard to the available operational data and previously published research that indicates those inputs are the most influential variables on the plant performance and characteristics. There are many parameters that affect the performance of the MPC. One of the most effective parameters is the prediction horizon (${\mathrm{H}}_{\mathrm{p}}$); in this model, the prediction horizon is reduced to 8 samples for the coal power plant and 6 samples for the gas turbine (GT), which achieves the best performance results. The load demand signal has been assumed to have a sudden increase followed by a sudden decrease in load demand. The choice of these parameters values has to be decided on the basis of many simulation scenarios [30].

The main objective of the advancement control strategy is to integrate 600-MW of clean coal supercritical (SC) power plant and 250-MW of dual-fuel gas turbine (GT) units into the grid to develop a feasible controller for the coal and gas unit for the load following capability, with integration of considerable wind energy penetration. The global model will be used to design predictive controllers for the clean coal supercritical (SC) power plant and dual-fuel gas turbine (GT) units, which allow different levels of wind energy penetration and cover the remaining amount of load that varies over a 12 h time window.

In summary of this section, the MPC algorithm computes the plant inputs so that the plant output follows the adopted reference signal. It utilizes an internal plant model to forecast the future production, uses optimization so that the plant’s output tracks the instructed reference, and calculates many steps in the optimum sequence. However, the MPC applies the first sample and ignores the others. Then, the prediction horizon is shifted forward one-time step, and the process of computation should be then repeated. The following section discusses the simulation results for testing the applied MPC strategy.

4. Simulation Results

4.1. Modeling Identification Results

For successful control system implementation, verified models must be available at earlier stage. This sub-section briefly reports the models’ features for the three resources under study, which are the wind, the coal, and the gas power generators, respectively. Because the wind resources embed higher variability than coal and gas, it is preferable to be simulated by nonlinear system identification, whereas the coal and gas tend to be more deterministic and could be simply described by linear system identification. Another reason that supports this argument is that the classical MPC needs linear models to act as an internal model for prediction of the MPC application of the coal and gas units. In this paper, the wind generator should be integrated into the grid such that it freely injects the harvested power to the network to cover the highest possible portion of the load with free energy. Therefore, the LFC should be the duty of the coal and gas units to attain the required balance. The designed MPCs have been applied to the coal and gas units for the purpose of LFC [31,32].

Regarding the wind model, because the wind is left uncontrollable, any model structure, no matter how overcomplicated, could be chosen in order to capture the practical dynamics with reasonable accuracy. The Hammerstein-Wiener (HW) model structure has been selected for the prediction of the harvested power of the wind farm. The HW model for the wind turbine generator system has been tuned and the parameters obtained are mentioned in

Table 1. The HW model description for the wind generator.

Input Nonlinearity Function: Type and Features	The Transfer Function: No, of Poles and Zeros	The Output Nonlinearity Function: Type and Feature
Piecewise function with 30 break-points for all inputs.	Data No. of poles = 13 No. of zeros = 2 For all Functions	Piecewise function with 30 break-points for all inputs.

.

The implemented models of the flexible generators are two multi-input single-output (MISO) discrete state-space mathematical models. The first one is for SCPP and has four inputs, U1, U2, U3, and U4: the air flow (AF), fuel flow (FF), feed water flow (FW), and digital electro-hydraulic (DEH) governor reference, respectively. It has one output, y, the SCPP output power. In addition, the SSM for GT has three inputs, Ug1, Ug2, and Ug3, for the natural gas valve, the pilot valve, and the compression ratio, respectively, and one output y, which is the GT output power. The parameters of the matrices A, B, C, and D have been identified to be as follows:

The parameters of the matrices for clean coal SCPP are as follows:

$$\mathrm{A}=\left[\begin{array}{cccccccc}0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 1\\ -0.2513& 1.182& -2.696& 4.288& -5.5& 5.681& -4.813& 3.108\end{array}\right]$$

$$B=\left[\begin{array}{cccc}0.2286& 0.324& -0.02447& -0.08589\\ -0.06089& 0.06483& 0.09362& 0.006508\\ -0.01787& -0.3207& 0.1021& 0.02841\\ 0.02413& 0.3035& -0.05365& -0.02026\\ -0.03907& 0.1484& -0.1002& 0.004572\\ -0.01455& -0.2376& 0.03089& 0.01863\\ 0.00596& 0.2219& 0.07073& -0.01692\\ -0.04258& 0.2435& -0.01026& 0.000967\end{array}\right]$$

$$C=\left[\begin{array}{cccccccc}1& 0& 0& 0& 0& 0& 0& 0\end{array}\right]$$

$$D=\left[\begin{array}{cccc}0& 0& 0& 0\end{array}\right]$$

Also, the parameters of the matrices for GT are as follows:

$$A=\left[\begin{array}{cccc}-0.01395& -0.8446& 0.4796& 0.07193\\ 0.6643& 0.4801& 0.1731& -0.0450\\ -0.06218& 0.2890& 0.4074& -0.8663\\ 0.05671& 0.07252& 0.1488& 0.4045\end{array}\right]$$

$$B=\left[\begin{array}{ccc}0.2365& 0.3046& -0.1499\\ 0.2378& 0.1942& -0.7624\\ -0.169& -0.03077& -0.4752\\ 0.05271& 0.04153& -0.1551\end{array}\right]$$

$$C=\left[\begin{array}{cccc}-10.8& -35.95& -25.41& 15.54\end{array}\right]$$

$$D=\left[\begin{array}{ccc}0& 0& 0\end{array}\right]$$

Figure 5 shows the performance of an identified model over 24 h of operation of the individual wind turbine generator system. The HW model of the wind energy conversion system follows the main dynamical trends of the real system, and the performance of the model has been quantified to be 80.88% accurate. The system has been augmented to be a typical wind farm by multiplying this output by the typical number of turbines in wind farms.

The linear models of the SCPP and GT have been identified using subspace identification that has been described in Section 2.1. The simulations have been depicted in Figure 6 and Figure 7. Fourth order models for both processes have reasonably represented the actual behavior of the SCPP and GT, with an accuracy for each model of 92.19% and 82.31%, respectively.

Every data set has been split to be 50% for identification and 50% for verification. The model quality has been ensured for the three sets of data as shown above, in which the first half of the samples are used for identification and the second half are used for verification. However, one must make sure that the overall sample time is unique for the whole system, as the gathered data samples differ from one system to another, which results in unifying the sample time of 5 s for every sample in the LFC simulator.

4.2. Controller Decisions for a Practical Wind Penetration

The proposed control strategy for the 600-MW clean coal supercritical (SC) power plant and 250-MW dual-fuel gas turbine (GT) units has been simulated to enhance the efficiency and reduce the negative effects on the frequency. This necessary for all stability classes (dynamic and steady-state stability) which allow different levels of wind energy penetration and cover the remaining amount of load.

The reason for developing a linear MPC strategy for both a coal and gas power plant, is to regulate them with focus on operational, safety and emissions constraints. In addition, the aim was to design a feasible controller for the coal and gas unit for the load following capability, with focus on prediction to obtain better results. Linear MPC provides a reasonable computational demand compared to the other nonlinear MPC. For this reason, the temperature, pressure, and humidity are not involved as model outputs. However, temperature, air pressure, wind speed, pitch angle and humidity are ensured by developing a Hammerstein–Wiener model, which is used as a safety detector for these five parameters.

The model has been developed and simulated in MATLAB Simulink on a PC environment with the assumed demand signal changes in a step from 640 MW to 600 MW (see Figure 8). Simulation results of Hammerstein–Wiener model for wind farm is reported and shows the maximum output of around $90 \mathrm{MW}$ as follows in Figure 9. The advancement control strategy with MPC is simulated, and the results are reported in Figures below.

The Hammerstein–Wiener model has been built to control five parameters as an input to get single output with different penetration. This output is used as a reference for coal and gas units. Figure 10 and Figure 11, respectively, show that the manipulated inputs of the plant, the air flow (AF), fuel flow (FF), feed water flow (FW), and digital electro-hydraulic (DEH) governor signal are maintained within their constraints, are restricted by the 600 MW SCPP. Figure 11 shows the output power at the maximum of approximately $440 \mathrm{MW}$.

As stated earlier in the paper, the Hammerstein–Wiener model has been built to ensure the integrity of the input parameters of the wind model, but this shall not affect the linear MPC performance of the thermal units for two reasons. Firstly, the linear MPC is preferable in practice due to its simplicity and lower computational burdens. Secondly, because the performance is tested on linear models for the thermal plants, for practical implementations, any glitches can be compensated by the testing of the parameters and calibration. After a wide range of experiments and trials, the parameters of the Hammerstein–Wiener models have been determined and selected with the most satisfactory results.

Figure 12 and Figure 13, respectively, show the manipulated inputs and the output shared by the GT plant, where the inputs are the pilot valve, natural gas valve (NGCTRL), and the compression ratio (COMPPO), are maintained within their constraints, those restricted by the 250-MW dual-fuel gas turbine (GT).

The next three figures show the different scenarios of total generation. The total generation for wind farm and clean thermal generation units is shown in Figure 14, the total generation compared with load following capability shown in Figure 15, and the total imbalance represented in Figure 16.

In order to enhance the efficiency and reduce the negative effects on the frequency, Figure 17 shows the actual frequency, approximately equal 50 $\mathrm{Hz}$.

It can be deduced that the feasible penetration of the wind energy results in considerable reductions in the fuel consumption of coal in the SCPP and the combined signals of fuel oil and natural gas injected to the GT. The water consumption is also reduced in the SCPP, with those indictors ensuring the economic viability of wind energy integration. In addition, there are enhancements to the environmental conditions, especially when it is combined with cleaner thermal units such as those proposed in this paper; these are the modern SCPP cleaner coal power plant and the GT, as a part of the combined-cycle energy-efficient unit. With a maximum penetration of approximately 90 MW, the total production follows the total demand signal with acceptable imbalance of around 6% in the load transition interval and 3% in the load steady interval. This can be investigated in an alternative way, meaning the acceptable imbalance cannot be standardized because it depends on some factors, such as the system strength or inertia, which is different from one grid to another. Here, we have considered three second s inertia and nine second inertia in order to show the effect of the domination of rotating machines in the power system in general, whether for generating or motoring/condensing actions. The frequency excursions for the worst-case inertia has been typically acceptable, and hence the model predictive load frequency control is a promising technology for enhancing the power system behavior in response to sudden load changes or load small disturbances.

5. Conclusions

This paper presents a feasible control strategy for clean thermal generation units, which are wind energy, clean coal SC power plant, and dual-fuel gas turbine (GT). This study has been based on a Hammerstein–Wiener Model for wind plants, the theory of subspace system identification, and MPC strategy for both coal and gas power plants. The practical advantages of the proposed strategy, in terms of stable and efficient operation, have been confirmed using validated simulations by analyzing the results of this study with emphasis on efficiency, frequency, and output shared power, regulated by the MPC algorithm. In addition, this study can be considered a general guideline for preliminary research into a further improvement in power system frequency responses to different levels of penetration from the wind plant [33].

In general, linear MPC is known to be preferred in the industry over non-linear, because of appropriate results and reasonable computational burdens. However, there is a subject that can be resolved in future research related to the computational burdens of the nonlinear MPC, due to its theoretical complication in comparison with linear MPC, to facilitate the application of nonlinear MPC in practical power plants. This can be achieved by suggesting more advanced optimization techniques. However, to use linear MPC in the future, the MPC algorithm can be modified with the extension of measured and unmeasured disturbances, and the linear specific model can be augmented by noise terms representing the practical noises that happen in power plants.

As a future recommendation, adaptive model predictive control could be a feasible suggestion for further improvements. The explicit MPC is not recommended to be applied to such studies as it exhibits longer computation time, which leads to an unsatisfactory online performance. Therefore, an adaptive MPC or multiple MPC are highly recommended, instead of the explicit MPC, to be a future point of research for further enhancements. In terms of modeling, it is suggested that the state-space linearized models are replaced with more accurate paradigms, such as nonlinear models, which are either derived from system physics, nonlinear system identification, or machine learning modeling. Further closer models to reality can be created through the inclusion of the tie lines between units, which form the multisource interconnected power system.