Design and robustness analysis of an Automatic Voltage Regulator system controller by using Equilibrium Optimizer algorithm

Highlights

• We propose the usage of the Equilibrium Optimizer (EO) algorithm for the determination of the optimal values of the Proportional – Integral – Derivative (PID) controller parameters of an Automatic Voltage Regulation (AVR) system.

• We propose a novel objective function for PID parameter determination.

• EO is found to be superior for PID parameters determination in the terms of the quality of the system response, in the time required for one iteration and whole optimization process, as well as in terms of the convergence speed in comparison with literature known methods for AVR PID parameters design.

Abstract

In this paper, a novel design method for the determination of the optimal values of the Proportional – Integral – Derivative (PID) controller parameters of an Automatic Voltage Regulation (AVR) system is proposed. This method is based on the usage of the Equilibrium Optimizer (EO) algorithm. In order to demonstrate the applicability and efficiency of the proposed algorithm, the literature review of papers dealing with AVR PID design has been considered first. After that, time responses of the AVR system (including PID controller optimized by EO algorithm) with and without different kinds of disturbances have been compared with corresponding results determined for different literature approaches. Furthermore, the comparisons in terms of the accuracy, requested time for one iteration, the total execution time of the algorithm, and convergence characteristics are performed, too. The results show that the EO algorithm significantly outperforms other techniques for all considered criteria.

Introduction

Synchronous generators (SG) are the most common type of electrical generators in power systems. They are used in both thermal power plant (as machines with round rotor) and hydropower plant (as machines with salient poles). The range of their power is very large – SG can produce micro – Watts to mega – Watts of electrical power [1].

The stability of the power system is measured by the values of the frequency and the voltage. The frequency is denoted as a global parameter because its nominal value is the same for the whole power system, whereas the voltage can be labeled as the local parameter since its nominal value is different for each node of the system. Keeping the terminal voltage of the generator on the desired value is related to reactive power flow, whose control is achieved by adjusting the generator's exciter voltage. The other important parameter, frequency, is linked with active power flow [1,2]. Frequency is kept constant by keeping rotor speed at the desired value, which is achieved with turbine control. This paper focuses on the voltage regulation of the SG.

As was mentioned before, maintaining the voltage of the node at a nominal value under different load conditions is one of the key problems in order to keep the power system stable. For this purpose, it is essential to control the generator's terminal voltage. A closed-loop system whose purpose is to keep the terminal voltage of the SG at the desired value is called an Automatic Voltage Regulation (AVR) system. To be more precise, the AVR system is used to define the exciter voltage of the synchronous generator, based on the measurements of the terminal generator's voltage [1]. The difference between the reference voltage and the measured voltage is used to determine the level of the exciter voltage which is applied to the field winding on the rotor of the generator. Therefore, the stator voltage (also called the terminal voltage) can reach the reference value, which presents the main role of the AVR system. A detailed description of the AVR system will be given in Section 2.

One of the main problems in designing the AVR system is the choice of the appropriate controller. The most used type of controller in the industry, as well as in AVR systems, is the Proportional – Integral – Derivative (PID) controller [3]. Adequacy of PID controller application in the AVR system is proved by a large number of journal papers that deal with this topic [4–39]. Concretely, a conventional PID controller, including real PID, is discussed in [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], the Fractional Order PID (FOPID) is concerned in [22], [23], [24], whereas the fuzzy PID controller is demonstrated in [25]. Also, a certain number of papers deal with unusual types of PID controllers, such as the 2DOF PI controller [26], PID plus second-order derivative (PIDD²) [27], and PID – Acceleration (PIDA) [7]. This paper deals with a conventional PID controller design for AVR systems.

In the available literature, in order to determine optimal values of PID controller parameters, large number of metaheuristic algorithms can be found, some examples are the Genetic Algorithms (GA) [11], Improved Kidney – inspired Algorithm (IKA) [4], Particle Swarm Optimization (PSO) [5,13], Whale Optimization Algorithm (WOA) [7], Cuckoo Search (CS) [8], Chaotic Ant Swarm (CAS) [21], Ant Colony Optimization (ACO) [9,10], Teaching – Learning Based Optimization (TLBO) [12,18], Chaos Optimization Algorithm (COA) [15], Symbiotic Organism Search (SOS) [16], Artificial Bee Colony (ABC) [17], Velocity Update Relaxation PSO (VURPSO) and Craziness PSO (CRPSO) [20], Local Unimodal Sampling (LUS) [18,19], Harmony Search (HS) [18], Many Optimizing Liaisons (MOL) [14] and similar. A large number of metaheuristic methods used in this field signifies that there is room for improvement by using other optimization techniques.

On the other side, the algorithms mentioned above introduce a large number of objective functions that are used during the optimization. The most commonly employed are: Integrated Absolute Error (IAE) [14], Integrated Squared Error (ISE) [7,14,18], Integrated Time multiplied Absolute Error (ITAE) [14] and Integrated Time multiplied Squared Error (ITSE) [14,17,19]. IAE, ISE, ITAE, and ITSE are known as error-based objective functions. Also, a popular objective function is proposed by Zwee Lee Gaing in [5] and it relies on achieving optimal transient response – minimum overshoot, steady-state error, rise time, and settling time. Similar to the work of Zwee Lee Gaing, in [11] D. H. Kim proposes an objective function that tends to minimize overshoot, steady-state error, rise time, and settling time. On the other side, many authors propose novel objective functions that are the combination of error-based functions and transient response characteristics. For example, [4] presents the combination of ITSE and Zwee Lee Gaing's function, [8] presents criterion based on ITAE, overshoot, steady-state error, and settling time, and [15,21] present the combination of only ITAE and overshoot. Four novel objective functions are proposed in [19]: all of them are based on the combination of the overshoot and settling time with ITAE, IAE, ITSE, and ISE, respectively. Certain objective functions take into account only transient response characteristic (but in different mathematical formulation than Z. L. Gaing): minimization of overshoot, rise time, settling time, and steady-state error is proposed in [6], a compromise between overshoot, rise time, and settling time is achieved in [13,20] only overshoot and settling time are considered. In the available literature, the objective functions dealing with frequency-domain parameters can be found [9,16]. One of the rarely used objective functions that take into account complex poles position in the complex plane is proposed in [16] and consists of three addends: first is related with ITAE, second is related with the number of complex poles computed from the characteristic equation and third is connected to the sum of the damping ratios of the complex poles. In [9] are also proposed two novel objective functions: the first of them takes into account only the phase margin and the gain cross – over frequency, while the second one combines eight different system characteristics: overshoot, rise time, settling time, steady-state error, IAE, field voltage, phase margin and gain margin. All these comments on previously published papers also state that we do not have the best objective function for AVR PID design. The term “PID design” stands for the determination of optimal values for PID controller parameters so a certain criterion is satisfied. Therefore, also here we can find that there is a field for improvement.

This paper demonstrates the usage of the Equilibrium Optimizer (EO) algorithm that has never been used before to determine the optimal values of PID parameters. Moreover, a novel objective function that is actually a modified objective function from [5] will be presented. Note, the EO is a novel optimization tool proposed in [28]. EO algorithm belongs to the group of metaheuristic algorithms, which are the most often used in the literature for solving the PID controller optimization problem. Precisely, this algorithm belongs to the population-based metaheuristics since it deals with the population of particles, where each particle represents a possible solution, as it will be briefly explained in Section 4. Unlike most of the mentioned metaheuristic algorithms which are inspired by the social behavior of the animals, the EO algorithm is based on the basic equations from physics and chemistry. The results presented in [28] prove that the EO algorithm outperforms many metaheuristic algorithms. Also, it is shown that the EO algorithm is superior compared to the other high-performance optimizers as Evolution Strategy with Covariance Matrix Adaptation (CMA-ES), Success-History Based Parameter Adaptation Differential Evolution (SHADE), and SHADE with linear population size reduction hybridized with the semi-parameter adaptation of CMA-ES (LSHADE-SPACMA). However, the testing of this algorithm is performed using 58 different functions and using different tests such as Friedman, Holm's, and Bonferroni-Dunn.

In this paper, the performance of the EO algorithm using the novel objective function will be compared to the other literature known algorithms that are used to determine the optimal PID parameters for the AVR system. Additionally, the convergence speed, the time for one iteration, as well as the total execution time for the proposed algorithm will be computed and compared to the other algorithms used in the literature. Finally, the robustness analysis of the EO algorithm will be conducted in order to show its’ strength and superiority to the other used algorithms. The main contribution of the paper is applying the algorithm which provides the best values of transient response parameters of the terminal voltage. The results obtained by the algorithm used in this paper are better than the corresponding results when any other algorithm from the available literature is applied. Furthermore, the proposed algorithm significantly outperforms other techniques for many criteria (accuracy, requested time for one iteration, the total execution time of the algorithm, and so on). Therefore, the obtained results can be used in real-time voltage control of the synchronous generator in real power plants.

The remaining part of this paper is organized as follows. Section 2 provides a detailed description of the complete AVR system. A literature review with the wide presentation of obtained results so far is given in Section 3. Section 4 presents the background and mathematical formulation of the proposed EO algorithm. Simulation results and comparative analyses are presented in Section 5. Finally, the conclusions are given in Section 6.