Optimal design of fractional order fuzzy PID controller using an intelligent hybrid algorithm for nonlinear power system

Abstract

Generally, increase in energy demand and consequently unwanted events in a current system will lead to instability. To increase the power system reliability, the current system can be expanded, resulting in a high financial burden. Therefore, one of the most effective options is appropriate controllers usage for stability. In order to overcome the defects of classical controllers such as non-linear, complex uncertain systems, an appropriate mathematical model needs to be designed in limited working conditions. In this paper, a new fractional-order fuzzy proportional-integral-deferential (FOPID) controller is proposed. The proposed controller in its structure is an integral, derivative gain with a fractional order. This controller is structurally adjustable with two fractional orders, performing the stability process in a short time. On the other hand, in the proposed controller, the optimal adjustment of the controller gain and membership functions has turned into an optimization problem, which is done by a hybrid algorithm based on the virus colony search (VCS) and artificial bee colony (ABC). Local and final search powers in the proposed hybrid algorithm reduce the possibility of local presence dramatically. Simulation results have shown that the proposed controller achieves the higher robustness, the lower fall time, and the lower frequency oscillations compared to the existing controllers.

Keywords

Fractional-order fuzzy controller intelligent hybrid algorithm virus serach colony bee colony optimization low frequency oscillation

Abbreviations

VCS:

virus colony search

FOPID:

fractional-order fuzzy proportional-integral-deferential

PID:

proportional-integral-deferential

ABC:

artificial bee colony

HVDC:

high voltage direct current

SVCs:

static variable compensators

PSO:

particle swarm optimization

HVCS-ABC:

hybrid VCS-ABC

QPSO:

quantum behaved particle swarm optimization

UPFC:

unified power flow controller

COA:

chaotic optimization algorithm

PNN:

probabilistic neural network

FLC:

fuzzy logic controller

FLPSS:

fuzzy logic power system stabilizer

NB:

negative big

NS:

negative small

zero

PS:

positive small

PB:

positive big

PD:

proportional-deferential

CA:

conclusion angle

CS:

conclusion space

CO:

conclusion optimize

CMA-ES:

evolution strategy with covariance matrix adaptation

FPID:

fuzzy PID

1 Introduction

Controlling systems play an important role in security maintenance of a system and its restoration to normal performance in respect to unplanned events. The frequency-based regulation of system performance, known as low-frequency stability, is one of the most important controlling issues in the design and operation of today’s systems [1]. The correct and safe operation of the system involves adapting the capacity of the production units to the system load and losses (i.e. the operating point). The disruption of this equilibrium causes a deviation in the system’s work point [2]. In this imbalance state, the frequency of the system is also deviated from its predetermined rated value and this systemic frequency disturbance can affect the performance of the entire system. In a way that the system’s equipment may be damaged, the protection plans may be defeated, resulted in the current system unstable. Therefore, appropriate control measures should be taken into account to restore the work point balance of the system in facing different disturbances [1].

In order to facilitate the investigation of stability and to obtain useful attitude toward the nature of stability issues, the rotor angle stability phenomenon should be divided into the following categories [2]:

- A small disturbance or small signal stability shows the system’s ability in maintaining a synchronized state due to small disturbances. These disturbances are constantly occurring due to small changes in load and production. The disturbances can be sufficiently small to allow the linearization of the system equations and to investigate stability. In spite of the existence of automatic voltage regulator, the system will be stable against small disturbances, only when system oscillations are ensured to have sufficient damping. Instability usually shows itself as oscillations with increasing ranges [3].

In today’s systems, small disturbance stability occurs mainly due to lack of oscillatory damping. Some stability oscillations are described as follows:

The local modes of the system that are related to the production unit oscillations compared to the rest of the system. Various equipment including rectifiers, controllers, turbines, and so on have typically a frequency range between 1 to 2 Hz [4].

Inter-regional modes that are related to the fluctuations of a number of electrical equipment in a system segment, compared to other parts that have typically frequency range between 0.1 to 1 Hz.

Controlling modes that are related to controllers. If the stimulation systems, governors, high voltage direct current (HVDC) converters and reactive power static variable compensators (SVCs) are not properly set, in this cases the instability of these modalities shall usually occur [1].

The torsional modes associated with the rotating components; the instability of these modes may occur due to the interaction of the components with the excitation system, the governor, and the HVDC controllers [1].

- Transient stability shows the ability of the system in maintaining a stable state due to the occurrence of a grave transient disturbance. In this case, the system shall usually change. In such a way that the operating point of the stable state of the system differs before and after disturbance [1].

Identification and segregation of these oscillations behavior can be used for applying control actions such as excitation system and power system stabilizer parameters tuning in order to control the oscillations damping. Determining the frequency of the system’s critical modes and participation factors of the generators in these modes are one of the conventional methods for analyzing these oscillations which are calculated using analytical methods such as modal analysis. In this paper, capability of the generators’ participation factor for determining the inter-area and local oscillations are identified by analytical methods. Conventional methods such as modal analysis and time domain simulations are challenging and time consuming tasks [5]. Thus, the generators’ participation factor and power system oscillations type in the critical mode are predicted using a hybrid method containing a feature selection and probabilistic neural network (PNN) part. The advantage of the proposed method is the accuracy, fast calculations, simultaneous prediction of oscillations type and their participation in the dominant state variable of the generators in the critical modes [6].

New methods for designing controllers are proposed along with significant advances in system theory and control, for example, controllers designed based on comparative control theories, robust control, artificial neural networks and fuzzy control [7 –9]. In all of these methods, it has been attempted to overcome the defects in the classical design. Thus, the controller affects the stability of the system and improves the damping of the fluctuations more effectively. In methods, the defects in the classical design are highlighted. Hence, the controller more effectively affects the stability of the system and improves the damping of the fluctuations.

In a general categorization, the provided methods can be classified in classical and optimization intelligence methods. In classical controller methods, speed and frequency are used as control signals alone or as multiple combinations to generate additional rotor torque for damping the frequency fluctuations [9]. The classic leg-lead stabilizer has been realized physically in many cases and has been widely used in the electric power industry. The operation principles of this controller are based on the damping and synchronous torque concepts in the generator [10]. It should be noted that for designing classical damper controllers, a linear model around a given point is utilized and he assumption is that a precise model of the system is available with constant parameters. Variant nature of the loading, production, consumption, and system configuration changes are among salient features of the system. Hence, if the system’s operating point changes, the classical controllers shall disrupt the performance of the system according to the constant parameters, and they will not have a suitable efficiency [11]. In other words, the major disadvantages of classical controllers are ignorance to the model’s uncertainty and the sensitivity to the work point which leads to lack of robust performance. Also, adaptive controllers and variable structure have not been prospering in practical applications due to the complexity of the control algorithm and the switching problem and the use of state variables [12].

The use of intelligent methods in controlling complex and nonlinear systems has been used specially in various fields of science and technology in recent years. Traditional optimization methods include differential-based techniques. Such methods are robust and their effectiveness has been proven in solving various types of optimization problems [13, 14]. In reference [15], a comparison between the classical controller of the power stabilizer and fuzzy stabilizer for a multi-machine system is presented using the particle swarm algorithm. In this method, the linearization structure is used. The presented algorithm is utilized to set up the classical controller parameters and membership functions. In [16], a method for promptly adjustment of the traditional power system stabilizer parameters is provided using a neural network with radial basis functions, which is taught based on the orthogonal-learning least square error algorithm. The basic function of this model is to gather inputs and, consequently, to create an output. Neuronal entrances enter through the donor, which connects to the other neurons through the synapse. In reference [17], the Quantum behaved Particle Swarm Optimization (QPSO) algorithm has been presented, which has fewer parameter than Particle Swarm Optimization (PSO), and is more powerful than PSO, and also provides better performance compared to PSO. In order to ensure that this method is sufficiently robust, it has been investigated in a wide range of exploitation positions and in various system configurations. Investigation has been done about the effect of the proposed controller in the nonlinear time domain and the examination of different indexes under different disturbances and a wide range of different load conditions. The results of the studies show that the QPSO-based unified power flow controller (UPFC) is capable of damping small frequency oscillations and enhancing the dynamic stability of the power system and is much more suitable than the PSO. In reference [18], another algorithm called chaotic optimization algorithm (COA) has been used instead of the PSO algorithm. Easy implementation, low execution time, strong mechanism to escape the optimal local values, can be indicated from among the characteristics of this algorithm which is promising for engineering work.

The fuzzy controller is a controller that does not require a detailed analysis of the network. In reference [19], the complete set of fuzzy controller progression named fuzzy logic power system stabilizer (FLPSS) is indicated. The fuzzy logic combination process has been applied quickly to a variety of controllers, including fuzzy logic controller (FLC), PID FLPSS, and FLPSS in reference [20]. In reference [21], the stability of the power system with the help of a fuzzy adaptive controller on a multi-machine system has been investigated using the particle swarm algorithm. Although the proposed method shows a suitable performance in damping, but in larger disorders, it cannot maintain its resistance. This problem is caused by incoming signal disturbances in the fuzzy controller. Recently, the use of the fractional order PID controller has been introduced due to its flexible and robust structure in controlling various systems in several papers and has been used in control of various processes. The FOPID controller has two more parameters in its structure compared to the integer order-value controllers in PID, one is the integral deferential fraction power and another one is the deferential fractional power, which leads to two more degrees of freedom in controller design and improvement of the system’s dynamics. Therefore, this controller has been utilized in recent years to control the frequency of different systems [22, 23].

In this paper, a fuzzy FOPID controller with structure (FP + FI^α + FD^β) is proposed to get a robust controller damping low frequency oscillations. On the other hand, the optimal adjustment of the parameters of this controller is turned into an optimization issue based on a square-integral resulted from the multiplication of time and frequency variations, which ultimately has been solved by a combinational algorithm based on the search colony of the virus and Bee colony. The search colony algorithm [24] and the Bee algorithm [25] are two successful methods for recent years, according to the study performed; it has been shown that the combination of these two can result in a higher local and final search.

In comparison with other similar optimization methods, the proposed method has the following key benefits:

Unlike a large number of methods, the possibility of locating in local point decreases dramatically.

Optimization of the proposed algorithm does not depend heavily on the objective function and will provide the best final search.

The parameters of this hybrid algorithm are few, which simplifies its execution.

In summary, the innovations of the article are:

The fractional-order design of fuzzy controller using combined algorithm.

Optimal adjustment of proposed fuzzy controller parameters in several steps, including: 1. adjusting coefficients and membership functions; 2. adjusting the weighting coefficients of the rules; and 3. regulating fuzzy rules in order to reduce the computational time.

Considering nonlinear parameters on a multi-machine model to approximate the behavior of the studied system to a real network.

Analysis of the performance of the system in different working conditions, based on the time and frequency domain, and its comparison with other methods available in recent papers

And providing a new high-featured hybrid algorithm to escape from local points and the suitable ability for the final search.

In Section 2, the structure of nonlinear equations governing the system studied is expressed and a test system is introduced for case study. In Section 3, the search virus colony, bee algorithm and the proposed combination method are identified. Section 4 describes how to apply the proposed method and the objective function of the study. In Section 5, the simulation results are compared qualitatively and quantitatively in different scenarios and eventually the final section is dedicated to the conclusion.

2 Modeling the problem under study

2.1 Non-linear modeling of multi-machine system

Nonlinear model of the single-machine system is described with a set of algebraic differential equations that is derived from models of generators, loads and other devices, such as control systems that are connected through algebraic equations of the network. In this paper, the system generator with a two-dimensional model [1] whose equations are expressed as follows is used to simulate the time domain. ${\dot{δ}}_{i} = ω_{b} (ω_{i} - 1)$ (1) ${\dot{ω}}_{i} = {\frac{1}{M}}_{i} (P_{mi} - P_{ei} - D_{i} (ω_{i} - 1))$ (2) ${\dot{E}}_{qi}^{'} = \frac{1}{T_{doi}^{'}} (E_{fdi} - (x_{di} - x_{di}^{'}) i_{di} - E_{qi}^{'})$ (3) ${\dot{E}}_{fdi} = \frac{1}{T_{Ai}} (K_{Ai} (v_{refi} - v_{i} + u_{i}) - E_{fdi})$ (4) $T_{ei} = E_{qi}^{'} i_{qi} - (x_{qi} - x_{di}^{'}) i_{di} i_{qi}$ (5) where T_e, is electric torque, δ is rotor angle, $T_{do}^{'}$ is constant time of excitation circuit, ω is rotor speed, K_A is regulator gain, P_m is mechanical input power, T_A is constant time regulator, P_e output electric power, v_ref is reference voltage, $E_{q}^{'}$ is generator internal voltage, v is terminal voltage, and $E_{q}^{'}$ are the internal voltage of the generator. Since production in a power system is based on synchronous machines (generators), it is a indispensable condition for the system to function properly that all of these machines remain in sync mode with each other. The mechanical equations based on the per-unit multi-machine can be expressed as follows [1]: $\frac{d δ_{i}}{dt} = ω_{B} (S_{mi} - S_{mi 0}) = ω_{i} - ω_{i 0}$ (6) $2 H_{i} \frac{{dS}_{mi}}{dt} = - D_{i} (S_{mi} - S_{mi 0}) + T_{mi} - T_{ei}$ (7) where H_i, T_mi and T_ei are respectively the inertia and the mechanical and electrical torque of ith machine. The per-unit based stator equations without regard for the zero sequence and the stator transient state is as follows [1]: $- (1 + S_{mi 0}) ψ_{qi} - R_{ai} i_{di} = v_{di}$ (8) $(1 + S_{mi 0}) ψ_{di} - R_{ai} i_{qi} = v_{qi}$ (9) where S_mo is the initial slip of the machine, which is considered to be zero in most cases. The index i is for the ith machine. The slip value for the above equations is expressed as follows: $S_{mi 0} = \frac{ω_{i 0} - ω_{B}}{ω_{B}}$ (10)

If the transient states of the stator are ignored and S_mo = 0, the multi-machine equations are rewritten as follows: $v_{di} = - i_{di} R_{ai} - x_{qi}^{"} i_{qi} + E_{di}^{"}$ (11) $v_{qi} = - i_{qi} R_{ai} + x_{di}^{"} i_{di} + E_{qi}^{"}$ (12)

2.2 A fractional-order fuzzy controller

The fractional calculus (derivative and integral with a fractional order) has been a matter of interest for mathematicians for a long time. But its application in engineering issues, especially in modeling and control issues, has not a long history. Physical systems can be modeled with great precision due to the practical freedom existing on the derivative and integral order [26]. There is a similar situation in the systems control area, that is, high efficiency controllers can be designed either for systems with a fractional order or for systems with a integer-order. In this section, the proposed FOPID fuzzy controller idea is presented for low frequency stability.

Before expressing a fuzzy model (FP + FI^α + FD^β), first, the PI^αD^β fractional-order controller model will be expressed. The fractional-order PI^αD^β controller is an appropriate structure with a fractional order used for control purposes. This controller was first introduced in 1999 [26]. Five parameters FOPID controller includes proportional gain, integral gain, derivative gain, integral order, and derivative order. The general operator for derivative, integral and proportional calculating can be expressed as follows: $α D_{t}^{α} = {\begin{matrix} \frac{d^{α}}{{dt}^{α}}, Re (α) > 0 \\ 1, Re (α) > 0 \\ \int_{α}^{t} (d τ)^{- α}, Re (α) > 0 \end{matrix}$ (13)

Considering the fractional order definition and the generalization of the deferential concept, the $\frac{d^{n} y (t)}{{dt}^{n}}$ integer n can be expressed in terms of the $\frac{d^{α} y (t)}{{dt}^{α}}$ non- integer α. Similarly, this concept can be expressed for the integral with the β-fractional-order. The fractional order is defined as follows according to Caputo concept. $α D_{t}^{α} f (t) = \frac{1}{Γ (n - α)} \int_{α}^{t} \frac{f^{(n)} (τ)}{(t - τ)^{α - n + 1}} d τ$ (14)

In the above equation n ≤ n - 1 ≤ α and n are an integer and $Γ (x) = \int_{0}^{\infty} e^{- 1} t^{(x - 1)} dt, x > 0$ is the gamma function, that in a special state n = x is equal to Γ (n) = (n - 1) !. The differential equation of a fractional-order PI^αD^β controller is presented by the transformation function as follows: $u (t) = K_{P} e (t) + K_{I} D_{t}^{- α} e (t) + K_{D} D_{t}^{β}$ (15)

Based on the Laplace transformation function, the continuous conversion equation is formulated as follows: $G (s) = K_{P} + \frac{K_{I}}{S^{α}} + K_{D} S^{β}$ (16)

To use this fractional-order operator in practical research, an approximation of it can be obtained by the integer-order transfer functions: $s^{β} = I \prod_{i = - P}^{P} \frac{s + ω_{i}^{'}}{s + ω_{i}}$ (17)

where 2P is the number of zeros and poles, and the gain (I), zeros ( $ω_{i}^{'}$ ) and poles (ω_i) in the specified frequency range [ω_l, ω_h] can be represented as follows: $\begin{matrix} ω_{i}^{'} = ω_{l} {(\frac{ω_{h}}{ω_{l}})}^{\frac{i + P + 0.5 (1 - β)}{2 P + 1}}, \\ ω_{i} = ω_{l} {(\frac{ω_{h}}{ω_{l}})}^{\frac{i + P + 0.5 (1 + β)}{2 P + 1}}, \\ I = {(\frac{ω_{h}}{ω_{l}})}^{- \frac{β}{2}} \prod_{i = - P}^{P} \frac{ω_{i}}{ω_{i}^{'}} \end{matrix}$ (18)

Now, the proposed fuzzy controller strategy is described by defining the FOPID fractional model. The proposed controller strategy is such that the PI^αD^β fuzzy controller coefficients in the FP + FI^α + FD^β structure are regulated by fuzzy logic. In order to improve the performance of the intended controller, the combined virus colony search (VCS) with the bee colony (ABC) known as (HVCS-ABC) is used. It has a good convergence rate and a good performance for optimization of nonlinear functions, in order to regulate it optimally. The parameters including coefficients, functions membership, FOPID weight coefficients and fuzzy rules are used in several steps. Figure 1 illustrates the design principles of the proposed controller, in which the first layer is FOPID controller and the second layer is fuzzy logic. A fuzzy system is a set of if – then fuzzy rules that describes how to select the PID gain under the specified performance conditions

Fig.1

General structure of the proposed controller which consists of two main stages; The first stage is a PID fraction controller that can be used for acceptable damping and fall time, and the second part is consisted of a fuzzy system to be able to cover uncertainty in the system and to create a resistant control.

A fuzzy set is defined in such a way as to cover the intended domain. For this purpose, a base station is initially generated randomly and then it is regenerated based on the operation of the base station of the fuzzy system in which the three CA, CS and CO parameters are used.

Conclusion line angle: In this reconstruction, CA is the angle of the conclusion line used to generate spaces, and is displayed with4 bits.

Conclusion area: In this reconstruction, CS is a dependent region with fixed distance between assumptions. This parameter is between 0.5 and 1.5, displayed with 4 bits.

Degree of Conclusion Line: In this reconstruction, CO is the angle of the conclusion line and is defined in two states: NB-NS-Z-PS-PB or PB-PS-Z-NS-NB and therefore has a bit.

In this paper, the hybrid algorithm HVCS-ABC is used to optimize the behavior of the proposed fuzzy controllers. The function of this method in the optimal adjustment of the proposed controller parameters is shown schematically in Fig. 2.

Fig.2

The proposed fractional fuzzy controller structure which adjusts the control parameters by using a frequency-based target function and the proposed algorithm.

The idea of this controller rises out of the view that the PD fuzzy controller speeds up the system response and reduces its maximum overshoot, and the integral controller eliminates the stable-state error.

Therefore, by combining the two types of controllers by a fuzzy key, all PD and integral controller properties can be achieved besides improving the controller performance. The controller also has a simple structure with two adjustable parameters and is designed simply. This controller performs the stability action in a short time. The response time can also be changed by adjusting the coefficient of time scale α and β.

3 HVCS-ABC combined algorithm

3.1 VCS standard virus search algorithm

In this section, the mathematical equations of the standard algorithm for virus search shall be exp-ressed. Further details have been described in [24].

- Virus diffusion: The walking algorithm expresses the behavior of the virus in detection of host cell. Gaussian Walks method is an appropriate method for modeling this behavior and avoiding the local optimal response formulated by the following equation: $V_{p o p_{i}^{'}} = G a u s s i a n (G_{b e s t}^{g}, τ) + (r_{1} . G_{b e s t}^{g} - r_{2} . V_{p o p_{i}})$ (19) where i is the index for random selection among the set {1, 2, … N} and N is the total population. $G_{best}^{g}$ is the best answer generated in the repetition of g. r₁ and r₂ are two random variables between 0 and 1. For the Gaussian parameter, τ standard deviation is obtained by this relation:

$log (g) / g . (V_{{pop}_{i}} - G_{best}^{g})$ . In formula, we use vector to avoid local points in which represent the ith place of the V_popi total population. log (g) / g is also intended to improve local search performance. This coefficient has higher oscillations in initial repetitions, which gradually leads to lower fluctuations by increasing the repetition of the program and leads to a better lead towards the optimal final result. $(G_{best}^{g}, τ)$ also guarantees the generation of better responses based on the guidance of the final solution that directs the $(r_{1} . G_{best}^{g} - r_{2} . V_{{pop}_{i}})$ vector.

- Impacting host cell: The host cell’s impact is modeled by the CMA-ES model based on the covariance matrix through following steps:

Step 1: Updating the Hpop with the following relationship:

H_{{pop}_{i}^{g}} = X_{mean}^{g} + σ_{i}^{g} \times N_{i} (0, C^{g})

(20) where N_i (0, C^g) is the normal distribution with the 0 mean and C^g covariance matrix with D×D dimensions, g is the current repetition of the program, D is the dimension of the problem, and σg>0.

X_{mean}^{g}

is expressed by the following initial values:

X_{mean}^{0} = \frac{1}{N} \sum_{i = 1}^{N} V_{{pop}_{i}}

(21)

Step 2: Selecting the best vector of γ from the previous section and considering it as parenting vector as follows:

\begin{matrix} X_{mean}^{g + 1} = \frac{1}{γ} \sum_{i = 1}^{γ} ω_{i} . V_{{pop}_{i}^{γ best}} | ω_{i} = \\ ln (γ + 1) / (\sum_{j = 1}^{γ} (ln (γ + 1) - \ln (j)) \end{matrix}

(22) where γ = [N/2] and w_i, is the coefficient of composition and i index i represents the best answer in the answer set. Therefore, we will have two evolutionary paths according to the following relations:

\begin{matrix} P_{σ}^{g + 1} = (1 - C_{σ}) P_{σ}^{g} + \\ \sqrt{C_{a} (2 - C_{σ}) γ_{ω}} \frac{1}{σ^{g}} {(C^{g})}^{- \frac{1}{2}} (X_{mean}^{g + 1} - X_{mean}^{g}) \end{matrix}

(23)

\begin{matrix} P_{C}^{g + 1} = (1 - C_{c}) P_{C}^{g} + \\ h_{σ} \sqrt{C_{C} (2 - C_{C}) γ_{ω}} \frac{1}{σ^{g}} (X_{mean}^{g + 1} - X_{mean}^{g}) \end{matrix}

(24) where

γ_{w}^{- 1} = \sum_{i = 1}^{γ} w_{i}^{2} . (C^{g})^{- 0.5}

is symmetric, positive, and sets for this condition: ( C ^g) ^-0.5 ( C ^g) ^-0.5 = ( C ^g) ^-1. The computational parameters are normally set with C_σ = (γ_ω + 2)/(N + γ_ω + 3)C_C = 4/(4 + N), and h_σ = 1. Of course, if hσ = 0 then

| | P_{σ}^{g + 1} | |

shall be a big amount.

Step 3: Updating the size of σ ^g +¹ and the covariance matrix with: $σ^{g + 1} = σ^{g} \times exp (\frac{C_{σ}}{d_{σ}} (\frac{| | P_{σ}^{g + 1} | |}{E | | N (0, 1) | |} - 1))$ (25) $\begin{matrix} C^{g + 1} = (1 - C_{1} - C_{γ}) C^{g} + C_{1} P_{C}^{g + 1} (P_{C}^{g + 1})^{T} \\ + C_{γ} \sum_{i = 1}^{γ} w_{i} \frac{(V_{{pop}_{i}^{γ best}} - X_{mean}^{g})}{σ^{g}} . \frac{(V_{{pop}_{i}^{γ best}} - X_{mean}^{g})^{T}}{σ^{g}} \end{matrix}$ (26)

In which, d_σ = 1 + C_σ + 2max ${0, (\sqrt{γ_{w} - 1} / \sqrt{N - 1}) - 1)}$ is usually close to 1 and C_γ and C_γ acts in accordance with the following relationship: $\begin{matrix} C_{1} = \frac{1}{γ_{w}} ((1 - \frac{1}{γ_{w}}) min {1, \frac{2 γ_{w} - 1}{{(N + 2)}^{2} + γ_{w}}} \\ + \frac{1}{γ_{w}} \frac{2}{{(N + \sqrt{2})}^{2}}), C_{γ} = (γ_{w} - 1) C_{1} \end{matrix}$ (27) where, 0 ≤ C_γ ≤ 1 is the update rate for the covariance matrix C.

- Operation of the Immunity System: The function of the immunity system of the body is formulated as follows:

Step 1: Calculating the P_r performance criteria for the V_pop population based on the objective function of the system under study:

{Pr}_{rank (i)} = \frac{N - i + 1}{N}

(28) where N is the total population of V_pop and rank (i) is the mean of the target function of ith of the V_pop population.

Step 2: The growth of each population individually from among the V_pop population with the following relationship:

{\begin{matrix} V_{{pop}_{i, j}^{"}} = V_{{pop}_{k, j}} - rand . (V_{{pop}_{h, j}} - V_{{pop}_{i, j}}) \\ i {f r > \Pr}_{rank (i)} \\ V_{{pop}_{i, j}^{"}} = V_{{pop}_{i, j}} & otherwise \end{matrix}

(29) where k, i, h indices are chosen randomly from the set [1, 2, 3, …, N] in such a way that i ≠ h ≠ k and j ∈ [1, 2, 3, … d . rand and r are random numbers between 0 and 1. The above formulas show that every single answer tries to save its best current value. Also, if the answer is digressed from the scope of the problem, it shall be generated again based on the upper up_ij and lower low_ij limits:

x_{ij} = {low}_{ij} + rand \times ({up}_{ij} - {low}_{ij})

(30)

3.2 Standard bee colony algorithm (ABC)

Non-Ferrum-based algorithms are generally based on the behavior of bees. The artificial bee colony algorithm is a technique for solving optimization problems based on the behavior of honey bees in the nature. In this section, its general relationship is identified; for further study, refer to reference [25].

The coding of this algorithm is as follows:

Initialization as the initial answer to Xij

Calculation of the initial answers in the target function

Repeat the first cycle = 1

Providing new answers based on finding new V_ij food supply in the X_ij vicinity, the following formula is used to generate new responses:

ν_{ij} = x_{ij} + φ_{ij} (x_{ij} - x_{k, j})

(31) where k is the answer obtained in the neighborhood of i and φ_ij and it is a random number between (0, 1).

The choice of the best source or the better answer between X_ij and V_ij.

Calculation of the probability for the X_ij answers based on the following formula:

P_{i} = \frac{{fit}_{i}}{\sum_{i = 1}^{SN} {fit}_{i}}

(32)

In fact, in order to obtain the correctness of the answers we use the following formula: ${fit}_{i} = {\begin{matrix} \frac{1}{1 + f_{i}} & f_{i} \geq 0 \\ 1 + abs (f_{i}) & f_{i} 〈 0 \end{matrix}}$ (33)

The answers to P_i are between (– 1, 1)

Production of new responses (new sources) of V_i based on spectators’ bees from among $X_{i}^{'}$ s answers and their probabilitydetermination, p.

Selection of the best answer (the most nutritious bee) from among the X_ij and V_ij responses.

Determination of the corrupted resources and replacing random resources with those random resources generated by the X_i scout bee using the following formula:

x_{ij} = min_{j} + rand (0, 1) \times (max_{j} - min_{j})

(34)

Saving the best answer (qualified power supply) that has been achieved to this step

Cycle = Cycle+1

Repetition of all previous steps until reaching the condition required for end of the program.

3.3 The HVCS-ABC proposed hybrid algorithm

The bee colony algorithm is inspired by the social behavior of a colony of bees in finding the nectar. In this algorithm, all members (bees) have a merit value that is determined in the objective function. The ABC algorithm starts with a random initialization. Then these values are arranged by new productions (new generations) based on the value of their objective function (suitability). In each repetition, two optimal values are obtained. One is regarded as the best answer in that repetition, and the other is the best obtained answer so far by any bee inthe population.

In the virus colony search algorithm, viruses require host cells to maintain survival, production and replication. That is, their movement shall occur at a time when they are replicated enough and also are adapted to changes in the new environment. The virus’s colonial strategy for survival includes replication and infection of the host cell and the function of the defense system.

Combining or hybridizing involves combining two or more different things to achieve a better result than their separate states. The bee colony algorithm and the virus search colony algorithm have similar characteristics. Including both have an initial random population and have merit value to assess the population. Therefore, the combination of these two methods can leads to the creation of an efficient combined algorithm. Figure 3 shows the overall structure of the proposed HVCS-ABC method. The combined VCS ABC algorithm starts with an initial population. If the problem is the N-dimension, algorithm has 4 N random population. The 4 N member is arranged according to the merit value, and the upper 2 N member enters into the VCS as the initial response, and a new population with 2 N member is created with the ABC algorithm. The mechanism for optimizing the ABC algorithm is applied to the 2 N lower member as a search bee. In applying the optimization mechanism, the new population created by the VCS algorithm is used as a regulator. Furthermore, the best member of this new population and each corresponding member are used as a neighbor. The population generated by application of the nectar search mechanism is combined with the population created by the VCS algorithm and the 4 N new member is regulated according to the suitability and the previous trend will be repeated until the achievementof convergence.

Fig.3

The structure of the proposed hybrid algorithm based on initial sample population, each of which is divided into two parts and improved by the proposed methods, continuous line (Virus Algorithm) and dashed line (bee Colony Algorithm).

4 Application of the proposed algorithm for designing the proposed controller

This section describes how to design a proposed fuzzy controller based on the developed model of the virus colony algorithm. The following steps are followed for the design:

Step 1: Sorting initial data such as the information of the system under study, applying constraints, generator information and related systems, bus information, and so on.

Step 2: Setting the initial parameters of the proposed algorithm, including number of population, number of program repetitions, and number of host cells, Gaussian function coefficients for damaging to the host cell, limit value which denotes the information of the system and imposing limits thesystem.

Step 3: Using the proposed hybrid algorithm to generate new answers in search space.

Step 4: To solve the problem, we need a target function, that through reducing its value, one can adjust the control parameters optimally. Since frequency variations in each region can be directly influenced by control parameters, a time-based and variation speed-based function is introduced. Accordingly, the objective function is based on the square of the frequency variations of each machine.

OBJ = \sum_{i = 1}^{N_{P}} \sum_{j = 2}^{N_{g}} \int_{0}^{t_{sim}} t^{2} (Δ ω_{ij})^{2} dt

(35) where t_sim is the simulation time for the optimization process, Δω is speed variations, Np the number of operating points of the system, and Ng is the number of generators. In this function, the goal is to reduce the overshoot and undershoot and the fall time. In the design of the proposed controller, the range of the parameters is as follows:

\begin{matrix} Minimize J Subject to \\ K_{i}^{P, min} \leq K_{i}^{P} \leq K_{i}^{P, max} \\ K_{i}^{D, min} \leq K_{i}^{D} \leq K_{i}^{D, max} \\ K_{i}^{I, min} \leq K_{i}^{I} \leq K_{i}^{I, max} \\ α_{i}^{min} \leq α_{i} \leq α_{i}^{max} \\ β_{i}^{min} \leq β_{i} \leq β_{i}^{max} \end{matrix}

(36)

Also, in the actual existing system, non-linear features of the system cause inadequate fuzzy controller performance. For this reason, in this paper, the structure of FOPID is added to the fuzzy system under study. In addition to membership functions in the above system, FOPID parameters are obtained as optimization variables using the proposed hybrid algorithm.

Step 5: Selecting the best answer and storing it in memory to upgrade later generations. It should be noted that this set of answers is replaced with the first ones which are considered to be null.

Step 6: Using the objective function introduced in equation (35) and calculating the suitability of the current answers and selecting the best answer.

Step 7: Comparing the best answer with the answer stored in the memory. If this answer be better than the response in the memory, it will replace it; otherwise the previous amount will be stored in memory itself.

Step 8: Upgrading responses based on Fig. 4 and performing an operations similar to step four

Step 9: Replacing the worst set of obtained responses with new random answers and using the proposed combined method for finding the best answer in each local search space.

Step 10: If the indispensable condition for completion is set, optimal answers will be displayed. Otherwise refer to step four

Fig.4

Flowchart of the proposed algorithm for the design of the proposed FOPID fuzzy controller.

Flowchart of the proposed algorithm is shown in Fig. 4.

5 Simulation results

5.1 Reviewing the performance of the proposed controller in the proposed nonlinear system

In this section, the performance evaluation of the proposed algorithm for optimal design of the proposed controller is provided. The performance of the proposed method has been studied in different scenarios, taking into account different working conditions.

In this paper, the 4-machine system of the 2-area system is considered as the system under study. This system is fully described in reference [1]. The reason for selecting this system is the existence of inter-regional fluctuations, which most of today’s research is focused on these fluctuations. The system under study is connected by two 220 kV amps line. Also, two 900 MW and 20 KW generators are situated in each area. The system studied is shown inFig. 5.

Fig.5

The two zone, four machines proposed system, in which the electric machine of each area is equipped with a proposed controller. Due to the power flow from one region to another, the possibility of inter-regional fluctuations becomes important.

Table 1

Shows the optimal parameters obtained for the proposed FOPID fuzzy controller

No	K_P	K_I	K_D	α	β
G₁	31.65	7.32	0.3139	1.091	0.887
G₂	34.12	7.23	0.2291	1.293	0.974
G₃	29.21	8.29	0.3112	1.102	0.987
G₄	33.87	8.23	0.4185	1.392	0.865

In order to design the proposed FOPID fuzzy controller, different operating conditions based on active power (P), reactive power (Q) in terminals’ generator and C1 (capacitive), C2, L1 (inductive) and L2 load points are considered. The following operating conditions are considered.

Load rated value

Increasing load by 25% as heavy load

Reducing load by by 25% as light load

Other working conditions based on the active power and reactive power of generators are shown in Fig. 6.

Fig.6

Operation conditions used for generators in each area based on active and reactive power changes.

Based on the optimization performed by changing the parameters, the fuzzy level along with the membership functions for two fuzzy logic used in Fig. 2 are shown in Figs. 7 –11, respectively.

Fig.7

3Dimentional surface of optimized fuzzy rules for FLC1 output in Fig. 2.

Fig.8

Final result for fuzzy inference for FLC1 output in Fig. 2.

Fig.9

The 3dimentional surface of optimized fuzzy rules for FLC2 output in Fig. 2.

Fig.10

Final result for fuzzy inference for FLC2 output in Fig. 2.

Fig.11

The convergence process process of the proposed algorithm in adjusting the fractional fuzzy controller parameters that has reached the final result by about 100 repetitions.

The convergence method for the algorithm is shown in Fig. 11 for 250 repetitions. As indicated by the convergence figure, the proposed method has a high speed.

In order to compare the performance of the proposed method, three aspects of the controller are taken into consideration according to the following descriptions:

Without a controller

Considering the classical reference controller [1]

The proposed controller

Figure 12 shows the result of the frequency variations of each machine considering the three controllers above. As can be seen, the proposed method has a good ability in reducing the fall time, and the amount of overshoot and undershoot is also reduced highly. Figure 13 shows a demonstration of real and imaginary values. As shown in the figure, the proposed method has been able to move to the left of the bold green line more successfully.

Fig.12

Output speed variations of generators without the line being lost from the grid and applying a three-phase error in normal loading conditions, the proposed method (continuous line), classic controller [1] (dashed-line), and without controller (dotted line).

Fig.13

Distribution of special values in a complex plane using the proposed method and classical controller without controller.

5.2 Analysis of the proposed algorithm

In this section, the performance of the proposed algorithm is compared with other optimization methods. The intended test function is in fact superior in several aspects; firstly, it has many local points and the distances between them are remarkably large. On the other hand, in spite of the existence of abundant valleys and peaks, there are smooth sections too that make the algorithm search much harder. In order to make a better evaluation of the performance of the proposed method compared with other methods, we have selected a larger range. This function is defined as: $\begin{matrix} f (x) = - \sum_{i = 1}^{n} {sin (x_{i}) (sin (\frac{i \times x_{i}^{2}}{π}))}^{2 m} \\ m = 10, i = 1, \dots, n, - 50 \leq x_{i} \leq 50 \end{matrix}$ (37)

In order to make a better comparison, the best control coefficients of the methods used in other papers have been gathered and we considered only the same initial population for them. Figure 14 shows the convergence of the proposed algorithms after 20 different implementations. As shown in the figure, the proposed method has a high speed and precision in finding the final solution.

Fig.14

The comparison of convergence of different methods and the proposed method for standard test function; as shown in the figure, the proposed method has reached convergence with a greater speed.

As shown in Fig. 13, in the stabilizer designed by proposed fuzzy method, the rotor angle reaches its final value faster than other methods. Also uplifting in fuzzy method is very low, while in other methods, in addition to the existence of great amounts of uplifting, we also have some down-lifting too. Meanwhile the final value of rotor angle is equal to 75 degree in all stabilizers. On the other hand, as shown in the figure, in fuzzy method, fall time is less than other methods and also in this method; the final speed is much closer to its initial amount. However the amount of down-lifting is more than other methods. Therefore by designing a fuzzy-based stabilizer, we were able to increases the speed of obtaining sustainability and to improve damping.

Figure 15 also shows the distribution of the final result obtained from 30 different implementations using proposed methods on the standard test function.

Fig.15

The amount of observed standard deviation for the introduced standard test function foe 30 different implementations; as it is shown, the proposed method has less standard deviation.

The proximity of the obtained answers from the proposed algorithm indicates its robustness and high performance. It also shows that the proposed method has a smaller standard deviation.

The proposed method is more successful in finding optimal response compared to other mentioned methods. The implementation time of the algorithm depends on the commuter structure and the coding performance of the program. At the same time, the multiple performance of the proposed method indicates an implementation time comparable to other methods (about few seconds). Also the mean and standard deviation of the obtained responses in multiple performance of the program are acceptable and indicates the efficiency of the method for future use. Considering the little time it takes to run the algorithm, it can be used to distribute economic loadings which are based on the load forecasting and will be performed in intervals of several minutes or hours.

In order to compare the proposed controller performance with fuzzy controller, following numerical criteria are used: $\begin{matrix} ITAE = 10000 . \int_{0}^{t_{sim}} t . ((ω_{1} - ω_{2}) + (ω_{1} - ω_{3}) \\ + (ω_{1} - ω_{4}) + (ω_{3} - ω_{4})) . dt \end{matrix}$ $\begin{array}{l} F D = \frac{\sum_{i = 1, j = 1}^{4} α_{1} . O S_{i, j}^{2} + α_{2} . U S_{i, j}^{2} + T_{i, j}^{2}}{4} \\ α_{1} = 4000, α_{2} = 1000 \end{array}$

The numerical comparison results are presented in Table 2.

Table 2

The comparison results in four machine systems for fuzzy controller and proposed controller

Controller	Scenario	Compare Index
		ITAE	FD
FOPID	Base Case	11.029	9.524
	20% increase	15.423	16.625
	20% decrease	20.011	19.712
	2 line tripe	11.324	11.312
FLC	Base Case	14.644	10.562
	20% increase	17.029	17.312
	20% decrease	22.524	21.413
	2 line tripe	15.029	15.839
FPID	Base Case	13.622	9.829
	20% increase	13.827	17.524
	20% decrease	19.435	20.311
	2 line tripe	12.091	13.089
PID	Base Case	14.928	10.423
	20% increase	15.837	18.029
	20% decrease	20.657	21.097
	2 line tripe	13.029	13.873

6 Conclusion

In this paper, a new controller based on fuzzy and fractional functions theory and PID controller is proposed for controlling nonlinear system considering different operating conditions. In other words, the proposed method consists of a (FP + FI^α + FD^β) fractional order PID fuzzy controller optimized with a combined algorithm of the virus colony and the bee colony to reduce the overhead time, the fall time, and the overhead voltage output time of the study system. The proposed fuzzy controller has a better performance comparing the PSS controller for damping system disturbances under hard operation conditions. One of the important points in the simultaneous application of fuzzy stabilizers in FOPID controller form is the coordinated design between these two types of controllers. With inadequate coordination between these two types of controllers, the system will be instable. For this purpose, a new hybrid algorithm has been proposed. Strong capabilities in local and final search compared to other methods are among its obvious feature. The proposed control method has been applied to the four-machine system of two zones. As shown in the results, the proposed method has well been able to conserve the stability of the nonlinear system studied.

It is clear from the numerical examples that the proposed algorithm has a lower standard deviation, thus it proves the robustness of this method in comparison with other methods. Also, the amount of overshot, undershoot, and the fall time are significantly optimized in the system equipped with the proposed controller. The developed algorithm has obtained a good optimal response time, which is improved by about 20% in comparison with other methods. In the future, with regard to the proposed controller, we will attempt to consider a system consisting of different equipment, including wind and photovoltaic, because the effect of this equipment is higher on output frequency and the controller’s function.

Footnotes

Appendix (Optimized adjustment of membership functions)

For an explanation of the performance of the proposed controller, suppose the following ranges: ‘ $K_{I}^{min} \leq K_{I} \leq K_{I}^{max}$ and $K_{D}^{min} \leq K_{D} \leq K_{D}^{max}$ . In this case, in order to describe the variables in a smaller range and to avoid any possible differences between them, these co-efficient are normalized to a range between zero and one according to the following equations: $\begin{array}{l} K_{P}^{'} = \frac{K_{P} - K_{P \min}}{K_{P \max} - K_{P \min}}, \\ K_{D}^{'} = \frac{K_{D} - K_{D \min}}{K_{D \max} - K_{D \min}}, \\ K_{I}^{'} = \frac{K_{I} - K_{I \min}}{K_{I \max} - K_{I \min}} \end{array}$

Now assume that the inputs of the fuzzy system to be e(t) and e′(t). So that the fuzzy system determining the parameters consisted of 3 fuzzy systems, 2 input and one output fuzzy system. We can deduce “if-then” rules.

Suppose that “if-then” fuzzy rules are as bellow. If e(t) belongs to A^L and e′(t) belongs to B^L, then ${\overset{´}{K}}_{p}$ belongs to C^L and ${\overset{´}{k}}_{d}$ belongs to D^L, α belongs to E^L where A^L, B^L, C^L, D^L and E^L are fuzzy sets and L = 1,2, ... M. suppose that the intended range for e(t) and e′(t) are respectively [ $e_{m}^{-}$ , $e_{m}^{+}$ ] and [ $e_{md}^{-}$ , $e_{md}^{+}$ ]. We define fuzzy set so that it is able to cover the intended range. For this purpose, a rule base is generated randomly and then rebuilt based on the function of fuzzy system of rule base in which three parameters of CA, CS and CO are used.

Conclusion line angle: in this reconstruction, CA is the angle of conclusion line used to generate spaces and is represented by 4 bits.

Conclusion region: in this reconstruction, CS is a dependent region with constant distance between assumptions. This parameter is between 0.5 and 1.5 and is represented by 4 bits. Conclusion line degree: in this reconstruction, CO is the line conclusion degree and is defined in both NB-NS-Z-PS-PB and PB-PS-Z-NS-NB situations and therefore possesses one bit. Basically, the behavior of the fuzzy controller depends on design information, including selection of member functions and control rules. In traditional design methods, design information is based on the experience of the exert individuals, which is determined through trial and error. So designing a suitable controller takes a lot of time. In this paper, an improved colony search algorithm is used to optimize the behavior of the proposed fuzzy controllers. The function of this method in an optimal adjustment of control parameters is shown in Fig. 16 as a schematic diagram.

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