Modeling and control of automatic voltage regulation for a hydropower plant using advanced model predictive control☆

0 Introduction

With water covering more than 70%of the earth’s surface, hydropower has long been used in electricity generation, where power is generated in bulk and transmitted to consumption points through long-distance transmission lines[1].As shown in Fig.1,a single generator can supply power to the expansive and complicated areas of a power system using connected plants[2].A tie-line electric power transfer device is used to provide electricity to nearby consumers.Synchronous generators are the most commonly used type of hydrogenators.A crucial aspect of electrical power generation is the ability of generating units to maintain synchronism with the grid during grid disturbances.In other words, the voltage and frequency of the generators must always be maintained within acceptable margins[3].Automatic voltage regulation is necessary in power systems to effectively manage voltage fluctuations (underand over-voltages) [4].The ability to track a steady reference by using controllers and actuators enables automatic voltage regulation (AVR) [5].

An automated voltage regulator (AVR) is critical in a generator excitation system because it stabilizes the generator voltage and controls the reactive power flow.AVR ensures that the terminal voltage magnitude of the synchronous generator remains at a specified level.Generally,the generator excitation systems in older systems used DC generators mounted on the same shaft as the rotor of the synchronous machines.Excitation was provided through slip rings and brushes.However, modern excitation systems employ brushless excitation, which uses AC generators with rotating rectifiers.A change in the real power demand primarily affects the frequency of the system [3];however, a change in the reactive power primarily influences the voltage magnitude.The interaction between the voltage and frequency controls is generally sufficiently weak to analyze them separately.The reactive power sources in power systems include generators, capacitors,and reactors.The reactive power of the generators can be controlled by adjusting the field excitation.Other methods for improving the voltage profile in electric transmission systems include transformer load-tap changers,step-voltage regulators, switched capacitors, and static VAR control equipment.However, the primary means of controlling the reactive power of the generators is the AVR.

Several controllers have been proposed in the literature for an adequate AVR, including the proportional integral derivative (PID) controllers, fractional order PID(FOPID), and nonlinear PID (NPID) [6].In [7], the authors used a FOPID controller, which has five tuning gains, in contrast to conventional PID controller, which has three tuning gains for the AVR of a synchronous generator.A marine predator algorithm (MPA) was used to tune the FOPID.Despite the effectiveness of the MPA in tuning the FOPID, it has poor search capabilities, premature convergence,and stagnation in the local optimum[8].

To achieve stable and efficient AVRs, the authors [9]integrated the Le´vy flight mechanism into the Runge-Kutta optimization algorithm to tune the PID and second-order derivative (PIDD2) using the integral of the squared error function.This was a master-slave approach,with Bode’s ideal reference model used as the master system, and the power plant was forced to follow the reference system.To improve the efficiency and robustness of AVRs, S.Ekinci et al in [10] used a quadratic waveletenhanced gradient-based optimization (QWGBO) algorithm coupled with a real proportional-integral -derivative with second-order derivative (RPIDD2) and fractional-order proportional-integral (FOPI) controllers.This method demonstrated enhanced precision, stability,and quick response.A modified artificial rabbit optimization (ARO) algorithm-based fractional-order proportional-integral derivative with double derivative(m-ARO-FOPIDD2) controller was proposed in [11] for AVR, which demonstrated its superiority over conventional methods in terms of stability, robustness, response time, and efficiency.

In [12], a nonlinear sine-cosine algorithm was used to tune the sigmoid PID controller for the AVR.However,the sine-cosine algorithm is known for the low accuracy of its solution,in addition to its poor global search ability[13].M.J.Lawal et al.in[14]enhanced the performance of AVRs using an adaptive neuro-fuzzy inference system(ANFIS),which was designed by training a fuzzy inference system(FIS)using a hybrid optimization learning scheme.In[15],the investigators proposed a novel optimized PIDbased model reference fractional adaptive controller followed by detailed mathematical models for AVRs.The results showed that the proposed controller was able to smoothly track the input signal with no observed overshoot and could also instantaneously react to changes in the reference voltage signal.In [16], a model-predictive control paradigm based on optimization algorithms was used for AVR resistance to uncertainties.An arithmetic optimization algorithm(AOA)was used to tune the model predictive controller (MPC).The simulation results demonstrate the effectiveness of the proposed method in controlling voltage fluctuations and uncertainties in parameters with little effort.In [17], the authors proposed the self-online tuning of PID controllers, referred as indirect design approach-2 (IDA-2), to maintain robustness in AVRs.The PID was indirectly tuned by shifting the frequency response.All PID parameters were simultaneously adjusted by modifying the frequency-shifting constant.Despite the superiority of the proposed method over other methods used in AVRs, it is only suitable for specific frequency-shifting constants.Using an equilibrium optimization algorithm, M.Micev et al.in [18] not only designed but also analyzed an AVR system.The goal of the design was to obtain optimal values for the AVR PID controller.The simulation results demonstrated the controller’s high speed and ability to provide a faster response than other PID controllers.

Micev et al.in [19] proposed a novel PI controller with two degrees of freedom (2DOF), in addition to antiwindup protection, to automatically regulate the voltage of synchronous generators.The objective function for determining the optimal parameters of the controller was defined by considering the disturbance, transient response characteristics, and noise rejection measurement features of the AVR system under investigation.The African vultures optimization algorithm (AVOA) was used to design the parameters of the controller.It is worth noting that although it is a high-performance optimization algorithm,the AVOA is known for its poor exploration capabilities in multimodal problems and limited population density [20].A whale optimization algorithm(WOA)tuned 2DOF fractional PI controller was proposed in [21] for AVRs.The research results showed that the proposed method performed better on mean disturbance rejection by 26 %and 20 % higher on average when the system parameter was altered from its rated values, based on its robustness.Similar to many optimization algorithms, the WOA has slow convergence, inclination toward local convergence,and low precision [22].Using a neurofuzzy controller, T.Weldcherkos et al.in [2] modeled and designed an automatic hydropower plant generation control system.Using a hybrid simulated annealing and Manta ray foraging optimization algorithm, M.Micev et al.in [23] optimized four types of PID controllers for AVRs: ideal PID,fractional-order PID, real PID, and PID with secondorder derivatives.The research results demonstrated the effectiveness of the hybrid optimization algorithm in tuning the four PID types.

This study focused on the modeling and control of AVR in hydropower plants.It investigated an advanced MPC and an ABC-based PID controller to achieve optimal voltage regulation.The contributions of this study to the literature are as follows.

- This study introduced a comprehensive mathematical model of the AVR of a hydroelectric power plant that captures its dynamic behavior under varying conditions.By integrating inherent dynamics, the model enhances the understanding of the AVR performance in hydropower applications, thereby contributing to the literature on modeling techniques for power systems.

- This study pioneered the application of MPC for voltage regulation in hydropower plants.By optimizing control actions over a finite horizon while considering operational constraints, this study demonstrated the potential of MPC for superior voltage regulation and stability, demonstrating its robustness in complex control scenarios.

- A comparative analysis of the MPC and ABC-PID controller (another widely used control architecture) was conducted.By evaluating performance metrics, such as voltage regulation accuracy, disturbance rejection, and resilience to uncertainties, this study provides valuable new insights into the strengths and limitations of each method, guiding practitioners in selecting appropriate control strategies for their applications.

Overall, this study contributes to the body of knowledge in the field of hydropower plant control by presenting a comprehensive study on the modeling and control of AVRs.It introduced an AVR controller with MPC, compared it with the ABC-PID controller, and provided insights into their individual contributions and comparative performances.The findings of this study enhance our understanding of and offer practical guidance for the design of efficient and robust control systems for hydropower AVR control, ultimately contributing to the advancement of sustainable power generation.

The succeeding section describes the methodology used in this study,followed by a presentation and discussion of the results, and then a conclusion.

1 Methodology

1.1 Mathematical model of automatic voltage regulation

An AVR system consists of various components;a simplified schematic diagram of an AVR is shown in Fig.2.The system includes a potential transformer that determines the voltage magnitude at the generator terminal.This voltage signal is rectified and compared with a preset DC point signal.The resulting error signal is amplified and used to control the exciter field,thereby increasing the terminal voltage of the exciter.Consequently, the generator field current increases,increasing the generated electromotive force (EMF).The reactive power generation is then adjusted to a new equilibrium by increasing the terminal voltage to the desired level.

The models used to represent the components of an AVR system can vary in complexity.However, simplified models are discussed in this paper.These models can facilitate in understanding and analyzing the behavior and components of AVR systems.

1.1.1 Amplifier modeling

Excitation system amplifiers play a crucial role in regulating the field current of synchronous generators,which in turn affects the terminal voltage and reactive power output of the generators.These amplifiers can be implemented using various technologies, including magnetic, rotating,and modern electronic amplifiers.The characteristics and parameters of the amplifiers can vary depending on their implementation.

An amplifier is typically represented by two parameters:the gain(ka)and the time constant(λa).The gain ka represents the amplification factor of the excitation system and determines the relationship between the input signal (generally the error between the desired and actual terminal voltages)and the resulting change in the field current.Typical values range from 10 to 400,depending on the specific system requirements and design considerations [24].

The time constant (λa) of an amplifier represents the rate at which the amplifier responds to changes in the input signal.It quantifies the time required for the amplifier output to reach a certain percentage (typically 63.2 %) of its final value after a step change in the input.In the context of excitation systems, the amplifier time constant is generally small,typically in the range of 0.02-0.1 s[13].In some cases,the time constant may be considered negligible compared to other significant time constants in the system,and therefore can be ignored in simplified models or analyses.

The transfer function of the excitation system amplifiers is denoted by Ga（s）, which represents the relationship between the Laplace transform of the input signal (typically the error signal) and that of the output signal (the change in the field current) [25].

The transfer function captures the dynamics and frequency-response characteristics of the amplifiers.It is worth noting that the specific implementation and characteristics of the amplifiers can vary depending on the technology used.Magnetic and rotating amplifiers have historically been used in excitation systems, whereas modern electronic amplifiers, such as operational amplifiers and power electronic devices, are commonly employed in contemporary systems because of their improved performance, reliability, and flexibility.

1.1.2 Exciter modeling

In modern excitation systems for synchronous generators, the most common approach is to use an AC power source rectified using solid-state devices, such as siliconcontrolled rectifiers (SCRs).This type of exciter provides AC voltage that is converted into DC voltage to supply the generator field winding.It is important to note that the output voltage of the exciter is a nonlinear function of the field voltage owing to saturation effects in the magnetic circuit.This saturation introduces nonlinearity into the excitation system, resulting in a relationship between the terminal and field voltages of the exciter complex.Consequently,establishing a simple relationship between these variables is challenging.

Various excitation system models with different sophistication levels have been developed to address this problem.The Institute of Electrical and Electronics Engineers(IEEE) has published recommendations that provide guidelines and models for excitation systems[24].In many cases, a reasonable approach is to employ a linearized model of a modern exciter.This model considers the major time constant while ignoring saturation effects and other nonlinearities.By simplifying the behavior of the excitation system, this linearized model provides a useful approximation for analysis and control.

In its simplest form, the transfer function (Ge（s）) of a modern exciter can be represented by a single time constant （λe） and a gain (ke).The time constant (typically extremely small for modern exciters)captures the dynamic response of the exciter and represents the rate at which the exciter output voltage responds to changes in the field voltage.The gain determines the amplification factor between the field voltage and exciter output voltage [24].

Although this linearized model provides a useful approximation, it is important to consider its associated limitations and assumptions.In some cases, more detailed models that account for saturation and other nonlinear effects may be necessary for an accurate analysis and control of the excitation system.

1.1.3 Generator modeling

In the linearized model of a synchronous machine, the relationship between the generator terminal voltage (vt)and its field voltage (vf) is represented by a transfer function (Gg（s）), which is expressed as follows:

where λg is the time constant and kg is the gain that represents the amplification factor between the field voltage and the generator terminal voltage.It quantifies the change in the terminal voltage for a unit change in the field voltage.These constants (kgandλg) are load-dependent, which indicates that their values vary based on the operating conditions of the generator

In the context of a linearized model, kg typically varies between 0.7 and 1, reflecting the load-dependent behavior of the synchronous machines.λg typically ranges from 1.0 to 2.0 s, depending on the load conditions.

1.1.4 Sensor modeling

In the AVR block diagram shown in Fig.2,the voltage sensing is accomplished using a potential transformer that reduces the generator terminal voltage to a suitable level for measurement.The sensed voltage, denoted as V s （s ），is subsequently rectified using a bridge rectifier.The voltage sensor, including the potential transformer and rectifier, can be modeled as a simple first-order transfer function, Gs （s ）.The transfer function represents the dynamic response of the sensor to changes in terminal voltage.The transfer function is expressed as [26]

In this transfer function,λs represents the time constant of the sensor and ks is the gain of the rectifier.λs is typically small, ranging from 0.01 to 0.06 s.The specific value of λs depends on the characteristics of the sensor and its response time.The output of the voltage sensor, V s （s ）, is subsequently connected to the AVR block diagram,which typically includes components such as amplifiers, controllers, and feedback loops to regulate the excitation of the generator system and maintain the desired terminal voltage.

1.2 AVR control

When modeling the AVR of a power system, the integrated structure of the amplifier, exciter, generator, and sensor must be developed and formulated appropriately.Combining Eqs.(1)-(4) in a block diagram results in an entire schematic representation of a simplified AVR, as shown in Fig.3.The system parameter values are listed in Table 1.

1.2.1 Transfer function model

The open-loop transfer function L（s ）shown in Fig.3 is described in Eq.(5), and the closed-loop transfer function W （s ） that establishes a relationship between the generator terminal voltage V t （s ） and the reference voltage V ref （s ） is expressed in Eq.(6) [27].

Substituting Eqs.(1),(2),(3),and(4)into Eqs.(5)and(6),the open-and closed-loop transfer functions of the system are as follows:

1.2.2 Steady-state analysis

For a unit step change in V ref （s ）， the accompanying Laplace transform is expressed as

The frequency response of the terminal voltage now becomes

When the final value theorem is considered, the steadystate response can be derived as

1.3 Control design of AVR

The objective of an AVR in a power system is to monitor and regulate the grid voltage, ensuring that it remains close to the desired reference value.The primary objective of an AVR controller is to robustly restore the voltage to its nominal value following any disturbance or perturbation.To achieve this,an AVR controller typically employs a feedback-control system.The controller can be implemented using various control techniques such as PID control, state feedback control, and advanced control algorithms.

1.3.1 PID controller design for AVR

PID controllers are one of the most prevalent controllers in the market.AVR systems frequently employ PID controllers to enhance their efficacy.The PID controllers enhance the dynamic response and reduce or eliminate the steady-state error.Using a proportional controller, the response rise time can be decreased; however, steady-state errors cannot be eliminated [24].The derivative controllers enhance the transient response by adding a finite zero to the open-loop plant transfer function, which reduces system overshoot, thereby increasing the stability margin of the system [3].The integral controllers add a pole at the origin,increasing the system type by one and reducing the steady-state error of the step function to zero; however, this may adversely influence the transient response.The Laplace domain function of a PID controller is defined as follows:

where kP, kI, and kD are the proportional, integral, and derivative gains of the controller, respectively.

Calculating the gains of a PID controller involves selecting closed-loop poles that satisfy the performance criterion by optimizing the controller using a model-based approach.This was performed irrespective of the type of the controller used.The use of three distinct gains in the PID controller design results in the introduction of two zeros and one pole at the origin of the system.This transforms the system into a Type 1 system, which eliminates the steady-state errors [3].Fig.4 shows an AVR system with a PID controller.

1.3.2 Artificial bee colony (ABC) optimization

The ABC algorithm, similar to other metaheuristic algorithms used to solve optimization problems [28-30],is a nature-inspired algorithm governed by the foraging behavior of honeybees, and is employed to enhance the performance of the PID controllers [31].The ABC algorithm is a nature-inspired optimization algorithm developed by Karaboga in 2005[32].In AVRs, the ABC algorithm is used as an adaptive-tuning mechanism for the PID controllers.By mimicking the foraging behavior of honeybees, the ABC algorithm explores the parameter space of the PID controllers and searches for optimal controller settings that result in improved voltage regulation performance.This adaptive-tuning capability enables the controller to adapt to changing operating conditions, system dynamics, and uncertainties inherent in hydropower plants.

Because bees are excellent in searching for food,any bee that discovers food informs other bees.Thus, other bees can deduce the quantity and location of their food sources.This assists other bees in their quest for food in the correct direction.These bees draw many other bees and continue to find food sources.The bee colony in the ABC comprises three groups of artificial bees: employed bees, onlookers,and scouts [33].Employed bees constitute the first half of the colony, whereas onlookers constitute the second half.There is only one employed bee per food source.This indicates that the number of employed bees in the hive is equal to the number of nearby food sources.An employed bee, whose food source is abandoned becomes a scout.In other words, when the location of a food source is not improved after the predetermined number of efforts known as the ‘‘limit,” an employed bee becomes a scout.In doing so,employed and onlooker bees perform the process of exploitation, whereas scouts explore solutions.

The specifics of the ABC algorithm described in[34]are as follows:

A.Initialization phase

The food source locations (xjk) are initialized randomly using Eq.(15), which represents the search space of the PID gain vector [kP，ki，kD].

B.Employed bee phase

Each bee is assigned a food source for deep exploitation.Therefore,Eq.(16)is used to obtain the food sources.

where l is the neighbor of j,l≠j,φ is a number randomly chosen between -1 and 1 to have an idea of the production of the neighbor solutions around xjk, and vjk is xjk’s new solution.

The fitness function of the new food source is expressed as

where fj is the objective function of each food source,and FITj is its fitness value.

From the original and new food sources, greedy selection was performed to select the superior source based on its fitness value.

C.Probabilistic selection phase

An onlooker bee chooses a food source based on the probability of each food source,which is determined using the following equation:

where FITj is the jth solution fitness value,and Pj is the jth solution selection probability.

D.Onlooker bee phase

Information about the sources of food is shared with the onlooker bees by the employed bees for deeper processing.Each onlooker bee chooses a food source to exploit based on its probability (better fitness and higher probability).Eq.(16) is used to better utilize the selected food sources,and Eq.(17)is used to determine their fitness values.The original and latest food sources are subjected to a greedy selection process, similar to the employed bee phase.

E.Scout bee phase

A food source is abandoned and the corresponding bee turns into a scout bee if it does not produce better results,up to a certain limit.Eq.(15) is used to generate a fresh food source in the search area, and the process continues until the termination requirement is satisfied.

The output is the best food source solution.The ABC algorithm is used to design the base controller based on its nature-inspired optimization capabilities, adaptive tuning characteristics, resilience to parameter uncertainties,and computational efficiency [35].These attributes align well with the objectives of achieving precise voltage regulation, stability, and adaptability in AVR systems of hydropower plants.The complete algorithm is illustrated in the flowchart shown in Fig.5.

1.3.3 MPC controller design for AVR

Model predictive control is a computer control algorithm belonging to the category of optimal control methods.It uses a mathematical model to predict future behaviors based on a sequence of control-variable manipulations.The key idea of MPC is to look ahead and consider the predicted response of the process.Utilizing optimization techniques, the control algorithm determines the optimal control actions that lead to the desired process output behavior.An optimization problem typically considers various constraints and objectives to determine the best possible control strategy.This ‘‘look ahead” strategy allows the MPC controller to make decisions based on future predictions and optimize the control actions accordingly.It considers long-term goals and criteria and considers the overall system performance over time.

In contrast, classical control techniques such as PID controllers focus on achieving short-term goals based on the current state of the system.Although PID controllers are effective for immediate control actions, they may not optimize the long-term performance of the system.This is the advantage of MPC because it considers the future behaviors of the process and optimizes control actions accordingly, thereby achieving better long-term performances and satisfying specific objectives [36].

While classical control methods can effectively address short-term goals, they may not be able to optimize the overall system performance in the long run.By contrast,MPC takes a more comprehensive approach by considering future predictions and optimizing control actions to achieve the desired long-term outcomes [37].

The fundamental paradigm of predictive control comprises two essential elements: system prediction and control optimization.Using the controlled system stateequation model, a critical component for predicting the controlled system, which is a power system with a controlled generator voltage, was formulated.Based on this system model, the MPC predicts the future system behavior by factoring it into the optimization,which determines the optimal trajectory of the manipulated variable u,with r and y as the reference input and output, respectively, as shown in Fig.6.

1.4 Formulation of the prediction model

a) State space model - continuous time

The prediction model of the controlled system, that is,the generator and its related power network, is essentially the state-equation model of the system.The state equation is composed of a series of first-order differential equations.The use of the MPC was demonstrated by applying it to a simplified AVR model, as shown in Fig.3.

After transforming the system from its s-domain into the time domain and writing the state equation in matrix form, the following was obtained:

The time domain state space model is expressed as

where x-（t）is the state vector,y-（t）is the output vector,u（t ）is the controlled input,At is the state matrix,Bt is the input matrix,Ct is the output matrix,δ（x- （t ））is the control input disturbance, and Dt and dt are feedthrough/feedforward matrix (which is zero) and output disturbance, respectively.dt was considered to be zero.

b) State space model - discrete time

In many practical scenarios that prioritize simplicity,robustness, and computational efficiency, the zero-orderhold (ZOH) method is preferred over the more complex and computationally intensive Tustin method.Although the Euler method offers a straightforward discretization,it may lack accuracy for systems with fast dynamics,making ZOH more suitable for precise discretization.Therefore, the system was discretized using ZOH.

Using the ZOH technique, Eq.(20) can be represented in a discrete-time form, as follows.Integrating Eq.(21) over time t after changing variables and simplifying the equation results in the following equation:

To calculate x- （t ） in discrete time at instant k, the following equation is defined:

where Ts is the sampling period.Thus,x- [k+1]can now be defined as

Decomposing the exponential and integral term into two parts, the following equations are obtained:

Evaluating Eq.(26), the following equation is obtained:

Because I is an identity matrix, the discrete-time statespace model can be represented as

The system state variables x- [k+1] at the future time step k+1 can be described by two components at the current time step k, as shown in Eq.(28), with the state variables x-[k] illustrating the system process continuity as one component, and the controlling input u[k] as another.

c) One-step predictive model

Based on the discrete-state equations (Eq.(28)), the conventional approach for defining the one-step model predictive control is as follows [39]:

where x-[k+1|k] indicates that the values of x- at time step k+1 are computed based on the information obtained at time step k.The model described above can be calculated using Eq.(28), as follows:

STEP-1: To determine y-[k], measurements are performed under controlled conditions.The state values x-[k]are determined starting with y-[k].

STEP-2:The prediction model is used to derive x-[k+1]from x- [k ] and the expected result y-[k+1] is calculated based on the input value x-[k+1].

STEP-3: To determine the optimal control input u, the predictive optimization performance index provided in Eq.(31) is used.

d) Improved predictive model

The next part concerns part (c) above, which describes how to determine the controlling input u.However,u[k|k]) in the normal form of Eq.(29) can be confusing because k|k indicates that the value of u at time step k is calculated from the data at the current time step k.Because all the data at time step k,including the driving input u,are already known, it is not possible to use an optimization process to determine the data that are already known.In this case, the controlling input u at the current time step is known and is applied to a controlled system.

It is not surprising that this kind of confusion could occur because the usual way to solve the state equation is to use a known input u to determine the value of x-.In predictive control, the controlling input to the controlled system for the succeeding time step must be predicted.This control is always performed to determine u at succeeding time step k+1 from current time k.

Therefore, this study suggested changing the form of the predictive control state equation such that the u[k ] in Eq.(29) becomes u[k+1|k], as shown below:

where u[k+1|k] is the control input to the controlled system,which in the case of AVR control is V e（s）.The data at time k are used to determine u[k+1|k] at time step k+1[36].

Using this prediction model, the system outputs[k+i|k]（i=1，2，...，N） are predicted over an interval of time N, starting at instant k.These predicted system outputs are calculated based on the measurements of y-[k]and future control inputs u[k+j|k]（j=0，1，...，n-1）.

These predicted outputs are then used in the MPC optimization process to compute the optimal control actions that lead to the desired output behavior of the system.

e) Formulation of the objective function

A straightforward optimization calculation in the form of least-squares minimization in terms of the system performance index J is the basis for optimal control of the AVR.The value of J is determined by computing the sum of a series of square values generated from the deviations of the expected system outputs from their references and the changes in the control inputs to the controlled system[36].For AVR control,MPC minimizes the cost function J denoted by Eq.(31) by tracking the error signal between the reference and measured output voltages.

where uLB and uUB are the lower and upper limit constraints on the inputs, respectively, and avd are the corresponding outputs.

The simulation diagram of an AVR with an MPC controller added to the forward path of the AVR loop is shown in Fig.7.

2 Results and discussion

2.1 Step input change

Fig.8 shows the voltage output response in relation to the step input change in the controlled and uncontrolled cases.The analysis of this response provided valuable insights into the performance and effectiveness of the control strategy.

In the analysis of the generator AVR system without a controller, simulations conducted using the Simulink model revealed significant insights.The Bode plot of the frequency response shown in Fig.8 indicates that the open-loop system remained stable when there was no feedback from the sensor.The change in the DC gain of this open-loop response was measured at 20.00 dB, which was attributed to the transfer function of the amplifier,which introduced a gain of 10.0.The phase margin (PM)and gain margin (GM) of the system were 5.69 dB and 18.59, respectively.The gain crossover frequency (GCF)and phase crossover frequency (PCF) were 4.4 rad/s and 6.12 rad/s, respectively.

The positive values of PM and GM, along with a PCF greater than the GCF, indicate that the open-loop system was stable, exhibiting no overshoot and being overly damped.However, the lack of feedback indicates that the generator voltage could not be regulated effectively.When the loop was closed through the sensor, the system became unstable around the desired voltage level, with GM and PM becoming negative, with values of-9.08 dB and -14.96, respectively.The change in the DC gain fell below the 0.0 dB line to-0.82 dB,as the previously stable PM of 18.59 degrees was compromised by unaccounted dynamics introduced by the sensor.This shift resulted in a PCF of 5.08 rad/s,which was less than a GCF of 6.06 rad/s.

Moreover, a sharp gain of 19.6738 dB is observed at 4.4707 rads/s,accompanied by a rapid drop in phase,indicating a loosely damped system with oscillations in the time response characteristics, as represented in Fig.9.The step response plot in Fig.9 has the illustration of the voltage deviation in the AVR system when simulated without an active controller,resulting in a significant voltage overshoot and oscillations.

To understand why overshoots,occur in the absence of a controller, we need to examine the dynamics of the system.In a power plant generator, the AVR system is responsible for regulating the output voltage by adjusting the excitation level of the field winding of the generator.The excitation level determines the strength of the magnetic field, which in turn affects the output voltage of the generator.Without a controller, the system cannot respond to changes in the input voltage quickly and accurately.When a disturbance, such as a step change in the input voltage,occurs,the system’s response is purely based on the inherent characteristics of the generator and its associated components.

During a voltage step change, the generator initially experiences a mismatch between the output voltage and the desired reference voltage.This mismatch creates an error signal that drives the excitation system of the generator to correct the deviation.However, without a controller, the response of the excitation system may be slow or insufficient to counteract the disturbance effectively.As a result, the generator’s output voltage overshoots the desired reference value before gradually settling down.This overshoot occurs because the excitation system may react too aggressively to the initial error, causing the voltage to exceed the desired setpoint.Furthermore, the lack of damping in the system’s response leads to oscillations and instability, as the system struggles to reach a stable equilibrium.

Incorporating a controller like the PID in the AVR system helps mitigate these issues.The proportional component of the PID controller provides immediate correction to reduce the error between the output voltage and the reference value.The integral component eliminates any steady-state error, ensuring long-term stability.The derivative component adds damping to the system’s response, suppressing oscillations and improving transient behavior.By incorporating a PID controller or any other controller, the AVR system can regulate the output voltage more effectively,minimizing overshoots,and maintaining stable operation even in the presence of disturbances.

The implementation of a PID controller, optimized using a PID tuner (as seen in Fig.7, Auto-tuned), significantly enhances voltage stability compared to the uncontrolled scenario.In power generation, it is essential for the time required to reach the desired voltage to be minimized without excessive overshoots,which is a fundamental requirement for effective control.This control system demonstrates satisfactory stability to a certain extent.

In this system,acceptable stability is achieved before the 2-second mark,with the voltage settling at a value of 1 per unit (pu) by 2.5 s.However, the system experiences a degree of overshoot of 8.7 %, which the controller was unable to eliminate although still achieving zero steadystate error.The controller design aimed to achieve the desired settling time while keeping the overshoot margin minimal.Table 2 below shows the result of the obtained PID gains and response characteristics.

While auto-tuned controllers may be suitable for some applications, the PID tuner offers a more comprehensive and efficient approach for tuning PID gains in AVR control systems.It provides a balance between transient response and robustness by systematically optimizing the gains based on the system’s performance criteria.It is important to note that,the PID tuner may not be particularly valuable or straightforward in complex systems where manual tuning may not yield satisfactory results due to the interdependencies between various control loops.However, in such cases, finding optimal gains may not yield conclusive results.Consequently, advanced design techniques become useful, as they offer robustness that may be lacking in an auto-tuned controller.

The transient response of the AVR system,as shown in Fig.9 (ABC-tuned PID), indicates that the ABC-designed PID controller performs better than the auto-tuned approach when subjected to a step input voltage change.The ABC-PID controller reduces the system’s overshoot in response to voltage fluctuations, and the maximum overshoot difference compared to the auto-tuned approach is approximately 5.2 %.As shown in Table 2, the PID gains obtained for the ABC-designed PID controller,along with the corresponding response characteristics have been presented.The ABC optimization algorithm is employed to tune the parameters of the PID controller.This algorithm explores the parameter space iteratively to find optimal values that minimize the objective function,which in this case is the integral time absolute error.Minimizing this error helps to reduce the deviation of the output voltage from its reference value while minimizing overshoots and ensuring a smooth transient response.

The use of ABC tuning provides an alternative and effective method for determining optimal PID gains.It enables a global search to find suitable gains,especially for complex systems with multiple control loops.The effectiveness of the ABC-tuned PID controller lies in its ability to achieve smooth and effective control in the AVR system.

Fig.7 (MPC control) illustrates that the MPC exhibits exceptional time response characteristics for the AVR system.It yields a stable step response with minimal overshoot.The MPC controller achieves superior stability within a relatively short timeframe, specifically before 0.4 s, surpassing the performance of other controllers in achieving stability during this period.Table 2 shows the results obtained from MPC control.

Due to the MPC’s optimal control design, the terminal voltage remains largely unaffected by changes occurring in the system parameters.Additionally, the maximum overshoot difference with the MPC controller is limited to approximately 0.5 %, outperforming other approaches.The MPC controller effectively handles dynamic systems characterized by rapidly changing dynamics and fast transients, showcasing its robustness.

2.2 Validation of proposed MPC-based controller for AVR control

a) Proposed MPC controller under ±50 % step change in voltage

MPC demonstrates its efficacy as an alternative and robust control method for an AVR system, particularly when compared to other control methods.Fig.10 clearly shows that the MPC-controlled AVR system exhibits a faster and smoother transient response compared to the PID-controlled model.The MPC controller achieves quicker adjustments and smoother transitions in response to a ± 50 % step change in rated voltage just before 0.5 s and 1.5 s respectively during the first and second change in reference voltage.This capability ensures rapid and accurate voltage regulation, minimizing settling time and transient oscillations.The performance and robustness of the proposed MPC strategy are assessed through comparisons with other control techniques using unit step references and different references for the terminal voltage.

Fig.10 also demonstrates that the MPC controller provides the best-damped voltage response compared to other control techniques,such as the auto-tuned PID and ABCbased PID, as well as the Integral of Time-Weighted Squared Error PID designed using Harris Hawks Optimization, particularly under varying reference values during transients [40,41].

b) Proposed MPC controller under disturbance to control input

Based on Fig.11,it can be observed that when a disturbance is injected into the control input, the MPC outperforms PID controllers in terms of minimizing oscillations.The peak voltage overshoot of MPC remains below 1.5 pu of the reference value, while the PID controllers exceeded 2.0 pu above the reference.Additionally,MPC maintains the settling time requirement as it achieves this overshoot value.Also,MPC is renowned for its ability to handle disturbances and uncertainties robustly.It achieves this by employing a predictive model of the system, which allows it to anticipate future behavior and make control decisions accordingly.This predictive capability enables the MPC controller to proactively respond to disturbances and adjust its control action, ensuring stable voltage regulation.In contrast, the PID control methods struggle to handle disturbances and uncertainties with the same level of robustness.

Considering the nonlinear dynamics and operational constraints of the AVR system, MPC is particularly wellsuited for addressing these complexities.Its predictive nature allows it to explicitly account for nonlinearities and incorporate constraints into the control algorithm.This capability empowers the MPC controller to optimize control actions while satisfying operational constraints, ultimately ensuring stable and reliable voltage regulations.

c) Proposed MPC controller under parameter variation

Following Fig.12, MPC once again demonstrates its adaptability and flexibility in control design.It possesses the capability to handle changes in the system by updating the predictive model and optimizing control decisions accordingly.This adaptability ensures that the control system can accommodate variations in system parameters,load conditions, and disturbances without the need for manual retuning or adjustment of control parameters.Fig.12 illustrates how MPC maintains excellent stability,with no steady-state error, even after a -5% shift in the system gains and time constants listed in Table 3 while the other controllers were unable to maintain the reference after parameter drift.

As a result, MPC emerges as a robust alternative for AVR control systems compared to PID and other classical control methods, especially when considering transient response and robustness.In comparison to similar works that fail to address PID controller performance under more complex scenarios such as parameter variation and control input uncertainties [40,42], MPC stands out.Its ability to provide faster and smoother transient responses,superior performance with different references, robustness to disturbances,handling of nonlinearities and constraints,and adaptability to system changes make it an effective choice for achieving accurate and stable voltage regulation in AVR systems.All of these aspects highlight the benefits and stability of the proposed algorithm,indicating that the MPC controller can effectively handle diverse operating conditions,and control input disturbances,and equipment wear and tear while maintaining stable and accurate voltage regulation.

The demonstrated control methods for AVR each come with notable disadvantages and limitations.PID controllers, while widely used, can be complex to tune and sensitive to noise,potentially leading to steady-state errors and overshoot.The ABC-tuned PID controller, though effective in optimization, may suffer from computational intensity and a tendency to converge on local minima,making it less adaptable to changing system dynamics.Meanwhile, MPC offers predictive capabilities but relies heavily on accurate system models, which can be complex to develop and computationally demanding to implement.Additionally, MPC’s performance is contingent on the correct specification of constraints, and inaccuracies in the model can lead to suboptimal control.Collectively,these challenges highlight the need for careful consideration when selecting and implementing control strategies for AVR systems.

3 Conclusions

The research presented in this paper addresses the importance of achieving stable voltage regulation in the power grid through the use of AVRs.The study focuses on the modeling and control of an AVR system in hydroelectric power plants.Traditional PID controllers are commonly employed for AVR control but their performance can vary depending on tuning.This research introduces a robust control strategy utilizing an MPC controller and an ABC-designed PID controller.Simulation results demonstrate the effectiveness and robustness of the MPC control technique compared to the ABC-designed PID controller and other optimization and PID control methods for AVR systems.The MPC controller exhibits superior performance in terms of settling time, peak overshoot, and stability under a step change in voltage.The controller’s robustness is further evaluated by subjecting the AVR system to varying step inputs within a range of -50 % to +50 % of the rated input value, disturbance injection to the control input signal, and a -5% variation in the system parameters.The MPC controller demonstrates superior performance in handling these variations.The implementation of the controller using Matlab/Simulink software showcases their ability to enhance the performance characteristics of the AVR system, including steady-state error, settling time, and overshoot.

This research provides a valuable contribution to power plant engineers in achieving state-of-the-art automatic voltage regulation.The proposed control strategy offers simplicity, versatility, and potential for application in various engineering domains.It surpasses complex mathematical and graphical approaches and shows promise for future work in automatic generation control.However,the MPC controllers can sometimes be computationally intensive due to the predictive nature of the control algorithm, which may require more computational resources compared to simpler control methods and may require specialized software and hardware resources, which can add complexity and cost to the entire system.Overall,the study demonstrates the effectiveness of the MPC in comparison to intelligent PID controllers in achieving stable and accurate voltage regulation, thereby improving the reliability and performance of the power grid.The future scope of the work presented in this paper will involve combining automatic generation control (AGC)with AVR.AGC is an essential aspect of power system operation that aims to maintain the balance between generation and load demand in real-time.By integrating AGC and AVR, the overall performance and stability of the power grid can be further enhanced.

Authors’ contributions

Willy Stephen Tounsi Fokui and Ebunle Akumpan Rene contributed to the investigation, data curation,establishing methodology, and formal analysis, and were major contributors to writing the manuscript.

The manuscript was read and approved by all authors.

Funding

The authors received no funding for this work.

Data availability

All data and materials have been included in the manuscript.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

The authors thank their various institutions for the exposure given to them.