Design of Optimal Controllers for Automatic Voltage Regulation Using Archimedes Optimizer

Automatic voltage regulators (AVRs) in electrical grids preserve the voltage at its nominal value. Regulating the parameters of proportional–integral– derivative (PID) controllers used for AVRs is a nonlinear optimization issue. The objective function is designed to minimize the settling time, rise time, and overshoot of step response of resultant voltage with subjugation to constraints of PID controller parameters. In this study, we suggest using an Archimedes optimization algorithm (AOA) to tune the parameters of the PID controllers for AVRs. In addition, using an AOA to optimize the parameters of a fractional-order PID (FOPID) controller and a PID plus second-order derivative (PIDD) controller for AVRs is also investigated to validate their effectiveness. The disturbance repudiation and robustness of the AOA-PID controllers are also examined and confirmed. To validate the results of the AOA-PID controllers, they are compared with those of other optimized controllers for convergence speed, the quality of the step response. The results indicate that the AOA functions perfectly and it has good potential for optimizing the PID controller parameters with better step response compared with the PID controller based on other approaches while preferring the results of the AOA– PIDD controller over other kinds of the AOA-PID controllers.


Introduction
An electrical power grid is a complex system with many electrical components that are responsible for electric power generation, transmission, and distribution. In power networks, maintaining the stability and constancy of the voltage level is a major problem. If the voltage is different from its rated level, the performance of this equipment will be deteriorated. Another reason for achieving this control is that the real losses in the transmission system depend on both active and reactive power flow [1][2][3].
However, the essential part of the excitation system is called the automatic voltage regulator (AVR), which is responsible for adjusting the voltage level at the terminals of the synchronous generators under different operating conditions [4]. Also, it is implemented in the power system network to control the reactive power flow as well as guarantee appropriate participation of the reactive power among the synchronous generators that are connected in parallel. So, the constancy of the AVR system against the varying exciter voltage affects the power system security [5].
The refinement of AVR performance is very important, because of the difficulties in achieving the stability and fast response of AVR due to the load variations and high inductance field windings of the alternator [6,7].
This study deals with the control and the operating system of the AVR. The proportional-integralderivative (PID) is the most used controller of AVR systems due to its robust performance under different operating conditions as well as the simplicity of structure [8][9][10][11]. The standard PID (SPID) controller encompasses three control parameters: the proportional gain (K p ), the integral gain (K i ), and the derivative gain (K d ) [12][13][14][15][16].
Furthermore, few studies used the real PID (RPID) controller presented in [17][18][19][20] in which the derived action was filtered, so the RPID controller adds another parameter called the filter coefficient N to the three conventional parameters. There is also other modification over the PID controller namely PID with the derivative of second order (PIDD 2 ) as developed in [21,22] where PIDD 2 controller involving four parameters namely K p , K i , K d , and K d2 , where K d2 is the gain of second-order derivative. Moreover, the fractional order PID controller (FOPID) was developed by adding the fractional calculus to the PID controller as introduced in [23][24][25][26][27][28] to improve the performance of the PID controller. The FOPID controller includes five parameters, viz. K p , K i , K d , λ, and µ, where λ and µ define the order of integration and differentiation, respectively [29,30].
Despite this brief literature survey, the no-free-launch theorem guides us that the estimation of the controller parameters is likely improved based on the recent optimization techniques. So, in this study, a new algorithm called Archimedes optimization algorithm (AOA) that has been presented in 2021 [41], is implemented to identify the optimal parameters of SPID, RPID, FOPID, and PIDD 2 controllers to establish their optimal setting. Also, to achieve this goal, the construction of AVR that was presented in the literature is used. The AOA is inspired using the Archimedes' principle that describes the forces acting on an object immersed in a fluid. The AOA is chosen because its published results are hopeful and outperform other optimizers. Utilization of the AOA has succeeded for optimum distributed generations [42,43].
Performance assessments are carried out to confirm the effectiveness of applying the AOA-based technique. However, all controllers tuned by the AOA are compared with other controllers that are optimized by other reported techniques for convergence speed and quality of step response to validate the superiority of the AOA-PID controllers. The superiority of the AOA-PID controllers is validated by comparing their convergence speed, their quality of step response with other controllers that are optimized by other reported techniques. The disturbance repudiation and the robustness of the AOA-PID controllers are analyzed and proved. In addition, to decide which is the most suitable controller as a regulator in the AVR systems, a comparative study is also carried out. The main contribution of this study is described as follows: Innovative use of the AOA for identification of the controller parameters. Four types of controllers (SPID, RPID, FOPID, and PIDD 2 ) are examined and analyzed for defining the most suitable controller. Comparison of the AOA-PID controllers with other controllers that are tuned by recent optimizers based on simulation results. Investigating the robustness of the AOA based on the controlled AVR system during parametric variation of the system model.
The paper is organized as follows: In Section 2, the modeling of the AVR system and the structure of four different kinds of PID controllers are introduced. The objective function is mathematically formulated in Section 4. Section 4 presents a short overview of the AOA. Section 5 presents the simulation results obtained using the AOA-PID controllers for the AVR and other compared techniques as well as discussion. Section 6 concludes this paper.

AVR System
Disturbances are the commonly found phenomena in electrical systems, which in turn lead to voltage fluctuations and oscillations. These voltage variations deteriorate the equipments in the power system. The AVR is used in these electrical systems to maintain their voltage level at a prescribed value by correcting their terminal voltage. It is installed at many places in the power system, such as generators, transformers, and feeders. Fig. 1 shows the main components of the AVR used to control the terminal voltage of a generator. The main parts of the AVR are controllers, amplifiers, exciters, and sensors. The plant voltage V P (s) is measured and compared with a prescribed value V ref (s) and the error signal V(s) is sent to the controller to calculate the required control signal U(s) that is amplified through the amplifier. The amplifier error signal is used to control the exciter of the generator, which in turn controls the generator terminal voltage V P (s).
The transfer functions of all AVR components, except the controller, are first order with dimensionless gain (K) and time constant (T). The utilized AVR in the literature has nominal values of gains and time constants of amplifier, exciter, generator, and sensor as follows: K A = 10, K E = 1, K G = 1, K S = 1, T A = 0.1, T E = 0.4, T G = 1, and T S = 0.01, respectively [18][19][20][21][22]. The aforesaid values of gains and time constants are used in this study to validate the comparison with previous approaches.
The controller is the pivotal VAR component that is used to improve the system dynamic performance. The PID controller is one of the well-known controllers with four kinds, viz. the SPID, the RPID, the FOPID, and the PIDD 2 whose transfer functions are given in (1)−(4), respectively [40]. Generally, the SPID controller possesses tuning three parameters K p , K i , and K d . There are additional parameters for other kinds of PID controllers as follows: N for the RPID controller, λ and µ for the FOPID controller, and K d2 for the PIDD 2 controller.

Objective Function Formulation
Generally, the parameters of the PID controller to be optimized are either 3-5, depending on its kind. The researchers used numerous objective functions for tuning the parameters of the SPID, RPID, and FOPID controllers, but we have found in the literature that using the following objective function (OF 1 ) yields the best results when compared with others since it guarantees the equilibrium among speed, overshoot, and steady-state error, so it is employed in this research.
where β is a constant whose value is 1, OV is the overshoot (%), E ss symbolizes the steady-state error (pu), t s is the settling time (s), and t r is the rise time (s).
Tuning the parameters of the PIDD 2 controller using OF 1 does not give the best results since its different construction due to the inclusion of additional second-order derivative, thus different objective function (OF 2 ) is innovatively suggested to tune its parameters.
where w 1 −w 5 are the weighting factors, whose values are 6, 4, 2, 1, and 1, respectively, and e is the error signal.
Both OF 1 and OF 2 are subjugated to constraints which are defined by the upper and lower limits of the PID controller parameters.

AOA
Lately, introduced metaheuristic techniques, especially those inspired using physics, have created an interesting result. AOA was inventive with inspirations based on the Archimedes' principle. It mimics the principle of buoyancy force that exerted upward on the object, completely or partially submerged in a fluid and is proportional to the weight of the fluid that has been displaced. However, when the weight of the object is greater than the weight of the fluid displaced, the object will sink. In contrast, the object will float when the object's weight is equal to the weight of the fluid displaced. In AOA, the objects submerged in the fluid are considered as the population individuals. These objects have volume, acceleration, and density which have a major impact on the buoyancy of the object. The concept of AOA is based on reaching the point at which the objects are neutrally buoyant, where the fluid force is equal to zero.
AOA ensures a notable balance between exploitation and exploration that makes AOA appropriate for solving the complex optimization problems that consist of many local solutions since it stores solutions population and investigates a wide extent to find the best optimal global solution.
Furthermore, AOA has a simple design with only a few control parameters; still it is powerful and robust as proved in [41]. In addition, it has the ability to adjust the candidate solutions pool to avoid the trap or located at the suboptimal locations.
Similar to other metaheuristic algorithms based on population, AOA also starts the searching process through initial objects (population) as a candidate solution with random densities, volumes, and accelerations. At this step, all objects are also initialized with their random positions in the fluid. After evaluation of the initial population fitness, AOA works in iterations until it meets the conditions. During each iteration, AOA updates the volume and density of each object. Then, based on the condition of its collision with other adjacent objects, the acceleration of each object is updated. The updated volume, density, and acceleration compute the new position of each object [42]. The mathematical formulation of AOA is described as follows.

Initialization
The position of each object is initialized as follows: where x k is the k-numbered object, N is the population, and ll k and ul k are the lower and upper limits of the search space, respectively. Volume (Vol k ), density (Den k ), and acceleration (Acc k ) for each object are initialized as below: where rand is defined as a dimensional vector that is randomly generated numbers between [0, 1]. In this step, after evaluating the initial population, the object with the best fitness (X best ) is selected.

Update of the Volumes and Densities
The volume and density of the k-numbered object for the iteration t + 1 is updated as follows: Den tþ1 k ¼ Den t k þ rand Á ðDen best ÀDen t k Þ where Den best and Vol best are the density and volume of the best-found object and rand is a random number that is uniformly distributed.

Transfer Factor and Density Operator
Initially, a collision occurs between objects and after some time, the objects endeavor to reach the equilibrium state. This is carried out in AOA by transfer factor TF that is used to transform the search from exploration to exploitation. It also increases progressively with time till reaching 1 and is described as below: where t and t max are the iteration number and the maximum number of iterations, respectively. Similarly, the density decreasing operator D helps AOA in global search.
where D t + 1 decreases with time, providing the ability to converge to the identified promising region. It is worth noting that proper treatment of this variable will guarantee the balance between exploitation and exploration within AOA.

The Existence of Collision Among Objects
The occurrence of collision among objects takes place at TF ≤ 0.5 [43]. After selecting the random material (rm), for iteration t + 1, the object's acceleration ðAcc tþ1 k Þ is updated as follows: where Vol rm , Den rm , and Acc rm are the volume, density, and acceleration of rm, respectively.

The Absence of Collision Among Objects
The collision among objects is blocked at TF > 0.5. Therefore, the object's acceleration is updated at iteration t + 1 as stated in the following equation: where Acc best is the acceleration of the best object.

Normalization of Acceleration
To calculate the percentage of change, the acceleration of the object is normalized as below: where L and U are the normalization range and set to 0.1 and 0.9, respectively.
However, Acc tþ1 kÀnorm computes the step percentage that every agent will change. So, the balance between exploitation and exploration is achieved using AOA.

Update of Positions
The used equations include four constants C 1 -C 4 and their values in engineering problems are 2, 6, 2, 0.5, respectively [41]. For iteration t + 1, the exploration stage (if TF ≤ 0.5), the position of k-numbered object ðx tþ1 k Þ is determined as follows: where x rand is the random object.
Furthermore, at the exploitation stage (TF > 0.5), the positions of objects are updated as below: where T is proportional to TF and increases with time and it is described as Also, F is the flag for changing the motion direction according to the following equation:

Evaluation
Each object is evaluated using the OF and the best solution is remembered, that is, x best , Vol best , Den best , and Acc best are assigned.

Results with Discussion
This section includes the obtained results of applying AOA to optimize the parameters of four kinds of the PID controller for AVR, namely SPID, RPID, FOPID, and PIDD 2 . Tab. 1 presents the upper and lower bounds for the parameters of the four kinds of the PID controller. Tab. 2 lists the parameters of AOA. All four kinds of the AOA-PID controller are compared with other optimizers, according to their results.

Step Response
The increase in reference voltage from 0 to 1 pu at t = 0 is simulated to examine the performance of the SPID, RPID, FOPID, and PIDD 2 controllers for AVR. Tab. 3 lists the optimized gains of the four kinds of the PID controller for AVR using AOA.
The convergence curve of the AOA-PIDD 2 controller with other kinds of the AOA-PID controller for AVR is displayed in Fig. 2a for comparison, and it is also displayed in Fig. 2b because its OF 2 is different. In general, the AOA-PIDD 2 controller attains the best convergence in terms of the minimum OF and convergence speed.
The performance of AOA in optimizing the PID controllers for AVR is validated by comparing with other approaches in the literature according to the time-domain analysis of step response in terms of t s , t r , and OV, as summarized in Tab. 4. We can see that both the AOA-SPID and AOA-RPID controllers result in smaller t s and t r , that is, faster response than the corresponding controllers which were optimized by other approaches via the same OF 1 in the literature, with maintaining the OV at the acceptable values.
The value of t s produced by the AOA-FOPID controller is the smallest in comparison with other tuned FOPID controllers regardless of t r of the AOA-FOPID controller being slightly higher than that of the CS-FOPID controller in [28] due to the obvious smallness of t s of the AOA-FOPID controller. The OV obtained using the AOA-FOPID controller is preserved at satisfactory values. Therefore, we can declare that the AOA-SPID, AOA-RPID, and AOA-FOPID controllers yield a better step response than the corresponding controllers based on other optimizers.  Furthermore, Tab. 4 also shows that the AOA-PIDD 2 controller produces a faster response than both the WOA-PIDD 2 controller in [21] and the PSO-PIDD 2 controller in [22], with maintaining the OV at the satisfactory values. The AOA-PIDD 2 controller results in lesser OV than the MRFO-PIDD 2 controller in [40] with almost the same t s and t r , that is, the response speed is approximately equal. Consequently, we can say that the AOA-PIDD 2 controller attains better equilibrium among speed and overshoot than other optimized PIDD 2 controllers. The following OFs are in [21,22,40] for the optimizers of the PIDD 2 controller with which the AOA-PIDD 2 controller is compared: Fig. 3 shows the resultant voltage of AVR, which is controlled by the AOA-PID controllers. It reveals that the kinds of the PID controller can be arranged according to the response speed and overshoot as following the PIDD 2 controller, FOPID controller, SPID controller, and then RPID controller, and Tab. 4 details this arrangement using a quantitative performance assessment.

Robustness Analysis
The time constants of the amplifier, exciter, generator, and sensor are changed from −50% to +50% of their nominal values to analyze the performance robustness of the AOA-PID controllers for AVR. The

Analysis of Disturbance Rejection
The capability of the AOA-PID controllers for AVR to reject disturbances is analyzed by testing the subjection to both the disturbed control signal and the disturbed load, as shown in Fig. 8.  The disturbed output of the controller is simulated as a step signal of 1 pu in the interval starting at t = 2 s and ending at t = 8 s. In addition, the disturbed load is also simulated as a step signal of 0.3 pu in the interval starting at t = 2 s and ending at t = 3.5 s. The disturbances and the corresponding resultant voltages are displayed in Figs. 9−12, which reveals the AOA-PID controllers' capability of disturbances rejection

Conclusions
In electrical networks, AVR is used to maintain the voltage at a prescribed value. We have used AOA for optimizing the parameters of the PID controllers for AVR. In addition, the AOA-FOPID controller and the AOA-PIDD 2 controller for AVR are also used. The objective function minimizes the settling time, rise time, and overshoot of step response of voltage with restriction of the PID controller parameters within the predefined limits. The results of the AOA-PID controllers for AVR are compared with those of the other algorithms, which indicate that AVRs with the AOA-PID controllers result in the best step response since they achieve better balance among speed and overshoot. The work of the AOA-PID controllers for AVR during abnormal status, for example, variations in time constants and disturbances, has been proved to be perfect in the simulation results. In addition, comparisons among the kinds of AOA-PID controllers revealed that the AOA-PIDD 2 controller has the best performance.