# Fractional Order PID Controller Design for an AVR System Using Chaotic Yellow Saddle Goatfish Algorithm

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

- Firstly, the recently proposed Yellow Saddle Goatfish Algorithm (YSGA) [23] is merged with Chaos Optimization Algorithm [24] in order to obtain novel Chaotic Yellow Saddle Goatfish Algorithm (C-YSGA). Original YSGA can improve the optimization process in terms of accuracy and convergence in comparison to several state-of-the-art optimization methods. The improvement is proven by applying this method on five engineering problems, while the comparison with other methods is carried out by using 27 well-known functions [23]. Additionally, in this paper, the superiority of the original YSGA over several other metaheuristic techniques will be demonstrated on the particular optimization problem. Moreover, an improvement of the YSGA by adding Chaotic Logistic Mapping is introduced. The purpose of merging two algorithms is to additionally improve the convergence speed of the YSGA algorithm. Therefore, the original optimization algorithm for optimal tuning of the FOPID controller will be presented in this paper.
- Afterward, the new objective function that tends to optimize time-domain parameters has been proposed. It is demonstrated that the usage of the proposed objective function provides significantly better results than the other functions proposed in the literature.
- Such an obtained FOPID controller has been compared with those tuned by different optimization algorithms in terms of transient response quality. The conducted analysis clearly demonstrates the superiority of the FOPID controller tuned by C-YSGA.
- Finally, different uncertainties have been introduced to the system in order to examine its behavior. Precisely, the robustness test that implies changing the AVR system parameters is carried out. Also, the ability of the system to cope with the different disturbances (control signal disturbance, load disturbance, and measurement noise) is investigated. During all of the mentioned tests, the FOPID controller tuned by C-YSGA shows significantly better performances compared to the FOPID controller, whose parameters are optimized by the other algorithms considered in the literature.

## 2. Description of the AVR System

- controller,
- amplifier,
- exciter,
- generator, and
- sensor.

_{A}, K

_{E}, K

_{G}

_{,}and K

_{S}stand for gains of an amplifier, exciter, generator, and sensor, respectively, while T

_{A}, T

_{E}, T

_{G}, and T

_{S}are time constants of the amplifier, exciter, generator, and sensor. Values that are considered in this paper are 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 [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. It is important to mention that the gain of the generator K

_{G}depends on the load of the generator. Namely, K

_{G}can take a value from 0.7 (non-loaded generator) to 1 (nominal loaded generator).

_{G}(0.7, 0.8, 0.9, and 1).

_{G}, frequency responses are shown in Figure 3.

_{r}), settling time (t

_{s}), overshoot in percentage (OS), steady-state error (E

_{ss}), frequency response parameters-gain margin (G

_{m}) and phase margin (P

_{m}), as well as the poles of the closed-loop system.

_{p}is the proportional gain, K

_{i}is the integral gain, and K

_{d}is the derivative gain. Integer-order is a specific case of the PID controller, where the integral and the derivative are first order.

## 3. About the Fractional Order Calculus

^{α}) is given by Equation (8):

_{b}, ω

_{h}}. In this study, the order of the filter is chosen to be 9 (N = 4), and the selected frequency range is {10

^{−4}, 10

^{4}} rad/s.

^{δ}, since s

^{n}is already an integer-order derivative.

## 4. Overview of the Literature

_{p}, K

_{i}, K

_{d}, λ, and μ so that the certain objective function achieves the minimum (or maximum) value. The most common performance indicators of the tuned FOPID controller are the transient response parameters of the closed-loop system: rise time, settling time, overshoot, and steady-state error. In order to present the results obtained by the recent studies that deal with FOPID tuning, Table 3 provides optimal values of the FOPID parameters and the corresponding transient response parameters of the acquired AVR system.

_{f}is the voltage of the generator field winding, ω

_{gc}is the gain crossover frequency, u is the control signal (the output of the controller), e

_{load}is the error signal when load disturbances are present, and max_dv is the maximum point of the voltage signal derivative. The weighting coefficients are marked as w

_{1}, w

_{2}, w

_{3}, ..., w

_{8}.

## 5. Proposed Chaotic-Yellow Saddle Goatfish Algorithm

_{1}, p

_{2}, ..., p

_{m}}), each goatfish is initialized randomly between the low boundary (b

^{L}) and the high boundary (b

^{H}) of the search space [23]:

_{i}is a vector that consists of n decision variables (variables that are being optimized). Furthermore, b

^{L}and b

^{H}are also vectors that represent lower and upper boundaries for each decision variable.

_{i}that are the products of Logistic Mapping are introduced as follows:

_{k}has a chaser fish Φ

_{l}and the blocker fish φ

_{g}. Clustering can be made using any of the clustering algorithms. However, the YSGA algorithm uses the K-means clustering algorithm in order to divide the population, as it is described in [23] in detail. The cluster organization of the population is depicted in Figure 10.

_{l}

^{t}, the updated position is Φ

_{l}

^{t}

^{+1}, and the best chaser fish from all clusters is Φ

_{best}

^{t}, where t represents the number of the iteration; the updated law is given by the following Equation:

_{max}stands for the maximum number of iterations):

_{g}

^{t}

^{+1}can be determined based on the following Equation:

_{g}

^{t}and p

_{g}

^{t}

^{+1}represent old and new positions of the goatfish, respectively. The whole described process is iteratively repeated until the maximum number of iterations is reached. The detailed description is provided with the pseudo-code presented in Table 5.

## 6. Simulation Results

#### 6.1. Formulation of the Optimization Problem

_{p}, K

_{i}, K

_{d}, λ, and μ, which need to be optimized so that the controller satisfies the desired performances. The optimization process is guided by the objective function that defines the performances of the AVR system.

_{1}= 1, w

_{2}= 0.02, w

_{3}= 1, and w

_{4}= 5. The values of the coefficients are chosen after many experiments with different combinations. It can be seen that w

_{2}has a significantly lower value than the other three weighting coefficients. The reason for this is that the overshoot in (21) is given in percentage, and its value is always larger than the values of ITAE, settling time, and steady-state error. Concretely, from Table 3, it can be observed that the highest value of overshoot can go to 45%, while the settling time and the steady-state error reach maximum values of 1.9 s and 0.17 pu, respectively. However, it is very important to highlight that the presence of the FOPID controller can make the closed-loop system unstable. In order to surpass that, this paper uses optimization with constraints. In other words, each solution (each set of FOPID parameters) is first tested to examine if the obtained closed-loop system remains stable. If a certain solution makes the system unstable, it is automatically removed, ignoring its fitness value. The size of the population in the C-YSGA algorithm is selected to be 40, and the maximum number of iterations is 50. Also, the lower and upper boundary must be defined for each of the optimization variables. Taking into account previous studies related to this topic, the chosen boundaries that are used in this paper are presented in Table 6.

**K**,

_{p}= 1.762**K**,

_{i}= 0.897**K**,

_{d}= 0.355**μ**

**= 1.26**, and

**λ = 1.032**. The proposed method is compared with all methods presented in Table 3, and the results are provided in Table 7 where the best value is in bold. Note, the best solutions of each method, as it is shown in Table 3, are applied with the proposed fitness function given by (21). It is clear that the new fitness function proposed in this paper has the lowest value when the FOPID parameters obtained by C-YSGA algorithm are used.

#### 6.2. Convergence Characteristics

#### 6.3. Step Response

#### 6.4. Robustness Analysis

_{A}, T

_{E}, T

_{G}, and T

_{S}from −50% to +50% of the nominal value, in steps of 25%. The step response of the AVR system is shown in Figure 13, Figure 14, Figure 15 and Figure 16.

#### 6.5. Rejection of the Disturbances

- One of the most common disturbances not only in the AVR system but generally in every control system is control signal disturbance. In this subsection, the obtained C-YSGA FOPID controller is compared with FOPID controllers tuned by PSO [11], CS [10], and GA [5] algorithms. Control signal disturbance is presented as a constant step signal in the first case, and in the second case as a step signal that lasts from t = 2 s to t = 8 s. Step responses of the AVR system are shown in Figure 18 for both cases.
- Afterward, the load disturbance that is specific mainly for AVR systems is presented. Similarly to control signal disturbance, it is modeled as a step signal that lasts from t = 2 s to t = 3.5 s. The obtained step responses, in this case, are shown in Figure 19.
- The last type of disturbance is measurement noise, which is modeled as white Gaussian noise with the power 0.0001 dBW. Figure 20 presents the step responses of the AVR system when the measurement noise is present.

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AVR | Automatic Voltage Regulation |

SVC | Static Var Compensator |

SG | Synchronous generator |

PID | Proportional-Integral-Derivative |

FOPID | Fractional Order Proportional-Integral Derivative |

YSGA | Yellow Saddle Goatfish Algorithm |

C-YSGA | Chaotic Yellow Saddle Goatfish Algorithm |

PSO | Particle Swarm Optimization |

GA | Genetic Algorithm |

CNC-ABC | Improved Artificial Bee Colony |

CAS | Chaotic Ant Swarm |

MOEO | Multi-Objective Extremal Optimization |

CS | Cuckoo Search |

SSO | Salp Swarm Optimization |

IAE | Integrated Absolute Error |

ITSE | Integrated Time Squared Error |

## Nomenclature

K_{A} | amplifier gain |

K_{E} | exciter gain |

K_{G} | generator gain |

K_{S} | sensor gain |

T_{A} | amplifier time constant |

T_{E} | exciter time constant |

T_{G} | generator time constant |

T_{S} | sensor time constant |

t_{r} | rise time |

t_{s} | settling time |

OS | overshoot |

E_{ss} | steady-state error |

G_{m} | gain margin |

P_{m} | phase margin |

K_{p} | proportional gain |

K_{i} | integral gain |

K_{d} | derivative gain |

λ | order of the integral |

μ | order of the derivative |

e | error signal |

V_{f} | voltage of the generator field winding |

ω_{gc} | gain crossover frequency |

u | control signal |

e_{load} | error signal when load disturbances are present |

max_dv | maximum point of the voltage signal derivative |

w_{1}, w_{2}, w_{3}, ..., w_{8} | weighting coefficients |

P | population |

m | number of goatfishes |

b^{L} | low boundary |

b^{H} | high boundary |

rand | vector of random numbers between 0 and 1 |

y_{i} | product vector of Logistic Mapping |

k | number of clusters |

c_{k} | cluster |

Φ_{l} | chaser fish |

φ_{g} | blocker fish |

Φ_{best} | best chaser fish |

t | number of the current iteration |

t_{max} | maximum number of iterations |

α | step size |

β | Levy index |

Γ | gamma function |

N | normal distribution |

a | exploitation factor |

r | random number between 0 and 1 |

ρ | random number between a and |

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**Figure 6.**Rise time for each method from Table 3.

**Figure 7.**Settling time for each method from Table 3.

**Figure 8.**Overshoot for each method from Table 3.

**Figure 9.**Steady-state error for each method from Table 3.

**Figure 18.**Step responses in the two different cases of the control signal disturbance. (

**a**) constant signal, (

**b**) step signal that lasts from t = 2 s to t = 8 s.

Component | Transfer Function | Range of the Parameters |
---|---|---|

Amplifier | K_{A}/(1 + sT_{A}) | 10 ≤ K_{A} ≤ 400, 0.02 s ≤ T_{A} ≤ 0.1 s |

Exciter | K_{E}/(1 + sT_{E}) | 1 ≤ K_{E} ≤ 10, 0.4 s ≤ T_{E} ≤ 1 s |

Generator | K_{G}/(1 + sT_{G}) | 0.7 ≤ K_{G} ≤ 1, 1 s ≤ T_{G} ≤ 2 s |

Sensor | K_{S}/(1 + sT_{S}) | 1 ≤ K_{S} ≤ 2, 0.001 s ≤ T_{S} ≤ 0.06 s |

Parameter | K_{G} = 1 | K_{G} = 0.9 | K_{G} = 0.8 | K_{G} = 0.7 |
---|---|---|---|---|

Overshoot (%) | 65.214 | 61.3825 | 55.9051 | 50.4818 |

Rise time (s) | 0.2613 | 0.2755 | 0.2945 | 0.3171 |

Settling time (s) | 7.0192 | 6.5237 | 5.4086 | 4.9012 |

Steady-state error (p.u.) | 0.0881 | 0.102 | 0.1108 | 0.1249 |

Closed-loop system poles | −0.51 ± 4.66i −12.48 −99.97 | −0.6 ± 4.46i −12.31 −99.97 | −0.69 ± 4.25i −12.12 −99.97 | −0.79 ± 4.01i −11.92 −99.97 |

Gain margin (dB) | 4.61 | 5.53 | 6.55 | 7.71 |

Phase margin (^{o}) | 16.1 | 19.56 | 23.56 | 28.26 |

Method Number | Reference | K_{p} | K_{i} | K_{d} | μ | λ | t_{r} (s) | t_{s} (s) | OS (%) | |Ess| (pu) |
---|---|---|---|---|---|---|---|---|---|---|

1 | [5] | 0.408 | 0.374 | 0.1773 | 1.3336 | 0.6827 | 1.0083 | 1.512 | 0.0221 | 0.0155 |

2 | [5] | 0.9632 | 0.3599 | 0.2816 | 1.8307 | 0.5491 | 1.3008 | 1.6967 | 6.99 | 0.0677 |

3 | [5] | 1.0376 | 0.3657 | 0.6546 | 1.8716 | 0.5497 | 0.0104 | 1.8796 | 30.8479 | 0.0595 |

4 | [6] | 1.9605 | 0.4922 | 0.2355 | 1.4331 | 1.5508 | 0.1904 | 1.0259 | 4.8187 | 0.0102 |

5 | [7] | 1.0537 | 0.4418 | 0.251 | 1.1122 | 1.0624 | 0.2133 | 0.6145 | 5.2398 | 0.0153 |

6 | [7] | 0.9315 | 0.4776 | 0.2536 | 1.0838 | 1.0275 | 0.2259 | 0.564 | 3.7006 | 0.0098 |

7 | [8] | 0.9894 | 1.7628 | 0.3674 | 0.7051 | 0.9467 | 0.1823 | 1.8835 | 58.315 | 0.0409 |

8 | [8] | 0.8399 | 1.3359 | 0.3511 | 0.7107 | 0.9146 | 0.1998 | 1.8727 | 44.8059 | 0.0146 |

9 | [8] | 0.4667 | 0.9519 | 0.2967 | 0.2306 | 0.8872 | 0.3041 | 1.986 | 45.2452 | 0.1768 |

10 | [9] | 2.9737 | 0.9089 | 0.5383 | 1.3462 | 1.1446 | 0.0769 | 0.388 | 8.6266 | 0.0086 |

11 | [10] | 2.549 | 0.1759 | 0.3904 | 1.38 | 0.97 | 0.0963 | 0.9774 | 3.5604 | 0.0321 |

12 | [10] | 2.515 | 0.1629 | 0.3888 | 1.38 | 0.97 | 0.0967 | 0.9849 | 3.5141 | 0.033 |

13 | [10] | 2.4676 | 0.302 | 0.423 | 1.38 | 0.97 | 0.0902 | 0.9933 | 3.2504 | 0.0283 |

14 | [11] | 1.5338 | 0.6523 | 0.9722 | 1.209 | 0.9702 | 0.0614 | 1.3313 | 22.5865 | 0.0175 |

15 | [12] | 1.9982 | 1.1706 | 0.5749 | 1.1656 | 1.1395 | 0.1011 | 0.5633 | 13.2065 | 0.0068 |

Objective Function | Reference |
---|---|

$OF={w}_{1}\cdot OS+{w}_{2}\cdot {t}_{r}+{w}_{3}\cdot {t}_{s}+{w}_{4}\cdot {E}_{ss}+{\displaystyle \int \left({w}_{5}\cdot |e(t)|+{w}_{6}\cdot {V}_{f}{(t)}^{2}\right)}dt+\frac{{w}_{7}}{{P}_{m}}+\frac{{w}_{8}}{{G}_{m}}$ | [4] |

${J}_{1}={\omega}_{gc},\text{}{J}_{2}={P}_{m}$ | [5] |

$IAE={\displaystyle \int |e(t)|dt}$ | [6] |

$ZLG=(1-{e}^{-\beta})\cdot (OS+{E}_{ss})+{e}^{-\beta}\cdot \left({t}_{s}-{t}_{r}\right)$ | [7,10] |

${J}_{1}={\displaystyle \int t{e}^{2}(t)dt},\text{}{J}_{2}={\displaystyle \int \mathsf{\Delta}{u}^{2}(t)dt},\text{}{J}_{3}={\displaystyle \int t{{e}_{load}}^{2}(t)dt}$ | [8] |

${J}_{1}=IAE,\text{}{J}_{2}=1000|{E}_{ss}|,\text{}{J}_{3}={t}_{s}$ | [9] |

$OF={w}_{1}\cdot OS+{w}_{2}\cdot {t}_{s}+{w}_{3}\cdot {E}_{ss}+{w}_{4}{\displaystyle \int |e(t)|}dt+{w}_{5}{\displaystyle \int {u}^{2}(t)dt}$ | [11] |

$OF={({w}_{1}\cdot OS)}^{2}+{w}_{2}{{t}_{s}}^{2}+\frac{{w}_{3}}{{\left(\mathrm{max}\_dv\right)}^{2}}$ | [11] |

$ITAE={\displaystyle \int t|e(t)|dt}$ | [12] |

Pseudo-Code of the C-YSGA |
---|

Enter the input data: m, k, t_{max}, λInitialize the population P using chaotic logistic mapping According to the fitness values determine Φ _{best}Split the population into k clusters and determine the chaser fish Φ _{l} for each clusterwhile (t < t _{max})for each cluster Update the position of the chaser fish and blocker fish Calculate the fitness value of every fish Exchange the roles if any blocker fish has better fitness value than the chaser fish Update the Φ _{best} if the chaser fish has better fitness valueIf the fitness value of the chaser fish has not improved, increase the counter q by 1 If the counter q is higher than λ then apply the formula for the change of the zone end for t = t + 1 endwhile Φ _{best} is the output result of the algorithm |

Parameter | Lower Bound | Upper Bound |
---|---|---|

K_{p} | 1 | 2 |

K_{i} | 0.1 | 1 |

K_{d} | 0.1 | 0.4 |

λ | 1 | 2 |

μ | 1 | 2 |

Method Number | Proposed | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|

OF value | 1.08 | 24.6 | 47.3 | 50 | 10.1 | 8.4 | 4.5 | 12.3 |

Method number | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |

OF value | 10.1 | 53.6 | 2.3 | 4.8 | 4.8 | 3 | 9.8 | 3.1 |

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## Share and Cite

**MDPI and ACS Style**

Micev, M.; Ćalasan, M.; Oliva, D.
Fractional Order PID Controller Design for an AVR System Using Chaotic Yellow Saddle Goatfish Algorithm. *Mathematics* **2020**, *8*, 1182.
https://doi.org/10.3390/math8071182

**AMA Style**

Micev M, Ćalasan M, Oliva D.
Fractional Order PID Controller Design for an AVR System Using Chaotic Yellow Saddle Goatfish Algorithm. *Mathematics*. 2020; 8(7):1182.
https://doi.org/10.3390/math8071182

**Chicago/Turabian Style**

Micev, Mihailo, Martin Ćalasan, and Diego Oliva.
2020. "Fractional Order PID Controller Design for an AVR System Using Chaotic Yellow Saddle Goatfish Algorithm" *Mathematics* 8, no. 7: 1182.
https://doi.org/10.3390/math8071182