A novel optimized fractional-order hybrid fuzzy intelligent PID controller for interconnected realistic power systems

Abstract

This article presents a novel hybrid fractional-order intelligent proportional-integral-derivative (PID) controller integrated with a fractional-order fuzzy PID controller for load frequency control of a realistic interconnected two-area power system, where each area is powered from a different source. The strategy presented here involves incorporating a damping controller based on a thyristor-controlled series capacitor in the system. In doing so, the parameters of the current controller are optimized via craziness-based particle swarm optimization. The effectiveness of the strategy presented here in load frequency control is validated by extensive simulation examples. The results are compared with a recently published technique based on a fractional-order PID controller. Finally, the sensitivity analysis of the plant is studied by varying the plant parameters and operating load conditions from their prescribed values. It is observed that the current controller is a robust controller and performs satisfactorily with variations in system parameters and load patterns.

Keywords

Two-area interconnected power system fractional-order intelligent PID controller fractional-order fuzzy PID controller thyristor-controlled series capacitor device particle swarm optimization

Introduction

Electrical power distribution is highly significant; power companies are responsible for providing an uninterrupted, reliable, efficient and effective power supply to their customers with an acceptable quality. A modern electrical power system network is made up of different controlled areas; for stable operation of power system utilities, the total generation of each controlled area must match total load demands plus associated plant losses; in addition, system frequency and power exchange must be regulated accordingly (Gomaa Haroun and Li, 2017). This strategy, termed load frequency control, plays a significant role in electrical power plant operation and control. Load frequency control involves continuously observing the system frequency, calculating net deviations of the system frequency from the prescribed value as the area control error (ACE) and accordingly adjusting the valve settings of the generators so as to minimize the ACE (Huang et al., 2017). Thus, the objective of load frequency control is to regulate power produced from various sources in each area so that the frequency of the power system and tie-line power are kept within stated values (Guha et al., 2016a; Huang et al., 2017; Zamani et al., 2016). The diverse generation units in the electrical power system are coherently interconnected by a stiff network; this is why the frequency deviations are supposed to be equal in an area. An adequate controller design for the power plant must cope with load demand and system perturbations, and it should provide an acceptable power quality while preserving both voltage and frequency within stated limits. With the ever increasing size and complexity of modern power systems, insufficient control might deteriorate the frequency and the system oscillation might spread into a wide area, resulting in a system blackout.

Hitherto, several control strategies, such as sliding-mode control, fuzzy logic control, optimal control, adaptive control, self-tuning control, robust and intelligent control, have been designed for complex power systems to enhance their dynamic performance under the occurrence of load perturbations. A survey of the literature reveals that, owing to its simple user-friendly architecture, most scientists are focusing on the proportional-integral-derivative (PID) controller or its alternative to handle load frequency control problems (Arya and Kumar, 2017; Ganguly et al., 2017; Gomaa Haroun and Li, 2017; Guha et al., 2016a, 2016b, 2017; Pradhan et al., 2016). Gomaa Haroun and Li (2017) suggest optimization based on a novel hybrid fuzzy logic intelligent PID (FLiPID) strategy for an interconnected complex power plant with nonlinearities. It has been shown that this FLiPID strategy surpasses other controllers in improving the system dynamic performance. Pradhan et al. (2016) designed a fuzzy PID controller for an interconnected complex power system, in which the controller gains are tuned using the firefly algorithm.

Recently, fractional-order PID (FOPID) has been introduced; this is a generalized type of the classical PID controller (Arya and Kumar, 2017; Morsali et al., 2017; Panda et al., 2012). Regarding this controller, the derivative and integrator parts have non-integer orders; therefore, the order should be determined by the designer. As a result, would have five scaling parameters would have to be determined for the controller (Arya and Kumar, 2017). Arya and Kumar (2017) and Morsali et al. (2017), have presented a FOPID controller based on optimization algorithms to develop a load frequency control strategy for a power system according to time domain performance indices.

In addition, the significance in power plant schemes of utilizing power electronic devices in the form of a flexible alternating current transmission system (FACTS) is extensively recognized (Morsali et al., 2016; Zare et al., 2015); this system provides improved resilience and adaptability. This extra flexibility allows the independent adjustment of certain system variables, such as tie-line power flows, which are not easily controllable. Owing to its fast dynamic performance, a number of FACTS devices, such as the thyristor-controlled series capacitor (TCSC) have been utilized in transmission lines of interconnected power systems to regulate tie-line power fluctuations (Morsali et al., 2016; Zare et al., 2015). Zare et al. (2015), presented a new TCSC structure and proved that the TCSC device greatly enhances the dynamic performance of a power plant when inserted in series along the tie-line.

In most existing load frequency controls applied to power systems, linear models are studied but non-linearity limitations are neglected. However, evaluation of the performance of load frequency control without consideration of the generation rate constraint and the governor dead band might not represent a realistic controlled area in complex power systems.

In this article, a novel hybrid fractional-order intelligent PID (FOiPID) controller integrated with a fractional-order fuzzy PID (FOFPID) controller is suggested as a secondary controller strategy for a load frequency control mechanism. The strategy presented here is incorporated into an interconnected multi-area power system with diverse sources in the presence of a TCSC-based damping controller. The FOiPID controller is a new strategy introduced in this field; it is considered effective in solving the load frequency control problem. This technique is based on an extended state observer, which estimates the uncertainties of the plant and guarantees that the trajectory tracking error will rapidly tend to zero. Furthermore, to compensate for the uncertainties of the plant, the FOFPID controller is designed and added in parallel with a FOiPID controller, which is subjected to the plant, thus the controller is termed a FOiPID-FOFPID controller. Accordingly, the performance of the interconnected multisource power system is evaluated using the FOiPID-FOFPID controller incorporated with a TCSC device along the tie-line. Several simulations are conducted on the plant to validate the performances of the presented strategy.

The following are the main contributions of this article:

A new structure of a fractional-order-based hybrid fuzzy intelligent PID controller coordinated with a TCSC device is proposed for load frequency control.

The controller’s gains are adapted using a craziness-based particle swarm optimization algorithm.

The interconnected complex power system is assessed using the suggested scheme, in which the physical constraints for nonlinearities are taken into account.

The dynamic performances of the current controller is confirmed by comparing the results with a recently published article based on a FOPID controller for an identical system (Morsali et al., 2017).

Sensitivity analysis is conducted to prove the robustness of the suggested controller.

The remainder of this article are organised as follows. Mathematical modelling of the system is addressed next, followed by TCSC modelling. The structures of the FOPID, FOiPID and FOFPID controllers are then elaborated on and the objective function is stated, followed by its optimization solution. Simulation results and comparative analysis are presented. Finally, conclusions are drawn.

Power system under testing and TCSC modelling

System modelling

In this article, a two-area interconnected power system comprising a different generation utility, such as thermal, hydro or gas, in each control area is studied. A schematic diagram of the proposed power plant is displayed in Figure 1; the detailed architecture for a realistic power plant is established in Figure 2.

Figure 1.

Interconnected two-area power plant.

Figure 2.

Transfer function model of multisource interconnected power system, comprising thermal, hydro and gas generation utilities.

All the power plants in each control area are combined to make a control area that can be represented by an equivalent plant dynamics. In Figure 2, $AC E_{1}$ and $AC E_{2}$ are area control errors; $u_{1}$ and $u_{2}$ are the signal outputs from the controllers; $R_{th}$ , $R_{hyd}$ and $R_{g}$ are the speed governor regulation parameters of thermal, hydro and gas, in Hz/p.u. MW; $Δ P_{L 1}$ and $Δ P_{L 2}$ are the load perturbations and $T_{12}$ denotes the synchronizing coefficients. More details of the parameters are described in Zare et al. (2015). The parameter values of the plant are obtained from Morsali et al. (2017) and Parmar et al. (2012), and are listed in the appendix. The appropriate generation rate constraint models for the hydro and thermal utilities and the governor dead band for the thermal plants in both control areas are addressed by Gomaa Haroun and Li (2017).

The governor dead band model of the dead band type can be linearized in terms of the change and the rate of change in speed. Following most recent articles (Morsali et al., 2016; Zare et al., 2015), the Fourier coefficients of $N_{1}$ and $N_{2}$ in the governor dead band transfer function model are taken as $N_{1} = 0.8$ and $N_{2} = - 0.2 / π$ , respectively. The appropriate generation rate constraint model of 10% per minute for the thermal utility of Figure 2 is addressed for both raising and falling rates. For the hydro utility, a typical generation rate constraint model of 270% per minute for the raising generation unit and 360% per minute for the falling generation unit is established (Zare et al., 2015). The power ratings of area 1 and area 2 are represented by $P_{r 1}$ and $P_{r 2}$ , respectively, and are measured in megawatts. For area 1, assume that the quantities $P F_{th}$ , $P F_{hyd}$ and $P F_{g}$ are the shares of power generation from thermal, hydro and gas sources, respectively. The values of these quantities depend on the total load and economic load dispatch. Suppose that $P_{Gth 1}$ , $P_{Ghyd 1}$ and $P_{Gg 1}$ signify the power generation in megawatts by thermal, hydro and gas utilities in area 1, respectively. Under normal operating conditions, there is no mismatch between generation and load; consequently, the power generations from thermal, hydro and gas power utilities are given by (Ibraheem et al., 2014)

\begin{matrix} P_{Gth 1} = P F_{th 1} P_{G 1} \\ P_{Ghyd 1} = P F_{hyd 1} P_{G 1} \\ P_{Gg 1} = P F_{g 1} P_{G 1} \end{matrix}

(1)

Under nominal operating conditions, the total power generated, $P_{G 1}$ for area 1 is given by

P_{G 1} = P_{Gth 1} + P_{Ghyd 1} + P_{Gg 1}

(2)

P_{G 1} = P F_{th 1} P_{G 1} + P F_{hyd 1} P_{G 1} + P F_{g 1} P_{G 1}

(3)

and

P F_{th 1} + P F_{hyd 1} + P F_{g 1} = 1

(4)

For a small load change in area 1, equation (2) can be modelled as

Δ P_{G 1} = Δ P_{Gth 1} + Δ P_{Ghyd 1} + Δ P_{Gg 1}

(5)

Similarly, equations (4) and (5) can be modelled for the area 2 as

P F_{th 2} + P F_{hyd 2} + P F_{g 2} = 1

(6)

Δ P_{G 2} = Δ P_{Gth 2} + Δ P_{Ghyd 2} + Δ P_{Gg 2}

(7)

As highlighted in Figure 2, the inputs to the controllers are their respective ACEs, and the outputs from the controllers are the signals $u_{1}$ and $u_{2}$ . The ACEs of both areas are presented by

AC E_{1} = B_{1} Δ f_{1} + Δ P_{tie 12}

(8)

AC E_{2} = B_{2} Δ f_{2} + Δ P_{tie 21}

(9)

where $Δ P_{tie 12}$ and $Δ P_{tie 21}$ are the perturbations in the tie-line power exchanges in areas 1 and 2, respectively, $B_{1}$ and $B_{2}$ are the frequency bias factors of areas 1 and area 2, respectively, and $Δ f_{1}$ and $Δ f_{2}$ are the frequency errors in areas 1 and 2, respectively.

Mathematical modelling of TCSC in load frequency control

The TCSC is a power electronic device that is incorporated in series with the tie-line between two interconnected power plants to regulate the power flow by varying the line reactance (Morsali et al., 2016; Zare et al., 2015). It provides inductive reactance as well as capacitive reactance compensation to modify the active power flow of the tie-line. This ameliorates the active power exchange for the interconnected power systems. The TCSC is a variable reactance, $X_{TCSC}$ , as mentioned; thus the parameter values are adjusted automatically to regulate the power flow across the tie-lines. When a TCSC is connected in series with the transmission-line between areas 1 and 2, for instance, the power flow between these areas can be expressed as (Zare et al., 2015)

Δ P_{Tie 12} (s) = Δ P_{12}^{0} (s) + Δ P_{TCSC} (s)

(10)

where

Δ P_{12}^{0} (s) = \frac{2 π T_{12}}{s} [Δ f_{1} (s) - Δ f_{2} (s)]

(11)

\begin{matrix} Δ P_{TCSC} (s) \\ = [Δ K_{C} + Δ {K_{C}}^{2} + Δ {K_{C}}^{3} + Δ {K_{C}}^{4} + Δ {K_{C}}^{5} + \dots] \end{matrix}

(12)

in which $Δ P_{12}^{0} (s)$ denotes the transmission-line power flow without the TCSC and $Δ P_{TCSC} (s)$ signifies the impact of the TCSC on the transmission-line power flow interchange. Equation (12) reveals that the power flow interchange can be regulated by controlling the compensation ratio $Δ K_{C}$ as (Zare et al., 2015)

Δ K_{C} (s) = \frac{K_{TCSC}}{1 + s T_{TCSC}} Δ Error (s)

(13)

To attain a more efficient TCSC dynamic-based damping strategy than the basic structure, a lead–lag block is proposed, as shown in Figure 3. Therefore, the frequency deviation in area 1, i.e. $Δ f_{1}$ , may be appropriately characterized as the control signal $Δ Error (s)$ for the TCSC device to handle the percentage deviation change in the system compensation. It should be noted that, according to the design objectives, only the five terms in equation (12) are employed in Figure 3. Thus, the model of the TCSC-based damping controller with phase lead–lag architecture can be stated as

Δ K_{C} (s) = \frac{K_{TCSC}}{1 + s T_{TCSC}} \frac{1 + s T_{1}}{1 + s T_{2}} \frac{1 + s T_{3}}{1 + s T_{4}} Δ f_{1} (s)

(14)

where $K_{TCSC}$ and $T_{TCSC}$ represent the adaptable gain and time constant of TCSC, respectively, $T_{1}$ and $T_{3}$ are the lead time constants, and $T_{2}$ and $T_{4}$ are the lag time constants, which are considered here as $T_{2} = T_{4} = 0.01$

Figure 3.

Structure of damping controller based on thyristor-controlled series capacitor (TCSC).

Controller structures

A review of fractional calculus

Fractional order is generally defined by the operator ${{}_{a}D}_{t}^{α}$ , which is composed of integer-order integral and differential parts, where a and t denote the confines of the operation and $α$ is the fractional order, which is a complex number (Çelik et al., 2016)

{{}_{a}D}_{t}^{α} = {\begin{matrix} d^{α} / d t^{α}, & ℜ (α) > 0 \\ 1, & ℜ (α) = 0 \\ \int_{a}^{t} {(d τ)}^{α}, & ℜ (α) < 0 \end{matrix}

(15)

Based on the Riemann–Liouville approach, the fractional-order differential and fractional-order integral can be stated as in equations (16) and (17), respectively

{{}_{a}D}_{t}^{α} f (t) = \frac{1}{Γ (n - α)} \frac{d^{n}}{d t^{n}} \int_{a}^{t} (t - τ)^{n - α - 1} f (τ) d τ

(16)

where $n - 1 > α \geq n$ , n is an integer number and $Γ (\cdot)$ signifies Euler’s gamma function, and

{{}_{a}D}_{t}^{α} f (t) = \frac{1}{Γ (α)} \int_{a}^{t} (t - τ)^{α - 1} f (τ) d τ

(17)

where $α$ $(0 < α < 1)$ . In general, the Laplace transformation of the Riemann–Liouville fractional derivative or integral is modelled by

L {{{}_{a}D}_{t}^{α} f (t)} = s^{α} F (s) - \sum_{k = 0}^{n - 1} {s^{k}}_{a} D_{t}^{α - k - 1} f (t) |_{t = 0}

(18)

Under zero initial conditions for $n - 1 \geq α < n$

L {{{}_{a}D}_{t}^{\pm α} f (t)} = s^{\pm α} F (s)

(19)

where s signifies the Laplace operator.

FOPID controller

The FOPID or $P I^{λ} D^{μ}$ controller can be formulated as

u_{FOPID} (t) = K_{p} e (t) + K_{i} \frac{d^{- λ} e (t)}{d t^{- λ}} + K_{d} \frac{d^{μ} e (t)}{d t^{μ}}

(20)

where $K_{p}$ , $K_{i}$ and $K_{d}$ represent the gains and $λ$ and $μ$ are the integration and differentiation of fractional order. Hence, the architecture of the $P I^{λ} D^{μ}$ controller has five independent knobs to be optimized (Chathoth et al., 2015; Oustaloup et al., 1995; Tepljakov et al., 2011).

Fractional-order intelligent PID controller

The dynamic structures for a general single-input single-output nonlinear system can be stated as (Fliess and Join, 2008)

y^{(n)} = F + \bar{α} \cdot u_{i}

(21)

where $n \geq 1$ denotes the derivative order of the output y, $u_{i}$ is the control signal, $\bar{α}$ is the input gain and F signifies the aggregate unknown dynamic disturbance. If $n = 2$ , the second-order system can be chosen to define the dynamics of the controlled plant. Consequently, if F and $\bar{α}$ are known, the FOiPID technique can be proposed as

\begin{matrix} u_{i} (t) \\ = \frac{1}{\bar{α}} (- \hat{F} + {\overset{\cdot\cdot}{y}}^{*} + K_{p} e (t) + K_{i} \frac{d^{- λ} e (t)}{d t^{- λ}} + K_{d} \frac{d^{μ} e (t)}{d t^{μ}}) \end{matrix}

(22)

where $y^{*}$ is the desired reference and $e = y^{*} - y$ is the output error $(y = AC E_{i})$ . The estimated value $\hat{F}$ will be obtained by using the extended state observer approach (Yang and Huang, 2009). The extended state observer is more advantageous in control systems, for example, it can be used to estimate the internal and external disturbance or uncertainty in real time (Wu et al., 2016). Therefore, the uncertainty can be estimated by employing an extended state observer tracked by a feedback compensation system to cancel the total disturbance. The extended state observer employed here is

{\begin{matrix} e_{1} = z_{1} - Δ f \\ {\overset{\cdot}{z}}_{1} = z_{2} - β_{1} fal (e_{1}, α_{1}, δ) \\ {\overset{\cdot}{z}}_{2} = z_{3} - β_{2} fal (e_{1}, α_{2}, δ) + b \cdot u_{i} \\ {\overset{\cdot}{z}}_{3} = - β_{3} fal (e_{1}, α_{3}, δ) \\ \hat{F} = z_{3} \end{matrix}

(23)

where $z_{1}$ , $z_{2}$ and $z_{3}$ represent the states of the extended state observer, $β_{1}$ , $β_{2}$ and $β_{3}$ represent the observer gains, $e_{1}$ is the observer error and $u_{i}$ is the control signal.

\begin{matrix} fal (e_{1}, α, δ) = {\begin{matrix} {| e_{1} |}^{α} sign (e_{1}), & | e | > δ \\ e_{1} / δ^{1 - α}, & otherwise \end{matrix} \\ 0 \leq α \leq 1, δ > 0 \end{matrix}

(24)

In the extended state observer architecture, $z_{3}$ signifies the estimate produced by the extended state observer. So, in this case, we have $z_{3} = \hat{F}$ .

According to the aforementioned, the performance of the controller depends on the parameters $K_{p}$ , $K_{i}$ , $K_{d}$ , $λ$ and $μ$ , as well as the estimated value of F. If the result of the observer is not precise, this approach will be ineffective.

Fractional-order hybrid fuzzy intelligent PID controller design

The FOiPID-FOFPID controller, shown in Figure 4, which is composed of a fractional-order fuzzy PID controller integrated with a fractional-order intelligent PID controller is proposed here as a secondary controller for an interconnected multi-area power system. In Figure 4, $K_{e}$ and $K_{d}^{\}$ signify the input factors and $K_{pi}$ and $K_{pd}$ are the output factors. The relationship between the input and the output variables of the typical integer-order fuzzy type PID controller, shown in Figure 4 with the product-sum inference and triangular membership function, as reported by Das et al. (2013), Engin et al. (2004) and Qiao and Mizumoto (1996), can be stated as

u = A + Be + C \overset{\cdot}{e}

(25)

where, the parameters A, B and C are synthesized from the error and control signal (Qiao and Mizumoto, 1996). The fuzzy logic controller (FLC) output will be a function of the fractional-order rate of error instead of the traditional integer-order derivative of the error signal. The output signal of the FOFPID design can be highlighted as

\begin{matrix} u_{FLC - FOPID} (t) & = u_{FLC - FOPI} (t) + u_{FLC - FOPD} (t) \\ = K_{pi} \frac{d^{- λ} u_{FLC} (t)}{d t^{- λ}} + K_{pd} u_{FLC} (t) \end{matrix}

(26)

where $u_{FLC - FOPI} (t)$ represents the output signal of the FLC-type FOPI controller and $u_{FLC - FOPD} (t)$ represents the output signal of the FLC-type FOPD controller. To obtain the final expression for the FOiPID-FOFPID controller as highlighted in Figure 4, equation (22) is combined with equation (26), to give

\begin{matrix} u_{FOiPID - FOFPID} (t) \\ = u_{FOiPID} (t) + u_{FLC - FOPID} (t) \\ = \frac{1}{\bar{α}} (- \hat{F} + {\overset{\cdot\cdot}{y}}^{*} + K_{p} e (t) + K_{i} \frac{d^{- λ} e (t)}{d t^{- λ}} + K_{d} \frac{d^{μ} e (t)}{d t^{μ}}) \\ + K_{pi} \frac{d^{- λ} u_{FLC} (t)}{d t^{- λ}} + K_{pd} u_{FLC} (t) \end{matrix}

(27)

Figure 4.

Fractional-order hybrid fuzzy intelligent PID controller architecture.

Fractional-order hybrid fuzzy intelligent PI+PD controller design

This type of controller is composed of the FLC based on FOPI and FOPD controllers, as shown in Figure 5, with the addition that the input–output scaling factors are chosen independently (Das et al., 2013). The two-part stages of the fractional-order fuzzy PI+PD controller employ the same fractional-order rate for fuzzy inference except after multiplication by diverse optimally optimized, input scaling factors, which modify FLC-1 and FLC-2 in various ways to give the architecture more flexibility. The control output of this scheme is modelled as

\begin{matrix} u_{FLC - FOPID} (t) & = u_{FLC - FOPI} (t) + u_{FLC - FOPD} (t) \\ = K_{pi} \frac{d^{- λ} u_{FLC - 1} (t)}{d t^{- λ}} + K_{pd} u_{FLC - 2} (t) \end{matrix}

(28)

Figure 5.

Fractional-order hybrid fuzzy intelligent PI+PD controller architecture

To obtain the final expression for the fractional-order hybrid fuzzy intelligent PI+PD (FOiPID-FOFPI+PD) controller, as highlighted in Figure 5, equation (22) is combined with equation (28) to give

u_{FOiPID - FOFPID} (t) = u_{FOiPID} (t) + u_{FLC - FOPI + PD} (t)

(29)

The error $(e (t))$ and its fractional derivative $(D^{μ} e (t))$ are selected as input variables of the proposed controllers, as highlighted in Figures 4 and 5, and $(u_{FLC})$ is selected as the output variable of the fuzzy controller. These variables are fuzzified into seven triangular membership functions, which are characterized by linguistic variables, as shown in Figure 6, followed by the rule base associated with the fuzzy controller, as shown in Table 1, which consists of 49 rules.

Figure 6.

Membership function of fuzzy logic controller: (a) input signal $(e (t))$ ; (b) input signal $D^{μ} (e (t))$ ; (c) output signal $u_{FLC}$ .

Table 1.

Fuzzy rule base.

$e (t)$	$D^{μ} (e (t))$
$e (t)$	NL	NM	NS	Z	PS	PM	PL
NL	PL	PL	PL	PM	PM	PS	Z
NM	PL	PM	PM	PM	PS	Z	NS
NS	PL	PM	PS	PS	Z	NS	NM
Z	PM	PM	PS	Z	NS	NM	NM
PS	PM	PS	Z	NS	NS	NM	LN
PM	PS	Z	NS	NM	NM	NM	LN
PL	Z	NS	NM	NM	LN	LN	LN

In this study, the proposed fractional-order-type fuzzy PID controller is designed with the same structure as the FLC, and the final output $(u_{FLC - FOPID})$ is represented as an extra controller output, as posted in Figures 4 and 5. The essential contribution of this work lies in the combination of the output of the fractional-order-type fuzzy PID controller $(u_{FLC - FOPID})$ with the output of the fractional-order intelligent PID controller, which are then tuned in parallel. Finally, the output of the proposed controller $(u_{FOiPID - FOFPID})$ is subjected to the interconnected multisource power system. In this way, the associated gains and the scaling parameters are optimized by employing a craziness-based particle swarm optimization algorithm.

The objective function and its solution

Objective function for controller design

In this article, the integral time-weighted absolute error (ITAE) and the integral time squared error (ITSE) are stated as objective functions to optimize the controller’s parameters (Sahu et al., 2015)

J = ITAE = \int_{0}^{T_{sim}} t \cdot [| Δ F_{1} | + | Δ F_{2} | + | Δ P_{tie, 12} |] \cdot d t

(30)

J = ITSE = \int_{0}^{T_{sim}} t \cdot [Δ F_{1}^{2} + Δ F_{2}^{2} + Δ P_{tie 12}^{2}] \cdot d t

(31)

where $T_{sim}$ signifies the simulation time. An optimization problem is solved using a craziness-based particle swarm optimization technique (Ho et al., 2005) to minimize the objective criterion to attain the optimal parameters, incorporated in the following constraints

\begin{matrix} K_{p}^{min} \leq K_{p} \leq K_{p}^{max}, K_{i}^{min} \leq K_{i} \leq K_{i}^{max} \\ K_{d}^{min} \leq K_{d} \leq K_{d}^{max}, K_{TCSC}^{min} \leq K_{TCSC} \leq K_{TCSC}^{max} \\ T_{TCSC}^{min} \leq T_{TCSC} \leq T_{TCSC}^{max}, T_{1}^{min} \leq T_{1} \leq T_{1}^{max} \\ T_{3}^{min} \leq T_{3} \leq T_{3}^{max}, K_{e}^{min} \leq K_{e} \leq K_{e}^{max} \\ K_{d}^{\ min} \leq K_{d}^{\} \leq K_{d}^{\ max}, K_{e 1}^{min} \leq K_{e 1} \leq K_{e 1}^{max} \\ K_{e 2}^{min} \leq K_{e 2} \leq K_{e 2}^{max}, K_{d 1}^{min} \leq K_{d 1} \leq K_{d 1}^{max} \\ K_{d 2}^{min} \leq K_{d 2} \leq K_{d 2}^{max}, λ^{min} \leq λ \leq λ^{max} \\ μ^{min} \leq μ \leq μ^{max} \end{matrix}

Craziness-based particle swarm optimization

Particle swarm optimization is one of a wide category of swarm intelligence techniques for solving optimization issues; it was first introduced by Kennedy and Eberhart (1995). It can deliver higher-quality solutions than other stochastic algorithms. Particle swarm optimization employs particles that signify potential solutions of the issue. Each particle flies over the space at a certain velocity that is dynamically tuned according to its individual flying experience. The modified position of the ith particle of the swarm $x_{i}$ , and the velocity of this particle $v_{i}$ at the $(t + 1)$ th iteration are defined and updated according to

v_{i}^{t + 1} = v_{i}^{t} + c_{1} r_{1} (p_{i}^{t} - x_{i}^{t}) + c_{2} r_{2} (g_{i}^{t} - x_{i}^{t})

(32)

x_{i}^{t + 1} = x_{i}^{t} + v_{i}^{t + 1}

(33)

with $i = 1, 2, \dots, n$ , where n is the size of the swarm, $c_{1}$ and $c_{2}$ represent the acceleration factors, $r_{1}$ and $r_{2}$ represent uniformly random numbers distributed in the range [0, 1], t denotes the iteration number, $p_{i}$ signifies the best previous position of the ith particle and $g_{i}$ signifies the best particle of all the particles in the swarm. As a result, at the end of the iterations, the best position of the swarm will signify the solution of the problem.

Since the standard particle swarm optimization technique can fall into premature convergence, particularly for intricate problems with many local optimum and optimization parameters (Gomaa Haroun and Li, 2017). The craziness-based particle swarm optimization technique is introduced, which is particularly effective in exploring the global optimum in an intricate search space. The main difference between particle swarm optimization and craziness-based particle swarm optimization is the propagation mechanism to define a new velocity for a particle, which is (Ho et al., 2005)

\begin{matrix} v_{i}^{t + 1} = r_{2} sign (r_{3}) v_{i}^{t} + (1 - r_{2}) c_{1} r_{1} (p_{i}^{t} - x_{i}^{t}) \\ + (1 - r_{2}) c_{2} (1 - r_{1}) (g_{i}^{t} - x_{i}^{t}) \end{matrix}

(34)

where $r_{1}$ , $r_{2}$ and $r_{3}$ denote random parameters uniformly taken from the interval $[0, 1]$ , and $sign$ represents a $sign$ function, modelled as

sign (r_{3}) = {\begin{matrix} - 1, & r_{3} \leq 0.05 \\ 1, & r_{3} > 0.05 \end{matrix}

(35)

In flocks of birds or schools of fish, a bird or a fish will often change direction suddenly (Ho et al., 2005). This is described using a ‘craziness’ factor and is modelled in the primary algorithm (pattern) by employing a craziness variable (Ho et al., 2005). However, this mechanism is eliminated in subsequent patterns by introducing a cornfield vector. To keep the diversity of the particles in an optimization technique, it is essential to retain the craziness operation in a particle swarm optimization algorithm. Therefore, a craziness factor is reintroduced in the suggested algorithm to guarantee that the particle will have a predefined craziness probability to keep the diversity of the particles. Thus, before updating its position using equation (33), the velocity of the particle is crazed by

v_{i}^{t + 1} = v_{i}^{t} + P (r_{4}) sign (r_{4}) v_{i}^{craziness}

(36)

where $r_{4}$ denotes a random parameter, which is chosen uniformly within the range [0, 1]; $v_{i}^{craziness}$ denotes a random parameter, which is uniformly chosen from the interval $[v_{i}^{min}, v_{i}^{max}]$ and $P (r_{4})$ and $sign (r_{4})$ are defined, respectively, as

P (r_{4}) = {\begin{matrix} 1, & r_{4} \leq P_{cr} \\ 0, & r_{4} > P_{cr} \end{matrix}

(37)

sign (r_{4}) = {\begin{matrix} 1, & r_{4} \leq 0.5 \\ - 1, & r_{4} > 0.5 \end{matrix}

(38)

where $P_{cr}$ represents the predefined probability of craziness and is selected to maintain the diversity of the particles; it is taken as 0.32 in this article.

Simulation results and discussion

In the following, the load frequency control designed in this article is tested, through computer simulation, on the complex interconnected two-area power system shown in Figure 2. The dynamic modelling of the system was developed in MATLAB/SIMULINK, while the MATLAB code of the proposed method was written separately in the .m file. The simulation results are compared with the FOiPID and FOPID (Morsali et al., 2017) controllers. In this study, three cases are considered. In case 1, the simulation is performed using the nominal values of the parameters given in the appendix; initially the system is subjected to 1% step load perturbations in area 1, then the simulations are conducted at different times, such that load perturbations give a 0.01 p.u. step change in the load demand of area 1 at $t = 0$ s, followed by a 0.01 p.u. step change in the load demand of area 2 at $t = 10$ s. In case 2, the proposed strategy is investigated for an applied sinusoidal load perturbation of area 1. Finally, in case 3, the system is tested under a wide change of system parameters and loading conditions.

Case 1: step load perturbations

In this scenario, the two-area interconnected power system is initially incorporated with the FOPID, FOiPID, FOiPID-FOFPI+PD and FOiPID-FOFPID controllers under a 1% step load demand in each area as a secondary controller. The optimum factors of the controllers are attained using a craziness-based particle swarm optimization algorithm, furnished in Table 2. The simulation results of the closed-loop control system for the scheduled parameters are shown in Figure 7 and a quantitative analysis of the performances is tabulated in Table 3.

Table 2.

Optimal values of the controller gains coordinated with the thyristor-controlled series capacitor controller.

	$K_{p}$	$K_{i}$	$K_{d}$	$K_{e}$	$K_{d}^{\}$	$K_{e 1}$	$K_{e 2}$	$K_{d 1}$	$K_{d 2}$	$K_{FACTS}$	$T_{FACTS}$	$T_{1}$	$T_{3}$	$λ$	$μ$
FOPID	0.7642	0.9234	0.5201	–	–	–	–	–	–	0.5129	0.6107	0.1207	0.0771	0.9715	0.8342
FOiPID	0.7342	0.9004	0.6201	–	–	–	–	–	–	0.5659	0.5107	0.1607	0.0871	0.9875	0.8352
FOiPID-FOFPID	0.7542	0.9734	0.6501	0.8440	0.9513	0.6029	0.5401	0.5328	0.8851	0.5959	0.2107	0.1107	0.0971	0.9715	0.7342
FOiPID-FOFPI+PD	0.7642	0.9234	0.5201	0.8540	0.8513	0.5029	0.5001	0.5328	0.7851	0.5459	0.1187	0.1187	0.0871	0.9215	0.7342

FOPID: fractional-order PID; FOiPID: fractional-order intelligent PID; FOFPID: fractional-order fuzzy PID; PID: proportional-integral-derivative.

Figure 7.

Scenario I: (a) $Δ f_{1}$ ; (b) $Δ f_{2}$ ; (c) area control error (ACE) deviations in area 1; (d) $Δ P_{tie 12}$ .

Table 3.

System dynamic characteristics with the optimized controllers.

Controller	Settling time (s)				Peak overshoot				Peak undershoot (−ve)				ITAE value
Controller	$Δ f_{1}$	$Δ f_{2}$	$Δ P_{tie 12}$	$AC E_{1}$	$Δ f_{1}$	$Δ f_{2}$	$Δ P_{tie 12}$	$AC E_{1}$	$Δ f_{1}$	$Δ f_{2}$	$Δ P_{tie 12}$	$AC E_{1}$	ITAE value
FOPID	9.94	8.91	11.41	7.89	0.0077	0.0069	0.0001	0.0026	0.0300	0.0371	0.0063	0.0189	1.1790
FOiPID	9.25	8.33	10.71	7.46	0.0064	0.0055	0.0001	0.0025	0.0294	0.0357	0.0061	0.0182	1.0420
FOiPID-FOFPID	2.86	3.50	8.06	4.54	0.0026	0.0025	0.0001	0.0001	0.0254	0.0310	0.0052	0.0163	0.7056
FOiPID-FOFPI+PD	4.12	3.51	10.12	4.55	0.0022	0.0015	0.0001	0.0	0.0254	0.0311	0.0056	0.0164	0.8593

FOPID: fractional-order PID; FOiPID: fractional-order intelligent PID; FOFPID: fractional-order fuzzy PID; PID: proportional-integral-derivative: ITAE: integral time-weighted absolute error.

The simulation results obtained for the suggested controller are compared with the FOiPID and fractional-order PID controllers (Morsali et al., 2017), for the identical power system. It is clear from Figure 7 that during the transient the load demand increases in area 1 but that this load is met by importing power from the area 2; this load then diminishes to zero at steady state. This highlights the fact that in the steady state the local power demand in all areas is met locally. It can also be observed from Figure 7 that the frequency errors $(Δ f)$ , ACEs and tie-line power deviations have been successfully driven to zero by the proposed controller in the presences of load demand changes. It can be seen from Table 3 that the minimum ITAE value is obtained by the FOiPID-FOFPID controller (ITAE = 0.7056); this compares well with the FOiPID-FOFPI+PD (ITAE = 0.8593), FOiPID (ITAE = 1.042) and FOPID (ITAE = 1.179) controllers.

Furthermore, to investigate the robustness of the proposed controller, the load demand is assumed to be 0.01 p.u. step load perturbation in area 1 at $t = 0$ s, followed by 0.01 p.u. step load perturbation in area 2 at $t = 10$ s. The responses of frequency errors and the control signal output regarding load perturbations at different times are depicted in Figure 8. Here again the current controller performs significantly better than the other controllers. Finally, it is obvious from Figure 7 and 8 and Table 3 that the hybrid coordinated FOiPID-FOFPID controller gives better transient performances than the FOiPID, FOPID controllers from the point of view of settling time, peak deviations and ITAE.

Figure 8.

Responses of the step load change: (a) frequency error in area 1; (b) frequency error in area 2; (c) Control signal inputs in area 1.

Case 2: sinusoidal load perturbations

To evaluate the efficiency of the strategy presented here under a continuous load pattern, the sinusoidal load change represented by equation (39) with varying amplitude is applied to area 1 (Pradhan et al., 2016)

Δ P_{L} = 0.03 \sin (4.36 t) + 0.05 \sin (5.3 t) - 0.1 \sin (6 t)

(39)

The responses of the frequency errors, ACEs and tie-line power deviations are shown in Figure 9. The simulations revealed that the frequency errors $(Δ f)$ , ACEs and tie-line power oscillations are restricted most effectively using the FOiPID-FOFIPD coordinated TCSC device. It is concluded that the FOiPID-FOFIPD coordinated TCSC controller coordinated provides excellent stabilizing performance to the oscillations in comparison with the other controllers.

Figure 9.

Scenario 2: (a) sinusoidal load perturbations; (b) $Δ f_{1}$ ; (c) area control error (ACE) deviations in area 1; (d) $Δ P_{tie 12}$ .

Case 3: sensitivity analysis

In this case, sensitivity analysis is performed for the system, under a wide change in system parameters and loading conditions (Gomaa Haroun and Li, 2017; Pradhan et al., 2016). To do this, the operating load conditions $T_{sg}$ , $K_{r}$ and $T_{12}$ are changed in the range of $- 50 %$ to $+ 50 %$ in steps of 25% from their nominal values, individually. Quantitative analysis of the suggested FOiPID-FOFPID controller in terms of ITAE, ITSE, settling times, peak overshoot and peak undershoots with these changed conditions are summarized in Table 4 for 0.01 p.u step load perturbations in area 1. It can be seen from Table 4 that, by applying $\pm 50 %$ with the aforementioned uncertainties, the values of the ITAE, ITSE, settling times, maximum overshoots and minimum overshoots of the oscillations deviate slightly from their prescribed values in Table 3. It can also be observed that by applying the $\pm 50 %$ extreme changes, the power plant is still dynamically stable, since the damping measures are close to the nominal values.

Table 4.

Sensitivity analysis of the system under different scenario uncertainties.

Controller parameter variation	% change	Settling time (s) for 5% band			Peak overshoot			Peak undershoot (−ve)			ITAE	ITSE
Controller parameter variation	% change	$Δ f_{1}$	$Δ f_{2}$	$Δ P_{tie, 12}$	$Δ f_{1}$	$Δ f_{2}$	$Δ P_{tie, 12}$	$Δ f_{1}$	$Δ f_{2}$	$Δ P_{tie, 12}$	ITAE	ITSE
FOiPID-FOFPID controller
Nominal	No change	2.86	3.50	8.06	0.0026	0.0025	0.0001	0.0254	0.0310	0.0052	0.7056	0.0023
Loading condition	$+ 50$	2.99	3.56	8.08	0.0030	0.0029	0.0001	0.0258	0.0355	0.0061	0.8716	0.0034
	$+ 25$	2.97	3.53	8.07	0.0029	0.0027	0.0001	0.0257	0.0332	0.0057	0.7794	0.0028
	$- 25$	2.84	3.49	8.05	0.0019	0.0025	0.0001	0.0218	0.0265	0.0045	0.5843	0.0016
	$- 50$	2.83	3.47	8.05	0.0018	0.0024	0.0001	0.0194	0.0232	0.0051	0.5038	0.0012
$T_{sg}$	$+ 50$	2.86	3.50	8.07	0.0027	0.0025	0.0001	0.0255	0.0310	0.0052	0.7110	0.0024
	$+ 25$	2.86	3.50	8.06	0.0026	0.0025	0.0001	0.0255	0.0310	0.0052	0.7088	0.0023
	$- 25$	2.86	3.50	8.06	0.0026	0.0025	0.0001	0.0254	0.0310	0.0052	0.7025	0.0023
	$- 50$	2.86	3.50	8.06	0.0026	0.0025	0.0001	0.0254	0.0310	0.0052	0.6982	0.0023
$K_{r}$	$+ 50$	2.87	3.50	8.07	0.0027	0.0026	0.0001	0.0254	0.0311	0.0053	0.7238	0.0024
	$+ 25$	2.86	3.51	8.06	0.0026	0.0025	0.0001	0.0254	0.0310	0.0052	0.7149	0.0023
	$- 25$	2.86	3.50	8.06	0.0026	0.0025	0.0001	0.0254	0.0310	0.0052	0.6991	0.0023
	$- 50$	2.86	3.50	8.06	0.0025	0.0025	0.0001	0.0254	0.0310	0.0051	0.6913	0.0023
$T_{12}$	$+ 50$	3.54	4.53	8.05	0.0035	0.0035	0.0001	0.0253	0.0324	0.0057	0.8419	0.0031
	$+ 25$	2.99	4.10	8.06	0.0029	0.0032	0.0001	0.0254	0.0320	0.0053	0.7117	0.0024
	$- 25$	2.88	3.42	8.07	0.0018	0.0024	0.0001	0.0261	0.0292	0.0049	0.6665	0.0021
	$- 50$	2.87	3.41	8.07	0.0018	0.0023	0.0001	0.0265	0.0272	0.0045	0.6576	0.0020
FOiPID-FOFPI+PD controller
Nominal	No change	4.12	3.51	10.12	0.0022	0.0015	0.0001	0.0254	0.0311	0.0056	0.8593	0.0029
Loading condition	$+ 50$	4.15	3.53	10.14	0.0026	0.0018	0.0001	0.0281	0.0352	0.0062	1.0740	0.0044
	$+ 25$	4.13	3.52	10.13	0.0024	0.0017	0.0001	0.0268	0.0340	0.0059	0.9541	0.0035
	$- 25$	4.11	3.49	10.10	0.0021	0.0014	0.0001	0.0221	0.0290	0.0054	0.6910	0.0020
	$- 50$	4.09	3.48	10.10	0.0020	0.0014	0.0001	0.0199	0.0280	0.0052	0.6039	0.0015
$T_{sg}$	$+ 50$	4.14	3.52	10.13	0.0023	0.0016	0.0001	0.0255	0.0312	0.0057	0.8639	0.0030
	$+ 25$	4.13	3.51	10.12	0.0022	0.0015	0.0001	0.0254	0.0311	0.0056	0.8621	0.0029
	$- 25$	4.12	3.50	10.11	0.0022	0.0015	0.0001	0.0254	0.0311	0.0056	0.8568	0.0029
	$- 50$	4.11	3.50	10.10	0.0022	0.0015	0.0001	0.0254	0.0311	0.0056	0.8531	0.0029
$K_{r}$	$+ 50$	4.12	3.51	10.11	0.0023	0.0016	0.0001	0.0254	0.0311	0.0056	0.8675	0.0030
	$+ 25$	4.12	3.51	10.12	0.0022	0.0015	0.0001	0.0254	0.0311	0.0056	0.8640	0.0030
	$- 25$	4.12	3.51	10.12	0.0022	0.0015	0.0001	0.0254	0.0311	0.0056	0.8536	0.0029
	$- 50$	4.12	3.51	10.13	0.0022	0.0015	0.0001	0.0254	0.0311	0.0056	0.8443	0.0028
$T_{12}$	$+ 50$	4.53	3.95	10.07	0.0029	0.0023	0.0001	0.0248	0.0330	0.0059	1.0070	0.0038
	$+ 25$	4.41	3.72	10.09	0.0024	0.0019	0.0001	0.0253	0.0312	0.0057	0.8643	0.0030
	$- 25$	4.00	3.50	10.12	0.0019	0.0013	0.0001	0.0259	0.0300	0.0053	0.8153	0.0026
	$- 50$	3.98	3.49	10.13	0.0018	0.0012	0.0001	0.0260	0.0298	0.0050	0.7874	0.0024

FOiPID: fractional-order intelligent PID; FOFPID: fractional-order fuzzy PID; PID: proportional-integral-derivative: ITAE: integral time-weighted absolute error; ITSE: integral time squared error.

For better insight into the results of the sensitivity analyses, the frequency errors $(Δ f)$ and tie-line power fluctuations under the considered uncertainty $T_{sg}$ and $T_{12}$ are displayed in Figures 10 and 11, respectively. Briefly, it can be concluded that the sensitivity analyses of the system demonstrated the robustness, meaningfully, of the proposed controller under uncertainties. Moreover, once the gains of the suggested controller are adjusted, there is no need to reset for the $\pm 50 %$ perturbations in the synchronizing coefficient, governor time constant of thermal plants, steam turbine reheats constant or loading conditions.

Figure 10.

Scenario IV: dynamic responses of the suggested controller under variation in $(T_{sg})$ .

Figure 11.

Scenario IV: dynamic responses of the suggested controller under uncertainties in $(T_{12})$ .

Conclusion

An attempt has been made in this article to diminish the area frequency and tie-line power oscillations after a sudden load perturbation using a FOiPID-FOFPID controller coordinated with a TCSC in an interconnected multi-area power system, to improve the responses of the power system areas. This approach utilizes an observer based on algebraic techniques to estimate the unknown uncertainties, which are then added to the controller as feedback. The physical constraints of the generation rate constraint nonlinearity and governor dead band impact are also considered in the plant, for a challenging investigation. The craziness-based particle swarm optimization technique has been employed to optimize the controller gains and other parameters in finding a global optimal solution. It is found that, in all the scenarios considered, the area frequency, tie-line power flow deviations and ACE become zero at the steady state.

Responses of the current controller are compared with FOPID and FOiPID controllers under different load perturbations, such as step and sinusoidal load change patterns. Simulations showed that the current controller gives higher dynamic performances than the other controllers in consideration of less settling time, minimum peak overshoot or undershoot of the frequency errors $(Δ f)$ , ACEs and transmission-line power oscillations. Furthermore, a sensitivity analysis shows that the power system subjected to the current controller is meaningfully robust even, when the system is conducted with a large uncertainty in parameters.

Footnotes

Appendix

System parameters

$P_{rt}$	2000 MW (rated capacity of each area)	$T_{w}$	1.1 s
f	60 Hz	$T_{rs}$	4.9 s
$K_{p}$	68.955 Hz/p.u. MW	$T_{rh}$	28.749 s
$T_{p}$	11.49 s	$T_{gh}$	0.2 s
$T_{sg}$	0.06 s	$X_{g}$	0.6 s
$T_{t}$	0.3 s	$Y_{g}$	1.1 s
$R_{th}$	2.4 Hz/p.u. MW	$R_{hyd}$	2.4 Hz/p.u. MW
$R_{g}$	2.4 Hz/p.u. MW	$B_{1}$	0.4312
$B_{2}$	0.4312	$T_{f}$	0.239 s
$C_{g}$	1	$B_{g}$	0.049 s
$T_{12}$	0.0433	$P F_{th}$	0.5747
$P F_{hyd}$	0.2873	$P F_{g}$	0.1380
$a_{12}$	$- 1$	$T_{cr}$	0.01 s
$T_{r}$	10.2 s	$K_{r}$	0.3
$T_{cd}$	0.2 s

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (grant numbers 61871221 and 61273076) and a Chinese governmental scholarship (grant number CSC 2015GXZT30).

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