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A new fuzzy sliding mode controller for load frequency control
of large hydropower plant using particle swarm optimization
algorithm and Kalman estimator
R. Hooshmand*
,†
, M. Ataei and A. Zargari
Department of Electrical Engineering, University of Isfahan, Isfahan, Iran
SUMMARY
The load frequency control (LFC) is very important in power system operation and control for supplying
sufficient, reliable, and high‐quality electric power. The conventional LFC uses an integral controller. In
this paper, a new control system based on the fuzzy sliding mode controller is proposed for controlling the
load frequency of nonlinear model of a hydropower plant, and this control system is compared with the
proportional–integral controller and the conventional sliding mode controller. To regulate the membership
functions of fuzzy system more accurately, the particle swarm optimization algorithm is also applied.
Moreover, because of the unavailability of the control system variables, a nonlinear estimator is suggested
for estimating and identifying the system state variables. This estimator provides the physical realization of
the method and will reduce the costs of implementation. The proposed control method is performed for the
LFC of hydropower plant of Karoon‐3 in Shahrekord, Iran. The simulation results show the capability of
the controller system in controlling local network frequency. Copyright © 2011 John Wiley & Sons, Ltd.
key words: hydropower plant; sliding mode control; extended Kalman estimator; PSO algorithm; fuzzy
control
1. INTRODUCTION
The utilization of running water energy sources in power systems has been increased because of
environmental concerns [1]. Hydropower is characterized by low‐cost, flexible commitment, and fast
load follow, making it potentially favorable to load frequency control in power system. Hydropower
plants convert potential energy of falling water into electricity. Keeping the parameters of power plant
output (such as the frequency and voltage) within permissible limits is necessary for the appropriate
performance and effectiveness.
Load frequency control (LFC) is very important in power system operation and control for supplying
sufficient, reliable electric, and high‐quality power. For this reason, the LFC should be able to control the
output power of each generator so as to keep the frequency and the tie‐line power within prespecified limits.
Most of the load frequency controllers initially comprised an integral controller because of the
simplicity and the feasibility of implementation [2]. The main drawback of this is that the dynamic
performance of the system is limited by its integral gain. A high‐gain controller may deteriorate the system
performance causing large oscillations and instability. Thus, the integral gain must be set to a level that
provides a compromise between a desirable transient recovery and low overshoot in the dynamic response
of the overall system [3]. To improve the transient function, in [4] an integral controller is presented to
achieve zero steady‐state error and a reasonable undershoot in the system’s response. In [5], an intelligent
proportional–integral–derivative (PID) controller is created based on principle of anthropomorphic
intelligent. The simulation studies and field test indicate that the intelligent PID controller can somewhat
*Correspondence to: R. Hooshmand, Department of Electrical Engineering, University of Isfahan, Isfahan, Iran.
†
E‐mail: Hooshmand_r@eng.ui.ac.ir
Copyright © 2011 John Wiley & Sons, Ltd.
EUROPEAN TRANSACTIONS ON ELECTRICAL POWER
Euro. Trans. Electr. Power (2011)
Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/etep.609
improve the dynamic performance and stability of the hydroturbine governing system when compared
with a conventional PID controller. In some methods, the fuzzy proportional–integral (PI) controller is
suggested [6–8]. A fuzzy gain schedule proportional and integral controller and a fuzzy gain schedule
integral and derivative controller were presented in [7] and [8], respectively. Also in [9,10], the designing
and the regulating of the PID controller, which are designed based on the nominal plant parameters, has
been suggested. Generally, despite the attempts for improving the conventional PID controllers, they do
not have good dynamical performance for a wide range of operating conditions and various load change
scenarios and in the presence of parameters’nonlinearity and uncertainties.
In addition, the previously mentioned methods are designed based on linear model and nominal
parameters. On the other hand, the power system’s components are naturally nonlinear, and the
operating conditions of power systems are usually changing, and also some parameter uncertainties
also occur. Therefore, the behavior of the actual system is different from that of the simulated model.
Consequently, the importance of the control method in the presence of parameters’nonlinearity and
uncertainties and system load changes is the main discussion in the design of a controller. In this
regard, some robust control methods are presented. In [11], a robust LFC was proposed based on
Riccati equation approach for the stabilization of the system with uncertainties. The controller
provided better performance in simulation, but their chance for real implementation is still unsure. The
robust H
∞
control and adaptive controller were suggested in [12] and [13], respectively. These
controllers not only identify parameter uncertainties but also regulate the area control error signal.
However, these methods require either information on the system states or an efficient online
identifier. Also, because the order of the power system is large, the model reference approach
presented in [13] may be difficult to apply. In [14], a μ‐synthesis control technique was introduced to
compensate modeling uncertainties based on the linearized model. Moreover, some methods using
intelligence algorithms are presented in [15,16].
In this paper, a nonlinear model of the large hydropower plant is introduced, and the parameters of an
actual hydropower plant of Karoon‐3 in Shahrekord/Iran are used in simulation results. In this paper, a PI
controller for controlling the load frequency, based on the supplementary control method of fuzzy sliding
mode is suggested. The sliding mode controller (SMC) can meet the control requirements of the system
against load changes and parameters’nonlinearity and uncertainties. The problem with this control
method is the chattering phenomenon that increases the control activities and excites the unmodeled high‐
frequency dynamic that may even destabilize the system [17]. To accelerate the law of reaching the sliding
surface and removing the chattering problem, a SMC is designed in which the fuzzy function has replaced
the sign function in the control law. By using particle swarm optimization (PSO) algorithm, the fuzzy
membership functions of the system are more accurately regulated to obtain a better system response.
Also, because of the unavailability of the control system variables, an extended Kalman estimator is
suggested for estimatingand identifying the system variables. Thisestimator willreduce the implementing
costs and increase the effectiveness of this controller. Simulation results show that the proposed controller
causes faster response with less settling time and less undershoot compared with PI controller and with
less chattering and less control effort in comparison with the conventional SMC achieved. Also, stability
of proposed controller against various changes in load and some parameters are evaluated.
2. THE LOAD FREQUENCY CONTROL MODELING OF A HYDROPOWER PLANT
In this section, the nonlinear model of the hydropower plant is introduced. First, the physical
characteristics of the power station are described. Then, the dynamic equations, considering a general
nonlinear model with surge tank effects, are presented. Finally, the model of the power system, the
measurement system, and the PI controller are combined to analyze the network frequency response.
2.1. Physical characteristics of the hydroelectric power station
Figure 1 presents a diagram of the main physical characteristics of a power plant. In this figure, the
main elements of the plant and some parameters are shown, and the surge tank effect is considered. In
this figure, the descriptions of parameters of this model are as follows:
R. HOOSHMAND, M. ATAEI AND A. ZARGARI
Copyright © 2011 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power (2011)
DOI: 10.1002/etep
H
0
: head in reservoir.
H
t
:head in turbine.
H
r
:head in riser of the surge tank.
H
l
:head loss in penstock.
H
l2
:head loss in tunnel.
U
c
:velocity of the water in the conduit or flow in tunnel.
U
s
:velocity of the water in the conduit or flow surge tank.
U
t
:velocity of the water in the conduit or flow turbine.
2.2. General nonlinear equations
The general dynamic equations of the hydropower plant of Karoon‐3 in Shahrekord, Iran are based on
the WG5 and WG4 models of IEEE working Group and the QR52 and QR51 models [18–20]. The
following are these equations with descriptions for different subsystems.
•Dynamics of the tunnel
d–
Uc
dt¼1−–
Hr−–
H12
TW
(1)
–
H12 ¼fp1⋅–
Uc⋅j–
Ucj(2)
where the parameter T
w
is called the rated water starting time. This parameter is a constant in the
nonlinear case, independent of loading, which is obtained as follows [21]:
TW¼L⋅Qbase
Hbase⋅G⋅A(3)
in which Gis the acceleration due to gravity, Ais the cross‐sectional area of the tunnel, H
base
is the
per‐unit base value of the water column head, Q
base
is the per‐unit base value for the flow, and Lis the
path length for each section in the water passage from draft tube intake to turbine outlet. Also, f
p1
is
the head loss coefficient for the tunnel.
•Equation of continuity
–
Ut¼–
Uc−–
Us(4)
•Dynamics of the surge tank
–
Hr¼1
Cs
⋅∫–
Us⋅dt−f0⋅–
Us⋅j–
Usj(5)
Figure 1. Main physical characteristics of a hydropower plant.
SLIDING MODE CONTROLLER FOR HYDROPOWER PLANTS
Copyright © 2011 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power (2011)
DOI: 10.1002/etep
where f
0
is the head loss coefficient for the surge chamber orifice. C
s
is the storage constant of the
surge tank.
•Dynamics of the penstock
–
H1¼fp2⋅–
U2
t(6)
–
Ht¼–
Hr−–
H1−zp⋅tanh Te⋅sðÞ⋅–
Ut(7)
–
Ut¼–
G⋅ffiffiffiffiffiffi
–
Ht
q(8)
where f
p2
is the head loss coefficient for the penstock and z
p
is the Hydraulic surge impedance of the
conduit of penstock. T
e
is the wave travel time that is given by
Te¼L
a(9)
in which Lis the Length of the conduit in penstock and ais the wave velocity. The guide vane
function Gin the existing model is assumed to vary linearly with the guide vane opening only. In
reality, the slope of this function will vary with flow coefficient and Reynolds number over the full
range of turbine operations, and it should properly be modeled as a nonlinear function.
The hyperbolic tangent function is given by
tanh Tep⋅s
¼1−e−2Tep⋅s
1þe−2Tep⋅s¼
s⋅Tep ⋅Q
n¼1
n¼∞
1þs⋅Tep
n:π
2
!
Q
n¼1
n¼∞
1þ2s⋅Tep
2n−1ðÞπ
2
!(10)
It should be noted that depending on the goal of the study and the request accuracy, some of the
terms of expansion in Equation (10) can be eliminated, and the phrase “tanh”can be summarized as a
parametric equation. If the expansion with n= 0 is considered, tanh(T
ep
·s)≈T
ep
·s[22].
•Mechanical power
–
pmechanical ¼At⋅–
Ht⋅–
Ut−–
UNL
(11)
In this equation, U
NL
is velocity of the water in the conduit or flow in no load condition. Also, A
t
is
the turbine gain with constant value, which is calculated using the turbine megawatt (MW) rating and
the generator megavolt ampere (MVA) base, as follows:
At¼Turbine MW rating
Generator MVA ratingðÞ⋅–
Hr⋅–
Qr−–
QNL
(12)
where H
r
is the per‐unit head at the turbine in the rated flow condition, Q
r
is the per‐unit flow at the
rated load, and Q
NL
is the turbine per‐unit no load flow accounting for turbine fixed power losses.
To adjust the output power (P
mechanical
), it is necessary to multiply the value given in Equation (11),
by a nonlinear function of Gthat represents the efficiency of the turbine. This function depends on the
gate opening, and its shape is similar to the efficiency curve of a Francis hydraulic turbine [23].
–
pmech ¼η–
G
⋅–
pmechanical ¼η–
G
⋅At⋅–
Ht⋅–
Ut−–
UNL
(13)
By combining the hydroelectric system model with the PI controller, the measurement system, and
the power system model, the block diagram of load frequency control of a hydropower plant is
generally achieved as shown in Figure 2.
R. HOOSHMAND, M. ATAEI AND A. ZARGARI
Copyright © 2011 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power (2011)
DOI: 10.1002/etep
The following state equations are acquired by considering the state variables, shown in the block
diagram of Figure 2.
˙x1¼−1
Tp
x1þKp
Tpðηx5
ðÞ⋅At⋅x2ðx5ffiffiffiffiffi
x2
p−UNLÞ−DPLÞ
˙x2¼2ffiffiffiffiffi
x2
p
zp⋅Teðx3−f0⋅ðx4−x5ffiffiffiffiffi
x2
pÞ⋅x4−x5ffiffiffiffiffi
x2
p−fp2⋅x2⋅x2
5−x2−
−ffiffiffiffiffi
x2
pð−1
Tm
x5−Km
Tm
x6−KmK
Tm
x1þKm
Tm
uÞ
Þ
˙x3¼1
Cs
⋅ðx4−x5ffiffiffiffiffi
x2
pÞ
˙x4¼1
TW
⋅ðH0−x3þf0⋅ðx4−x5ffiffiffiffiffi
x2
pÞ⋅x4−x5ffiffiffiffiffi
x2
pj−fp1⋅x4⋅jx4Þ
˙x5¼−1
Tm
x5−Km
Tm
x6−KmK
Tm
x1þKm
Tm
u
˙x6¼KIx1
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
(14)
The nonlinearity property of the model and variable parameters create problems in the performance of
the PI controller at various operation points. So, it is appropriate to use the advanced control systems.
3. DESIGN OF A PROPOSED FUZZY SLIDING MODE CONTROLLER
In this section, after introducing classical SMC and its features, the suggested fuzzy sliding control
method is presented. In this regard, the manner of designing a fuzzy control, the membership
functions, and its fuzzy rules are described.
f
Abs
s
W
T.
1
∏
Abs
s
s
C.
1
∏
+
−
+−
+
f
)
.
tanh(. sTZ
∏
+
−
∏
A
∏
∏
)(G
H
−
H
U
H
U
−
U
H
H
−
G
f
+−
U
P
P
H
Power System
P
x
m
sT
m
K
+1
K
s
K
Measurment
System
u
x
+
+
+
−
−x
x
x
x
Dynamics of the
Surge Tank
Dynamics
of the tunnel
Dynamics
of the Penstock
r
pe
t
2
t
mechanical mech
6
Figure 2. Block diagram of load frequency control of a hydro power plant.
SLIDING MODE CONTROLLER FOR HYDROPOWER PLANTS
Copyright © 2011 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power (2011)
DOI: 10.1002/etep
3.1. Conventional sliding mode controller
In recent years, the use of variable structure strategy by the SMC in the control system types has
attracted attention. The main advantages of this method are its robustness against the parameter
changes and external disturbances and uncertainties, its quick dynamic response, and its simplicity of
design and implementation [24].
The control laws in the SMC comprise two separate parts. The first part is responsible for
conducting the state trajectory to the sliding surface. The second part makes the system output
convergent to the desirable output, based on the desired dynamics. In fact, the limit of high switching
frequency and the presence of uncertainties in the system prevent the system states from remaining on
the sliding surface, and they oscillate around it. These oscillations are called chattering, which is
usually an undesirable phenomenon, as it may increase the control activities and excite the high‐
frequency unmodeled dynamics and destabilize the system.
As described, according to the features and properties of thiscontroller, it is usable in controlling various
nonlinear systems. The nonlinear system is considered in the following canonical dynamic equations:
xnðÞ¼fxðÞþbxðÞutðÞþDtðÞ (15)
In this equation, f(x) and b(x) are uncertain nonlinear functions with known uncertainty bounds.
Furthermore, the D(t) function relates to the disturbance entering the system.
In Equation (1), xtðÞ¼ xtðÞ˙xtðÞ…xn−1ðÞ
tðÞ
T∈Rnis the system state vector. By assigning the
desirable state vector x
d
(t), the error state vector can be shown as follows:
˜xtðÞ¼xtðÞ−xdtðÞ¼ ˜xtðÞ
˙
˜xtðÞ…˜xn−1ðÞ
tðÞ
hi
T
(16)
The first step in designing the SMC is to define an appropriate sliding surface in the state space.
This sliding surface, which is called the switching function, is considered as follows:
stðÞ¼ d
dtþλ
n−1ðÞ
˜xtðÞ and λ>0 (17)
The second step is to determine the control law for conducting the system to the selected sliding
surface. In this method, the control law always consists of two parts, shown in this equation:
utðÞ¼ueq tðÞ−untðÞ (18)
Here, u
eq
(t) is a part of the input. If this is implemented when the system and the system states are
on the sliding surface, then the x(t) become convergent to the desirable state vector of system, which is
the same as x
d
(t).
The equivalent control signal is obtained by setting zero for the derivative of the switch function not
considering the terms of uncertainty and disturbance. Moreover, u
n
(t) is also part of the input. If the system
mode is not on the sliding level, this can make the system state convergent to the sliding level with each
primary condition in the limited time, in the presence of uncertainties and external disturbances. This part
of control law is designed by the switch sign function in the conventional method as follows:
untðÞ¼KxðÞsign sðÞ (19)
To satisfy the sliding condition s˙s<0 and ensure the system’s stability, the Kfunction is selected
rather than the uncertainties of the upper bound in the system. The problem here is to define the upper
bound of the uncertainties in the system. As seen in Equation (19), K(x) is the measure of control gain
that is applied to conduct the system states toward the switching surface. Hence, the irregular bulk of
this parameter produces chattering, even if the continuous function replaces the sign function in the
control signal.
3.2. Design procedure of fuzzy sliding mode controller
Based on tentative information about designing the SMCs, it is deduced that a big switch factor K(x)
can make the system state convergent to the sliding level rapidly, but at the same time, it increases the
chattering. Therefore, for having the proper response, the switch factor must be increased in the same
proportion when the system state is away from the sliding level, and vice versa. Therefore, in this
R. HOOSHMAND, M. ATAEI AND A. ZARGARI
Copyright © 2011 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power (2011)
DOI: 10.1002/etep
paper, in order to accelerate the law of reaching the sliding surface and of removing the chattering
problem, an SMC is designed in which the fuzzy function has replaced the sign function in the control
law. So in the suggested system, a fuzzy controller is used in the discontinuous part of the control
signal. The block diagram of an fuzzy SMC (FSMC) structure is shown in Figure 3.
As seen in Figure 3, the control law is indicated by Equation (20) in which the fuzzy controller
u
f
=K
fs
·u
fs
(t) replaces the sign function K(x)sign(s).
utðÞ¼ueq tðÞ−Kfs ⋅ufs tðÞ (20)
To design the FSMC for this system, the sliding surface and the equivalent control law will first be
obtained based on Equations (17) and (20) and state Equation (14) as follows:
stðÞ¼x1þ˙x1
utðÞ¼ueq tðÞ−Kfs⋅ufs
ueq tðÞ¼
Tm:½1−1
TP
˙x1þ2KP⋅ffiffiffiffiffi
x2
p
TP⋅zp⋅Te
⋅ηx5
ðÞ⋅At⋅3
2x5ffiffiffiffiffi
x2
p−UNL
⋅ðx3−f0x4−x5ffiffiffiffi
x2
pÞjx4−x5ffiffiffiffiffi
x2
pj−fp2⋅x2⋅x2
5−x2
Km:KP
TP
⋅ηx5
ðÞ⋅At⋅3
2x5ffiffiffiffiffi
x2
p−UNL
−x
3
2
!"#
þ1
Km
x5þx6þK⋅x1
8
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
:(21)
where u
eq
(t) is obtained by setting zero for the derivative of the switch function irrespective of the
terms of uncertainty and disturbance.
3.2.1. Variables and membership functions. Fuzzy control system used in this paper has the input
variables sand ˙sand the output variable u
fs
. Also, five fuzzy sets are considered for each input
variable, and seven fuzzy sets are assigned to the output variable. The pattern of the membership
functions of these input and output variables are shown in Figure 4, which must be optimally
determined by the PSO algorithm in the next section. The parameters N, P, S, B and Z in Figure 4 are
defined as negative, positive, small, big and zero, respectively.
3.2.2. Fuzzy rules of the controller. According to the fuzzy logic, a lingual and mental control
strategy changes into an automatic control strategy, and all the control laws are constructed based on
the expert experiences. To extract the fuzzy rules (based on the switch function’s diagram shown in
Figure 5), it is performed as follows:
(1) At Fand Bpoints, the switch function and its derivative have the same sign, and the switch
function is moving away from the sliding level. Therefore, it is necessary to prevent the s
distance from the sliding surface with further input application (at point Bwith the positive sign
and at point Fwith the negative sign)
(2) Also, at points Dand H, the sign of the switch function and its derivative are opposite to each
other, and the switch function is moving toward the sliding level. So, in this mode, it is essential to
prevent the switch function from passing the sliding level again by reducing the input.
(3) Furthermore, despite the zero derivative of the switch function at points Cand G, the amount of
deviation from the sliding surface is extensive. Therefore, it is necessary to lead the switch
function toward the sliding surface by applying the additional input.
Figure 3. Block diagram of fuzzy sliding controller.
SLIDING MODE CONTROLLER FOR HYDROPOWER PLANTS
Copyright © 2011 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power (2011)
DOI: 10.1002/etep
(4) At points I,E, and A, the switch function is on the sliding level. But because the derivative of
switch function at these points is considerable, it can be avoided to move away from the sliding
surface by applying a small input.
By considering these conditions, the fuzzy rules base in the FSMC may be obtained as in Table I.
4. OPTIMIZATION OF THE PROPOSED CONTROLLER BY PARTICLE SWARM
OPTIMIZATION ALGORITHM
4.1. Particle swarm optimization algorithm application
Achieving the desirable features in the suggested method requires that the parameters of fuzzy sets in the
fuzzy controller be optimally determined. The parameters of fuzzy sets are {a,b,c,d,e,f,g,h,i,j, k},
s
µ
ufs
kjih
g-g
-h-i-j-k
cba-a-b-c
µ
s
fed-d-e
Figure 4. The fuzzy sets in the input and output variables of fuzzy controller.
(seconds)
s
Figure 5. The diagram of switch function.
R. HOOSHMAND, M. ATAEI AND A. ZARGARI
Copyright © 2011 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power (2011)
DOI: 10.1002/etep
which are shown in Figure 4. If the fuzzy sets are not regulated, the sliding fuzzy controller cannot reduce
the chattering, it may even increase it. So the parameters of fuzzy sets must be optimally computed by
intelligent methods such as PSO algorithm. Indeed, fuzzy rules are based on Table I, and fuzzy sets are
regulated using PSO algorithm. In the algorithm, the fitness function is considered to obtain the best
response for switch function s. In other words, PSO algorithm will optimize the position of fuzzy sets in
the output and input variables of the fuzzy controller. In this direction, the fuzzy sets will be
symmetrically considered toward the coordinate origin. As a result, the PSO algorithm will perform
faster as the parameters will be half.
4.2. Particle swarm optimization algorithm structure
The PSO algorithm, which is a kind of evolutionary algorithm, was proposed by Kennedy and
Eberhart in 1995 [25]. The PSO algorithm is a set of particles (as the optimization variables) that
diffuse in the research space. Each particle may be a potential solution. It is obvious that some
particles have better position than others and they try to promote their position to the superior
particle’s position. At the same time, the superior particle’s position is also changing. The change of
each particle’s position is based on the particle’s experience in the previous movements and the
neighboring particles’experience. Indeed each particle is aware of its superiority and nonsuperiority
with respect to the neighboring particles and also toward the total group.
The PSO concept involves changing the velocity of each particle toward its pbest and gbest
positions at each time step. Velocity is weighted by a random term, with separate random numbers
generated for velocity toward pbest and gbest positions. The process of PSO algorithm can be
described as follows:
(1) Initialize a population of particles with random positions and velocities on ddimensions in the
problem space.
(2) For each particle, evaluate the desired optimization fitness function in dvariables.
(3) Compare particle’sfitness evaluation with particle’spbest. If current value is better than pbest,
then set pbest position equal to current position in ddimensional space.
(4) Compare fitness evaluation with the population’s overall previous best. If current value is better
than gbest, then reset gbest to the current particle’s array index and value.
(5) Change the velocity and the position of the particle according to Equations (22) and (23),
respectively:
vkþ1
i¼wv k
iþc1rand ⋅ðÞpbest−xk
i
þc2rand ⋅ðÞgbest−xk
i
(22)
xkþ1
i¼xk
iþvkþ1
i(23)
where vk
i,vkþ1
i, and xk
iare velocity vector, modified velocity, and positioning vector of particle iat
generation k, respectively. c
1
and c
2
are the cognitive and social coefficients, respectively, that
influence particle velocity. In these equations, it should be noted that the largeness of v
max
may
make the particles pass over the minimum point and its smallness may also make the particle rotate
around its position and cannot search the testing space. The amount of v
max
is typically selected
between 10 and 20% of the range of variables .Besides, an appropriate selection of wcauses that
the algorithm is less repeated for achieving the optimized point. In usual algorithms of PSO,
Table I. Fuzzy rules in the fuzzy sliding mode controller.
u
fs
˙s
NB NS Z PS PB
sNB NS PM PB PB PB
NS NM PS PS PM PM
ZNB NS Z PS PB
PS NM NM NS NS PM
PB NB NB NB NM PS
SLIDING MODE CONTROLLER FOR HYDROPOWER PLANTS
Copyright © 2011 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power (2011)
DOI: 10.1002/etep
coefficient wreduces from 0.9 to 0.4 while implementing the algorithm and reduced according to
the following equation:
w¼wmax −wmax −wmin
itermax
iter (24)
where iter
max
and iter are the maximum and the current number of iterations (or generation),
respectively. Another problem in implementing this algorithm is the selection of the appropriate c
1
and c
2
. In many algorithms, the amounts of c
1
and c
2
are selected in a way that c
1
+c
2
≤4.
(6) Return to step 2 until a criterion is met, usually a sufficiently good fitness or a maximum
number of iterations (generations).
4.3. Fitness function
The fitness function in the PSO algorithm comprises a summation of three parts: (i) the steady state
error (Ess) of switch function; (ii) the amount of undershoot (OS) of switch function; and (iii) the
integral of the absolute value of the switch function (∫
t
0jstðÞjdt) as Equation (25).
Fobj ¼W1:Ess þW2:OS þW3:∫t
0jstðÞjdt(25)
where W
1
,W
2
, and W
3
are the weighting coefficients of the evaluation function and s(t) is the switch
function introduced in Equation (21).
5. THE PRACTICAL STRUCTURE OF THE PROPOSED FUZZY SLIDING
MODE CONTROLLER
Thus far, a new control system based on the FSMC has been proposed for controlling the load
frequency of the nonlinear model of a hydropower plant. But in practice, the required measurement
instruments for measuring the state variables are usually expensive and it is sometimes inaccessible at
the power plant. Therefore, an appropriate, cheap solution could be the estimation of the state variable
of the system by a good estimator such as the extended Kalman estimator [26].
The block diagram of the proposed FSMC structure has been shown with the extended Kalman
estimator in Figure 6. In this figure, the state variables are estimated by the Kalman estimator block.
Then, the Equivalent control input, u
eq
, is made according to Equation (21). In the sliding function
block, the switch function is constructed by using the desired values and estimated ones of state
variables according to Equation (21). In the FSMC block, the u
fs
output is made by implementing
fuzzy rules on the switch function (as input) that the fuzzy rules are defined based on Table I. By
multiplying u
fs
in K
fs
, the control output u
f
is computed. At the end, based on Equation (18), the u
input is applied to the system to control the frequency response. In the following, the related equations
and the flowchart of estimation algorithm are presented.
y
y
ˆ
X
ˆ
Figure 6. Block diagram of fuzzy sliding mode controller with Kalman estimator.
R. HOOSHMAND, M. ATAEI AND A. ZARGARI
Copyright © 2011 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power (2011)
DOI: 10.1002/etep
5.1. Extended Kalman estimator algorithm
The nonlinear process model (from time kto time k+ 1) is described as
Xkþ1¼fXk;Uk
ðÞ
þWk(26)
Zkþ1¼hXkþ1
ðÞþVkþ1(27)
where X
k
and X
k+1
are the system state (vector) at time kand k+ 1. Also, f is the system transition
function, U
k
is the control signal, and h is the observation function. Meanwhile, W
k
and V
k+1
are the
process and observation noises that are both assumed to be zero mean multivariate Gaussian noise
with covariance Q
k
and R
k
, respectively. The estimated process model is described as
ˆ
Xkþ1¼f
ˆ
Xk;Uk
(28)
ˆ
Zkþ1¼h
ˆ
Xkþ1
(29)
To obtain the
ˆ
Xkþ1state as a proper estimation of the X
k+1
state, it should be updated by the
optimal Kalman gain Kas follows:
Ykþ1¼Zkþ1−ˆ
Zkþ1(30)
ˆ
Xkþ1jkþ1¼
ˆ
Xkþ1jkþKkþ1Ykþ1(31)
where Kis calculated by Equation (32):
Kkþ1¼Pkþ1jkHT
kþ1S−1
kþ1(32)
In the above equation, the Pand Smatrices are calculated as follows:
Pkþ1jk¼FkPkjkFT
kþQ(33)
Skþ1¼Hkþ1Pkþ1jkHT
kþ1þR(34)
in which the Hand Fmatrices are linearized functions of h and f around the present estimate as
follows:
Fk¼∂f
∂xjˆ
xk;uk(35)
Hkþ1¼∂h
∂xjˆ
xkþ1jk(36)
In any stage of Kalman estimator algorithm, the Pmatrix must be updated by Equation (37):
Pkþ1jkþ1¼I−Kkþ1Hkþ1
ðÞPkþ1jk(37)
The related flowchart of estimation algorithm of Kalman estimator in the k‐th stage is shown in
Figure 7.
6. STABILITY ANALYSIS OF PROPOSED CONTROLLER
In SMC, for maintaining state trajectory on switching surface, the switching surface must be an
absorbent surface. It means that the system states always move towards the sliding surface and hit it.
The Lyapunov’s theorem is often used for discussing about the stability in the sliding mode control
design. The candidate for the Lyapunov’s function is considered as follows [27]:
VsðÞ¼1
2s2(38)
According to Lyapunov’s theorem the stability condition is presented as follows:
˙
VsðÞ¼1
2
d
dts2
¼s˙
s<0 (39)
This equation is known as sliding condition. By using the Lyapunov’s direct method, since V(s)is
clearly positive definite and also radially unbounded, satisfying the negative definiteness of
SLIDING MODE CONTROLLER FOR HYDROPOWER PLANTS
Copyright © 2011 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power (2011)
DOI: 10.1002/etep
˙
VsðÞimplies that the equilibrium at the origin s= 0 is globally asymptotically stable, and therefore, s
tends to zero as the time tends to infinity. Moreover, all trajectories starting off the sliding surface s=0
must reach it in finite time and then will remain on this surface. The control signal must be realized as
this sliding condition is provided in the presence of load changes, parameters’nonlinearity and
uncertainties. In other words, the coefficient Kin Equation (19) must be determined as sliding
condition is provided. This aim is realized as follows:
s˙s¼x1þ˙x1
ðÞ˙x1þ˙˙x1
ðÞ¼x1
˙x1þ˙x2
1þ˙x1
˙˙x1(40)
where x
1
is one of state variables, ˙x1is introduced in Equation (14), and ˙˙x1is obtained by derivative
of the ˙x1as follows:
˙˙x1¼−1
Tp
˙x1þKpηx5
ðÞ⋅At
Tp
x2ffiffiffiffiffi
x2
p˙x5þ3
2x5
˙x2ffiffiffiffiffi
x2
p−UNL
˙x2Þ
¼−1
Tp
˙x1þKpηx5
ðÞ⋅At
Tp
x2ffiffiffiffiffi
x2
p−1
Tm
x5−Km
Tm
x6−KmK
Tm
x1þKm
Tm
u
þ3
2x5
˙x2ffiffiffiffiffi
x2
p−UNL
˙x2Þ
¼−1
Tp
˙x1þKpηx5
ðÞ⋅At
Tp
x2ffiffiffiffi
x2
p−1
Tm
x5−Km
Tm
x6−KmK
Tm
x1þKm
Tm
ueq−K⋅sign sðÞ
þ3
2x5
˙x2ffiffiffiffiffi
x2
p−UNL
˙x2
(41)
Finally, for realizing the sliding condition (Equation (39)), the following condition should be satisfied:
K>
ueq−Tm
Km½−Tp
KpηAt
x1
x2ffiffiffi
x2
p−Tp−1
ðÞ
KpηAt
˙x1−3
2
x5
˙x2
x2þTP
KPUNL
˙x2þ1
Tmx5þKm
Tmx6þKmK
Tmx1
sign sðÞ (42)
Pk|k
Pk+1|k
Kk+1
Y S
S-1
k
X
ˆ
Pk+1|k+1
1|1
ˆ
++ kk
X
Figure 7. Flowchart of estimation algorithm of extended Kalman estimator in the k‐th stage.
R. HOOSHMAND, M. ATAEI AND A. ZARGARI
Copyright © 2011 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power (2011)
DOI: 10.1002/etep
In practice and in the presence of load changes, parameters’nonlinearity and uncertainties, the u
eq
can be different from Equation (21), and Kmust be selected as the sliding condition is provided
considering changes of variables. To derive a reasonable value for lower bound of K, it is supposed
that in u
eq
, the time constant and the coefficients are varied ±25% with respect to nominal values and
the state variables are varied as Table II in worst status. By considering these change ranges and
applying a searching numerical algorithm, the safe minimum Kis obtained as K= 234.
By using this value of K, the sliding condition is guaranteed. But in proposed controller in this
paper, the sign function K·sign(s) is replaced by fuzzy controller as u
f
=K
fs
·u
fs
(t). For providing the
stability condition in proposed controller, it is sufficient to keep the value of u
f
within range (−K,+K)
in the worst status. This is accomplished by regulating the membership functions of u
f
. These
membership functions are optimized using PSO algorithm to obtain best response.
Also, it should be noted that in proposed controller, the estimations of variables are used for
constructing the control signals u
eq
and u
f
. Because the estimation error is guaranteed to be small by
using extended Kalman filter, this does not affect on the stability analysis. This is due to that in the
proposed PSO‐FSMC controller, the control signal u
f
has been created with a high safety margin with
respect to the parameters errors.
7. SIMULATION RESULTS
In this section, the results of simulating the proposed control method in the hydropower plant of
Karoon‐3 in Shahrekord, Iran are studied. The model of this power plan is based on Figure 2 whose
parameters are included in Appendix. Simulation results are based on four cases as follows:
Case 1: System’s response in the presence of integral controller and for the 0.2 pu change in load
(conventional PI controller).
Case 2: System’s response by applying conventional SMC.
Case 3: System’s response by applying PSO‐FSMC without use of extended Kalman estimator.
Case 4: System’s response by PSO‐FSMC and use of extended Kalman estimator (proposed
controller).
Indeed, the simulation results begun from conventional PI controller, and its deficiencies are
evaluated. Finally, the control method is improved using SMC and fuzzy logic. Then by using Kalman
Table II. Change range of state variables in the worst status.
State variable x
1
x
2
x
3
x
4
x
5
x
6
Change range (−10,+10) (0–200) (0–200) (0–100) (−1,+1) (−10,+10)
0 20 40 60 80 100 120 140 160 180 200
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
time (seconds)
x1(t)
Figure 8. Frequency response of power plant for 0.2 pu change in load.
SLIDING MODE CONTROLLER FOR HYDROPOWER PLANTS
Copyright © 2011 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power (2011)
DOI: 10.1002/etep
estimator, the control method becomes practically possible as it is shown in simulation results. It
should be noted that the fuzzy rules in all cases are defined based on Table I throughout simulations.
7.1. System’s responses for case 1
The step response Δω(t) of the system for the 0.2 pu change in load is as shown in Figure 8.
As can be seen, the settling time of response is 170 s, and the amount of undershoot is 2.9. So the
system does not have an appropriate response in the settling time and undershoot, and we need to have
a better control over the power plant.
7.2. System’s responses for cases 2 and 3
The diagram of the system frequency response Δω(t) by PSO‐FSMC in comparison with the
conventional SMC is shown in Figure 9. The diagram of the switch function s(t) by PSO‐FSMC in
comparison with the conventional SMC is shown in Figure 10. Also, the diagram of control signal u(t)
by PSO‐FSMC in comparison with the conventional SMC is shown in Figure 11.
As can be seen, by using the conventional SMC, the settling time and the undershoot are reduced
noticeably, but the control effort is very high and the chattering problem still exists. The control signal,
0 5 10 15
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
time (seconds)
x1(t)
Figure 9. System’s response Δω(t) by particle swarm optimization‐fuzzy sliding mode controller (PSO‐
FSMC) in the comparison with conventional SMC.
0 2 4 6 8 10 12
-2
-1
0
1
2
time (seconds)
s(t)
0 2 4 6 8 10 12
-6
-4
-2
0
2
time (seconds)
s(t)
PSO-FSMC
SMC
Figure 10. Switch function s(t) by particle swarm optimization‐fuzzy sliding mode controller (PSO‐FSMC)
in comparison with conventional SMC.
R. HOOSHMAND, M. ATAEI AND A. ZARGARI
Copyright © 2011 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power (2011)
DOI: 10.1002/etep
which is very high, may excite the high‐frequency unmodeled dynamics and even make the system
unstable. Moreover, providing such a control signal is impossible or very difficult in practice. Ideally, an
infinitely fast switching mechanism is needed. Practically, it is impossible to realize this infinite rate of
switching. This is due to the physical limits of the actuators and the time delays in the control systems.
Therefore, it is necessary to remove the chattering problem so that the control effort is smoothed.
Although, by applying the proposed PSO‐FSMC method, the settling time and the undershoot are
increased a little, the system’s response will be acceptable, whereas the control effort is reduced and
the chattering problem will be eliminated. In this case, the obtained settling time is 10 seconds, and the
amount of undershoot is 0.71. It should be noted that the {a, b, c, d, e, f, g, h, i, j, k} parameters of
fuzzy sets, which are indicated in Figure 4, are obtained as {1.73, 5.34, 8.53, 0.63, 4.04, 8.12, 1.02,
3.06, 5.18, 7.23, 9.09}. The fuzzy rules in this case are defined based on Table I. Also, the range of
parameters of fuzzy sets and optimizing algorithm PSO are presented in the Appendix.
7.3. System’s responses for case 4
Because the access to the system states is difficult and sometimes impossible, the simulation results in
the previous sections are practically unavailable. Therefore, it is suitable to use the estimators for
estimating the system state variables. By using the suggested control system in the presence of
designed extended Kalman estimator, the system frequency response Δω(t), control signal u(t), and
switch function s(t) are shown in Figure 12. It should be noted that the parameters of Kalman
estimator are R= 0.0001 and Q= 0.0001. The {a, b, c, d, e, f, g, h, i, j, k} parameters of fuzzy sets are
similar to Section 7.2, and the fuzzy rules are defined based on Table I. It should be noted that after
adding the Kalman estimator to PSO‐FSMC, there is no requirement to reform the fuzzy rules and the
fuzzy sets. This is because of the well‐acceptable performance of the designed Kalman estimator that
estimates the state variables well.
As can be seen, in PSO‐FSMC and using Kalman estimator, the settling time and the undershoot are
equal to PSO‐FSMC nearly and the system’s response is proper, the settling time is 10 seconds, and
the undershoot is 0.71. It is worthwhile that the accuracy of the obtained results is without using the
actual state variables of the system.
7.4. Comparison of performance indices for all cases
The comparisons of performance indices, obtained from simulating, are presented in Table III. In this
case, the control effort of this supplementary controller is evaluated with respect to other
supplementary controllers. So, the value of control effort provided in the first column of Table III
0 5 10 15
-100
0
100
200
300
u(t)
0 5 10 15
0
20
40
60
80
time (seconds)
u(t)
SMC
PSO-FSMC
Figure 11. Control signal u(t) by particle swarm optimization‐fuzzy sliding mode controller (PSO‐FSMC)
in comparison with conventional SMC.
SLIDING MODE CONTROLLER FOR HYDROPOWER PLANTS
Copyright © 2011 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power (2011)
DOI: 10.1002/etep
is related to the case in which only PI controller has been used with no supplementary control effort.
As the results show, in PI controller, the settling time and the undershoot are big, and so the system’s
response is not proper, but it can be improved by applying the SMC. By using conventional SMC, the
settling time and the undershoot are reduced noticeably, but the control effort is quite high. By using
PSO‐FSMC, the control effort is reduced. Although the settling time and the undershoot, in PSO‐
FSMC control, are increased a little, still the system’s response is acceptable, the control effort is
reduced, and the chattering problem is eliminated. In PSO‐FSMC, using Kalman estimator, the
settling time and the undershoot are almost equal to PSO‐FSMC, and the system’s response is proper,
but the control effort is a little increased.
7.5. Numerical evaluation of the system stability
In this subsection, the stability of proposed controller is evaluated for various changes in load and
some parameters.
7.5.1. The evaluation against various changes in load. Here, the load is changed in steps 0.3, 0.4,
−0.1, −0.2 pu, and the stability of proposed controller is evaluated. Simulation results are shown in
Figure 13. As it is shown in this figure, the proposed control method has well‐acceptable performance
in the settling time and overshoot points of view in the face of load changes.
0 2 4 6 8 10 12
-1
-0.5
0
0.5
time (seconds)
x1(t)
0 2 4 6 8 10 12
-5
0
5
time (seconds)
s(t)
0 2 4 6 8 10 12
0
50
100
time (seconds)
u(t)
(a)
(b)
(c)
Figure 12. System’s response by using the suggested control system in the presence of Kalman estimator:
(a) response of Δω(t), (b) switch function, and (c) control signal.
Table III. Comparison of performance indices of conventional and the proposed methods.
Index Settling time
in second
Undershoot Control effort (∑|u(t)|) of
supplementary signal
Method of control
PI controller 170 2.9 0
Conventional SMC 6 0.34 2.473 × 10
6
PSO‐FSMC 10 0.71 1.282 × 10
5
PSO‐FSMC using Kalman estimator
(proposed controller)
10 0.71 1.301 × 10
5
PI, proportional–integral; SMC, sliding mode controller; PSO‐FSMC, particle swarm optimization‐fuzzy SMC.
R. HOOSHMAND, M. ATAEI AND A. ZARGARI
Copyright © 2011 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power (2011)
DOI: 10.1002/etep
7.5.2. The evaluation against changes in some parameters. Here, some parameters such as f
p1,
T
m
,T
e
are changed as 25% with respect to nominal value, and the stability of proposed controller is
evaluated. The result is shown in Figure 14 for the 0.2 pu change in load. As it is shown in this figure,
the proposed control method also performs well against the change of parameters. The obtained
settling time is 12 seconds, and the amount of undershoot is 0.74; they have small difference with
respect to Figure 9.
8. CONCLUSION
A proposed PSO‐FSMC system has been used to improve the frequency response of a hydropower
plant. The simulation results show that applying the PI controller does not have an appropriate
response, based on the settling time and the undershoot and that a better control over the plant is
needed. By using the supplementary control based on the conventional SMC, the settling time and the
undershoot are reduced noticeably, but the control effort is quite high and the chattering problem
continues to exist, which increases the control activities and excites the unmodeled high‐frequency
dynamic and may even destabilizes the system. By applying the proposed PSO‐FSMC in comparison
with the conventional SMC, the settling time and the undershoot are increased a little, whereas the
system’s response remains acceptable, the control effort is reduced and smoothed, and the chattering
010 20 30 40 50 60
-0.5
0
0.5
time (seconds)
Change in load in pu
010 20 30 40 50 60
-2
-1
0
1
2
3
time (seconds)
x1(t)
Figure 13. Evaluating the proposed controller for various changes of load in pu.
0 5 10 15
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
time (seconds)
x1(t)
Figure 14. Evaluating the proposed controller for 25% change of some parameters such as f
p1
,T
m
, and T
e
.
SLIDING MODE CONTROLLER FOR HYDROPOWER PLANTS
Copyright © 2011 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power (2011)
DOI: 10.1002/etep
problem is eliminated. To regulate the fuzzy logic parameters of this controller, the PSO algorithm is
used to attain the best frequency response. Moreover, the Kalman estimator is applied to estimate the
actual state variables until the implementation of the control method becomes practically possible. In
addition to the unavailability problem of some state variables, the implementing cost is reduced in
comparison with the case of using the measurement tools. The simulation results show the
effectiveness of the estimator.
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R. HOOSHMAND, M. ATAEI AND A. ZARGARI
Copyright © 2011 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power (2011)
DOI: 10.1002/etep
APPENDIX
PARAMETERS OF HYDROPOWER PLANT
Capacity of the small hydropower plant, P
R
= 250 MW.
Nominal load, P
0
= 230 MW.
System nominal frequency, f
0
= 50 Hz.
Model of power system:
D¼P0=PR
f0¼230
250*50 ¼0:0184 pu=Hz
KP¼1
D¼54:347 Hz=pu
TP¼2H
f0D¼2*7
50*0:0184 ¼15:217 s
Parameters:
Parameters KKI A
t
f
0
f
p1
f
p2
C
s
T
e
T
w
Z
p
T
m
K
m
U
NL
H
0
Value 5 0.06 1.89 0 0.04 0.02 30.75 0.194 1.475 2.289 0.03 0.5 0.14 179.4
PARAMETERS OF FUZZY SETS AND OPTIMIZING ALGORITHM
The values of parameters which are used in this optimization algorithm:
Parameter c
1
c
2
w
min
w
max
Iteration of algorithm Number of particles w
1
w
2
w
3
Value 2 2 0.4 0.9 30 50 0.5 0.5 1
Range of variables {a, b, c, d, e, f, g, h, i, j, k}:
Variable a,db,ec,fg h i j k
Range of variable 0–64–88–10 0–32–54–76–98–10
SLIDING MODE CONTROLLER FOR HYDROPOWER PLANTS
Copyright © 2011 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power (2011)
DOI: 10.1002/etep
