# An Efficient Design of Adaptive Model Predictive Controller for Load Frequency Control in Hybrid Power System

Authors: Muhammad Majid Gulzar; Daud Sibtain; Arslan Ahmad; Imran Javed; Sadia Murawwat; Imran Rasool; Aamir Hayat
Journal: International Transactions on Electrical Energy Systems
 (2022)

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## Abstract

The technology has proceeded so much that the power system should be substantial and explicit to give optimal results. Ever-increasing complexities of the power system and load disparity cause frequency fluctuations leading to efficiency degradation of the power system. In order to give a suitable real power output, the system entails an extremely perceptive control technique. Consequently, an advanced control method, that is, an adaptive model predictive controller (AMPC), is suggested for load frequency control (LFC) of the series power system which comprises photovoltaic (PV), wind, and thermal power. The suggested method is considered to enhance the power system execution as well as to decrease the oscillations due to a discrepancy in the system parameters and load disturbance under a multi-area power system network. The AMPC design verifies the constant frequency by maintaining a minimum steady state error under varying load conditions. The proposed control approach pledge that the steady-state error of frequencies and interchange of tie line powers is maintained in a given tolerance constraint. The effectiveness of the proposed controller is scrutinized by conventional controllers like genetic algorithm-tuned PI (GA-PI), firefly algorithm-tuned PI (FA-PI), and model predictive controller (MPC) to show the competence of the proposed method.


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## Body

## 1. Introduction

### 1.1. Literature Review

In the past decade, conventional energy resources were utilized to fulfill the demand of energy requirements. Unfortunately, the natural energy resources like coal, gas, oil, and so on are decreasing and there is a critical need for some other measures to compete with the necessity of increasing energy demands. This substantial situation induces the use of renewable resources, for instance PV, wind, hydro, and so on, which became the popular topic of research during the past few years. As renewable resources play an indispensable role in attaining the goal of energy demand, around18% of energy need is currently being satisfied through these resources worldwide [1]. Renewable resources are sustainable and will never run out. These energy resources require less maintenance, and their cost of operation is also reduced because they depend on natural and available resources like sun, wind, heat, and so on. Renewable resources produce one-fifth of the world’s electric power in accordance with the Renewables Global Status Report (GSR) [2].The concatenation of nonconventional sources with conventional sources is complex. In order to meet certain requirements of current and voltage for the balanced power system, different techniques are adopted in the renewable resources to minimize the disturbances caused by them during grid connection [3]. Controlling the frequency deviation within a specified limit is necessary for the operation of a balanced power system, and for these, different techniques and controllers have been used. The penetration of renewable energy resources into grid power systems causes frequency deterioration. Many factors like load perturbation, faults, expeditious change in load and operational unreliability, and so on deteriorate the implementation of the electrical power system [4].Power grids are composed of a number of different interconnected areas that are connected to each other via a transmission line (tie line). In a single area, thermal generator is usually affixed to a set of renewable energy sources to satisfy its own energy requirements and control power switching within adjoining areas. However, the changes occurring in the generation and load side lead to the deviation of voltage and frequency from its limits [5]. The infusion of renewable resources in this research article consists of solar PV, wind turbine, and thermal energy generators.PV system is rapidly growing technology and is the most abundant renewable source due to its numerous advantages like no release of harmful air, eco-friendly, and locally available renewable resources [6, 7]. The incorporation of renewable energy into the power grid gives rise to perturbation in system frequency. The speed of wind is wavering for the whole year in particular places which results in the disparity of output power and frequency of wind generators. The photovoltaic output is dependent upon the climatic condition that varies with the change of climate. So, it is hard to manage the frequency of the hybrid generating system without any latest control method because of the unpredictable nature of renewable sources. The grid power may be affected by the outcome of renewable energy sources, thus originating a serious issue from the frequency deviation of the grid. Also, an unexpected change of load demands precipitates for power imbalance and frequency fluctuations. The two-area PV-wind connected thermal grid system is depicted in Figure 1.Figure 1
Two-area hybrid grid connected system.A reliable and working power system should be balanced irrespective of inconsistent load and system disruption. In wind power generation under uncertainty of load requirements and arbitrary input of wind power, the pitch controller will no longer be able to control the frequency of the system constructively due to their inactive response [8]. Therefore, it is essential that the power system frequency is unchanged during any discrepancy in the load demand. That is why the significance of LFC in the design and working of the electric power system has come into consideration. The core function of LFC is to regulate the frequency of each area within limits and to make the flow of the tie line power suitable during the changing load requirements [9]. To counter the challenge of LFC various control methods are studied, like modern or advanced approach with linearity and nonlinearity approaches are analyzed [9].To counter the challenge of LFC, various control methods are studied, like modern or advanced approach with linearity and nonlinearity approaches are analyzed [10,11]. LFC plays a very important role in the working and designing of a system. Due to uncertain and unpredictable changes in load, the system frequency gets affected which is not required. However, by using the LFC method either by escalating the capacity of generation or by reducing the frequency fluctuations, this problem can be resolved by keeping the values within certain limits. This phenomenon is known as load frequency control [12]. An appropriately working power system should resist the load perturbation and must provide productive power quality by maintaining both frequency fluctuation and production capacity within normal limits. Research work, for instance [13, 14], shows fundamental information about different frequency regulations of an interconnected power system. Because of the complication of the modern power system, these frequency oscillations when open to any unwanted interference may expand to a wide interlinked system of the various control area. It originates unwanted interference that causes uncertainty which will result in the blackout of the complete power system [15–17].Different controllers are proposed in the literature to mitigate LFC problems, such as artificial neural network (ANN) [18], hybrid artificial neural network [19] to track maximum power point tracking, fuzzy approach [20], cascade fuzzy-proportional integral derivative incorporating filter (PIDN)-fractional order PIDN (FPIDN-FOPIDN) [21], and wavelet neural network with data filtering [22] to escalate the execution of LFC methodology. Classical control techniques are used to overcome the LFC issue. In [23], a proportional integral (PI) controller and dual-mode PI controller were used for the design of frequency changing problems with communication delays, and it also discussed the hydropower system with the use of PI control. PI with MPC was used in [24] to attain better closed-loop results. Similarly, a lot of work was carried out with integral derivative (ID), and the cascaded design was proposed in [25], where (1 + PD)-PID was formulated to counter frequency oscillation under uncertain conditions. Similarly, cascade-fractional order ID with filter (C − IλDμN) was proposed in [26]. In [27], the authors highlighted the idea about scheduling the integral gains using GA-based fuzzy control algorithm. In [28], fuzzy logic-based LFC regarding high penetration of wind turbines was presented. In [29], sliding mode control (SMC) was proposed, JAYA optimization was proposed in [30], and a PSO-based sliding mode controller was discussed in [31]. In [32], a MPC-based PV connected thermal two-area system was presented for the LFC problem. Similarly, in [33], a MPC-based thermal-thermal connected system for the LFC problem was presented. The flowchart of the survey conducted on LFC depending on the controller and area is shown in Figure 2.Figure 2
Review on LFC.Figure2 presents the fundamental execution of various power system models and includes the load forecasting technique in LFC. Here, the classification of LFC is studied like soft technique and other various algorithms. Moreover, it also illustrates the application of different FACTS devices, energy storage systems (ESSs), and HVDC links incorporation with LFC [13].


### 1.2. Research Gap and Motivation

After considering the detailed literature, the survey has identified void sections that need to be dealt with the proposed methodology. The research gap and motivation are as follows:(i)
The literature survey sheds light on the studies performed by the researchers in the field of LFC, and they have presented exclusive ideas to deal with frequency in order to maintain stable operation of the power system under the load disturbance.(ii)
An advanced controller in the form of adaptive model predictive controller (AMPC) is proposed for multi-area sets to be a unique control technique having the ability to counter under uncertainty and controlling attenuation due to external disturbance.


### 1.3. Contribution and Paper Organization

The motivation has paved the way towards the following contributions and novelties:(i)
In this paper, the multi-area PV, wind connected thermal systems are developed, where the importance of penetration level of renewable sources and its impacts on the system has been kept in consideration during designing.(ii)
Designing of the PV, wind, and thermal system is carried out where doubly fed induction generator (DFIG)-based wind turbine model is designed, where the pitch actuator is controlled by MPC.(iii)
Adaptive model predictive controller (AMPC) is presented to control the LFC problem in a multi-area power system to mitigate frequency fluctuations.(iv)
AMPC is designed in such a manner that it fulfills two requirements ensuring the LFC requirement under uncertainty and controlling attenuation due to external disturbance. The scaling factor is introduced in the output which is adjusted in each sampling time by an adaptive mechanism.This paper is organized as follows. Section2 establishes the system modeling of PV, wind, and thermal power systems. Section 3 represents the overall model of a hybrid power system. In Section 4, the controller based on AMPC is proposed. Section 5 is regarding the simulations and results analysis of the proposed controller. Finally, Section 6 concludes the paper.


## 1.1. Literature Review

In the past decade, conventional energy resources were utilized to fulfill the demand of energy requirements. Unfortunately, the natural energy resources like coal, gas, oil, and so on are decreasing and there is a critical need for some other measures to compete with the necessity of increasing energy demands. This substantial situation induces the use of renewable resources, for instance PV, wind, hydro, and so on, which became the popular topic of research during the past few years. As renewable resources play an indispensable role in attaining the goal of energy demand, around18% of energy need is currently being satisfied through these resources worldwide [1]. Renewable resources are sustainable and will never run out. These energy resources require less maintenance, and their cost of operation is also reduced because they depend on natural and available resources like sun, wind, heat, and so on. Renewable resources produce one-fifth of the world’s electric power in accordance with the Renewables Global Status Report (GSR) [2].The concatenation of nonconventional sources with conventional sources is complex. In order to meet certain requirements of current and voltage for the balanced power system, different techniques are adopted in the renewable resources to minimize the disturbances caused by them during grid connection [3]. Controlling the frequency deviation within a specified limit is necessary for the operation of a balanced power system, and for these, different techniques and controllers have been used. The penetration of renewable energy resources into grid power systems causes frequency deterioration. Many factors like load perturbation, faults, expeditious change in load and operational unreliability, and so on deteriorate the implementation of the electrical power system [4].Power grids are composed of a number of different interconnected areas that are connected to each other via a transmission line (tie line). In a single area, thermal generator is usually affixed to a set of renewable energy sources to satisfy its own energy requirements and control power switching within adjoining areas. However, the changes occurring in the generation and load side lead to the deviation of voltage and frequency from its limits [5]. The infusion of renewable resources in this research article consists of solar PV, wind turbine, and thermal energy generators.PV system is rapidly growing technology and is the most abundant renewable source due to its numerous advantages like no release of harmful air, eco-friendly, and locally available renewable resources [6, 7]. The incorporation of renewable energy into the power grid gives rise to perturbation in system frequency. The speed of wind is wavering for the whole year in particular places which results in the disparity of output power and frequency of wind generators. The photovoltaic output is dependent upon the climatic condition that varies with the change of climate. So, it is hard to manage the frequency of the hybrid generating system without any latest control method because of the unpredictable nature of renewable sources. The grid power may be affected by the outcome of renewable energy sources, thus originating a serious issue from the frequency deviation of the grid. Also, an unexpected change of load demands precipitates for power imbalance and frequency fluctuations. The two-area PV-wind connected thermal grid system is depicted in Figure 1.Figure 1
Two-area hybrid grid connected system.A reliable and working power system should be balanced irrespective of inconsistent load and system disruption. In wind power generation under uncertainty of load requirements and arbitrary input of wind power, the pitch controller will no longer be able to control the frequency of the system constructively due to their inactive response [8]. Therefore, it is essential that the power system frequency is unchanged during any discrepancy in the load demand. That is why the significance of LFC in the design and working of the electric power system has come into consideration. The core function of LFC is to regulate the frequency of each area within limits and to make the flow of the tie line power suitable during the changing load requirements [9]. To counter the challenge of LFC various control methods are studied, like modern or advanced approach with linearity and nonlinearity approaches are analyzed [9].To counter the challenge of LFC, various control methods are studied, like modern or advanced approach with linearity and nonlinearity approaches are analyzed [10,11]. LFC plays a very important role in the working and designing of a system. Due to uncertain and unpredictable changes in load, the system frequency gets affected which is not required. However, by using the LFC method either by escalating the capacity of generation or by reducing the frequency fluctuations, this problem can be resolved by keeping the values within certain limits. This phenomenon is known as load frequency control [12]. An appropriately working power system should resist the load perturbation and must provide productive power quality by maintaining both frequency fluctuation and production capacity within normal limits. Research work, for instance [13, 14], shows fundamental information about different frequency regulations of an interconnected power system. Because of the complication of the modern power system, these frequency oscillations when open to any unwanted interference may expand to a wide interlinked system of the various control area. It originates unwanted interference that causes uncertainty which will result in the blackout of the complete power system [15–17].Different controllers are proposed in the literature to mitigate LFC problems, such as artificial neural network (ANN) [18], hybrid artificial neural network [19] to track maximum power point tracking, fuzzy approach [20], cascade fuzzy-proportional integral derivative incorporating filter (PIDN)-fractional order PIDN (FPIDN-FOPIDN) [21], and wavelet neural network with data filtering [22] to escalate the execution of LFC methodology. Classical control techniques are used to overcome the LFC issue. In [23], a proportional integral (PI) controller and dual-mode PI controller were used for the design of frequency changing problems with communication delays, and it also discussed the hydropower system with the use of PI control. PI with MPC was used in [24] to attain better closed-loop results. Similarly, a lot of work was carried out with integral derivative (ID), and the cascaded design was proposed in [25], where (1 + PD)-PID was formulated to counter frequency oscillation under uncertain conditions. Similarly, cascade-fractional order ID with filter (C − IλDμN) was proposed in [26]. In [27], the authors highlighted the idea about scheduling the integral gains using GA-based fuzzy control algorithm. In [28], fuzzy logic-based LFC regarding high penetration of wind turbines was presented. In [29], sliding mode control (SMC) was proposed, JAYA optimization was proposed in [30], and a PSO-based sliding mode controller was discussed in [31]. In [32], a MPC-based PV connected thermal two-area system was presented for the LFC problem. Similarly, in [33], a MPC-based thermal-thermal connected system for the LFC problem was presented. The flowchart of the survey conducted on LFC depending on the controller and area is shown in Figure 2.Figure 2
Review on LFC.Figure2 presents the fundamental execution of various power system models and includes the load forecasting technique in LFC. Here, the classification of LFC is studied like soft technique and other various algorithms. Moreover, it also illustrates the application of different FACTS devices, energy storage systems (ESSs), and HVDC links incorporation with LFC [13].


## 1.2. Research Gap and Motivation

After considering the detailed literature, the survey has identified void sections that need to be dealt with the proposed methodology. The research gap and motivation are as follows:(i)
The literature survey sheds light on the studies performed by the researchers in the field of LFC, and they have presented exclusive ideas to deal with frequency in order to maintain stable operation of the power system under the load disturbance.(ii)
An advanced controller in the form of adaptive model predictive controller (AMPC) is proposed for multi-area sets to be a unique control technique having the ability to counter under uncertainty and controlling attenuation due to external disturbance.


## 1.3. Contribution and Paper Organization

The motivation has paved the way towards the following contributions and novelties:(i)
In this paper, the multi-area PV, wind connected thermal systems are developed, where the importance of penetration level of renewable sources and its impacts on the system has been kept in consideration during designing.(ii)
Designing of the PV, wind, and thermal system is carried out where doubly fed induction generator (DFIG)-based wind turbine model is designed, where the pitch actuator is controlled by MPC.(iii)
Adaptive model predictive controller (AMPC) is presented to control the LFC problem in a multi-area power system to mitigate frequency fluctuations.(iv)
AMPC is designed in such a manner that it fulfills two requirements ensuring the LFC requirement under uncertainty and controlling attenuation due to external disturbance. The scaling factor is introduced in the output which is adjusted in each sampling time by an adaptive mechanism.This paper is organized as follows. Section2 establishes the system modeling of PV, wind, and thermal power systems. Section 3 represents the overall model of a hybrid power system. In Section 4, the controller based on AMPC is proposed. Section 5 is regarding the simulations and results analysis of the proposed controller. Finally, Section 6 concludes the paper.


## 2. Thermal Generator Prototype

The thermal generator prototype comprises a generator, governor, reheater, steam turbine, and droop. Governor is utilized to handle the speed of the turbine that is based on the error, which is the difference of power and frequency (∆P − ∆f). Moreover, the governor manages the mechanical motion of the turbine by supervising the working of the valve and by recovering the lost energy subsequently going through reheater, which supports to attain the mandatory goal. The speed of prime mover is controlled with the help of droop, which is attached to the generator. The generator is the last unit in the thermal power system, which transforms the mechanical motion into electricity, which is our fundamental demand [34]. The thermal power plant is modeled for the capacity of 4000MW operating at 2200MW at 50HZ. The speed governor equation has two inputs, the reference power ΔPref and change in frequency Δf and governor output ΔPgs is given by:(1)ΔPgs=ΔPrefs−1RΔfs,where 1/R represents the droop which acts like a valve to control the flow.Furthermore, in the two-area case, a bias factor is interconnected with a tie line, which specifies the quantity of power transferred within two areas which is described in the following equation:(2)ΔPij=TijΔfi−Δfj,i≠j.For theith area, the area central error (ACE) and tie line power variation are stated as(3)ACEi=BiΔfi+ΔTieij,i≠j,where Bi is the frequency bias factor parameter.The output of the generator isΔfi which has two inputs ΔPRs and ΔPds, where ΔPds is the load disturbance in the system and the output Δfi is given in the following equation:(4)Δfis=Gps∑j=1nΔPRij−ΔPd,i−ΔPtie,i,where Gps=1/Mis+Di, and output of the generator is dependent upon the disturbance in the load. The generator will always adjust its output power to meet the power demand.


## 3. Photovoltaic Prototype

The most extensively utilized renewable source of energy is the PV system. The current source is in parallel with the diode, making it a nonlinear current source, which exactly transforms the solar irradiation into electricity by propagating the movement of electrons and holes within the PV cell. As it is a nonlinear current source, it is necessary to form the PV source to inevitably work on its MPPT so that maximum power can be achieved [35].Figure3 demonstrates the I-V curve attributes at 25°C. As irradiance reduces from 1000W/m2 to 200W/m2, Ipv also reduces correspondingly. At 500W/m2, the current value is half of its max value, and there is a slight effect on voltage Vpv. In this paper, the PV for the integrated system is reviewed to work at MPPT, at 1000W/m2 at 25°C. PV system is designed for 150 to 30kW connect with a constant voltage source of 6kV at PV array side with penetration level set to 40 percent.Figure 3
PV characteristics.Integrating PV array with a thermal power system needs to go through various steps to get interconnected. The designing of PV system composed of various parts like converter, inverter, and calculation of average power actually fed to the grid as shown in Figure4.Figure 4
PV farm connected grid prototype.Here, the designing of the PV system is carried out. AC voltage can be estimated using(5)q=vdcvac,where q is the gain between AC and DC voltage in the system and its value is generally less than 0.86. For this prototype, its value is 0.7.The transfer function of the boost converter can be estimated by using equations (6) and (7).(6)m1=v2v1=i1i2,(7)g1s=1m1,where 1/m1 is the gain of the boost converter. The transfer function of the inverter is mentioned in the following equation:(8)g2s=IacsI2s=s2s2+ω2,where ω=2πf=2π50=314.12rad/sec. For instantaneous power, the transfer function is given in the following equation, where vm/im is the impedance.(9)Ps=vmim2s+vmim2ss2+2ω2.The gain in instantaneous power is mentioned in the following equation:(10)g3s=psIacs=vms2+ω2s2+2ω2s2s2+4ω2.The Laplace domain equation for average power is given in the following equation:(11)pavgs=vmim2s.The gain for average power is described in the following equation:(12)g4s=pavgsps.


### 3.1. Wind System

The contribution of wind energy in the power system is the main concern. Wind energy variation urges auxiliary power imbalance to the power system and causes frequency fluctuations from its nominal value. Owing to this fact, the interconnection of wind into the power system causes LFC problem [36, 37]. To deal with the LFC problem, an optimum controller is needed which addresses the increasing complexity of the power system and the involvement of renewable resources into the power system. This paper introduces a decentralized AMPC to resolve the LFC problem, and this controller will show its superiority as compared to other controllers.


### 3.2. Modeling for a Single Wind Turbine

The wind power system is divided into two main sections: (1) electrical and (2) mechanical. The electrical system consists of an MPPT calculator, which calculates the maximum power based onω. Furthermore, it also contains the inverter which generates the reference power, according to which output power is generated and here the generator converts mechanical power into electrical power and then feds it to the grid, and overall, the wind system is designed at 150 kW rating. The mechanical portion consists of a rotor speed controller which senses the power and torque of the turbine. On the basis of these factors, it gives feedback to the pitch controller which controls the pitch actuator angle to control the orientation of the wind turbine. The overall wind model is shown in Figure 5.Figure 5
Block diagram of LFC DFIG-based wind farm.To carry out the modeling of the wind system for the LFC problem, the state-space equation is formulated and designing of each part of the wind model is carried out.Rotation of wind turbine is given by wind powerPw represented by(13)Pw=12ρAVw3,where ρ represents air density, cross-sectional area is represented by A, and wind speed is represented by Vm.The aerodynamics of the wind turbine allow to absorb mechanical powerPm which is expressed as cube of wind speed.(14)Pm=Cpλ,βPw.Further, it can be elaborated under per unit value system.(15)Pm=CpCp−nVVn3,where Cp is related to λ (TSR) and β represents the pitch angle of the turbine. For calculation of Cp, the simplified formula is introduced as [38](16)Δλ=λnVnwr−nV∘Δwr,where wr−n=1.2 pu from a case study.From equation (16), ΔCp can be modified as(17)ΔCp=∂Cp∂ωrΔωr+∂Cp∂βΔβ.So, to rewritePm, ΔPm can be written as(18)ΔPm=Xω∂Cp∂ωrΔωr+∂Cp∂βΔβ,where Xω=V∘3/Cp−nVn3.The following condition can be addressed here; ifβ=0, rated speed equation (18) becomes(19)ΔPm=Xω∂Cp∂ωrΔωr.Ifβ > 0 above rated speed, the pitch angle is changed for capturing wind power. The diagram for the pitch angle is shown in Figure 4. ΔPm is represented as(20)ΔPm=Xω∂Cp∂ωrΔωr+Kp∂Cp∂βΔωr+Ki∂Cp∂βΔθ.The origination of equation (20) takes place by describing the pitch angle as(21)Δβ=KpΔωr+KiΔθ,where Kp and Ki are gains of the PI controller to change the angle dθ/dt=Δωr.Furthermore, considering the importance of the gearbox, designing of maximum power tracking calculation is carried out for rotational speed [39].(22)ωr=12Hws+FTm−Te,where Tm=Pm/ωr and Te=Pe/ωr and the above equation can be modified for the above-rated speed as(23)dΔωrdt=Xω∂Cp/∂ωr+Kp∂Cp/∂β−Fωr∘2Hωωr∘Δωr−12Hωωr∘ΔPe+XωKi∂Cp/∂β2Hωωr∘Δθ,Pωr=Xcωrωr−n3.During the LFC process,ΔPωr can be calculated as(24)ΔPωr=3Xcωr−n3ωr∘2Δωrθ.Another section is a generator from [40]; the transfer function for first-order active power control for wind form is determined by the response time of generator current control, which can be written as(25)Pe=1τes+1Pref,where Pref=Pωr+Pf.Change in electrical power can be written as(26)∂ΔPe∂t=1τe−ΔPe+ΔPωr+ΔPωf.The above presented equations for the model of wind system can be compiled into a state-space model for the above-rated wind speed, and it can be written as(27)X˙=Ax+Bu,y=Cx+Du.The generating matrix forA, B, and C is as follows:(28)A=Xω∂Cp/∂ωr+Kp∂Cp/∂β−Fωr∘2Hωωr∘XωKi∂Cp/∂β2Hωωr∘−12Hωωr∘1003Kcωr∘2τeωr−n30−1τe,(29)B=001τeT,(30)C=001.


## 3.1. Wind System

The contribution of wind energy in the power system is the main concern. Wind energy variation urges auxiliary power imbalance to the power system and causes frequency fluctuations from its nominal value. Owing to this fact, the interconnection of wind into the power system causes LFC problem [36, 37]. To deal with the LFC problem, an optimum controller is needed which addresses the increasing complexity of the power system and the involvement of renewable resources into the power system. This paper introduces a decentralized AMPC to resolve the LFC problem, and this controller will show its superiority as compared to other controllers.


## 3.2. Modeling for a Single Wind Turbine

The wind power system is divided into two main sections: (1) electrical and (2) mechanical. The electrical system consists of an MPPT calculator, which calculates the maximum power based onω. Furthermore, it also contains the inverter which generates the reference power, according to which output power is generated and here the generator converts mechanical power into electrical power and then feds it to the grid, and overall, the wind system is designed at 150 kW rating. The mechanical portion consists of a rotor speed controller which senses the power and torque of the turbine. On the basis of these factors, it gives feedback to the pitch controller which controls the pitch actuator angle to control the orientation of the wind turbine. The overall wind model is shown in Figure 5.Figure 5
Block diagram of LFC DFIG-based wind farm.To carry out the modeling of the wind system for the LFC problem, the state-space equation is formulated and designing of each part of the wind model is carried out.Rotation of wind turbine is given by wind powerPw represented by(13)Pw=12ρAVw3,where ρ represents air density, cross-sectional area is represented by A, and wind speed is represented by Vm.The aerodynamics of the wind turbine allow to absorb mechanical powerPm which is expressed as cube of wind speed.(14)Pm=Cpλ,βPw.Further, it can be elaborated under per unit value system.(15)Pm=CpCp−nVVn3,where Cp is related to λ (TSR) and β represents the pitch angle of the turbine. For calculation of Cp, the simplified formula is introduced as [38](16)Δλ=λnVnwr−nV∘Δwr,where wr−n=1.2 pu from a case study.From equation (16), ΔCp can be modified as(17)ΔCp=∂Cp∂ωrΔωr+∂Cp∂βΔβ.So, to rewritePm, ΔPm can be written as(18)ΔPm=Xω∂Cp∂ωrΔωr+∂Cp∂βΔβ,where Xω=V∘3/Cp−nVn3.The following condition can be addressed here; ifβ=0, rated speed equation (18) becomes(19)ΔPm=Xω∂Cp∂ωrΔωr.Ifβ > 0 above rated speed, the pitch angle is changed for capturing wind power. The diagram for the pitch angle is shown in Figure 4. ΔPm is represented as(20)ΔPm=Xω∂Cp∂ωrΔωr+Kp∂Cp∂βΔωr+Ki∂Cp∂βΔθ.The origination of equation (20) takes place by describing the pitch angle as(21)Δβ=KpΔωr+KiΔθ,where Kp and Ki are gains of the PI controller to change the angle dθ/dt=Δωr.Furthermore, considering the importance of the gearbox, designing of maximum power tracking calculation is carried out for rotational speed [39].(22)ωr=12Hws+FTm−Te,where Tm=Pm/ωr and Te=Pe/ωr and the above equation can be modified for the above-rated speed as(23)dΔωrdt=Xω∂Cp/∂ωr+Kp∂Cp/∂β−Fωr∘2Hωωr∘Δωr−12Hωωr∘ΔPe+XωKi∂Cp/∂β2Hωωr∘Δθ,Pωr=Xcωrωr−n3.During the LFC process,ΔPωr can be calculated as(24)ΔPωr=3Xcωr−n3ωr∘2Δωrθ.Another section is a generator from [40]; the transfer function for first-order active power control for wind form is determined by the response time of generator current control, which can be written as(25)Pe=1τes+1Pref,where Pref=Pωr+Pf.Change in electrical power can be written as(26)∂ΔPe∂t=1τe−ΔPe+ΔPωr+ΔPωf.The above presented equations for the model of wind system can be compiled into a state-space model for the above-rated wind speed, and it can be written as(27)X˙=Ax+Bu,y=Cx+Du.The generating matrix forA, B, and C is as follows:(28)A=Xω∂Cp/∂ωr+Kp∂Cp/∂β−Fωr∘2Hωωr∘XωKi∂Cp/∂β2Hωωr∘−12Hωωr∘1003Kcωr∘2τeωr−n30−1τe,(29)B=001τeT,(30)C=001.


## 4. Adaptive Model Predictive Controller

MPC is an advanced control algorithm used in many industries to resolve the industrial process to satisfy a number of constraints to deal with linear and nonlinear problems. It was first proposed by Richalet in 1978 [41]. This MPC has a great potential in dealing with the system having disturbance and also dealing with optimization requirements of the system. MPC mainly contains three blocks: rolling optimization, prediction model, and feedback correction. The rolling optimization block is used to optimize a number of constraints online which differentiate it from traditional optimal control. The next part is the prediction model which predicts the future value in real time, and this model is basically working on the history of the controlled object. Moreover, it is designed for multi-input/output constraints. The last block is the feedback correction which is actually calculating the deviation from the predicted value and the actual output value [42].As from its name, it is obvious that feedback correction compensates the future error while utilizing the feedback information to develop the closed-loop optimization operation. This paper presents an adaptive model predictive controller (AMPC). This controller is designed to deal with the constraints of the LFC problem. AMPC is presented in Figure6.Figure 6
The block diagram for AMPC strategy.The designing of this controller is established in such a way that the one section is dealing with the LFC requirements in presence of uncertaintyΔFik. The other part of MPC is designed to deal with the attenuation effect due to external disturbance and to counter tie line variation. The other important element between the two areas is the frequency bias factor (β) whose value is between 0 and 1 and is used in balancing the ACE.. The last thing which is the output scaling factor (G) is set constant. The output scaling factor is adjusted in each sampling time by the adaptive mechanism.


### 4.1. Rolling Optimization

The designing of the controller is carried out in such a way to minimize the oscillation in the frequency due to disturbance in load. The controller should have the ability to restore the system frequency as quickly as possible to ensure the safe operation of the hybrid power system. The AMPC has the capability to show less oscillations with minimal undershoot and overshoot in frequency and prevent the violation of input and output constraints. The optimization is based on a steady-state model of the process, and it is typically a linear model used to minimize a profit function, minimizing cost function and maximizing production rate. The objective function is presented as follows.(31)Jik=qiΔfik+12+r1,iΔPeb,ik2+r1,iΔPeu,ik2.Minimize the objective function. MinJik for safe operation following constraints must be fulfilled.(32)Δfimink≤Δfik≤Δfimaxk,ΔPeb,imink≤ΔPeb,ik≤ΔPeb,imaxk,ΔPeu,imink≤ΔPeu,ik≤ΔPeu,imaxk,0<β<1.Considering all these constraints in equation (32), the LFC efficiency is achieved and the transfer of active power is controlled accurately.


## 4.1. Rolling Optimization

The designing of the controller is carried out in such a way to minimize the oscillation in the frequency due to disturbance in load. The controller should have the ability to restore the system frequency as quickly as possible to ensure the safe operation of the hybrid power system. The AMPC has the capability to show less oscillations with minimal undershoot and overshoot in frequency and prevent the violation of input and output constraints. The optimization is based on a steady-state model of the process, and it is typically a linear model used to minimize a profit function, minimizing cost function and maximizing production rate. The objective function is presented as follows.(31)Jik=qiΔfik+12+r1,iΔPeb,ik2+r1,iΔPeu,ik2.Minimize the objective function. MinJik for safe operation following constraints must be fulfilled.(32)Δfimink≤Δfik≤Δfimaxk,ΔPeb,imink≤ΔPeb,ik≤ΔPeb,imaxk,ΔPeu,imink≤ΔPeu,ik≤ΔPeu,imaxk,0<β<1.Considering all these constraints in equation (32), the LFC efficiency is achieved and the transfer of active power is controlled accurately.


## 5. Results and Discussion

This section highlights the performance of the proposed model and control technique. The proposed technique is verified on MATLAB/Simulink. The two-area hybrid power system is proposed where the control of the PV, wind, and thermal power system is controlled by AMPC. Each area contains a set of PV, wind, and thermal power plant, and they are connected via a tie line as shown in Figure7.Figure 7
Two-area hybrid interconnected thermal system.The output generation of power by a wind turbine under varying wind conditions is presented in Figure8. The control of the wind farm is carried out by MPC where the output pattern of power generation is obtained by applying varying wind patterns. The output generation of power is averaged around 200MW, and wind speed varies between 9 and 14ms−1.Figure 8
Wind speed and power: (a) wind speed; (b) wind power generation (MW).
(a)(b)To verify the proposed technique, the two-area model is tested under different scenarios. Two different tests are applied to validate the efficiency of the proposed controller. (1) This step is applied for a step change in load at worst-case scenario that rarely occurs, and it is applied for both areas. (2) This step is very close to real-world scenarios, and the system load is varying throughout the day, so a varying test of load change is applied to monitor the performance of AMPC. Furthermore, the obtained results are compared with the state-of-the-art controllers. Figure9 verifies the step response and the robustness of the proposed controller. In Figures 10 and 11, the validation for the varying load is monitored. The exchange of power between the two areas is also monitored as shown in Figure 12.Figure 9
Frequency response and change in power: (a) Δf1; (b) Δf2; (c) ΔPTie.
(a)(b)(c)Figure 10
Controller response under varying load conditions for area 1.Figure 11
Controller response under varying load conditions for area 2.Figure 12
Exchange of power between two areas.The power output response of thermal generator is shown in Figure13, while the integration of wind with thermal is depicted in Figure 14. The collective response of thermal, PV, and wind is exhibited in Figure 15.Figure 13
Thermal output power response.Figure 14
Thermal and wind power output response.Figure 15
Collective power response of thermal, PV, and wind.From the results, it is quite clear that AMPC is leading in superiority as compared to other controllers in terms of oscillation and response time which justify its robustness. Furthermore, the communication delay can originate in a system that is presented by time delayτ, and it can deteriorate the frequency from its nominal value. Time delay usually occurs in the governor section as shown in Figure 16, and that for turbine is represented in Figure 17.Figure 16
Response under uncertainty in the governor system.Figure 17
Response of controller under uncertainty in the turbine unit.The stability response of the proposed controller is tested as shown in Figure18.Figure 18
Stability response of controllers.If there is any uncertainty in the governing units of the thermal system, this might lead to a sudden breakdown of the overall system. To validate the effectiveness of the proposed controller, an uncertainty test is carried out where the proposed controller validates its robustness. The validation of the proposed technique can be verified by their performance indices; in this research work, three commonly used performance indices are presented: integral time absolute error (ITAE), integral time square error (ITSE), and integral square error (ISE), to verify the performance of the controllers. Table1 defines the performance indices of different controllers.Table 1
Performance analysis of different controllers.
ControllerITAEITSEISEAMPC0.18120.00010990.0001269MPC0.20120.00013480.0001721GA0.79080.0018910.0009201FA0.80110.0020040.0009316The obtained performances of the controllers are analyzed in bar chart format to get a better idea. Figure19 presents the ITAE, ITSE, and ISE-based controller performances.Figure 19
ITAE, ITSE, and ISE-based controller performance.


## 6. Conclusion

(i)
This paper has presented a control strategy for a hybrid thermal power system in which the fluctuation in frequency is controlled by a newly adaptive-based MPC controller. The two-area system is presented which is connected via a tie line. Modeling of PV and wind farms is presented where the impact of renewable sources is visualized.(ii)
The wind turbine system is controlled by MPC while the thermal section is controlled by AMPC to mitigate the effect of frequency fluctuation under load disturbance.(iii)
The AMPC deals with two main controlling components: initially, it deals with LFC requirements and later it deals with attenuation effect due to external disturbance to achieve zero tracking for tie line variation in frequency.(iv)
The overall proposed controlling mechanism quite effectively adapts to the unpredictable situation in the power system. Moreover, simulation results are presented under step, practical scenarios, and under uncertainty in the thermal power system which has shown the superiority of the proposed technique.



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*Source: 7894264-2022-04-18.xml*