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Adaptive self-tuning PID fuzzy sliding mode control for mitigating power system oscillations
Author’s Accepted Manuscript Adaptive self-tuning PID fuzzy sliding mode control for mitigating power system oscillations Ali Reza Tavakoli, Ali Reza Seifi
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PII: DOI: Reference:
S0925-2312(16)30957-2 http://dx.doi.org/10.1016/j.neucom.2016.08.061 NEUCOM17495
To appear in: Neurocomputing Received date: 4 December 2015 Revised date: 23 May 2016 Accepted date: 17 August 2016 Cite this article as: Ali Reza Tavakoli and Ali Reza Seifi, Adaptive self-tuning PID fuzzy sliding mode control for mitigating power system oscillations, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.08.061 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Adaptive self-tuning PID fuzzy sliding mode control for mitigating power system oscillations Ali Reza Tavakolia, Ali Reza Seifib aDepartment
of Power and Control Engineering, Shiraz University, Shiraz, Iran
bDepartment
of Power and Control Engineering, Shiraz University, Shiraz, Iran
[email protected] [email protected]
Abstract In this paper, an adaptive self-tuning PID fuzzy sliding mode controller (ASPFSM) with a PID switching surface is proposed to damp power system oscillations. To overcome the problems in the design of a sliding-mode controller, which are the supposition of known uncertainty bounds and the chattering phenomenon in the control effort, an adaptive self-tuning PID based Lyapunov theory is studied. An adaption law is obtained from the Lyapunov stability theory, so the stability of the closed-loop system can be guaranteed. Then, the effectiveness of the ASPFSM is studied under different situations of a two-area four-machine power system. The simulation results confirm that performance of ASPFSM is much better than conventional power system stabilizer (CPSS). Keywords: Self-tuning PID; fuzzy sliding mode; Lyapunov theory, power system oscillation 1. Introduction
In the 1960s, many power plants were utilized continuously acting automatic voltage regulators (AVRs). As the number of power plants with AVRs increased, it became clear that the high performance of the AVRs negatively affected the power system stability. Small magnitude power oscillations having low frequency often persisted for long periods of time. In some cases, this led to constraints on the amount of the power to be transmitted by transmission lines. Power system stabilizers (PSSs) that are auxiliary control devices installed on synchronous generators are used in conjunction with their excitation systems to generate control signals toward improving the system damping and removing power transfer limits. However, some insufficiencies of this controller such as fixed parameters, need to an accurate mathematical model to tune its parameters cause other controllers to be suggested. The modern power system operates adjacent to their restraints as a result of the growing energy consumption and impediments of various kinds, and the extension of existing electric transmission lines. The operation of this situation needs a significantly less conservative power system operation and control regime which, in turn, is possible exclusively by monitoring the system moods in much more detail than was necessary previously. Sophisticated computer tools have become prevalent in finding a solution for the difficult problems that appear in the areas of Power System planning, operation, control, diagnosis and design of the systems. Among these computer tools, artificial intelligence has grown and used widely in recent years and has been applied in the areas of the power systems [1-7]. The most widely used and important ones of artificial intelligent tools, applied in the field of electrical power systems are the artificial neural networks (ANN) and the so-called fuzzy systems. In general, we can divide the power system stabilizers into two major groups. The first consists of PSSs with constant parameters such as PID [8] and lead-lag compensators [9-13]. This
category of PSSs are frequently adjusted by optimization algorithms such as PSO [9], genetic algorithm [10], chaotic algorithm [11], simulated annealing [12], bacteria foraging [13] and other tuning algorithms such as H∞ [14] and sequential linear programming [15]. In the second group, neural networks (NNs) [16-19], wavelet neural networks (WNNs) controllers [20-22], self-tuning PID controllers [23], neuro-fuzzy controller [24] and fuzzy logic (FL) controllers [25] are used as adaptive power system stabilizers. Both groups have some disadvantages and advantages. The first group is easy to implement, because of their simple structure. However, their disadvantages are the time-consuming adjusting, needing an accurate model and its non-optimal damping in different situations. The PSS can sometimes be designed by using an adaptive control law. In spite of the adaptive feature, the implementation of these PSSs is difficult in a practical application. Some of these PSSs require an offline training which is problematic in practice. The convergence problem can be mentioned as next problem of these PSSs. So, a suitable PSS can be selected by considering the drawbacks and benefits of both groups. However, if the drawbacks of the adaptive controllers can be resolved, using this kind of PSSs is logical and preferable. From the control point of view, since the dynamic characteristic of power systems is highly nonlinear and the exact model are difficult to find; the model-based control methods are difficult to be implemented. If the exact model of the controlled system is well known, an ideal controller scheme can be used to achieve favorable control performance by canceling all the system dynamics [26].Nevertheless, a compromise between stability and accuracy is essential for the performance of ideal controller. To surmount this requirement, a sliding-mode control strategy suggests several attractive properties for the tracking control, such as lack of sensitivity to parameter variations, external disturbance rejection and fast dynamic responses [26]. However,
the chattering phenomena of the sliding mode control will wear the bearing mechanism and excite unmodelled dynamics. The motivation of this study is to use an adaptive fuzzy sliding-mode control as a power system stabilizer. This controller (1) needs no exact mathematical model of the system; (2) is implemented in online mode; and (3) guarantees stability of the resulting closed-loop system in the sense of Lyapunov stability theorem. An adaptive fuzzy sliding mode control system is used in this study as power system stabilizer to mitigate power system oscillations. First, an adaptive fuzzy sliding-mode control system is introduced. In the proposed sliding-mode control system, the upper bounds of uncertainties are assumed to be known, and the stability analysis is developed in the Lyapunov sense. The total sliding-mode control is composed of the adaptive fuzzy controller design and the curbing controller design. In the fuzzy control design, a fuzzy controller is designed to mimic a feedback linearization (FL) control law and in the curbing controller design an additional controller is designed using a sliding surface to compensate the approximation error between the FL control law and the fuzzy controller. Subsequently, to overcome the difficulties in the design of a sliding-mode controller, which are the supposition of known uncertainty bounds and the chattering phenomenon in the control effort, a self-tuning PID sliding-mode control system is studied. In the self-tuning PID sliding-mode control system, a self-tuning PID bound observer is utilized to adjust the bound of uncertainties real time. Additionally, the theoretical analyzes for the proposed self-tuning PID sliding-mode control system are comprehensively explained. In addition, to ensure that the control error converges to zero, a discrete Lyapunov theory is used
to obtain an updating law for the parameters of self-tuning PID. At the end, simulated results are presented to demonstrate the efficiency of the proposed control system.
2. Power system under study The power system shown in Fig. 1 contains two areas linked together by two 230 kV lines of 220 km length [27]. It was specifically designed in [27] to study low frequency electromechanical oscillations in large interconnected power systems. Despite its small size, it mimics very closely the behavior of typical systems in actual operation. Each area is equipped with two identical round rotor generators rated 20 kV/900 MVA. The synchronous machines have identical parameters [27], except for inertias which are H = 6.5s in area 1 and H = 6.175s in area 2 [27]. Thermal plants having identical speed regulators are further assumed at all locations, in addition to fast static exciters with a 200 gain [27]. The load is represented as constant impedances and split between the areas. Since the surge impedance loading of a single line is about 140 MW [27], the system is somewhat stressed, even in steady-state. In order to damp the oscillations in the example power system, generators G1 to G4 are equipped with an ASPFSM. The detailed bus data, line data, and the dynamic characteristics for the machines, exciters and loads can be found in [27]. Modal analysis of acceleration powers of the four machines after a disturbance shows three dominant modes:
An inter area-mode (frequency of oscillation = 0.64 Hz , damping ratio = -0.026) involving the whole area 1 against area 2: this mode is clearly observable in the tie-line power (power exchanged between two areas). This mode is unstable because of having negative damping.
Local mode of area 1 (frequency of oscillation = 1.12Hz, damping ratio = 0.08) involving this area's machines against each other.
Local mode of area 2 (frequency of oscillation= 1.16Hz, damping ratio = 0.08) involving machine G3 against G4 (i.e.: the smaller the inertia, the greater the local natural frequency).
More detail on the power system shown in Fig. 1 can be found in [27].
10 25km km
Line-1
10 km 25km
Line-2
G1
G4
220km
Area 1
Area 2 G2
G3
Fig. 1. Four-machine, two-area power system under study. 3. Sliding mode control The dynamic of the power system is described as
Bu t t
x t f x t
(1)
where x t Rn is a state vector, u t Rm is a control vector, t Rn is a bounded signal that represents uncertainty or disturbance, B Rn is a constant matrix, f(x(t)) is a map
R
x t Rn f x t
n
and t denotes time. The control objective is to find a suitable control
law so that the trajectory state x can track a trajectory command xd. Define a tracking error as e x xd
(2)
The first phase of sliding-mode control design is to choose a sliding surface which models the desired closed-loop performance in state variable space. Then, the controller should be designed in such a way that the system state trajectories are forced in the direction of the sliding surface and stay on it. Now, assume that an integral operation sliding surface is presented as t
s t x t xd k1e k2e d (3) 0
where k1 and k2 are non-zero positive constants. If the system dynamic function is well-known, there is an ideal controller as:
u*
1 f x xd k1e k2e t B
(4)
Substituting the ideal controller (4) into (1), we obtain e k1e k2e 0 (5)
If the control gains k1 and k2 are appropriately selected such that the characteristic polynomial of (5) is strictly Hurwitz, that is a polynomial whose roots lie strictly in the open left half of the complex plane, then it implies that lim e t 0 . Given that the system dynamic and the external t
load disturbance are always unknown or perturbed, the control law u∗ is not implementable in practical applications. Therefore, an ASPFSM system is used to mimic the control law in this paper. 4. ASPFSM system design 4.1.
Structure of intelligent control system
Fig. 2 displays the block diagram of intelligent control system that is used to modulate the output voltage of generator. The intelligent control system contains the blocks of sliding surface,
fuzzy controller, adaption law, hitting controller and bound estimation. As seen in this figure, the input of sliding surface is error between desired rotor speed deviation Δωd, rotor speed deviation Δω. In this paper ,Δωd and Δω are selected trajectory command and trajectory state. The desired value of rotor speed deviation is selected to be zero, since this signal in steady-state and no fault conditions is zero. As shown in Fig. 2, the controller output is u uˆfz uvs
(6)
where the fuzzy controller uˆfz is the head tracking controller to mimic the control law u∗ and the hitting control uvs is used to compensate the difference between the control law and the fuzzy controller. Moreover, as seen in Fig. 2, the output of the ASPFSM after being clipped is summed with the exciter system's input. In fault conditions, the excitation system of generator regulates the output voltage of generator based on the output of the ASPFSM.
e=Δω-Δωd
Sliding
S t
surface (3)
Fuzzy controller (10)
u fz
0.2
uvs
-0.2
Curbing controller (22) Ew
Self-tuning PID
AFSMC
(23)
Fig. 2. The block diagram of ASPFSM
4.2.
Description of fuzzy controller
If αi is selected as an adjustable parameter, we can write
Excitation system
Vfd
Gen
u fz s, T (7)
where α=[α1; α2; : : : ; αm]T is a parameter vector and ξ=[ ξ1; ξ2; : : : ; ξm]T is a regressive vector with ξi described by
i
wi
(8)
m
wi i 1
Where wi is the firing weight of the ith rule. Regarding the universal approximation theorem
[31], there is an optimal fuzzy control system u fz* s, * in the form of (7) such that
u * t u fz* s, * *T (9)
where is the approximation error and is supposed to be limited by E . Using a fuzzy control system uˆfz s, ˆ to approximate u∗(t)
uˆfz s, ˆ ˆT (10)
where ˆ is the estimated vector of * . By substituting (10) into (1), it is shown that
B uˆ
x t f x t
fz
uvs t
(11)
After some straightforward manipulation, the error equation governing the closed-loop system can be obtained from (3), (4) and (11) as follows:
e t k1e t k2e t B u * uˆfz uvs s t
(12)
And, u fz is denoted as u fz uˆfz u * uˆfz u fz*
(13)
To simplify discussion, define ˆ * to acquire a rephrased form of (13) via (9) and (10) as u fz T
(14)
In fact, the basic idea of Lyapunov stability theory is the mathematical extension of a fundamental physical observation: if the total energy of a system is continuously dissipated, then the system must finally stay in equilibrium point. Thus, the stability of a system is the descent variation of an energy function (Lyapunov function) for introducing a suitable control law and associated adaptation rules. To force s(t) and tend to zero, define a Lyapunov function as [32,33]:
Va t
1 2 B T s t 2 21
(15)
where 1 is a positive constant. Differentiating (15) with respect to time, we can obtain
B T B T s t B uˆfz uvs u * 21 21 B T =s t B T uvs 21 1 =B T s t s t B uvs 1
Va t s t s t
(16)
To achieve Va t 0 , the following adaptation law and hitting controller are used. ˆ 1s t
(17)
uvs E sgn s t
(18)
where sign(.) is a sign function. Then (15) can be rewritten as
Va t E s t B s t B E s t B s t B
= E s t B 0
(19)
This shows thatV t is a negative semi-definite function. Define the following equation
Q t E s t B Va t
(20)
SinceVa t is bounded andVa t is non-increasing and bounded, then t
Q d V t V t a
0
1
a
2
(21)
Moreover, since is bounded by Barbalat’s Lemma [26], limQ t 0 . That is, s t 0 as t . t
Accordingly, the stability of the ASPFSM can be guaranteed. 4.3.
Structure of self-tuning PID
To implement ASPFSM system, the approximation error should be bounded. However, the bound of approximation error E cannot be measured simply for practical applications in industry. If E is chosen too large, we will observe large chattering in the control effort. If E is chosen too small, the control system may be destabilized. To surmount the requirement for the bound of approximation error, therefore, a self-tuning PID is adopted in this study to facilitate adaptive control gain adjustment. The structure of the Self-tuning PID sliding-mode control system is shown in Fig. 2. In this paper, we select e as the input of the Self-tuning PID, and the output of the Self-tuning PID is Ew. If the uncertainties are nonexistent, once the switching surface is reached initially, a very small positive value of Ew would be adequate to retain the trajectory on the sliding surface, and the chattering amplitude would be small. Nevertheless, as soon as the uncertainties are existent, deviations from the sliding surface will require an incessant updating of Ew produced by the Selftuning PID to direct the system trajectory quickly back into the sliding surface. As soon as the tracking error is zero, the tuning of Ew is stopped. If the tracking error e t 0 as t implies S(t) and S t 0 as t . Replacing by in (18), the curbing controller can be represented as follows:
uvs Ew sgn s t
(22)
The control law of the self-tuning PID is given by [28] U PID k U PID k 1 1 k u1 k 2 k u2 k 3 k u3 k
(23)
Where Θ1(k), Θ2(k) and Θ3(k) are proportional, integral and derivative gains of PID controller at the kth sampling period, and Ew U PID . Parameters of u1(k), u2(k) and u3(k) are represented by [12]: u1 k e k e k 1 , u2 k e k , u3 k e k 2e k 1 e(k 2)
(24)
To make straightforward, the following vector related to the controller gains is presented:
k 1 k 2 k 3 k
T
(25)
To update the parameter Θi(k), we consider the following discrete Lyapunov function: V k e 2 k (26)
where γ is a positive constant. The changes in the control error is as follows: e k e k 1 e k (27)
The changes in the Lyapunov function is written by V k V k 1 V k 0 = e 2 k 1 e 2 k 0
(28)
In accordance with (27), (28) can be rewritten as follows V k 2 e k e k e 2 k 0
(29)
The Δe(k) can be rewritten as follows:
e k
e k
k
k
e k
k U PID k
k U PID k
k
k
(30)
Where
U PID k k
eu k u1 k u2 k u3 k
In (30), the term
k
U PID k
(31)
is the Jacobian of controlled system. In most papers, a neural
network has been proposed to approximate this Jacobian [29,30]. This scheme has some disadvantages such as time limitation of online training, because of the presence of a number of neurons in hidden layer and how to determine the structure of neural network. To overcome these problems, an auto-tuning neuron which is composed of only one neuron is used to approximate the dynamic of power system in this study. To overcome this problem, the Jacobian of the plant is replaced by sign(e).
Theorem 1. Let 0<μi<2 be the learning rates of self-tuning PID's parameters and parameter vector Θ(k) be updated by the following equation:
k 1 k
1
k
U PID k
k
eu k e k T
eu k eu k
T
(32)
Where 1 0 0 2 0
0
0 0 3
(33)
According to (32), the e(k) is guaranteed to converge to zero asymptotically.
Proof 1. Using (30) and (32), (29) can be written as:
V k 2 e k 2 e k
k U PID k
U PID k
k
k U PID k
U PID k
k U PID k k k U PID k k 1
k
k
k
U PID k
eu k e c k T
eu k eu k
T k U PID k eu k e k 1 k T k U PID k k eu k eu k U PID k 2 e
2
k eu k k
eu k
T
eu k eu k
T
T
2
T eu k e eu k k T eu k eu k 2
2
2
1u12 k 2u 22 k 3u 32 k 2 e k e k u12 k u 22 k u 32 k u12 k u 22 k u 32 k 2 2 2 2 2 2 1u1 k 2u 2 k 3u 3 k 1 2 u1 k 2 2 u 2 k 3 2 u 3 k e 2 k 2 2 2 u12 k u 22 k u 32 k u1 k u 2 k u 3 k (34) 2
1u12 k 2u 22 k 3u 32 k
2
2
According to (34), it is clear that ΔV(k) is always negative, if 0<μi<2 holds, it means that tracking error e(k) converges to zero by using the updating algorithm given in (32), to design Θ(k+1). This completes the proof 1.
5. Simulation and discussion At this time, the performance of the ASPFSM is evaluated under different situations of the power system. In this regard, different loading conditions presented in Table 1 are used. Simulations are performed for the example power system subjected to various disturbances. To validate the effectiveness of the proposed controller in damping the power system oscillations, the results obtained from the ASPFSM are compared with the results obtained from a lead-lag
compensator (or CPSS) proposed in [27]. Lead-Lag compensators are widely utilized in power systems for different objectives, because of their simple structure and acceptable performance. It should be noted that each generator is equipped with a separate ASPFSM. The parameters of the ASPFSM for areas 1 and 2 are presented in Table 2. If the values of k1 and k2 are properly selected, the desired system dynamics such as rise time, overshoot, and settling time can be easily achieved. Moreover, the gains 1 and 2 are chosen to achieve the best transient responses by trial and error in the experimentation taking into consideration the constraint of stability and the control effort. In addition, the fixed bound E in the ASPFSM system can be obtained roughly regarding the possible operating conditions. It should be noted that the bound value enhances as the amplitude or the frequency of reference trajectory rises because it is related to the kind of reference trajectory. Furthermore, all simulations are carried out for two situations of fixed bound and adaptive bound Ew. In addition, the following performance index which is based on the system performance characteristics is defined. tsim
ITAE 1000
t 1,3 1,4 2,3 2,4 1,2 3,4 dt 0
(34)
where tsim is the time period of simulations. A smaller value of the ITAE index demonstrates more satisfactory performance and dynamical response.
Table 1. Operating conditions used for the simulations (in p.u.).
Operating condition Case 1
Case 2
P1
0.78
0.78
Q1
0.10
0.21
G1
P2
0.78
0.56
Q2
0.13
0.26
P3
0.80
0.80
Q3
0.10
0.07
P4
0.78
0.89
Q4
0.10
0.22
G2
G3
G4
Table 2. The parameters of the ASPFSM for areas 1 and 2.
Parameter
5.1.
1
2
k1
k2
E
200
0.5
5
1
1
Case 1
As first simulation, a 220ms three-phase to ground fault is occurred at beginning of line-2. The inter-area oscillations are shown in Fig. 3(a). As seen in this figure, the performance of proposed controller is better than CPSS so that the oscillations are quickly damped out. Figs. 3(b) and (c) show the control effort of ASPFSM for fixed and adaptive E. Although, ASPFSM with fixed bound E is able to damp oscillations better than adaptive E, however, it causes a large chattering in the control effort. Fig. 3(d) presents numerical results of the ITAE performance index for two controllers. Clearly, the minimum value belongs to the ASPFSM which means better dynamical response and faster damping of oscillations.
d
150 100 ITAE index
50 0 AFSMC with
AFSMC with
fixed E
adaptive E
CPSS
Fig. 3. (a) inter-area oscillations for a 220ms three-phase to ground at the beginning of line-2, solid line (ASPFSM with adaptive E), dotted line (ASPFSM with fixed E), dashed line (CPSS); (b) control effort at presence of ASPFSM with adaptive E; (c) control effort at presence of ASPFSM with fixed E; (d) ITAE index
For the next simulation, it is assumed that a two-phase to ground fault is occurred at the middle of line-1 at t= 0.1s and cleared 220ms later. Fig. 4(a) displays the inter-area oscillations for this fault. It is obvious from this figure; the best performance belongs to the proposed controller. Fig. 4(b) presents Numerical results of the ITAE performance index for two controllers. As predicted, the ITAE index in the presence of ASPFSM has a smaller value.
b
60 40 ITAE index
20 0 ASPFSM
CPSS
Fig. 4. (a) inter-area oscillations for a 220ms two-phase to ground at the middle of line-1, solid line (ASPFSM with adaptive E), dashed line (CPSS); (b) ITAE index.
5.2.
Case 2
In order to verify the effectiveness of ASPFSM, the performance of proposed controller is evaluated on case 2 given in Table 1. Fig. 5 (a) illustrates the inter-area oscillations for when a 5% increase and 5% decrease are applied to the reference voltage of generators 1 and 4 at t=1 and t=4, respectively. Clearly, the ASPFSM can suppress all oscillations satisfactorily. Numerical results of the ITAE performance index for two controllers can be seen from Fig. 5 (b).
b
100 80 60 ITAE index
40 20 0 ASPFSM
CPSS
Fig. 5. (a) inter-area oscillations for a 5% increase and 5% decrease in reference voltage of generators 1 and 4, solid line (ASPFSM with adaptive E), dashed line (CPSS); (b) ITAE index.
For completeness, a single-phase to ground fault is applied to the end of line-2 during 220ms. The response of example power system to this fault is shown in Figs. 6 (a). As seen in this figure, the performance of the proposed controller is still satisfactory. A numerical result of the ITAE performance index for two controllers is drawn in Fig. 6 (b). Accordingly, the oscillations are more effectively damped out by the ASPFSM.
b
100 50
ITAE index
0 ASPFSM
CPSS
Fig. 6. (a) inter-area oscillations for a 220ms single-phase to ground at the end of line-2, solid line (ASPFSM with adaptive E), dashed line (CPSS); (b) ITAE index.
5.3.
A comparative study
In order to demonstrate the successful performance of the proposed controller in damping oscillations, the results obtained from the ASPFSM are compared with the results obtained from an adaptive and online controller which has been proposed in [21]. This controller is an online trained wavelet neural network
controller (OTWNNC) with adaptive learning rates derived by the Lyapunov stability. Figs. 7 displays the inter-area of oscillations for tripping out line-1. As it can be seen from these results, the ASPFSM provides a better dynamical situation for the power system such that inter-area of oscillations is damped out faster.
Fig. 7. A comparative study between ASPFSM and OTWNNC for tripping out line-; solid(ASPFSM), dashed(OTWNNC).
6. Conclusion This study proposes an adaptive self-tuning PID fuzzy sliding-mode control (ASPFSM) with bound estimation as a power system stabilizer to damp the power system oscillations. This paper has successfully validated the adaptive technique applied to the design of the stable fuzzy controller. The Lyapunov stability theorem is used to obtain adaption law to automatically adjust the ASPFSM parameters. Thus, the overall closed-loop system is global stable. Several
simulations with different disturbances are used to validate the effectiveness of ASPFSM in damping all oscillations. In general, the performance of the ASPFSM makes it ideal to implement in real applications.
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[23] Hsu, Yuan-Yih, Liou, Kan-Lee,“ Design of Self-tuning PIDPower System Stabilizers for Synchronous Generators”,IEEETransEnergyConversion1987,Vol.EC-2, No. 3, pp. 343-348. [24] Z. Barton, " Robust control in a multimachine power system using adaptive neuro-fuzzy stabilisers ", IEE Proc., Gener. Transm. Distrib. 2004, Vol. 151, No. 2, pp.261–267. [25] T.Hussein,M.S.Saad,A.L.Elshafei,A.Bahgat,“Damping inter-area modes of oscillation using an adaptive fuzzypowersystemstabilizer”,Elec Power Syst Res 2010, Vol. 80, No. 12, pp. 1428-1436. [26] J.J.E. Slotine, W.P. Li, Applied Nonlinear Control, Prentice-Hall, Englewood Cliffs, NJ, 1991. [27] Kundur P. Power system stability and control. New York: McGraw Hill; 1994. [28] Junghui Chen, Tien-Chih Huang, "Applying neural networks to on-line updated PID controllers for nonlinear process control", Journal of Process Control 2004, Vol. 14, pp. 211–230. [29] P. Shamsollahi, O.P. Malik,“ Design of a neural adaptive power system stabilizer using dynamic backpropagation method”, Electrical Power and Energy Systems 2000, Vol. 22, pp. 29–34. [30] J. He, O.P. Malik, “An Adaptive Power System Stabilizer Based on Recurrent Neural Networks”, IEEE Transactions on Energy Conversion 1997, Vol. 12, No. 4, pp. 413-418. [31] L.X. Wang, Adaptive Fuzzy Systems and Control: Design and Stability Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1994. [32] Rong-Jong Wai, Chih-Ying Lin, Yu-Chih Huang, Yung-Ruei Chang, "Design of High-Performance StandAlone and Grid-Connected Inverter for Distributed Generation Applications," IEEE Trans. Ind. Electron., vol.60, no.4, pp. 1542-1555, Apr. 2013. [33] Rong-Jong Wai, "Adaptive sliding-mode control for induction servomotor drive", IEE Proc. Electr. Power Appl., vol. 147, no. 6, pp.553-562, Nov. 2000.
Ali Reza Tavakoli , was born in Ahvaz, Iran, on September 16, 1984. He received his B.S. degree in Electrical Engineering from azad University, Tehran, Iran, in 2006, and , M.S. degree in Electrical Engineering from The Islamic azad University, dezfoul branch , Iran, in 2010 . He is currently as a PhD student in Department of Power and Control Eng, School of Electrical and Computer Engineering, Shiraz University, Shiraz, Iran. His research areas of research interest are power
system stability, Power plant simulation, Power systems operation, intelligent control, Power electronic and Fuzzy optimization.
Ali Reza Seifi, was born in Shiraz, Iran, on August 9, 1968. He received his B.S. degree in Electrical Engineering from Shiraz University, Shiraz, Iran, in 1991, and ., M.S. degree in Electrical Engineering from The University of Tabriz, Tabriz,Iran, in 1993 and PhD degree in Electrical Engineering from Tarbiat Modarres University (T.M.U), Tehran, Iran, in 2001, respectively. He is currently as a professor in Department of Power and Control Eng, School of Electrical and Computer Engineering, Shiraz University, Shiraz, Iran. His research areas of research interest are Energy,Energy Managment, Power plant simulation, Power systems, Electrical machines simulation, Power electronic and Fuzzy optimization.
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PII: DOI: Reference:
S0925-2312(16)30957-2 http://dx.doi.org/10.1016/j.neucom.2016.08.061 NEUCOM17495
To appear in: Neurocomputing Received date: 4 December 2015 Revised date: 23 May 2016 Accepted date: 17 August 2016 Cite this article as: Ali Reza Tavakoli and Ali Reza Seifi, Adaptive self-tuning PID fuzzy sliding mode control for mitigating power system oscillations, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.08.061 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Adaptive self-tuning PID fuzzy sliding mode control for mitigating power system oscillations Ali Reza Tavakolia, Ali Reza Seifib aDepartment
of Power and Control Engineering, Shiraz University, Shiraz, Iran
bDepartment
of Power and Control Engineering, Shiraz University, Shiraz, Iran
[email protected] [email protected]
Abstract In this paper, an adaptive self-tuning PID fuzzy sliding mode controller (ASPFSM) with a PID switching surface is proposed to damp power system oscillations. To overcome the problems in the design of a sliding-mode controller, which are the supposition of known uncertainty bounds and the chattering phenomenon in the control effort, an adaptive self-tuning PID based Lyapunov theory is studied. An adaption law is obtained from the Lyapunov stability theory, so the stability of the closed-loop system can be guaranteed. Then, the effectiveness of the ASPFSM is studied under different situations of a two-area four-machine power system. The simulation results confirm that performance of ASPFSM is much better than conventional power system stabilizer (CPSS). Keywords: Self-tuning PID; fuzzy sliding mode; Lyapunov theory, power system oscillation 1. Introduction
In the 1960s, many power plants were utilized continuously acting automatic voltage regulators (AVRs). As the number of power plants with AVRs increased, it became clear that the high performance of the AVRs negatively affected the power system stability. Small magnitude power oscillations having low frequency often persisted for long periods of time. In some cases, this led to constraints on the amount of the power to be transmitted by transmission lines. Power system stabilizers (PSSs) that are auxiliary control devices installed on synchronous generators are used in conjunction with their excitation systems to generate control signals toward improving the system damping and removing power transfer limits. However, some insufficiencies of this controller such as fixed parameters, need to an accurate mathematical model to tune its parameters cause other controllers to be suggested. The modern power system operates adjacent to their restraints as a result of the growing energy consumption and impediments of various kinds, and the extension of existing electric transmission lines. The operation of this situation needs a significantly less conservative power system operation and control regime which, in turn, is possible exclusively by monitoring the system moods in much more detail than was necessary previously. Sophisticated computer tools have become prevalent in finding a solution for the difficult problems that appear in the areas of Power System planning, operation, control, diagnosis and design of the systems. Among these computer tools, artificial intelligence has grown and used widely in recent years and has been applied in the areas of the power systems [1-7]. The most widely used and important ones of artificial intelligent tools, applied in the field of electrical power systems are the artificial neural networks (ANN) and the so-called fuzzy systems. In general, we can divide the power system stabilizers into two major groups. The first consists of PSSs with constant parameters such as PID [8] and lead-lag compensators [9-13]. This
category of PSSs are frequently adjusted by optimization algorithms such as PSO [9], genetic algorithm [10], chaotic algorithm [11], simulated annealing [12], bacteria foraging [13] and other tuning algorithms such as H∞ [14] and sequential linear programming [15]. In the second group, neural networks (NNs) [16-19], wavelet neural networks (WNNs) controllers [20-22], self-tuning PID controllers [23], neuro-fuzzy controller [24] and fuzzy logic (FL) controllers [25] are used as adaptive power system stabilizers. Both groups have some disadvantages and advantages. The first group is easy to implement, because of their simple structure. However, their disadvantages are the time-consuming adjusting, needing an accurate model and its non-optimal damping in different situations. The PSS can sometimes be designed by using an adaptive control law. In spite of the adaptive feature, the implementation of these PSSs is difficult in a practical application. Some of these PSSs require an offline training which is problematic in practice. The convergence problem can be mentioned as next problem of these PSSs. So, a suitable PSS can be selected by considering the drawbacks and benefits of both groups. However, if the drawbacks of the adaptive controllers can be resolved, using this kind of PSSs is logical and preferable. From the control point of view, since the dynamic characteristic of power systems is highly nonlinear and the exact model are difficult to find; the model-based control methods are difficult to be implemented. If the exact model of the controlled system is well known, an ideal controller scheme can be used to achieve favorable control performance by canceling all the system dynamics [26].Nevertheless, a compromise between stability and accuracy is essential for the performance of ideal controller. To surmount this requirement, a sliding-mode control strategy suggests several attractive properties for the tracking control, such as lack of sensitivity to parameter variations, external disturbance rejection and fast dynamic responses [26]. However,
the chattering phenomena of the sliding mode control will wear the bearing mechanism and excite unmodelled dynamics. The motivation of this study is to use an adaptive fuzzy sliding-mode control as a power system stabilizer. This controller (1) needs no exact mathematical model of the system; (2) is implemented in online mode; and (3) guarantees stability of the resulting closed-loop system in the sense of Lyapunov stability theorem. An adaptive fuzzy sliding mode control system is used in this study as power system stabilizer to mitigate power system oscillations. First, an adaptive fuzzy sliding-mode control system is introduced. In the proposed sliding-mode control system, the upper bounds of uncertainties are assumed to be known, and the stability analysis is developed in the Lyapunov sense. The total sliding-mode control is composed of the adaptive fuzzy controller design and the curbing controller design. In the fuzzy control design, a fuzzy controller is designed to mimic a feedback linearization (FL) control law and in the curbing controller design an additional controller is designed using a sliding surface to compensate the approximation error between the FL control law and the fuzzy controller. Subsequently, to overcome the difficulties in the design of a sliding-mode controller, which are the supposition of known uncertainty bounds and the chattering phenomenon in the control effort, a self-tuning PID sliding-mode control system is studied. In the self-tuning PID sliding-mode control system, a self-tuning PID bound observer is utilized to adjust the bound of uncertainties real time. Additionally, the theoretical analyzes for the proposed self-tuning PID sliding-mode control system are comprehensively explained. In addition, to ensure that the control error converges to zero, a discrete Lyapunov theory is used
to obtain an updating law for the parameters of self-tuning PID. At the end, simulated results are presented to demonstrate the efficiency of the proposed control system.
2. Power system under study The power system shown in Fig. 1 contains two areas linked together by two 230 kV lines of 220 km length [27]. It was specifically designed in [27] to study low frequency electromechanical oscillations in large interconnected power systems. Despite its small size, it mimics very closely the behavior of typical systems in actual operation. Each area is equipped with two identical round rotor generators rated 20 kV/900 MVA. The synchronous machines have identical parameters [27], except for inertias which are H = 6.5s in area 1 and H = 6.175s in area 2 [27]. Thermal plants having identical speed regulators are further assumed at all locations, in addition to fast static exciters with a 200 gain [27]. The load is represented as constant impedances and split between the areas. Since the surge impedance loading of a single line is about 140 MW [27], the system is somewhat stressed, even in steady-state. In order to damp the oscillations in the example power system, generators G1 to G4 are equipped with an ASPFSM. The detailed bus data, line data, and the dynamic characteristics for the machines, exciters and loads can be found in [27]. Modal analysis of acceleration powers of the four machines after a disturbance shows three dominant modes:
An inter area-mode (frequency of oscillation = 0.64 Hz , damping ratio = -0.026) involving the whole area 1 against area 2: this mode is clearly observable in the tie-line power (power exchanged between two areas). This mode is unstable because of having negative damping.
Local mode of area 1 (frequency of oscillation = 1.12Hz, damping ratio = 0.08) involving this area's machines against each other.
Local mode of area 2 (frequency of oscillation= 1.16Hz, damping ratio = 0.08) involving machine G3 against G4 (i.e.: the smaller the inertia, the greater the local natural frequency).
More detail on the power system shown in Fig. 1 can be found in [27].
10 25km km
Line-1
10 km 25km
Line-2
G1
G4
220km
Area 1
Area 2 G2
G3
Fig. 1. Four-machine, two-area power system under study. 3. Sliding mode control The dynamic of the power system is described as
Bu t t
x t f x t
(1)
where x t Rn is a state vector, u t Rm is a control vector, t Rn is a bounded signal that represents uncertainty or disturbance, B Rn is a constant matrix, f(x(t)) is a map
R
x t Rn f x t
n
and t denotes time. The control objective is to find a suitable control
law so that the trajectory state x can track a trajectory command xd. Define a tracking error as e x xd
(2)
The first phase of sliding-mode control design is to choose a sliding surface which models the desired closed-loop performance in state variable space. Then, the controller should be designed in such a way that the system state trajectories are forced in the direction of the sliding surface and stay on it. Now, assume that an integral operation sliding surface is presented as t
s t x t xd k1e k2e d (3) 0
where k1 and k2 are non-zero positive constants. If the system dynamic function is well-known, there is an ideal controller as:
u*
1 f x xd k1e k2e t B
(4)
Substituting the ideal controller (4) into (1), we obtain e k1e k2e 0 (5)
If the control gains k1 and k2 are appropriately selected such that the characteristic polynomial of (5) is strictly Hurwitz, that is a polynomial whose roots lie strictly in the open left half of the complex plane, then it implies that lim e t 0 . Given that the system dynamic and the external t
load disturbance are always unknown or perturbed, the control law u∗ is not implementable in practical applications. Therefore, an ASPFSM system is used to mimic the control law in this paper. 4. ASPFSM system design 4.1.
Structure of intelligent control system
Fig. 2 displays the block diagram of intelligent control system that is used to modulate the output voltage of generator. The intelligent control system contains the blocks of sliding surface,
fuzzy controller, adaption law, hitting controller and bound estimation. As seen in this figure, the input of sliding surface is error between desired rotor speed deviation Δωd, rotor speed deviation Δω. In this paper ,Δωd and Δω are selected trajectory command and trajectory state. The desired value of rotor speed deviation is selected to be zero, since this signal in steady-state and no fault conditions is zero. As shown in Fig. 2, the controller output is u uˆfz uvs
(6)
where the fuzzy controller uˆfz is the head tracking controller to mimic the control law u∗ and the hitting control uvs is used to compensate the difference between the control law and the fuzzy controller. Moreover, as seen in Fig. 2, the output of the ASPFSM after being clipped is summed with the exciter system's input. In fault conditions, the excitation system of generator regulates the output voltage of generator based on the output of the ASPFSM.
e=Δω-Δωd
Sliding
S t
surface (3)
Fuzzy controller (10)
u fz
0.2
uvs
-0.2
Curbing controller (22) Ew
Self-tuning PID
AFSMC
(23)
Fig. 2. The block diagram of ASPFSM
4.2.
Description of fuzzy controller
If αi is selected as an adjustable parameter, we can write
Excitation system
Vfd
Gen
u fz s, T (7)
where α=[α1; α2; : : : ; αm]T is a parameter vector and ξ=[ ξ1; ξ2; : : : ; ξm]T is a regressive vector with ξi described by
i
wi
(8)
m
wi i 1
Where wi is the firing weight of the ith rule. Regarding the universal approximation theorem
[31], there is an optimal fuzzy control system u fz* s, * in the form of (7) such that
u * t u fz* s, * *T (9)
where is the approximation error and is supposed to be limited by E . Using a fuzzy control system uˆfz s, ˆ to approximate u∗(t)
uˆfz s, ˆ ˆT (10)
where ˆ is the estimated vector of * . By substituting (10) into (1), it is shown that
B uˆ
x t f x t
fz
uvs t
(11)
After some straightforward manipulation, the error equation governing the closed-loop system can be obtained from (3), (4) and (11) as follows:
e t k1e t k2e t B u * uˆfz uvs s t
(12)
And, u fz is denoted as u fz uˆfz u * uˆfz u fz*
(13)
To simplify discussion, define ˆ * to acquire a rephrased form of (13) via (9) and (10) as u fz T
(14)
In fact, the basic idea of Lyapunov stability theory is the mathematical extension of a fundamental physical observation: if the total energy of a system is continuously dissipated, then the system must finally stay in equilibrium point. Thus, the stability of a system is the descent variation of an energy function (Lyapunov function) for introducing a suitable control law and associated adaptation rules. To force s(t) and tend to zero, define a Lyapunov function as [32,33]:
Va t
1 2 B T s t 2 21
(15)
where 1 is a positive constant. Differentiating (15) with respect to time, we can obtain
B T B T s t B uˆfz uvs u * 21 21 B T =s t B T uvs 21 1 =B T s t s t B uvs 1
Va t s t s t
(16)
To achieve Va t 0 , the following adaptation law and hitting controller are used. ˆ 1s t
(17)
uvs E sgn s t
(18)
where sign(.) is a sign function. Then (15) can be rewritten as
Va t E s t B s t B E s t B s t B
= E s t B 0
(19)
This shows thatV t is a negative semi-definite function. Define the following equation
Q t E s t B Va t
(20)
SinceVa t is bounded andVa t is non-increasing and bounded, then t
Q d V t V t a
0
1
a
2
(21)
Moreover, since is bounded by Barbalat’s Lemma [26], limQ t 0 . That is, s t 0 as t . t
Accordingly, the stability of the ASPFSM can be guaranteed. 4.3.
Structure of self-tuning PID
To implement ASPFSM system, the approximation error should be bounded. However, the bound of approximation error E cannot be measured simply for practical applications in industry. If E is chosen too large, we will observe large chattering in the control effort. If E is chosen too small, the control system may be destabilized. To surmount the requirement for the bound of approximation error, therefore, a self-tuning PID is adopted in this study to facilitate adaptive control gain adjustment. The structure of the Self-tuning PID sliding-mode control system is shown in Fig. 2. In this paper, we select e as the input of the Self-tuning PID, and the output of the Self-tuning PID is Ew. If the uncertainties are nonexistent, once the switching surface is reached initially, a very small positive value of Ew would be adequate to retain the trajectory on the sliding surface, and the chattering amplitude would be small. Nevertheless, as soon as the uncertainties are existent, deviations from the sliding surface will require an incessant updating of Ew produced by the Selftuning PID to direct the system trajectory quickly back into the sliding surface. As soon as the tracking error is zero, the tuning of Ew is stopped. If the tracking error e t 0 as t implies S(t) and S t 0 as t . Replacing by in (18), the curbing controller can be represented as follows:
uvs Ew sgn s t
(22)
The control law of the self-tuning PID is given by [28] U PID k U PID k 1 1 k u1 k 2 k u2 k 3 k u3 k
(23)
Where Θ1(k), Θ2(k) and Θ3(k) are proportional, integral and derivative gains of PID controller at the kth sampling period, and Ew U PID . Parameters of u1(k), u2(k) and u3(k) are represented by [12]: u1 k e k e k 1 , u2 k e k , u3 k e k 2e k 1 e(k 2)
(24)
To make straightforward, the following vector related to the controller gains is presented:
k 1 k 2 k 3 k
T
(25)
To update the parameter Θi(k), we consider the following discrete Lyapunov function: V k e 2 k (26)
where γ is a positive constant. The changes in the control error is as follows: e k e k 1 e k (27)
The changes in the Lyapunov function is written by V k V k 1 V k 0 = e 2 k 1 e 2 k 0
(28)
In accordance with (27), (28) can be rewritten as follows V k 2 e k e k e 2 k 0
(29)
The Δe(k) can be rewritten as follows:
e k
e k
k
k
e k
k U PID k
k U PID k
k
k
(30)
Where
U PID k k
eu k u1 k u2 k u3 k
In (30), the term
k
U PID k
(31)
is the Jacobian of controlled system. In most papers, a neural
network has been proposed to approximate this Jacobian [29,30]. This scheme has some disadvantages such as time limitation of online training, because of the presence of a number of neurons in hidden layer and how to determine the structure of neural network. To overcome these problems, an auto-tuning neuron which is composed of only one neuron is used to approximate the dynamic of power system in this study. To overcome this problem, the Jacobian of the plant is replaced by sign(e).
Theorem 1. Let 0<μi<2 be the learning rates of self-tuning PID's parameters and parameter vector Θ(k) be updated by the following equation:
k 1 k
1
k
U PID k
k
eu k e k T
eu k eu k
T
(32)
Where 1 0 0 2 0
0
0 0 3
(33)
According to (32), the e(k) is guaranteed to converge to zero asymptotically.
Proof 1. Using (30) and (32), (29) can be written as:
V k 2 e k 2 e k
k U PID k
U PID k
k
k U PID k
U PID k
k U PID k k k U PID k k 1
k
k
k
U PID k
eu k e c k T
eu k eu k
T k U PID k eu k e k 1 k T k U PID k k eu k eu k U PID k 2 e
2
k eu k k
eu k
T
eu k eu k
T
T
2
T eu k e eu k k T eu k eu k 2
2
2
1u12 k 2u 22 k 3u 32 k 2 e k e k u12 k u 22 k u 32 k u12 k u 22 k u 32 k 2 2 2 2 2 2 1u1 k 2u 2 k 3u 3 k 1 2 u1 k 2 2 u 2 k 3 2 u 3 k e 2 k 2 2 2 u12 k u 22 k u 32 k u1 k u 2 k u 3 k (34) 2
1u12 k 2u 22 k 3u 32 k
2
2
According to (34), it is clear that ΔV(k) is always negative, if 0<μi<2 holds, it means that tracking error e(k) converges to zero by using the updating algorithm given in (32), to design Θ(k+1). This completes the proof 1.
5. Simulation and discussion At this time, the performance of the ASPFSM is evaluated under different situations of the power system. In this regard, different loading conditions presented in Table 1 are used. Simulations are performed for the example power system subjected to various disturbances. To validate the effectiveness of the proposed controller in damping the power system oscillations, the results obtained from the ASPFSM are compared with the results obtained from a lead-lag
compensator (or CPSS) proposed in [27]. Lead-Lag compensators are widely utilized in power systems for different objectives, because of their simple structure and acceptable performance. It should be noted that each generator is equipped with a separate ASPFSM. The parameters of the ASPFSM for areas 1 and 2 are presented in Table 2. If the values of k1 and k2 are properly selected, the desired system dynamics such as rise time, overshoot, and settling time can be easily achieved. Moreover, the gains 1 and 2 are chosen to achieve the best transient responses by trial and error in the experimentation taking into consideration the constraint of stability and the control effort. In addition, the fixed bound E in the ASPFSM system can be obtained roughly regarding the possible operating conditions. It should be noted that the bound value enhances as the amplitude or the frequency of reference trajectory rises because it is related to the kind of reference trajectory. Furthermore, all simulations are carried out for two situations of fixed bound and adaptive bound Ew. In addition, the following performance index which is based on the system performance characteristics is defined. tsim
ITAE 1000
t 1,3 1,4 2,3 2,4 1,2 3,4 dt 0
(34)
where tsim is the time period of simulations. A smaller value of the ITAE index demonstrates more satisfactory performance and dynamical response.
Table 1. Operating conditions used for the simulations (in p.u.).
Operating condition Case 1
Case 2
P1
0.78
0.78
Q1
0.10
0.21
G1
P2
0.78
0.56
Q2
0.13
0.26
P3
0.80
0.80
Q3
0.10
0.07
P4
0.78
0.89
Q4
0.10
0.22
G2
G3
G4
Table 2. The parameters of the ASPFSM for areas 1 and 2.
Parameter
5.1.
1
2
k1
k2
E
200
0.5
5
1
1
Case 1
As first simulation, a 220ms three-phase to ground fault is occurred at beginning of line-2. The inter-area oscillations are shown in Fig. 3(a). As seen in this figure, the performance of proposed controller is better than CPSS so that the oscillations are quickly damped out. Figs. 3(b) and (c) show the control effort of ASPFSM for fixed and adaptive E. Although, ASPFSM with fixed bound E is able to damp oscillations better than adaptive E, however, it causes a large chattering in the control effort. Fig. 3(d) presents numerical results of the ITAE performance index for two controllers. Clearly, the minimum value belongs to the ASPFSM which means better dynamical response and faster damping of oscillations.
d
150 100 ITAE index
50 0 AFSMC with
AFSMC with
fixed E
adaptive E
CPSS
Fig. 3. (a) inter-area oscillations for a 220ms three-phase to ground at the beginning of line-2, solid line (ASPFSM with adaptive E), dotted line (ASPFSM with fixed E), dashed line (CPSS); (b) control effort at presence of ASPFSM with adaptive E; (c) control effort at presence of ASPFSM with fixed E; (d) ITAE index
For the next simulation, it is assumed that a two-phase to ground fault is occurred at the middle of line-1 at t= 0.1s and cleared 220ms later. Fig. 4(a) displays the inter-area oscillations for this fault. It is obvious from this figure; the best performance belongs to the proposed controller. Fig. 4(b) presents Numerical results of the ITAE performance index for two controllers. As predicted, the ITAE index in the presence of ASPFSM has a smaller value.
b
60 40 ITAE index
20 0 ASPFSM
CPSS
Fig. 4. (a) inter-area oscillations for a 220ms two-phase to ground at the middle of line-1, solid line (ASPFSM with adaptive E), dashed line (CPSS); (b) ITAE index.
5.2.
Case 2
In order to verify the effectiveness of ASPFSM, the performance of proposed controller is evaluated on case 2 given in Table 1. Fig. 5 (a) illustrates the inter-area oscillations for when a 5% increase and 5% decrease are applied to the reference voltage of generators 1 and 4 at t=1 and t=4, respectively. Clearly, the ASPFSM can suppress all oscillations satisfactorily. Numerical results of the ITAE performance index for two controllers can be seen from Fig. 5 (b).
b
100 80 60 ITAE index
40 20 0 ASPFSM
CPSS
Fig. 5. (a) inter-area oscillations for a 5% increase and 5% decrease in reference voltage of generators 1 and 4, solid line (ASPFSM with adaptive E), dashed line (CPSS); (b) ITAE index.
For completeness, a single-phase to ground fault is applied to the end of line-2 during 220ms. The response of example power system to this fault is shown in Figs. 6 (a). As seen in this figure, the performance of the proposed controller is still satisfactory. A numerical result of the ITAE performance index for two controllers is drawn in Fig. 6 (b). Accordingly, the oscillations are more effectively damped out by the ASPFSM.
b
100 50
ITAE index
0 ASPFSM
CPSS
Fig. 6. (a) inter-area oscillations for a 220ms single-phase to ground at the end of line-2, solid line (ASPFSM with adaptive E), dashed line (CPSS); (b) ITAE index.
5.3.
A comparative study
In order to demonstrate the successful performance of the proposed controller in damping oscillations, the results obtained from the ASPFSM are compared with the results obtained from an adaptive and online controller which has been proposed in [21]. This controller is an online trained wavelet neural network
controller (OTWNNC) with adaptive learning rates derived by the Lyapunov stability. Figs. 7 displays the inter-area of oscillations for tripping out line-1. As it can be seen from these results, the ASPFSM provides a better dynamical situation for the power system such that inter-area of oscillations is damped out faster.
Fig. 7. A comparative study between ASPFSM and OTWNNC for tripping out line-; solid(ASPFSM), dashed(OTWNNC).
6. Conclusion This study proposes an adaptive self-tuning PID fuzzy sliding-mode control (ASPFSM) with bound estimation as a power system stabilizer to damp the power system oscillations. This paper has successfully validated the adaptive technique applied to the design of the stable fuzzy controller. The Lyapunov stability theorem is used to obtain adaption law to automatically adjust the ASPFSM parameters. Thus, the overall closed-loop system is global stable. Several
simulations with different disturbances are used to validate the effectiveness of ASPFSM in damping all oscillations. In general, the performance of the ASPFSM makes it ideal to implement in real applications.
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Ali Reza Tavakoli , was born in Ahvaz, Iran, on September 16, 1984. He received his B.S. degree in Electrical Engineering from azad University, Tehran, Iran, in 2006, and , M.S. degree in Electrical Engineering from The Islamic azad University, dezfoul branch , Iran, in 2010 . He is currently as a PhD student in Department of Power and Control Eng, School of Electrical and Computer Engineering, Shiraz University, Shiraz, Iran. His research areas of research interest are power
system stability, Power plant simulation, Power systems operation, intelligent control, Power electronic and Fuzzy optimization.
Ali Reza Seifi, was born in Shiraz, Iran, on August 9, 1968. He received his B.S. degree in Electrical Engineering from Shiraz University, Shiraz, Iran, in 1991, and ., M.S. degree in Electrical Engineering from The University of Tabriz, Tabriz,Iran, in 1993 and PhD degree in Electrical Engineering from Tarbiat Modarres University (T.M.U), Tehran, Iran, in 2001, respectively. He is currently as a professor in Department of Power and Control Eng, School of Electrical and Computer Engineering, Shiraz University, Shiraz, Iran. His research areas of research interest are Energy,Energy Managment, Power plant simulation, Power systems, Electrical machines simulation, Power electronic and Fuzzy optimization.