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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
Mehrshad Khosraviani
Mohsen Jahanshahi
Amir Reza Zare Bidaki
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
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
International Journal of Fuzzy Systems
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
M. Khosraviani et al.: Load–Frequency Control Using Multi-objective Genetic Algorithm…
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
International Journal of Fuzzy Systems
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
M. Khosraviani et al.: Load–Frequency Control Using Multi-objective Genetic Algorithm…
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
International Journal of Fuzzy Systems
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
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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
International Journal of Fuzzy Systems
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
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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
Method proposed in [16] 12.68 19.03 19.03 18.69 15.12 15.36
Method proposed in [22] 6.00 6.03 5.70 7.44 3.80 6.81
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
International Journal of Fuzzy Systems
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
M. Khosraviani et al.: Load–Frequency Control Using Multi-objective Genetic Algorithm…
123
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International Journal of Fuzzy Systems
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.
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123