load–frequency control using multi-objective genetic algorithm...

Load–Frequency Control Using Multi-objective GeneticAlgorithm and Hybrid Sliding Mode Control-Based SMES

Mehrshad Khosraviani1 • Mohsen Jahanshahi2 • Mohsen Farahani3 •

Amir Reza Zare Bidaki4

Received: 22 September 2016 / Revised: 21 March 2017 / Accepted: 30 May 2017

� Taiwan Fuzzy Systems Association and Springer-Verlag GmbH Germany 2017

Abstract This paper aims to better the dynamic response

of interconnected power systems following any load

change using the combination of multi-objective opti-

mization algorithm-based PID and a hybrid adaptive fuzzy

sliding mode. In the proposed method, a hybrid sliding

surface including two subsystems’ information is intro-

duced to produce a control effort to move both subsystems

toward their related sliding surface. A feedback lineariza-

tion control law is mimicked by an adaptive fuzzy con-

troller. To compensate the error between the feedback

linearization and adaptive fuzzy controller, a hitting con-

troller is developed. The design of PID controller is for-

mulated into a multi-objective optimization problem. The

performance of suggested method is assessed on two

interconnected power systems. These results validate that

the suggested method confirms better disturbance rejection,

keeps the control quality in different situations, reduces the

frequency deviations preventing the overshoot and has

more robustness to uncertainties and change in parameters

in the power system.

Keywords Load–frequency control (LFC) � Multi-

objective optimization algorithm � Genetic algorithm �Pareto-set � SMES � Hybrid adaptive fuzzy sliding mode

control

1 Introduction

Frequency can be mentioned as one of the key conditions

for the stability of large-scale interconnected power sys-

tems. In power systems, frequency variations depend on

active power variations. If active power demand/generation

at power systems changes, this is reflected all over the

power system by frequency changes so that by increasing

the active power consuming in an area, the frequency of

power systems will decrease and vice versa [1]. In multi-

area interconnected power systems, any change in fre-

quency can cause severe stability difficulties. To prevent

such a situation, designing a LFC system for controlling the

output active power of generators and tie-line power is

essential. In the traditional LFC, PI controllers are almost

employed. Numerous approaches have been offered in the

papers to adjust the gain of the PI controller [2]. To over-

come the disadvantages of traditional PI controllers, inno-

vative control methods such as fuzzy approach [3], variable

structure [4], adaptive [5], IMC [6] and robust [7] were

recommended for the LFC. Nevertheless, these approaches

are dependent on either knowledge about the states of

system or an effectual online identifier thus may be prob-

lematic to implement in practice. In general, every con-

troller has its own benefits and drawbacks. Linear optimal

& Mohsen Farahani

[email protected]

Mehrshad Khosraviani

[email protected]

Mohsen Jahanshahi

[email protected]

Amir Reza Zare Bidaki

[email protected]

1 Department of Computer Engineering and IT, Parand Branch,

Islamic Azad University, Parand, Tehran, Iran

2 Young Researchers and Elite Club, Central Tehran Branch,

Islamic Azad University, Tehran, Iran

3 Young Researchers and Elite Club, East Tehran Branch,

Islamic Azad University, Tehran, Iran

4 Young Researchers and Elite Club, Buinzahra Branch,

Islamic Azad University, Buinzahra, Iran

123

Int. J. Fuzzy Syst.

DOI 10.1007/s40815-017-0332-z

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controllers are sensitive to parameter change. Training a

neural network (NN) and adaptive neuro-fuzzy inference

system (ANFIS) is a main problem, for the reason that it is

subject to different factors such as the perfect training data,

number of neurons and proper training algorithm. Design-

ing a fuzzy controller necessitates considering the number,

shape and overlap of membership functions, rules and so on.

Hence, the main problem in using ANFIS, NN and fuzzy

controllers is the ability of the user in mathematical sever-

ities, design and implementation.

Furthermore, numerous stabilization methods are used

to efficiently mitigate frequency deviations by developing

the traditional PI controller. In [8], an expanded integral

(I) control has been proposed to acquire zero steady-state

error in addition to having a limited overshoot in dynamic

response after any change in the load. In [9, 10], fuzzy PI

controllers have been suggested for the LFC. In the intro-

duced works, the derivative gain does not exist in LFC

owing to the effect of noise on its performance. However,

investigations confirmed a positive impact of a differential

feedback in the load–frequency control on the mitigation of

frequency deviations [11]. Thus, there exists a compromise

between a suitable mitigation and noise. To lessen the

effect of environment noise, a different derivative structure

with less effect noise was proposed [12]. From that day

forward, researchers focused on PID-LFC. In [13], a PID-

LFC for a single-machine infinite-bus (SMIB) system was

proposed by using the tuning method of PID controller

proposed in [14, 15], and the approach is used to inter-

connected two-area power system [16].

Among methods offered for the LFC, optimization algo-

rithms are popular methods to adjust parameters of LFC so

that different kinds of algorithms such as PSO [17], genetic

[18, 19], bacteria foraging [20], chaotic [21], pattern search

[22], cuckoo algorithm [23] imperialist algorithm [24], firefly

[25] have been proposed for this purpose so far. In all of these

methods, parameters are obtained by use of a weighted sum

method. In this method, the objective function consists of a

weighted sum of the objectives. However, the problem stays

behind the accurate choice of the weights. Recently, non-

inferior (non-dominated, Pareto-optimal) solutions are found

by use of the multi-objective problems (MOPs). In order to

generate such non-inferior solutions, the most widely used

techniques are the weighed min–max method, weighting

method and e-constraint method. The finest solution from the

acquired solution set is chosen by the decision maker.

Reviewing published papers demonstrates that in the

majority works proposed for the LFC, in spite of con-

verging errors of area control to zero efficiently, the

deviations of tie-line power and frequency continue for a

fairly long time. Therefore, a long settling time in these

responses can be expected. In this status, the governor

system may not control the frequency changes, as a

consequence of its slow dynamic [26]. Thus, overcoming

the sudden changes in the load requires a fast response

active power source such as SMES units, FACTS devices.

Using an SMES for every area of an interconnected two-

area system has been proposed in some papers [27, 28]. In

spite of mitigating the deviations of tie-line power and the

frequency usefully, from the point of view of economic,

using an SMES for each area of a power system is not

possible. Thus, a high capacity SMES was located in one of

the areas so that it is available for controlling other areas

[29]. Since mitigating the frequency deviations was not

desirable, the combination of SMES with FACTS devices

such as solid-state phase shifters [28] and SSSC [30] was

proposed. By doing this, notwithstanding the agreeable

mitigation of deviations and oscillations, the economic

efficiency is raised as a major difficulty.

In [31–33], sliding mode controller (SMC) is suggested

for the LFC which confirms suitable transient mitigation of

deviations and oscillations in addition to robustness per-

formance of controller compared to traditional controller.

To design an SMC, feedback gains and switching vectors

are optimized by a genetic algorithm [31]. In order to

optimize the SMC, linear state feedback control is con-

sidered by the author in [31]; in practice, accessing entire

state variables is restricted and measuring all of them is not

feasible [32]. To surmount the difficulty, optimal output

feedback controller is proposed in [32]. In [33], an output

feedback SMC is considered for a LFC system and

teaching and learning-based optimization (TLBO) algo-

rithm is employed to adjust feedback gains and switching

vector. The main drawback of these methods is to tune the

feedback gains and switching vector.

In some papers, the LFC problem is taken into account as

a MOP. In [38, 39], a multi-objective genetic algorithm is

used to optimize PID controllers. Tammam et al. [40] used a

multi-objective algorithm for tuning a fuzzy like PID con-

troller. In spite of advantages of this method such as

robustness to variations and simple structure, this method

suffers from a controller with fixed parameters whose good

performance may be weaken in all operating conditions.

In this paper, an HAFSMC with integral–proportional–

derivative surface is proposed for controlling an SMES for

the power system load–frequency control. A hybrid sliding

surface including two subsystems’ information is devel-

oped to produce a control effort to force both subsystems

toward their related sliding surface. To achieve a maximum

damping of frequency deviations, this method is combined

with a multi-objective optimization algorithm-tuned PIDs.

The objective is to better the dynamic response of an

interconnected power system following changing load

demand. In the suggested approach, a FLC law is mim-

icked by an adaptive fuzzy controller. A hitting controller

is used to balance the compensation error between the

International Journal of Fuzzy Systems

123

feedback linearization and adaptive fuzzy controller. An

adaption law is obtained by the Lyapunov stability theory.

Hence, the SMES unit controlled by the suggested method

in online mode will be able to quickly damp out any

oscillation in the power system. Three separate objective

functions are simultaneously minimized by the suggested

method in order to achieve an optimum LFC. A fuzzy-

based technique is employed to select the finest solution

from the attained Pareto-set [35]. The results of simulation

are provided and compared with a traditional PID con-

troller deigned based on GA, the results obtained from the

tuning method of LFC proposed in [16] and the methods

proposed in [22, 30, 34].

The major difference of this paper with other papers

published in the field of application fuzzy controller and

SMC to power systems is to use a hybrid sliding mode

control. In those controllers, only an objective function is

minimized by the controllers, while in the proposed con-

troller, two objective functions (even more) are minimized

by using the proposed controller.

2 Realistic Load–Frequency Control

Figure 1 displays an interconnected two-area single-source

power system [2]. In Fig. 1, the description of every block

is represented in [21]. A dead zone is also considered in the

speed governor control mechanism. In thermal power sta-

tions, generation rate constraint (GRC) determines maxi-

mum/minimum value of generating power.

To prove the potential of suggested approach, this study

is further applied to a realistic power system with high-

voltage direct current (HVDC) link as shown in Fig. 2. The

realistic power system consists of generating units of gas,

thermal and hydro. Figure 3 displays the linearized model

of governor thermal, hydro turbine and gas turbine.

3 Overview of SMES

The control of the converter firing angle of SMES unit

displayed in Fig. 4 can change the DC voltage across the

inductor constantly. In the beginning, a small positive

voltage charges the inductor to its nominal current Id0. By

2fΔ

+

2LPΔ

Ts

tiePΔSMPΔ

1fΔ1LPΔ

-

+

-

+ +

-

+ -

+

ACE2

Area-2

Load & Machine

B2

Droop Characteristic

PID-2 GRC Turbine Governor

SMES

Dead Zone +

-

- -

- ACE1+

-

Area-1

PID-1 Load & Machine

B1 Droop Characteristic

Governor Turbine GRC

Fig. 1 Block diagram of an interconnected power system along with the SMES unit

Area-1

Thermal,

Hydro, Gas

Area-2

Thermal,

Hydro, Gas Convertor Convertor

AC tie-line

HVDC tie-line

Fig. 2 A realistic power system interconnected by AC–DC tie lines

M. Khosraviani et al.: Load–Frequency Control Using Multi-objective Genetic Algorithm…

123

disregarding the losses of converter and the transformer,

the DC voltage across the inductor can be written as [22]:

Ed ¼ 2Vd0 cos a� 2IdRC ð1Þ

where a denotes the firing angle (in degree); Id is the

current through the inductor (in kA); RC is the equivalent

commutation resistance (in KX); and Vd0 is the maximum

bridge circuit voltage (in KV).

−

+

-

11

R R

R

T K sT s

++

SMES

-

+

+

+

-

-

11 R

31 R

21 R

11 GT s+

11

R R

R

T K sT s

++

11 TT s+ TKGRC

11

RS

RH

T sT s

++

11 GHT s+

11 0.5

W

W

T sT s

−+ GRC HK

1

gcg b s+ GK11

G

G

X sY s

++

11

CR

F

T sT s

++

11 CDT s+

2LPΔ

−

−

+

+

+

−

1DC

DC

KT s+

PID-2

HVDC Link

1DC

DC

KT s+

Turbine dynamic

-

-

Characteristics of unitsDroop

+

+

+

-

-

-

Power system

ACE

Speed govenor Reheat turbine Steam turbine

1LPΔ

Participation

factor

Mechanical-hydraulic govenor Hydro turbineParticipation

factor

Speed governorValve positionerParticipation

factor

11

R R

R

T K sT s

++

−

+

+

+

−

−

11

RS

RH

T sT s

++

11 GHT s+

11 0.5

W

W

T sT s

−+ GRC

21 R

HK

11 R

11 GT s+

11

R R

R

T K sT s

++

11 TT s+ TKGRC

31 R

1

gcg b s+ GK11

G

G

X sY s

++

11

CR

F

T sT s

++

11 CDT s+

PID-1

+−

122 Ts

π

1fΔ

2fΔ

1B

2B

SMPΔtiePΔ

Fig. 3 Configuration of a realistic power system along with HVDC link


123

In the LFC, the input signal of the SMES control loop

controls the Ed continuously. As affirmed in [22], an

instantaneous reaction to the next change in the load needs

the fast restoration of inductor current to its nominal value

after any change in load. To attain this goal, the inductor

current deviation (DId) is employed as a negative feedback

signal in the SMES control loop. Accordingly, the con-

verter voltage applied to the inductor (DEd) and inductor

current deviations (DId) can be written as follows:

DEd sð Þ ¼ 1

1 þ sTc

u sð Þ � kf

1 þ sTc

DId sð Þ ð2Þ

DId sð Þ ¼ 1

sLDEd sð Þ ð3Þ

where UFSMC is the control effort of FSMC; Tc denotes the

converter time constant (in sec); kf represents the feedback

gain of DId; L indicates the coil inductance (in H).

The deviation of SMES unit active power can be written

as:

DPSM ¼ DEd � DId þ DEd� Id0 ð4Þ

The block diagram of SMES loop control with

HAFSMC is illustrated in Fig. 5.

4 SMC

The dynamic of the power system is described as

€x tð Þ ¼ f x tð Þð Þ þ Bu tð Þ þ c tð Þ ð5Þ

where x tð Þ 2 Rn is a state vector, u tð Þ 2 Rm is a control

vector,c tð Þ 2 Rn is a bounded signal that represents uncer-

tainty or disturbance, B 2 Rn is a constant matrix, f(x(t)) is a

map x tð Þ 2 Rn ! f x tð Þð Þ 2 Rn and t represents time. The

control objective is to achieve a suitable control law with the

purpose of the trajectory state x being capable of tracing a

trajectory command xd. A tracking error can be defined as

e ¼ x� xd ð6Þ

The first phase of the design of SMC is to choose a

sliding surface. Then, the controller should be designed in

such a way that the state trajectories of system are moved

in the direction of the sliding surface and remain on it. At

this time, assume that an integral operation sliding surface

is presented as

s tð Þ ¼ k1e sð Þ þ k2

Z t

0

e sð Þdsþ k3 _e sð Þ ð7Þ

where k1 and k2 are positive constants. If the function of the

system dynamic is well known, an ideal controller can be

represented by:

u� ¼ �f xð Þ þ €xd þ k1 _eþ k2e ð8Þ

Substituting the ideal controller (8) into (5), we obtain

€eþ k1 _eþ k2e ¼ 0 ð9Þ

With proper selection of the control gains k1 and k2, the

characteristic polynomial of (9) is strictly Hurwitz, so the

roots of (9) lie strictly in the left half of the complex plane,

and it means that limt!1 e tð Þ ¼ 0. Since the external load

disturbance and the system dynamic are always unknown

or perturbed, the control law u* is not imple-

mentable practically. Hence, an adaptive fuzzy controller

system is employed to mimic the control law in this paper.

The states of system are speedily forced to the sliding

mode surface mostly by sliding mode controller at transient

once the system is away from the sliding manifold. At

steady state once the sliding manifold is moved toward, the

act of sliding mode controller should be replaced slowly by

the integral controller to decrease the chattering of the

sliding mode controller.

5 The Proposed Approach

5.1 Strategy of Control

The proposed strategy of control is composed of two sep-

arate parts: PID controllers tuned by a multi-objective

optimization algorithm and a fuzzy sliding mode con-

troller-based SMES. As shown in Figs. 1 and 3, frequency

Fig. 4 SMES unit circuit

1

2

PSM

Id+

Id0+

Id0+ Id

Ed-+HAFSMC

fk

11 csT+

1sLu

Fig. 5 Loop control of SMES along with HAFSMC


123

deviations Df1 and Df2 are considered as the input signal to

the SMES. Also, the output of the SMES unit is concur-

rently joined to areas with positive and negative signs. In

the arrangement shown in Figs. 1 and 3, to attain u1 and u2,

the PID controllers are utilized simultaneously with area

control error, ACE1 and ACE2 in (10) and (11), as the input

signal, respectively.

ACE1 ¼ DPtie þ B1 � Df1 ð10ÞACE2 ¼ DPtie þ B2 � Df2 ð11Þ

In the strategy of control, the control signals u1 and u2

are represented by:

u1 tð Þ ¼ Kp1 � ACE1 tð Þ þ Ki1

Z t

0

ACE1 sð Þds

þ Kd1

dACE1 tð Þdt

ð12Þ

u2 tð Þ ¼ Kp2 � ACE2 tð Þ þ Ki2

Z t

0

ACE2 sð Þds

þ Kd2

dACE2 tð Þdt

ð13Þ

5.2 HAFSMC System

5.2.1 Intelligent Control System

Figure 6 shows the block diagram of control system that is

used to modulate the active power of SMES unit. The

control system comprises the blocks of sliding surface,

fuzzy controller, adaption law, hitting controller and bound

estimation. As shown in this figure, the input of sliding

surface sh is errors between the frequency deviations Df1and Df2 with desired values DPd, i.e., ec1 = (Df1 - DPd1)

and ec2 = (Df2 - DPd2). In this paper, Df and DPd are

selected as trajectory command and trajectory state given

in (6). The desired values of frequency deviation are zero,

since these signals in steady state are zero. As shown in

Fig. 6, the output of controller can be written as:

u ¼ ufz þ uvs ð14Þ

where the fuzzy controller ufz is the main controller to

mimic u* and uvs compensates the difference between the

fuzzy controller and the control law. Moreover, as

observed in Fig. 6 the output of the HAFSMC is added to

the SMES control loop. In disturbance conditions, the

SMES unit regulates its output power based on the output

of the HAFSMC.

5.2.1.1 Hybrid Sliding Surface A hybrid sliding mode is

suggested for the LFC with suitable transient responses. In

this regard, a hybrid sliding surface is described by

sh ¼ s1 þ khs2 ð15Þ

where s1 and s2 are sliding surfaces computed by Eq. (7). It

should be noted that sliding surfaces s1 and s2 are corre-

sponding to ec1 and ec2. From the standpoint of the SMC,

the transient response of the errors is governed by the slope

of the hybrid sliding surface kh. The control goal is to force

the hybrid sliding surface sh to zero. If so, the sliding

surfaces s1 and s2 will instantaneously converge to zero,

and then, the errors will also converge to zero

simultaneously.

5.2.1.2 HAFSMC Design By selecting the sliding surface

as the input variable of fuzzy rules, the rules are repre-

sented by:

Rule i : IF sh isFis; THEN u is ai;

where Fis denotes the fuzzy set label and ai, i = 1; 2,…, m

symbolize the singleton control actions. The singletons and

triangular-typed functions are utilized to describe the

membership functions of THEN part and IF part, respec-

tively. The method of center of gravity is used for the

defuzzification of the control output:

ufz shð Þ ¼Pm

i¼1 wi � aiPmi¼1 wi

ð16Þ

If ai is selected as an adjustable parameter, we can write

Δ SMP( )hs tfzu

vsu

E

-0.1

0.1 f1, f2 Sliding

surface (sh)

Hitting

controller

Bound

Estimation

Fuzzy

controller Power system

SMES Loop ++

HAFSMC

u

Fig. 6 Configuration of intelligent control system


123

ufz sh; að Þ ¼ aTn ð17Þ

where a = [a1; a2; : : :; am]T is a parameter vector and

n = [n1; n2; : : :; nm]T is a regressive vector with nidescribed by

ni ¼wiPmi¼1 wi

ð18Þ

where wi is the firing weight of the ith rule. Regarding the

universal approximation theorem [36], there is an optimal

fuzzy control system u�fz sh; a�ð Þ in the form of (11) such that

u� tð Þ ¼ u�fz sh; a�ð Þ þ e ¼ a�Tfþ e ð19Þ

where e represents the approximation error and supposed to

be limited by ej j\E. Using a fuzzy control system

ufz sh; að Þ to approximate u*(t)

ufz sh; að Þ ¼ aTn ð20Þ

where a is the estimated vector of a�. ~ufz is indicated as

~ufz ¼ ufz � u� ¼ ufz � u�fz þ eh i

ð21Þ

To simplify, define ~a ¼ a� a� to acquire a rephrased

form of (21) via (19) and (20) as

~ufz ¼ ~aTf� e ð22Þ

To force s(t) and ~a tend to zero, define a Lyapunov

function as:

Va tð Þ ¼ 1

2s2h tð Þ þ B

2g1

~aT ~a ð23Þ

where g1 is a positive value. Differentiating Eq. (23) with

regard to time, we have

_Va tð Þ ¼ sh tð Þ _sh tð Þ þ B

g1

~aT _~a

¼ sh tð ÞB ufz þ uvs � u��

þ B

2g1

~aT ~a

¼sh tð ÞB ~aTfþ uvs � e� �

þ B

2g1

~aT _~a

¼B~aT sh tð Þfþ 1

g1

_~a

� �þ sh tð ÞB uvs � eð Þ

ð24Þ

To achieve _Va tð Þ� 0, the following equations are used.

_~a ¼ _a ¼ �g1sh tð Þf ð25Þuvs ¼ �Esgn sh tð Þð Þ ð26Þ

where sgn(.) is a sign function. Then, Eq. (24) can be

rephrased as

_Va tð Þ ¼ �E sh tð Þj jB� esh tð ÞB� � E sh tð Þj jBþ ej j sh tð Þj jB¼� E � ej jð Þ sh tð Þj jB� 0

ð27Þ

This shows that _V tð Þ is a negative semi-definite func-

tion. Define the following equation

Q tð Þ ¼ E � ej jð Þ sh tð Þj jB� � _Va tð Þ ð28Þ

Since Va(t) is bounded and Va(t)is non-increasing and

bounded, then

Z t

0

Q sð Þds�Va t1ð Þ � Va t2ð Þ\1 ð29Þ

Moreover, since is bounded by Barbalat’s Lemma [37],

limt!1 Q tð Þ ¼ 0, that is, sh tð Þ ! 0 as t ! 1. Accord-

ingly, the stability of HAFSMC can be ensured.

To implement FSMC system, the approximation error

should be bounded. Nevertheless, the approximation error

bound E cannot be measured simply for industrial appli-

cations. If the error bound is selected too large, we will

observe large chattering in the control effort. If the error

bound is selected too small, the system possibly will be

destabilized. To surmount the requirement for the bound of

approximation error, we use the FSMC system with bound

estimation. Replacing E by E tð Þ in Eq. (26), we have:

uvs ¼ �E tð Þsgn sh tð Þð Þ ð30Þ

where E tð Þ is the estimated approximation error. Consider

the following estimated error as

~E tð Þ ¼ E tð Þ � E ð31Þ

To force the sh(t), ~a and ~E tð Þ tend to zero, define the

following Lyapunov function.

Vb tð Þ ¼ 1

2s2h tð Þ þ B

2g1

~aT ~aþ B

2g2

~E2 ð32Þ

where g2 is a positive value. Differentiating Eq. (32) with

regard to time and using Eqs. (25) and (30), we can get:

_Vb tð Þ ¼ sh tð Þ _sh tð Þ þ B

g1

~aT _~aþ B

g2

~E _~E

¼B~aT sh tð Þfþ 1

g1

_~a

� �þ sh tð ÞB uvs � eð Þ þ B

g2

~E _~E

¼� E tð Þ sh tð Þj jB� esh tð ÞBþ B

g2

E tð Þ � E� � _

E tð Þ

ð33Þ

To achieve _Vb tð Þ� 0, the following estimation law is

used.

_E tð Þ ¼ g2 sh tð Þj j ð34Þ

Then, we can rewrite Eq. (33) as

_Vb tð Þ¼�E sh tð Þj jB� esh tð ÞBþ E�E� �

sh tð ÞB¼� esh tð ÞB�E sh tð Þj jB� ej j sh tð Þj jB�E sh tð Þj jB¼� E� ej jð Þ sh tð Þj jB�0

ð35Þ


123

Using Barbalat’s lemma [33], it is concluded that

sh(t) ? 0 as t ? ?.

5.3 Optimization Problem

It should be that the combination of PID controllers and

HAFSMC-based SMES is designed to damp the frequency

oscillations and steady-state error after any change in the

demanded load. These objectives are formulated as the

minimization of multi-objective functions J provided by:

J ¼ J1; J2; J3ð Þ ð36Þ

where

J1 ¼Z t

0

s Df1 sð Þj jds ð37Þ

J2 ¼Z t

0

s Df2 sð Þj jds ð38Þ

J3 ¼Z t

0

s DPtiej jds ð39Þ

where t is the simulation period. The following constraints

are considered into the design problem:

KminP �KP �Kmax

P

Kmini �Ki �Kmax

i

Kmind �Kd �Kmax

d

ð40Þ

6 Multi-objective Optimization Algorithm

6.1 Multi-objective Optimization Problem

and Pareto-solutions

Unlike single-objective optimization problem (SOP), an

MOP can optimize several objectives. In the SOP, the

purpose is to acquire the finest single solution, while in

MOPs with numerous and probably incompatible objec-

tives, there exists more than single optimal solution. So, the

decision maker is obligated to choose a solution from

solution set. A typical formulation of an MOP contains

numerous objectives with numerous inequality and equal-

ity constraints. More detail on the MOP can be found in

[35].

7 Simulations and Discussions

In this paper, MATLAB/SIMULINK is employed to exe-

cute the optimization algorithm and simulate the cases. At

this time, the effectiveness of the proposed method is

evaluated under different disturbances. To implement the

HNFSM, gains k1, k2 and k3 should be selected. Moreover,

the gains g1 and g2 are selected to attain the best perfor-

mance by trial and error in the experimentation taking the

constraint of stability and the control effort into consider-

ation. Table 1 presents these gains for the sliding surfaces

s1 and s2. To validate the effectiveness of the suggested

approach in damping the deviations, the results obtained

from the suggested method are compared with other con-

trollers proposed in [16] and [22, 30, 34]. The design

procedure of suggested method can be summarized as

follows:

1. Tuning PID controllers without SMES using multi-

objective genetic optimization algorithm.

2. Adding HAFSMC-based SMES to model.

3. Online updating of the parameters of HAFSMC.

7.1 Generation of Pareto-solution Set

In this paper, Pareto-solutions produced by GA for the PID

gains in each area are minimized the objective function J.

The parameters used for the multi-objective genetic algo-

rithm (MGA) are provided in Table 2. The objective

function J is assessed for each MGA with the simulation of

the both power systems, considering a DPL1 = 0.2 and

DPL2 = -0.2 at t = 0 and t = 25 s, respectively. It is

worth noting that a fuzzy-based approach is used to select

the finest compromise solution from the obtained Pareto-

set. The jth objective function of MGA Jj is represented by

a membership function lj defined as [30]:

lj ¼

1 Jj � Jminj

Jmaxj � Jj

Jmaxj � Jmin

j

; Jminj \Jj\Jmax

j

0 Jj � Jmaxj

8>>>><>>>>:

ð41Þ

Table 1 Parameters used in the HAFSMC

s1 s2 sh

k1 k2 k3 g1 g2 g1 g2 k1 k2 k3 kh

24 0.1 24 10 0.5 12 1 24 0.1 16 0.1

Table 2 Parameters utilized in MOP

Parameter Value/type

Maximum generations 100

Population size 50

Mutation rate 0.01

Number of Pareto-surface individuals 11


123

where Jminj and Jmax

j denote the maximum and minimum

values of the jth objective function, respectively.

For every solution i, the membership function is:

li ¼Pn

j¼1 lijPm

i¼1

Pnj¼1 l

ij

ð42Þ

where n and m denote the number of objectives functions

and the number of solutions, respectively. The solution

possessing the maximum value of li is the best compro-

mise solution. Tables 3 and 4 present the results of opti-

mization for two power systems under study. In Tables 3

and 4, Pareto-solutions are shown by MGA-x; x = 1,

2,….,11. As shown in these tables, maximum membership

function value belongs to MGA-1 (l9 = 0.1149) and

MGA-3 (l3 = 0.1292). Hence, results obtained in MGA-9

and MGA-3 are the best compromise solution and should

be selected as optimal gains of PID controllers for the

power systems under study.

7.2 Simulation Results

7.2.1 Two-Area Single-Source Power System

To validate the impressiveness of the suggested design

approach, simulations are performed for the model dis-

played in Fig. 4. In order to verify the suggested method,

the results attained from the suggested method are com-

pared with the responses attained from [16] and [22]. The

frequency deviations Df1, Df2, tie-line power flow and DPsm

for DPL1 = 0.2 are shown in Fig. 7a–e. It is noteworthy

that signals Df1 and Df2 represent a deviation from the

fundamental frequency of an interconnected power system.

It is evident that whatever deviation and frequency drop-

ping may be more, the power system is at risk of instability

and a loss of synchronization between its different areas. It

is obvious that the suggested method provides a better

dynamical response compared to the conventional LFC and

method proposed in [16] in damping deviations effectively

Table 3 Results of MOP for

two-area single-source power

system

Solution PID-1 PID-2 J1 J2 J3 li

Kp Ki Kd Kp Ki Kd

MGA-1 3.0000 3.0000 1.7500 3.0000 3.0000 3.0000 0.0588 0.0663 0.0765 0.0828

MGA-2 3.0000 2.0000 1.7500 3.0000 3.0000 2.0000 0.0705 0.0692 0.0530 0.1085

MGA-3 0.1660 0.2802 0.6095 1.5184 0.5779 0.5746 0.4742 0.4746 0.0423 0.0472

MGA-4 0.2634 0.2171 0.3935 1.6481 0.6019 0.8858 0.5172 0.5179 0.0403 0.0416

MGA-5 3.0000 2.9416 1.7500 3.0000 3.0000 2.0000 0.0596 0.0637 0.0584 0.1037

MGA-6 0.6563 0.9858 1.0097 1.9401 2.0482 1.9047 0.1412 0.1408 0.0428 0.1072

MGA-7 1.2886 1.8956 1.3678 2.7521 2.8336 1.9419 0.0885 0.0905 0.0461 0.1128

MGA-8 2.9867 2.9800 1.5115 2.8698 2.9876 1.4537 0.0500 0.0770 0.0754 0.0833

MGA-9 2.8820 2.2866 1.6631 2.4331 2.9474 1.2892 0.0660 0.0716 0.0475 0.1149

MGA-10 2.7500 2.9219 2.0000 3.0000 3.0000 3.0000 0.0634 0.0664 0.0562 0.1057

MGA-11 2.9704 2.8578 1.4844 2.7066 2.9491 1.3172 0.0582 0.0789 0.0673 0.0922

Bold values indicate the optimal values

Table 4 Results of MOP for

realistic power systemSolution PID-1 PID-2 J1 J2 J3 li

Kp Ki Kd Kp Ki Kd

MGA-1 3.0000 3.0000 3.0000 3.0000 3.0000 2.0000 0.26179 0.75909 0.2345 0.09355

MGA-2 2.5575 2.8852 2.7475 2.7685 2.8102 2.715 0.2667 0.2677 0.2677 0.0661

MGA-3 1.1795 2.6332 2.6136 2.7456 2.8653 1.8526 0.2482 0.6270 0.2476 0.1292

MGA-4 2.5305 2.7268 2.0187 2.9290 2.8921 1.9589 0.2655 0.4448 0.2518 0.0826

MGA-5 1.2894 2.7052 2.1133 2.7849 2.8268 1.8277 0.2598 0.6972 0.2353 0.1063

MGA-6 2.1442 2.9634 2.9325 2.8919 2.8737 2.8837 0.2632 0.2728 0.2645 0.0847

MGA-7 2.6826 2.9477 2.9975 2.7685 2.8727 2.9651 0.2684 0.1877 0.2652 0.0754

MGA-8 2.8385 2.7400 1.8660 2.9775 2.8909 1.9625 0.2619 0.3633 0.2521 0.1042

MGA-9 2.8551 2.8554 2.5609 2.9333 2.9222 1.9697 0.2668 0.5903 0.2447 0.0752

MGA-10 2.8700 2.9981 3.0000 2.8981 3.0000 1.8711 0.2617 0.7639 0.2349 0.0924

MGA-11 2.9447 2.8184 2.1542 2.9780 2.9032 1.9682 0.2637 0.5143 0.2471 0.0903

Bold values indicate the optimal values


123

and reducing settling time so that less deviation and fre-

quency dropping can be observed in the responses with the

suggested controller. Hence, compared to the other meth-

ods, suggested method greatly increases the stability of

power system and provides an improved damping of the

frequency deviations.

As we know, the frequency drop means that the con-

sumed power is greater than generated power. As shown in

Fig. 7a, b, a frequency drop is occurred in both areas, since

an increase load is occurred at area 1. Hence, to compen-

sate power shortage in area 1, the power is to be transferred

from area 2 to area 1. As shown in Fig. 7c, the tie-line

power flow is negative which means power flows from area

2 to area 1. Figure 7e compares the output of turbine sys-

tem for different methods. Clearly, the suggested method

generates the lowest output which means the energy

required for damping deviations is decreased.

For the second simulation, it is presumed that a 20%

increase in demand of area at t = 0.2 is occurred. The

frequency deviations Df1, Df2, tie-line power flow and DPsm

for this disturbance are shown in Fig. 8a–d. As shown in

these figures, the suggested method has again provided an

improved dynamic response than the other methods. To

compensate the power shortage in area 2, the power is

(a) (b)

(c)

(e)

(d)

Fig. 7 Responses of two-area single-source power system to a DPL2 = 0.2 applied to area 1 a frequency deviation in areas 1, b frequency

deviation in area 2, c tie-line power flow deviation, d the output of SMES unit, e the output of turbine system


123

(a) (b)

(c) (d)

Fig. 8 Responses of two-area single-source power system to a DPL2 = 0.2 applied to area 1 a frequency deviation in areas 1, b frequency

deviation in area 2, c tie-line power flow deviation, d the output of SMES unit

(a)

(c)

(b)

Fig. 9 Responses of two-area single-source power system to a DPL1 = -0.1 applied to area 1 a frequency deviation in areas 1, b frequency

deviation in area 2, c tie-line power flow deviation


123

transmitted from area 1 to area 2. As shown in Fig. 8c, tie-

line power flow is positive which confirms the above

statement.

To demonstrate the usefulness of suggested method,

results obtained from suggested approach are compared

with results obtained from the controller proposed in [34].

To do so, we assume that a DPL1 = -0.1 is occurred at

t = 0. The results of this comparison are shown in Fig. 9a–

c. By observing this result, it can be concluded that the

suggested approach is more successful in damping and

removing frequency deviation.

As displayed in Fig. 9a, b, with decreasing load in area

1, an increase frequency is occurred at both areas. As

expected, the surplus power is transmitted from area 1 to

area 2 and tie-line power flow is positive.

A comparative study between different methods is pro-

vided in Table 5. As shown in this table, the proposed

provides a less settling time compared to the other meth-

ods. The results presented in this table show that the fre-

quency deviation in the presence of the proposed controller

quickly converges to zero. This means that the power

system is stabilized quickly.

Table 5 A comparative study

between different methodsType of method Settling time (s)

DPL1 = 0.2 DPL2 = 0.2

Df1 Df2 DPtie Df1 Df2 DPtie

Proposed approach 5.94 6.01 5.70 7.41 3.77 6.81

Method proposed in [30] 6.12 6.11 5.73 7.68 3.83 6.89



Conventional PID 14.35 22.97 24.70 22.85 19.30 23.24

(a) (b)

(c)

Fig. 10 Responses of two-area single-source power system to a series of random variations applied to area 1 a frequency deviation in areas 1,

b frequency deviation in area 2, c tie-line power flow deviation


123

As the final simulation for system illustrated in Fig. 1

and to show the advantages of suggested approach, a series

of random variations (between -0.4 and ?0.4 p.u) in

demand are applied to area 1. To confirm the robustness of

suggested method against changes in the model parameters,

a 10% change is applied to the parameters of turbine and

governor. The results of this simulation are shown in

Fig. 10. As observed in Fig. 10, the performance of the

suggested approach compared to the method proposed in

[34] is better so that less power and the frequency deviation

can be seen in its response.

7.2.2 Realistic Power System

In this subsection, the simulation results of the realistic

power system shown in Fig. 3 are provided. In Fig. 11a–d,

the frequency deviations, tie-line power flow deviations

and output of SMES unit for DPL1 = 0.1 at t = 0 and

DPL2 = -0.1 at t = 10 s are shown. These results verify

the satisfactory performance of the suggested approach.

8 Conclusion

In this study, a combination of an HAFSMC with integral–

proportional–derivative switching surface-based SMES

and PID tuned by a multi-objective optimization algorithm

is suggested to solve the LFC in interconnected power

systems. A hybrid sliding surface including two subsys-

tems’ information is developed to produce a control effort

to force both subsystems toward their related sliding sur-

face. In order to make the dynamical response of an

interconnected power system better, in the suggested con-

troller is added to the control loop of an SMES. Obtaining

the optimal PID controller problem is formulated into a

multi-objective optimization problem. A fuzzy-based

membership method is used to find the finest compromise

solution from the generated Pareto-solution set. Simula-

tions are provided and compared with traditional PID

controller and the other controllers. These results demon-

strate the effectiveness and robustness of suggested

method.

Appendix

SMES control loop:

Tc ¼ 0:03; Id0 ¼ 20 kA; L ¼ 3H; kf ¼ 0:001

The parameters used for power system shown in Fig. 1

are provided in [2]:

The parameters used for power system shown in Fig. 3

are provided in [33].

(a) (b)

(c) (d)

Fig. 11 Responses of realistic power system to a DPL1 = 0.1 and DPL2 = -0.1 applied to area 1 and area 2 a frequency deviation in areas 1,

b frequency deviation in area 2, c tie-line power flow deviation, d the output of SMES unit


123

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123

Mehrshad Khosraviani re-

ceived his B.Sc. degree in

Computer Engineering from

Islamic Azad University, Cen-

tral Tehran Branch, in 2004, and

the M.Sc. and Ph.D. degrees in

Computer Engineering from the

Amirkabir University of Tech-

nology, Iran, in 2008 and 2016,

respectively. He is now a lec-

turer of the faculty of Computer

Engineering at the Islamic Azad

University, Parand Branch. He

is currently working on bio-de-

sign automation, and his other

research interests include quantum computing and artificial intelli-

gence-based optimization algorithms.

Mohsen Jahanshahi completed

his B.Sc. and M.Sc. studies in

Computer Engineering in Iran,

dated 2002 and 2005, respec-

tively. He joined the department

of Computer Engineering at

Islamic Azad University (Cen-

tral Tehran Branch) in 2005. He

also achieved his Ph.D. degree

in Computer Engineering from

Islamic Azad University (Teh-

ran Science and Research

Branch) in 2011. Since 2012, he

has been as head of both Com-

puter Engineering and Informa-

tion Technology departments. He was promoted to associate professor

in 2015 and currently is an IEEE Senior member. In addition,

Jahanshahi has been a member of Young Researchers and Elite Club

since 2012. His research interests include performance evaluation,

multistage interconnection networks, wireless mesh networks,

wireless sensor networks, cognitive networks, learning systems,

mathematical optimization and soft computing.

Mohsen Farahani was born in

Iran on October 15, 1985. He

received the B.Sc. degree from

the University of Birjand, Bir-

jand, Iran, in 2008, and the

M.Sc. degree from the Univer-

sity of Bu-Ali Sina, Hamedan,

Iran, in 2011, both in electrical

engineering, respectively. He is

currently with Young

Researchers and Elite Club,

East Tehran Branch, Islamic

Azad University, Tehran, Iran.

He has published several papers

in ISI journals in different areas

of electrical power engineering so far. His major field of interest

includes control of power systems and fuzzy and neural networks.

Amir Reza Zare Bidaki has

master certificate in control

engineering, and at present he is

student PHD in science and

research university of Tehran.

His papers published in IFAC,

IJSEE and MJMS. His works

are in design fuzzy controller

and identification of models.


123

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