Wavelet neural adaptive proportional plus conventional integral-derivative controller design of SSSC for transient stability improvement

Engineering Applications of Artificial Intelligence ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Contents lists available at ScienceDirect

Engineering Applications of Artificial Intelligence journal homepage: www.elsevier.com/locate/engappai

Wavelet neural adaptive proportional plus conventional integral-derivative controller design of SSSC for transient stability improvement Mojtaba Alizadeh a, Soheil Ganjefar b,n, Morteza Alizadeh c a

Faculty of Electrical Engineering, K.N. Toosi University of Technology, Shariatei Street, Seyed Khandan Bridge, P.O. Box 16315-1355, Tehran, Iran Department of Electrical Engineering, Faculty of Engineering, Bu-Ali Sina University, Shahid Fahmideh Street, P.O. Box 65178-38683, Hamedan, Iran c Faculty of Electrical Engineering, Babol Nooshirvani University of Technology, Shariatei Street, Babol, Iran b

art ic l e i nf o

a b s t r a c t

Article history: Received 21 October 2012 Received in revised form 30 April 2013 Accepted 27 June 2013

Although the PI or PID (PI/PID) controllers have many advantages, their control performance may be degraded when the controlled object is highly nonlinear and uncertain; the main problem is related to static nature of fixed-gain PI/PID controllers. This work aims to propose a wavelet neural adaptive proportional plus conventional integral-derivative (WNAP+ID) controller to solve the PI/PID controller problems. To create an adaptive nature for PI/PID controller and for online processing of the error signal, this work subtly employs a one to one offline trained self-recurrent wavelet neural network as a processing unit (SRWNN-PU) in series connection with the fixed-proportional gain of conventional PI/PID controller. Offline training of the SRWNN-PU can be performed with any virtual training samples, independent of plant data, and it is thus possible to use a generalized SRWNN-PU for any systems. Employing a SRWNN-identifier (SRWNNI), the SRWNN-PU parameters are then updated online to process the error signal and minimize a control cost function in real-time operation. Although the proposed WNAP+ID is not limited to power system applications, it is used as supplementary damping controller of static synchronous series compensator (SSSC) of two SSSC-aided power systems to enhance the transient stability. The nonlinear time-domain simulation and system performance characteristics in terms of ITAE revealed that the WNAP+ID has more control proficiency in comparison to PID controller. As additional simulations, the features of the proposed controller are compared to those of the literature while some of its promising features like its fast noise-rejection ability and its high online adapting ability are also highlighted. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Conventional PI/PID controller Wavelet neural network Adaptive control Static synchronous series compensator (SSSC) Power system stability and control

1. Introduction A power system can be considered as a highly nonlinear and large scale multi-input multi-output dynamical system with configurations and parameters that change with time (Sauer and Pai, 1998). Low frequency electro-mechanical oscillations may occur when a disturbance is applied to the power system. Power system damping can be achieved by applying supplementary controllers to control devices of transmission lines and/or generating units. Flexible AC transmission system (FACTS) devices as control devices of transmission lines are based on the application of power electronics as well as high voltage and high power converters, which are in a series, parallel, or a combination of both. These devices increase the power handling capacity of the line and improve transient stability and damping performance of the power system (Hingorani and Gyugyi, 2000). FACTS controllers

n

Corresponding author. E-mail address: [email protected] (S. Ganjefar).

such as UPFC, SVC, TCSC, IPFC, etc., have been applied for damping oscillations of power systems (Shayeghi et al., 2009, 2010a, 2010b; Vilathgamuwa et al., 2000; Banaei and Kami, 2010; Panda, 2009, 2011; Kazemi and Sohrforouzani, 2006). These devices can improve transient stability, if a suitable supplementary control scheme is considered. Conventionally, these devices employ PI or PID controller as supplementary controller (Hingorani and Gyugyi, 2000; Acha and Agelidis, 2001). Both the analytical and trial-anderror based approaches are used in the literature for PI/PID controller tuning (Astrom and Hagglund, 1995). The analytical methods are based on the frequency response method (Ziegler and Nichols, 1942). Because of their complexity, these methods are not suitable choices for tuning the PI/PID controller in high-order nonlinear plants. Trial-and-error based methods are used to deal with the problem. In this regard, such optimization techniques as evolutionary algorithm (Vlachos et al., 2002), particle swarm optimization (Kao et al., 2006), chaotic ant swarm optimization (Hong, 2010), and chaotic optimization (Coelho, 2009) are proposed. These techniques result in fixed-gain PI/PID controllers. Due to their static property, control performance of the fixed-gain

0952-1976/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engappai.2013.06.018

Please cite this article as: Alizadeh, M., et al., Wavelet neural adaptive proportional plus conventional integral-derivative controller design of SSSC for transient stability improvement. Eng. Appl. Artif. Intel. (2013), http://dx.doi.org/10.1016/j.engappai.2013.06.018i

2

M. Alizadeh et al. / Engineering Applications of Artificial Intelligence ∎ (∎∎∎∎) ∎∎∎–∎∎∎

PI/PID controllers may be degraded when the controlled object is highly nonlinear and uncertain like FACTS equipped power systems (Kazemi and Sohrforouzani, 2006). Adaptive controllers are thus proposed as supplementary damping controllers of FACTS devices to deal with the problem. Employing feed-forward neural networks (FFNNs), a continually online trained neuro-controller with the aid of a neuro-identifier, was proposed as a supplementary controller of FACTS devices (Venayagamoorthy and Kalyani, 2005). Based on this proposal, both the neural controller and identifier must be trained offline before placed in the system. The FFNN with feedback loop is also used as the supplementary damping controller of interline power flow controller (IPFC) to damp power system oscillations (Banaei and Kami, 2011). However, the FFNN has some disadvantages. It is a global network in which all the network weights are active for any given point in the input space, and may be updated with each training sequence. This global nature of the weight effects tends to blur the details of local structure, slows the rate of learning, and results in the existence of local minima. Combining wavelet transform theory with FFNNs architecture (Zhang and Benveniste, 1992), wavelet neural network (WNN) has been proposed as an alternative to FFNNs and is used to identify and adaptive control of nonlinear systems (Zhang et al., 1995; Oussar et al., 1998; Oussar and Dreyfus, 2000; Wai and Chang, 2002; Ashraf, 2005). The WNN is a local network in which output function is well localized in both time and frequency domains. The learning speed of the local network is generally much faster than the global networks like FFNNs (Lekutai, 1997). Besides these advantages, the WNN has a shortcoming: due to its feed-forward network structure, it can only be used for static problems. Hence, self-recurrent wavelet neural networks (SRWNNs), which combine features such as multi-resolution of wavelets, learning ability of FFNNs, fast convergence of WNNs, and attractor dynamics of Recurrent Neural Networks (RNNs), were proposed to identify nonlinear systems (Yoo et al., 2005, 2007). Considering advantages of the SRWNN, it was also used to design supplementary damping controllers. A multi-objective neural inverse damping controller with the aid of an inverse plant-emulator was proposed as the supplementary controller of the SSSC (Ganjefar and Alizadeh, 2012). By incorporating SRWNNs into Takagi–Sugeno–Kang fuzzy model, an adaptive fuzzy wavelet neural damping controller was also proposed to control FACTS devices (Alizadeh and Tofighi, in press). In this work, the SRWNN was employed to construct an adaptive self-recurrent consequent part for each fuzzy rule of a Takagi–Sugeno–Kang fuzzy model. A BP algorithm with the aid of a SRWNN-identifier was then employed to adjust fuzzy rules in real-time operation. With the aim of combining the benefits of the PI/PID and adaptive controllers, and to overcome the PI/PID controller problems, a new control scheme called wavelet neural adaptive proportional plus conventional integral-derivative (WNAP+ID) controller is proposed by this paper. To add an adaptive essence to PI/PID controller, a one to one trained SRWNN is subtly employed as a processing unit (PU) in series with the fixedproportional gain of conventional PID (or PI) controller. Employing a Lyapunov-based back-propagation (BP) training algorithm with adaptive learning rates (ALRs), the SRWNN-PU parameters are then updated online using system sensitivity provided by the SRWNNI. Therefore, the error signal processing is performed online in proportional branch of PID (or PI) controller to minimize a pre-specified control cost function. The major merits of this control system are:


Thanks to high adapting ability of the WNAP+ID, it is possible






to use a generalized proposed controller for different plants, in some special cases. Due to the use of a powerful stable adaptation law based on Lyapunov stability theorem, and, likewise, due to local nature of both the SRWNN-PU and the SRWNNI, and consequently, reach to a double-local network, the WNAP+ID can adapt very fast to changes in the system configuration and/or condition, in a stable manner. In fact, it is able to control highly nonlinear and uncertain systems. The proposed approach synthesizes the advantages of a SRWNN and a conventional PID (or PI) controller. The proposed scheme is not limited to power system applications and covers area of interest to a large part of the intelligence communities; the SRWNN-PU can be added to proportional branch of PI or PID controller of any traditional PI/PID aided control systems to increase their control performance.

To assess the effectiveness and robustness of the proposed WNAP+ID controller, it is applied to two high-order non-linear SSSC-equipped power systems for the purpose of stabilization. The nonlinear time-domain simulation and system performance characteristics in terms of ITAE revealed that the WNAP+ID has more control proficiency in comparison to PID controller, thus significantly improving the transient stability performance of the example power systems and well damping out the inter-area oscillations. As additional simulations, the features of the proposed controller are then compared to those of the literature while some of its promising features like its fast noise-rejection ability and its high online adapting ability are also highlighted.

2. Power system under study To assess effectiveness of the proposed approach a two-machine two-area power system with SSSC, shown in Fig. 1, is considered. The power grid consists of two generators and one major load center at Bus 3. The load center of approximately 2200 MW is modeled using a dynamic load model where the active and reactive power absorbed by the load is a function of the system voltage. Also, power system stabilizers (PSSs) are installed with each machine. The SSSC, located at Bus 1, is in series with line L1. It has a rating of 100 MVA and is capable of injecting up to 10% of the nominal system voltage. The relevant data for the system are given in Appendix A.

3. Self-recurrent wavelet neural network (SRWNN) 3.1. Description of the SRWNN structure A schematic diagram of the SRWNN structure (Yoo et al., 2005, 2007) shown in Fig. 2 has Ni inputs, one output and NinNw mother Bus 4 T2

L1

Bus 3 L3

Bus 1 T1

SSSC

G2

G1

L2-1

L2-2

Load 1

Load 3 Dynamic Load

Bus 2


The offline training of SRWNN-PU is very simple and can be done with any virtual training samples, independent of plant data. Therefore, it is possible to use a generalized SRWNN-PU for any system without any re-training process.

Load 2

Fig. 1. Two-machine power system with SSSC.

Please cite this article as: Alizadeh, M., et al., Wavelet neural adaptive proportional plus conventional integral-derivative controller design of SSSC for transient stability improvement. Eng. Appl. Artif. Intel. (2013), http://dx.doi.org/10.1016/j.engappai.2013.06.018i

M. Alizadeh et al. / Engineering Applications of Artificial Intelligence ∎ (∎∎∎∎) ∎∎∎–∎∎∎

wavelets. Its structure consists of four layers include: input layer, mother wavelet layer, product layer and output layer. Each node of mother wavelet layer has a mother wavelet function and a selffeedback loop. In this study the first derivative of a Gaussian function φ(x) ¼x exp(  0.5  2) is selected as a mother wavelet function. A wavelet φij of each node is derived from its mother wavelet φ as follows: φij ðzij Þ ¼ φðzij Þ;

∀zij ¼ ððuij t ij Þ=dij Þ

ð1Þ

where for discrete time n uij ðnÞ ¼ xj ðnÞ þ φij ðn1Þθij

ð2Þ

where tij and dij are the wavelets translation factor and dilation factor respectively. Also θij denotes the weight of the self-feedback loop in mother wavelet layer. The subscript ij indicates the jth input term of the ith wavelet. The nodes in the product layer are given by the product of the mother wavelets as follows: 
 Ni Ni 1 ψ i ¼ ∏ φðzij Þ ¼ ∏ ðzij Þexp  z2ij 2 j¼1 j¼1

ð3Þ

Output layer

Network output y is finally rendered by the output layer: Nw

yðnÞ ¼ ∑ wi ψ i

where wi is the connection weight between the product nodes and the output nodes. 3.2. Training of the SRWNN The training is based on the minimization of the following quadratic cost function: JðnÞ ¼

1 1 ðy ðnÞy^ ðnÞÞ2 ¼ eðnÞ2 2 d 2

ð5Þ

^ where yd(n) is the desired output and y(n) is the current output of SRWNN for the discrete time n. Using the GD method, the weight values of the SRWNN are adjusted so that the error is minimized after a given number of training cycles. The GD method may be defined as ! ∂JðnÞ i i i W ðn þ 1Þ ¼ W ðnÞ þ η  ð6Þ ∂W i ðnÞ where, for W ¼[tij, dij, θij, wi] and η¼ [ηt, ηd, ηθ, ηw]; Wi and ηi, respectively, represent an arbitrary weight and the corresponding learning rate in the SRWNN. The partial derivative of the cost function with respect to Wi is i

¼ eðnÞ

∂eðnÞ i

∂W ðnÞ

¼ eðnÞ

∂y^ ðnÞ ∂W i ðnÞ

ð7Þ

Product layer

∂W ðnÞ

Mother wavelet layer

The components of the weighting vector are also as follows:

 ^ ∂yðnÞ 1 1 ¼ wi ψ i ð8Þ zij ∂tij ðnÞ dij zij
 ∂y^ ðnÞ ∂y^ ðnÞ ¼ zij ∂dij ðnÞ ∂tij ðnÞ

1

1

ð4Þ

i¼1

∂JðnÞ

1

3

Input layer

ð9Þ


 ∂y^ ðnÞ ∂y^ ðnÞ ¼ ðφij ðn1ÞÞ ∂θij ðnÞ ∂t ij ðnÞ

Fig. 2. The SRWNN structure.

ð10Þ

WNAP+ID S

Kd

1 S

Ki

PU

Kp

PLANT

1/Z

Identifier

Mux

Mux one-to-one trained

TDL

SRWNN

(1-1/Z) Processing Unit (PU)

TDL

ALRs

SRWNNI

ALRs

Identifier Fig. 3. Detailed structure of the proposed controller.

Please cite this article as: Alizadeh, M., et al., Wavelet neural adaptive proportional plus conventional integral-derivative controller design of SSSC for transient stability improvement. Eng. Appl. Artif. Intel. (2013), http://dx.doi.org/10.1016/j.engappai.2013.06.018i

M. Alizadeh et al. / Engineering Applications of Artificial Intelligence ∎ (∎∎∎∎) ∎∎∎–∎∎∎

4

∂y^ ðnÞ ¼ ψi ∂wi ðnÞ

where Kp, Ki and Kd, respectively reperesent the proportional, integral and derivative gains of the PID controller, and, PE(n) is the processed error (PE) signal value in time step n. The SRWNN-PU can be reperesented as

ð11Þ

4. Wavelet neural adaptive proportional plus conventional integral-derivative (WNAP+ID) controller design

xc ¼ ½Δω12 ðnÞ; ðΔω12 ðnÞΔω12 ðn1ÞÞ

ð13Þ

The proposed control structure is shown in Fig. 3(a) which consists of a controller and a identifier. The controller is composed of a fixed integral-derivative term in parallel with a adaptive proportional term including a fixed-proportional gain and a PU. Detailed structure of both the “PU” and the “Identifier” is also shown in Fig. 3(b) and (c), respectively. It should be noted that all the input and output signals of both the SRWNNI and SWRNN-PU must be normalized to the range of [  1,1]. However, for simplicity, this is not considered in design procedure of the proposed controller. The control problem is to design the controller so that the control error signal e (here rotor speed deviation difference of two area Δω12) can be eliminated (damped) quickly. The controller output u(n) is applied to the plant (here, a SSSC equipped power system) so as to eliminate the error signal e (here Δω12). The error signal is considered as the input signal of the controller. Considering Δω12 as the error signal, the controller output can therefore be reperesented as

PEðnÞ ¼ f c ðxc Þ

ð14Þ

uðnÞ ¼ K p ⋅PEðnÞ þ K i

Δω12 ðnÞ þ K d SΔω12 ðnÞ S

where xc is the input vector to the SRWNN-PU. To optimally error signal processing, consequently achieve the optimal control signal u(n); let us define a cost function as   J c ðnÞ ¼ 12 e2c ðnÞ þ βu2 ðnÞ

ð15Þ

where ec(n) ¼(Δω12  d(n) Δω12(n)), and Δω12  d(n) is the desired speed deviation difference of two areas at the time step n which is identically zero. The parameter β is a penalty factor which is used to improve the dynamic trajectory and optimize the overshoot and the settling time of response curve via creating a good reconciliation between the control signal energy and the control performance. Using (15), the gradient of cost function Jc(n) with respect to an arbitrary weight Wic of the SRWNN-PU is   ∂J c ðnÞ ∂Δω12 ðnÞ ∂PEðnÞ þ βuðnÞ ¼ K p ec ðnÞ ð16Þ i ∂uðnÞ ∂W c ðnÞ ∂W ic ðnÞ

ð12Þ

-3

2.5

25

Open

x 10

PLANT

2

Conventional PID 20

SRWNNI

1.5

WNAP-ID (p.u) 2

15

0

1

1

-

-

2

(deg)

1 0.5

10

-0.5

5

-1.5

-1

-2 0

1

2

3

4

5

6

7

8

0

1

2

3

4

5

6

7

8

0

1

2

3

4

5

6

7

8

0

1

2

3

4

5

6

7

8

0

1

2

3

4

5

6

7

8

2553

w,max

3 2.5

I

c

w,max

3.5

2 2

3

4

5

6

7

8 ,max

,max

1

0.12

3.5 3

=

I

c

t,max

0.1 0.08

2.5

I

= t,max

2551 2550

0

c

2552

2 0

1

2

3

4

5

6

7

8

0.02

d,max

0.015

I

c

d,max

0.45 0.4 0.35 0.3 0.01 0

1

2

3

4 Time (sec)

5

6

7

8

Time (sec)

Fig. 4. System responses to removeing the line L2 of 10 cycle duration: (a) controller performance and (b) identifier performance.

Please cite this article as: Alizadeh, M., et al., Wavelet neural adaptive proportional plus conventional integral-derivative controller design of SSSC for transient stability improvement. Eng. Appl. Artif. Intel. (2013), http://dx.doi.org/10.1016/j.engappai.2013.06.018i

M. Alizadeh et al. / Engineering Applications of Artificial Intelligence ∎ (∎∎∎∎) ∎∎∎–∎∎∎

and then W ic ðn þ 1Þ ¼ W ic ðnÞ þ ηic 

then

!

∂J c ðnÞ

ð17Þ

∂W ic ðnÞ

where Wci and ηci respectively represent an arbitrary weight and the corresponding learning rate in the SRWNN-PU. In Eq. (16), ∂PE (n)/∂Wci(n) can be calculated by using (8)–(11), and ∂Δω12(n)/∂u(n) is the system sensitivity. In this paper a real-time plant tracker SRWNNI with series–parallel structure is considered to computing the system sensitivity based on indirect adaptive control theory (Yoo et al., 2007; Narendra and Parthasarathy, 1990). The SRWNNI can be represented as xI ¼ ½ðuðnÞ; uðn1Þ; :::; uðnpÞÞ; ðΔω12 ðnÞ; Δω12 ðn1Þ; :::; Δω12 ðnqÞÞ

ð18Þ

^ 12 ðn þ 1Þ ¼ f I ðxI Þ Δω

ð19Þ

^ 12(n+1) is the predicted inter-area speed deviation at the where Δω time step n+1, and xI is the input vector of the SRWNNI. In fact, the SRWNNI by identifying the generator in terms of its weights predicts generator’s output at the next step. Eqs. (8)–(11) can be used for minimizing the following cost function in both online and offline modes: ^ 12 ðnÞÞ2 ¼ 12e2I ðnÞ J I ðnÞ ¼ 12 ðΔω12 ðnÞΔω

ð20Þ

W iI ðn þ 1Þ ¼ W iI ðnÞ þ ηiI eI ðnÞ

ð21Þ

∂W iI ðnÞ

ð22Þ

From (18), ∂xI /∂u(n) can be written as ∂xI ¼ ½ð1 f 1 ðzÞ ∂uðnÞ

; :::; f p ðzÞÞ

ð0 0 ; :::; 0ÞT

ð23Þ

-5

5

Open Conventional PID WNAP-ID

x 10

PLANT SRWNNI

4 3 (p.u.)

2 0

2 1 0

1

1

-

-

2

(p.u.)

!

^ 12 ðn þ 1Þ ∂xI ∂Δω12 ðn þ 1Þ ∂Δω ≈ ∂uðnÞ ∂xI ∂uðnÞ

x 10

4

2

^ 12 ðnÞ ∂Δω

where WIi and ηIi, respectively, represent an arbitrary weight and the corresponding learning rate in the SRWNNI. As, in the proposed control scheme, the SRWNNI play a significant role to enhance the prosperity of the SRWNN-PU, it has to be initially trained offline. Later on, it ought to be placed in the system configuration and trained in each sampling period of online operation. In each sampling time the generator input and output are sampled and the input vector is formed to the SRWNNI as in (18). The error between the output of the plant and the SRWNNI is back-propagated through the SRWNNI in order to make the necessary updates to the weights of the SRWNNI. Hence, the SRWNNI, now an accurate plant tracker, can be used for calculating the system sensitivity in each time step n as

-5

6

5

-2

-1 -2

-4 -3 -6 1

2

3

4

5

6

7

8

9

-4

10

2553

1.875

2552

w,max

1.88

1.87

1

2

3

4

5

6

7

8

9

10

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5 Time (sec)

6

7

8

9

10

2.005

I

,max

0.074 0.0735

t,max

=

=

,max

2

2550 0

c

1

2551

1.865

2

I

0.073

c

t,max

0

I

c

w,max

0

1.995 0.0725 0

1

2

3

4

5

6

7

8

9

10

-3

x 10

0.2715

I

c

d,max

d,max

10

9.9

0.271 0.2705 0.27

9.8 0

1

2

3

4

5 Time (sec)

6

7

8

9

10

Fig. 5. System responses to a7 10% step changes in voltage reference of G1: (a) controller performance and (b) identifier performance.

Please cite this article as: Alizadeh, M., et al., Wavelet neural adaptive proportional plus conventional integral-derivative controller design of SSSC for transient stability improvement. Eng. Appl. Artif. Intel. (2013), http://dx.doi.org/10.1016/j.engappai.2013.06.018i

M. Alizadeh et al. / Engineering Applications of Artificial Intelligence ∎ (∎∎∎∎) ∎∎∎–∎∎∎

6

where fi(z)¼z  i, and also

 N I;w ^ 12 ðn þ 1Þ ∂Δω 1 1 ¼ ∑ wI;i  ψ I;i zij ∂xI;j dij zij i¼1

ð24Þ

where the subscript I indicates the parameters of SRWNNI and NI,w is the number of nodes in the SRWNNI product layer. This process is done in every sampling period and makes the training online, therefore (15) can be minimized having the system sensitivity which in turn results in an adaptive approach to plant control. Overall the adaption process is described as training cycle in Appendix B.

5. Stability analysis via ALRs

The errors difference can be represented by " #T ∂eI ðnÞ eI ðn þ 1Þ≈eI ðnÞ þ ΔW iI ∂W iI

ð27Þ

From Eqs. (20) and (21) ! ∂J I ðnÞ ∂Δω ^ 12 ðnÞ i i ΔW I ¼ ηI  ¼ ηiI eI ðnÞ ∂W iI ∂W iI

ð28Þ

Then we have following convergence theorem. Theorem 1. Let ηIi be an arbitrary learning rate for the SRWNNI weights and giI,max be defined as giI,max ¼ maxn||giI(n)|| where ^ 12(n)/∂WIi, and ||.|| is the usual Euclidean norm in Rn, then giI(n)¼ ∂Δω the convergence is guaranteed if ηIi chosen as 2 ð29Þ 0 o ηiI o i ðg I;max Þ2

5.1. Stability analysis for identifier Proof. From (25) to (28), ΔVI(n) can be represented as

A discrete-type Lyapunov function can be given by V I ðnÞ ¼ 12e2I ðnÞ

ð25Þ

where

Thus, the variation of the Lyapunov function is obtained by   ΔV I ðnÞ ¼ V I ðn þ 1ÞV I ðnÞ ¼ 12 e2I ðn þ 1Þe2I ðnÞ

ΔV I ðnÞ ¼ λI :e2I ðnÞ

ð26Þ

λI ¼ 0:5ηiI ∥g iI ðnÞ∥2 ð2ηiI ∥g iI ðnÞ∥2 Þ

3

35

SRWNNI

(p.u)

1

25

0

2

(deg)

2

-

2

20

1

1

-3

PLANT

Open Conventional PID WNAP-ID

30

x 10

ð30Þ

-1 -2

15 -3 -4

10 0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

2553

w,max

3 2.5

I

c

w,max

4 3.5

2552 2551

2 2550 2

3

4

5

6

7

8

9

10

,max

1

0.16 0.14

I

t,max

=

0.12 0.1 0.08 0.06

I

= t,max c

4 3.5

c

,max

0

0

1

2

3

4

5

6

7

8

9

3 2.5 2

10 0.6

d,max

0.5

0.015

0.4

I

c

d,max

0.02

0.3 0.01 0.2 0

1

2

3

4

5

Time (sec)

6

7

8

9

10

Time (sec)

Fig. 6. System responses to a three-phase to ground fault at Bus 2: (a) controller performance and (b) identifier performance.

Please cite this article as: Alizadeh, M., et al., Wavelet neural adaptive proportional plus conventional integral-derivative controller design of SSSC for transient stability improvement. Eng. Appl. Artif. Intel. (2013), http://dx.doi.org/10.1016/j.engappai.2013.06.018i

M. Alizadeh et al. / Engineering Applications of Artificial Intelligence ∎ (∎∎∎∎) ∎∎∎–∎∎∎

if λI 40 is satisfied ΔVI(n)o0 thus, the asymptotic convergence of the SRWNNI is guaranteed. Let ηα ¼ηIi(giI,max)2, then λI ≥0:5ηiI ∥g iI ðnÞ∥2 ð2ηα Þ 4 0

 2 1 jdI jmin NI;w N I;i jwI jmax  2 exp ð0:5Þ

ð35Þ

¼ ηθ;max I

 2 1 jdI jmin N I;w N I;i jwI jmax  2 exp ð0:5Þ

ð36Þ

where NI,w and NI,i are the number of nodes and inputs in the product layer and the input layer of SRWNNI, respectively.

Remark 1. The maximum learning rate which guarantees the most rapid or optimal convergence corresponds to 1 ð32Þ ηiI ¼ i ðg I;max Þ2

5.2. Stability analysis for controller The stability analysis of controller is mainly related to its adaptive segment, i.e., the SRWNN-PU. A discrete-type Lyapunov function can be given by h i V c ðnÞ ¼ 12 e2c;1 ðnÞ þ e2c;2 ðnÞ ð37Þ

This is the half of the upper limit in Theorem 1 (Yoo et al., 2007). Considering the SRWNN structure, the value of giI,max corresponds to each learning rate can be easily calculated (Ganjefar and Alizadeh, 2012). Then we have following theorem.

where ec,1(n)¼ (0  Δω12(n)) and ec,2(n)¼ β0.5u(n) represent the errors in the learning process. Thus, the variation of the Lyapunov function due to training process is obtained by

Theorem 2. Let ηIw, ηIt, ηId and ηIθ be the learning rates for the SRWNNI, respectively. The maximum learning rate which guarantees the most rapid or optimal convergence of the SRWNNI corresponds to 1 ηw;max ¼ ð33Þ I N I;w ¼ ηt;max I

ηd;max ¼ I

ð31Þ

From (31), we obtain 0 oηα o 2 and (29) follows. This completes the proof. □

 2 1 jdI jmin N I;w N I;i jwI jmax  2 exp ð0:5Þ

ΔV c ðnÞ ¼ V c ðn þ 1ÞV c ðnÞ ¼

ð34Þ

1

18

ð38Þ

x 10

-3

PLANT SRWNNI

0.5

(p.u)

16

0

2

14

1

1

-

-

(deg)

1 2 2 ∑ ½e ðn þ 1Þe2c;j ðnÞ 2 j ¼ 1 c;j

The errors difference due to learning can be represented by " #T ∂ec;j ðnÞ ΔW ic ð39Þ ec;j ðn þ 1Þ ¼ ec;j ðnÞ þ Δec;j ðnÞ≈ec;j ðnÞ þ ∂W ic

20

2

7

-0.5

12

Open Conventional PID WNAP+ID

10

-1

-1.5

8 0

1

2

3

4

5

6

0

1

2

3

4

5

6

0

1

2

3

4

5

6

0

1

2

3

4

5

6

0

1

2

3

4

5

6

2553

w,max

2.2 2.1

I

c

w,max

2.3

2

2551 2550

1

2

3

4

5

6

,max

0

I

2.2

0.08

t,max =

2.1 2

c

0.085

I

2.3

c

0.09

t,max =

,max

2552

0.075 0

1

2

3

4

5

6 0.32

d,max

0.011

0.3

I

c

d,max

0.012

0.28 0.01 0

1

2

3

Time (sec)

4

5

6

Time (sec)

Fig. 7. System responses to a two-phase to ground fault at Bus 4: (a) controller performance and (b) identifier performance.

Please cite this article as: Alizadeh, M., et al., Wavelet neural adaptive proportional plus conventional integral-derivative controller design of SSSC for transient stability improvement. Eng. Appl. Artif. Intel. (2013), http://dx.doi.org/10.1016/j.engappai.2013.06.018i

M. Alizadeh et al. / Engineering Applications of Artificial Intelligence ∎ (∎∎∎∎) ∎∎∎–∎∎∎

8

where

where ∂ec;1 ðnÞ ∂W ic ∂ec;2 ðnÞ ∂W ic

¼ K p S

∂PEðnÞ

ð40Þ

∂W ic

pffiffiffi ∂PEðnÞ ¼ Kp β ∂W ic

ð41Þ

From Eqs. (16) and (17) ! h i ∂PEðnÞ pffiffiffi ∂J ðnÞ ΔW ic ¼ ηic  c i ¼ ηic K p Sec;1 ðnÞK p βec;2 ðnÞ ∂W c ∂W ic

λc ¼ ηic ∥g ic ðnÞ∥2 ð112ηic ∥g ic ðnÞ∥2 K 2p ðS2 þ βÞÞ

If λc 40 is satisfied, ΔVc(n)o0, thus, the asymptotic convergence of the SRWNN-PU is guaranteed. Let ηα ¼ηci(gic,max)2, then ! ‖g ic ðnÞ‖2 2 2 i i 2 λc ¼ 0:5ηc ∥g c ðnÞ∥ 2ηα K p ðS þ βÞ ðg ic;max Þ2 ≥0:5ηic ∥g ic ðnÞ∥2 ð2ηα K 2p ðS2 þ βÞÞ 4 0

ð42Þ

^ 12(n+1)/∂u, and then we have following convergence where S≈∂Δω theorem: Theorem 3. Let ηci be the learning rate for the SRWNN-PU weights and gic,max be defined as gic,max ¼maxn||gic(n)|| where gic(n) ¼∂PE(n)/ ∂Wci, and ||.|| is the usual Euclidean norm in Rn, then the convergence is guaranteed if ηci chosen as 2 ð43Þ 0 o ηic o 2 2 K p ðS þ βÞðg ic;max Þ2

ð45Þ

ð46Þ

From (46), we obtain 0 oηα o2/Kp2(S2+β) and (43) follows. This completes the proof. □

7 1

5

6 25 km

G1

Line 1 9 8 110 km

B_SSSC

110 km

10 km

SSSC 10 km

Line 2 220 km C9

C7

L7

10

3 G3

4

G2

G4

Area2

Area1

ð44Þ

11

L9

2

Proof. From (37) to (42), ΔVc(n) can be represented as 
 2 1 ΔV c ðnÞ ¼ ∑ Δec;j ðnÞ ec;j ðnÞ þ Δec;j ðnÞ 2 j¼1 pffiffiffi ¼ ðK p Sec;1 ðnÞK p βec;2 ðnÞÞ2  λc

25 km

Fig. 9. Multi-machine power system with SSSC.

-4

6.5

8

x 10

Open

PLANT 6

Conventional PID

6

SRWNNI

4

(pu)

5.5

2

2

5

1

1

-

-

2

(deg)

WNAP-ID

0

4.5 -2 4

-4

3.5

-6 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

2553

w,max

2.05

I

c

w,max

2.1 2552 2551

2 2550 0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

,max

0.085

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

t,max =

2.05

I

0.075

c

t,max

2.1

I

0.08

=

c

,max

0

2

5

d,max

0.0105

I

c

d,max

0.285 0.011

0.01

0.28 0.275 0.27

0

0.5

1

1.5

2

2.5

Time (sec)

3

3.5

4

4.5

5

Time (sec)

Fig. 8. System responses to a three-phase to ground fault at middle of line L1: (a) controller performance and (b) identifier performance.

Please cite this article as: Alizadeh, M., et al., Wavelet neural adaptive proportional plus conventional integral-derivative controller design of SSSC for transient stability improvement. Eng. Appl. Artif. Intel. (2013), http://dx.doi.org/10.1016/j.engappai.2013.06.018i

M. Alizadeh et al. / Engineering Applications of Artificial Intelligence ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Corollary 1. From Remark 1, the SRWNN-PU learning rates for the maximum convergence are as follows: 1 ηic ¼ 2 2 ð47Þ K p ðS þ βÞðg ic;max Þ2 This is the half of the upper limit in Theorem 3. Considering the SRWNN structure, the value of giC,max corresponds to each learning rate can be easily calculated (Ganjefar and Alizadeh, 2012). Then we have following theorem. Theorem 4. Let ηcw,ηct, ηcd and ηcθ be the learning rates for the SRWNN-PU. The maximum learning rate which guarantees the most rapid or optimal convergence of SRWNN-PU corresponds to 1 ηw;max ¼ 2 2 ð48Þ c K p ðS þ βÞ N c;w ηt;max ¼ c

ηd;max ¼ c

¼ ηθ;max c



1

jdc jmin jwc jmax  2 exp ð0:5Þ

K 2p ðS2 þ βÞ N c;w Nc;i 

1 K 2p ðS2 þ βÞNc;w N c;i



1 K 2p ðS2 þ βÞN c;w Nc;i

3

x 10

jdc jmin jwc jmax  2 exp ð0:5Þ

2 ð49Þ

2

jdc jmin jwc jmax  2 exp ð0:5Þ

ð50Þ 2 ð51Þ

Proof. See Appendix C. 6. Simulation studies 6.1. Preliminary A third-order identifier that is sufficient enough to conduct a study of transient stability is selected in this paper, i.e., p and q in (18) are all set to 2. The SRWNNI has 6, 48, 8 and 1 neurons at the input, mother wavelet, product, and output layers, respectively. Also the SRWNN-PU has 2, 12, 6 and 1 neurons at the corresponding layers, i.e., the SRWNN-PU has a simple structure with only two inputs. The initialization of network parameters described in Oussar et al. (1998) and Oussar and Dreyfus (2000) is adopted to initialize the parameters of the proposed SRWNN-PU and SRWNNI in this study. The value of penalty factor β is set to 0.000005. For adaptive control implementation, a sampling rate of 40 Hz is selected in this study. It should be noted that all the above-mentioned parameters

x 10

Open Conventiona PID WNAP-ID

Open Conventiona PID WNAP-ID

1

(p.u.)

1

-3

2

0

-

4

0 -1

1

(p.u.) 3

1

where Nc,w and Nc,i are the number of nodes and inputs in the product layer and the input layer of SRWNN-PU, respectively. The system ^ 12(n+1)/∂u, which is provided by the SRWNNI, must sensitivity S≈∂Δω be replaced by Smax pffiffiffiffiffiffiffiffiffi 2 expð0:5Þ Smax ¼ N I;w wI jmax ð52Þ jdI jmin

-3

2

9

-2

-1

-2

-3 -3 0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

3474

w,max

1.5 1.4

I

c

w,max

1.6

1.3

3473 3472

1.2 3471 2

3

4

5

6

7

8

9

10

,max

1

0.06

I

0.055

1.6 1.5

t,max

=

0.05 0.045 0.04

I

c

t,max =

c

,max

0

1.4 1.3 1.2

0 x 10

1

2

3

4

5

6

7

8

9

10

-3

0.22

d,max

0.2

I

7

c

d,max

8

0.18

6

0.16

5 0

1

2

3

4

5

Time (sec)

6

7

8

9

10

Time (sec)

Fig. 10. System responses to a three-phase to ground fault of 8 cycle duration at line 1, near the Bus 9.

Please cite this article as: Alizadeh, M., et al., Wavelet neural adaptive proportional plus conventional integral-derivative controller design of SSSC for transient stability improvement. Eng. Appl. Artif. Intel. (2013), http://dx.doi.org/10.1016/j.engappai.2013.06.018i

M. Alizadeh et al. / Engineering Applications of Artificial Intelligence ∎ (∎∎∎∎) ∎∎∎–∎∎∎

10

as well as the network structure were found through simulation and trial-and-error. 6.2. Training process It should be noted that to achieve an acceptable control performance, both the identifier and the controller must initially be trained offline. Later on, they can be placed in the final configuration (Fig. 3) for online training implementation. 6.2.1. Offline training of the SRWNNI Offline training process is performed when the system, subjected to various type of disturbances, is operated around the several stable pints while the SSSC is equipped with a conventional controller (here a conventional PI controller is used). During this phase, the input and the desired output of the SRWNNI are [{u(n 1),…, u(n (p+1))}, {Δω12(n 1),…,Δω12(n (q+1))}] and Δω12(n), respectively. First the training samples are accumulated. Then, offline training is performed using BP algorithm. Further, the SRWNNI is placed in the system and online training is applied to each sampling period using BP algorithm with ALRs. During this phase, the inputs and the desired output of the SRWNNI are [{u(n), …,u(n  p)}, {Δω12(n),…,Δω12(n  q)}] and Δω12(n+1), respectively. 6.2.2. Offline training of the controller The controller also needs to be trained and tuned offline before being placed in the system. First of all, a conventional PID controller, as a traditional PID aided control system, is finely tuned for the best possible performance. The conventional PID controller x 10


Without controller (open).
With the fixed-gain finely tuned conventional PID controller.

Open Conventional PID WNAP-ID (p.u.)

2

-3

Open Conventional PID WNAP-ID

1

0

4

(p.u.)

In this section simulation studies are carried out on the example power system subjected to various disturbances under different operating conditions. The results are presented in three cases under three operating points (Table 1). These cases are as follows:

x 10

4

-2

0.5 0

1

-

-

3

6.3. Nonlinear time-domain simulation

-4

6

1

parameters are presented in Appendix D. Later on, the SRWNN-PU must be added to proportional-term of such traditional PID aided control system in order to enhance its control performance. This must be performed in such a way which does not corrupt the initial performance of the traditional PID aided control system. This means that the SRWNN-PU must be trained offline with an aim to reach to a one to one network which does not corrupt the PID control performance. During this phase, the input and the desired output of the SRWNN-PU are [Δω12(n), {Δω12(n) Δω12(n  1)}] and Δω12(n), respectively. After offline training, the SRWNN-PU can successfully added to proportional-term of PID controller, in a series connection. This means that the performance of the controller, after offline training and without online training, is similar to that of the conventional PID controller. Eventually, the controller is hooked up in the final configuration, as shown in Fig. 3, and online training is performed in each sampling period using BP algorithm to enhance the control performance of the PID controller.

-4

-0.5

-6 -1

-8 0

1

2

3

4

5

6

7

8

9

10

w,max

1.23

I

w,max c

1.24

1.22 1

1

2

3

4

5

6

7

8

9

10

4

5

6

7

8

9

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

3473 3472 3471

10

,max

1.25

t,max =

I

1.24 1.23 1.22

I

t,max =

0 x 10

1

2

3

4

5

6

7

8

9

10

-3

0.169

d,max

6.6 6.4 6.2

0.168 0.167

I

d,max

3

0.048

0.044

c

2

0.05

c

c

,max

0

0.046

0 3474

1.25

0.166

6 0

1

2

3

4

5

Time (sec)

6

7

8

9

10

0.165

Time (sec)

Fig. 11. System responses to a three-phase to ground fault of 8 cycle duration at middle of line 2.

Please cite this article as: Alizadeh, M., et al., Wavelet neural adaptive proportional plus conventional integral-derivative controller design of SSSC for transient stability improvement. Eng. Appl. Artif. Intel. (2013), http://dx.doi.org/10.1016/j.engappai.2013.06.018i

M. Alizadeh et al. / Engineering Applications of Artificial Intelligence ∎ (∎∎∎∎) ∎∎∎–∎∎∎


With the proposed controller (WNAP+ID). The conventional PID parameters given in Appendix D were kept unchanged all through the performed tests. 6.3.1. Nominal loading Fig. 4 provides the system responses to 10-cycle removing of line L2. It is evident from the results that WNAP+ID provides better damping performance than PID controller, significantly improving the transient stability performance of the example power system while the SRWNNI provides a promising tracking performance. Intelligent changes in the learning rates are also evidented from the results. To assess the effectiveness of the proposed controller for damping the oscillations, raised by small disturbances, a710% step changes in voltage reference of G1 is selected as second proposed disturbances. The results are provided by Fig. 5. It is observed that the optimal learning rates for both the SRWNNI and SRWNN-PU are found via ALRs and, consequntly, both the SRWNNI and SRWNN-PU provide a promising performance. Also it can be seen that the WNAP+ID has better dapmping prformance in comparison to PID controller. 6.3.2. Light loading The loading condition was changed to light-loading, to test the robustness of the controller to the operating condition, and the type and location of the fault. The system responses to a threephase to ground fault at Bus 2 cleared 10 cycle later by the

4

x 10

6.4.1. Four-machine, two-area power system with SSSC A four-machine, two-area study system installed with SSSC, shown in Fig. 9, is considered. Each area consists of two generator x 10

-3

Open Conventional PID WNAP-ID

3

(p.u.)

2

1

1

4

(p.u.)

6.4. Design problem for Kundur’s four-machine, two-area system

Open Conventional PID WNAP-ID

3

-

0

0

1

-

6.3.3. Heavy loading To complete the simulation studies, the performance of the proposed controller was verified at heavy loading condition. Fig. 8 provides system responses to a three-phase to ground fault of 10 cycle duration at the middle of line L1. Again, from the results presented in this section, it is clear that the proposed approach provides the best control preformance, significantly improving the stability performance of the system despite a considerable change in system’s operating point. It can therefore be concluded that the SRWNN-PU and the SRWNNI (and consequently the WNAP+ID) have acceptable robustness to the operating condition, and to the type and location of the faults.

4

2

1

disconnection of the faulted line L2 and successful reclosure after 4 s is shown in Fig. 6. It can be observed that the WNAP+ID minimizes the variation of power angle deviation difference of the two areas after the fault, helping the system to reach the new operating point with a hight speed. System responces to a twophase to ground fault of 10 cycle duration at terminal of transformer T2, at Bus 4 is shown in Fig. 7. Here again, it is evident that the WNAP+ID provides the best damping preformance.

-3

3

11

-1

-1

-2

-2 -3

-3 0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

3474

w,max I

1.4

c

w,max

1.6 3473 3472

1.2 3471 2

3

4

5

6

7

8

9

10

,max

1

I

0.06

t,max =

0.05

I

0.04

c

t,max

=

c

,max

0

1.7 1.6 1.5 1.4 1.3 1.2

0 x 10

1

2

3

4

5

6

7

8

9

10

-3

d,max

7

0.2

I

c

d,max

0.22 8

6

0.18

5 0.16 0

1

2

3

4

5

Time (sec)

6

7

8

9

10

Time (sec)

Fig. 12. System responses to a single-phase to ground fault of 8 cycle duration at Bus 7.

Please cite this article as: Alizadeh, M., et al., Wavelet neural adaptive proportional plus conventional integral-derivative controller design of SSSC for transient stability improvement. Eng. Appl. Artif. Intel. (2013), http://dx.doi.org/10.1016/j.engappai.2013.06.018i

12

M. Alizadeh et al. / Engineering Applications of Artificial Intelligence ∎ (∎∎∎∎) ∎∎∎–∎∎∎

units. The rating of each generator is 900 MVA and 20 kV. Each of the units is connected through transformers to the 230 kV transmission line. There is a power transfer of 413 MW from area 1 to area 2. Each generator is equipped with PSS. The detailed data are given in Hingorani and Gyugyi (2000). 6.4.2. Parameters setting It is aimed to apply the WNAP+ID to the SSSC in four-machine, two-area study system in order to assess its effectiveness for damping of the transient oscillations. The difference of speed deviations of generators G1 and G3, that is observable for the inter-area oscillation modes, is selected as the input signal of the WNAP+ID. Here again, all the PID gains are set to the values as in previous control peroblem, presented in Appendix D. As stated before, the offline training of SRWNN-PU can be done independent of plant data, and, consequently, it does not need re-training depending on the controlled plant. This means that, here again, the SRWNN-PU with its initial weights, used in previouse control problem (transient stability enhancement of two-machine system), is directly employed in series with fixed-proportional gain of the PID controller. Howevere, the SRWNNI needs to be re-trained offline before placed in the final configuration. All network structure, and training processes for SRWNNI is same to that of previous system. Again, the value of β is set to 0.000005. A sampling rate of 40 Hz is also selected.

6.4.3. Nonlinear time-domain simulation System responses to a three-phase to ground fault of 8 cycle duration at line 1, near the Bus 9, are shown in Fig. 10. It can be seen that the proposed controller provides good damping characteristics to inter-area oscillations and stabilizes the system by modulating the SSSC-injected voltage. Intelligent variations of learning rates for both the SRWNN-PU and SRWNNI are also provided by this figure while the identifier performance is shown by Fig. 14(a). System responses to a three-phase to ground fault of 8 cycle duration at middle of line 2 are shown in Fig. 11. The identifier performance is also provided by Fig. 14 (b). It is evident from the results that WNAP+ID provides better damping performance than PID controller, significantly improving the transient stability performance of the example power system thanks to intelligent variations of learning rates for both the SRWNN-PU and the SRW NNI, while the SRWNNI provides a promising tracking performance. Fig. 12 shows system responses to a single-phase to ground fault of 8 cycle duration at Bus 7 while the identifier performance is presented by Fig. 14(c). Fig. 13 shows system responses to 8 cycle outage of line 1 at 0.5 s. The identifier performance is also shown in Fig. 14(d). Form these results, it can be observed that the proposed controller provides better performance for inter-area oscillations damping in comparison to PID controller, significantly improving the transient stability performance of the system subjected to a wide ranges of disturbances. Accep-

Fig. 13. System responses to 8 cycle outage of line 1 at 0.5 s.

Please cite this article as: Alizadeh, M., et al., Wavelet neural adaptive proportional plus conventional integral-derivative controller design of SSSC for transient stability improvement. Eng. Appl. Artif. Intel. (2013), http://dx.doi.org/10.1016/j.engappai.2013.06.018i

M. Alizadeh et al. / Engineering Applications of Artificial Intelligence ∎ (∎∎∎∎) ∎∎∎–∎∎∎

-4

-3

x 10

x 10 1.5

SRWNNI

(p.u.)

4

-

0

1

-

1

2

3

0.5

-0.5

-4 0

1

2

3

4

5

6

7

8

9

10

0

3

2

PLANT

1

3

4

5

6

7

8

9

10

x 10

PLANT

(p.u.)

1.5

-

3

0

SRWNNI

2

-1

1

(p.u.)

2

2.5

SRWNNI

3

1

-3

-3

x 10

1 0.5

-2 -3

0 -2

-1

-

PLANT

SRWNNI

3

(p.u.)

6

PLANT

1

1

13

0 0

1

2

3

4

5

6

7

8

9

10

-0.5

0

1

2

3

4

Time (sec)

5

6

7

8

load

9

10

Time (sec)

Fig. 14. The identifier performance for four different proposed faults.

table performance of the SRWNNI for different types of faults can also be deduced from the results.

6.4.4. Quantify the improvement of proposed approach In this sub-section, to assess the performance of the WNAP+ID controller in comparison to that of PID controller in terms of timedomain characteristics of inter-area oscillations, an integral time absolute error (ITAE) performance index is used. The ITAE is defined as ITAE ¼ 1000

P1

Q1

P2

Q2

PDynamic

Light Nominal Heavy

0.6667 0.7619 0.8589

 0.1031  0.0478 0.0539

0.5223 0.7509 1

0.0381 0.0513 0.0750

1.2143 1.5711 1.9981

ð53Þ

where tsim is the time range of the simulation. Numerical results of the ITAE for all four proposed disturbances are shown in Table 2. It is worth mentioning that the lower value of ITAE shows the better system responses in terms of time-domain characteristics and damping performance. From “Total” results presented in Table 2, it is evident that the total value of ITAE for PID controller is decreased by 49.15% to 26.9 for the WNAP+ID controller. This means that the WNAP+ID controller, compared to PID controller, has the better responses (about 50%) in terms of time-domain characteristics and damping performance.

load

0.1071 0.0718 0.1076

ITAE (tsim ¼ 10)

Controller type

t:ðjω1 ω3 j þ jω1 ω4 j þ jω2 ω3 j þ jω2 ω4 jÞ dt;

QDynamic

Table 2 Numerical results of the ITAE for four proposed faults.

OPEN PID WNAP+ID

First fault

Second fault

Third fault

Fourth fault

Total

56.4 16.3 06.7

38.1 16.2 08.8

55.0 16.9 08.8

11.5 03.5 02.6

161.0 052.9 026.9

22 MOWNAIC WNAP+ID

20

- 2 (deg)

0

t sim

Cases

18 16

1

Z 

Table 1 Operating conditions.

14 12 10

6.5. Additional simulation studies

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (sec)

Here again, the first test system shown by Fig. 1 is considered to conduct some additional simulation studies.

Fig. 15. System response to 10-cycle removing of line L2, at nominal loading condition.

Please cite this article as: Alizadeh, M., et al., Wavelet neural adaptive proportional plus conventional integral-derivative controller design of SSSC for transient stability improvement. Eng. Appl. Artif. Intel. (2013), http://dx.doi.org/10.1016/j.engappai.2013.06.018i

M. Alizadeh et al. / Engineering Applications of Artificial Intelligence ∎ (∎∎∎∎) ∎∎∎–∎∎∎

14

speed. Its superiority over the MOWNAIC can also be clearly deduced.

6.5.1. Comparative study Comparative study is carried out by this sub-section with an aim to emphasize the advantages of the proposed WNAP+ID controller. In this regard, damping performance of the proposed controller is compared to that of a multi-objective wavelet neural adaptive inverse controller (MOWNAIC) proposed by (Ganjefar and Alizadeh, 2012). The test system shown by Fig. 1 is exposed to 10cycle removing of line L2 at its nominal loading condition. Fig. 15 provides the system response to this fault. It is evident from the results that the WNAP+ID provide better damping performance than the MOWNAIC. This is because of the fact that, unlike the MOWNAIC, the proposed controller is a combination of both PID controller and the SRWNN, and thus synthesizes the advantages of an SRWNN and a conventional PID controller. As the second proposed fault at current loading condition, a 710% step changes in voltage reference of G1 is applied to the system. The result is provided in Fig. 16. Here again, it can be observed that the proposed controller minimizes the variation of power speed deviation difference of the two areas after the fault, helping the system to reach the new operating point with a high

4

6.5.2. Promising features of the WNAP+ID In this sub-section the additional simulation is conducted with the aim to demonstrate the promising features of the proposed controller in a more sensible manner. During the simulation studies in the previous sections, the desired speed deviation difference of the two areas Δωd,12 was set to zero; in this subsection, however, this value is changed during the simulation study so as to test the adapting ability of the WNAP+ID. Fig. 17 shows system responses to 10-cycle removing of line L2 with the proposed controller while Δωd,12 has a fixed value of zero (Δωd,12 ¼0) for Case B, and is changed online as a small noise, for Case A as follows: 8 t o 0:88 > <0 Δωd;12 ðtÞ ¼ 0:5e3 0:88≤t o1:2 ð54Þ > :0 t≥1:2 It can be seen that despite the sudden change in Δωd,12 at the critical time 0.88 s, the learning rates adapted quickly via the ALRs algorithm and the WNAP+ID consequently followed the reference very fast because of its high online adapting ability. At 1.2 s according to the change of the output reference Δωd,12 to zero, control system adapted to follow the new output reference very quickly, and the optimal learning rates, too, were found to quickly use the ALRs algorithm. It can hence be concluded that the WNAP +ID has fast noise-rejection as well as high online adapting abilities, thanks to its double-local structure and its powerful adaptation process.

x 10-5 MOWNAIC WNAP-ID

3

(p.u.)

2

1

-

2

1 0 -1 -2 -3 -4 0

1

2

3

4

5

6

7

8

9

10

7. Conclusions

Time (sec)

Combining the PID controller and a new local network structure, this paper proposed a wavelet neural adaptive proportional

Fig. 16. System responses to a7 10% step changes in voltage reference of G1, at nominal loading condition.

-3

2.5

x 10

4.5 Case A

2

4

w,max

Case B

c

3

1 0.5

0

,max

0

0.88 1.2

2.5

5

0.2

c

-1 -1.5 0.88 1.2

2.5

5

c

0

t,max =

1

Case B

2

-0.5

0.15

0.15 Case A 0.1

Case B

Case A 0

Case B

0.88 1.2

2.5

5

d,max

0.03

0

0.025 0.02

c

SSSC Injected Voltage V q (p.u)

Case A

2.5

-

2

(p.u.)

1.5

3.5

Case A

0.015

Case B 0.01 0

0.88 1.2

2.5

5

Time (sec)

-0.15 0

0.88 1.2

2.5

5

Time (sec) Fig. 17. System responses to 10-cycle removing of line L2.

Please cite this article as: Alizadeh, M., et al., Wavelet neural adaptive proportional plus conventional integral-derivative controller design of SSSC for transient stability improvement. Eng. Appl. Artif. Intel. (2013), http://dx.doi.org/10.1016/j.engappai.2013.06.018i

M. Alizadeh et al. / Engineering Applications of Artificial Intelligence ∎ (∎∎∎∎) ∎∎∎–∎∎∎

plus conventional integral-derivative controller to solve the PI/PID controller problems. A one to one trained SRWNN was subtly employed in series with the fixed-proportional gain of a conventional PID controller. The offline training can be performed by virtual training samples, regardless of the plant data. Using a stable BP algorithm with ALRs, the SRWNN-PU parameters were then updated online based on the information provided by the SRWNNI with the aim of control cost minimization through the online error signal processing. The proposed controller synthesized therefore the advantages of the SRWNN and PID controller. The WNAP+ID controller was applied to an SSSC in a two-machine two-area power system to transient stability improvement. The design problem was finally extended to a four-machine two-area benchmark power system in order to damp inter-area modes of oscillations. The nonlinear time-domain simulation and system performance characteristics in terms of ITAE revealed that the WNAP+ID has more damping proficiency in comparison to conventional PID controller, thus significantly improving the transient stability performance of the example power system and well damping out the inter-area oscillations. Promising features of the WNAP+ID and its superiority over a multi-objective wavelet neural inverse controller are demonstrated through the additional simulations.

D. If n ¼1, then go to step E, else: ^ 12(n) from previous time step and (a) Calculate eI (n) having Δω using current value of the plant output. (b) Train the SRWNNI having “Data I-c” and “Data I-d” from previous time step, using (20) and (21) E. Go to (A) for next time step. Appendix C

S¼


 NI;w ^ 12 ðn þ 1Þ ∂Δω12 ðn þ 1Þ ∂Δω ∂ψ i ≈ ¼ ∑ wI;i ∂u ∂u ∂u i¼1

8 9 < NI;i ∏NI;i φðzij Þ
∂φðz Þ ∂z ∂x = j¼1 ij ij I;j ¼ ∑ wI;i ∑ :j ¼ 1 ∂zij ∂xI;j ∂u ; φðzij Þ i¼1 N I;w

where, in order to simply computing; ∂xI/∂u≈[{1 0,…., 0} {0 0,…., 0}], so


 N I;w ∂φðzij Þ ∂zij o ∑ wI;i max ∂zij ∂xI;j i¼1


 NI;w 2 exp ð0:5Þ o ∑ wI;i max dI i¼1 pffiffiffiffiffiffiffiffiffi 2 exp ð0:5Þ N I;w wI jmax jdI jmin

Appendix A. System date

o

Generators: SB1 ¼2100 MVA, SB2 ¼1400 MVA, H ¼3.7 s, VB ¼13.8 kv, f ¼60 Hz, RS ¼2.8544e 3, Xd ¼ 1.305, X′d ¼ 0.296, X″d ¼ 0.252, Xq ¼ 0.474, X′q ¼0.243, X″q ¼0.18, Td ¼1.01 s, T′d ¼ 0.053 s, T″d ¼0.1 s. Loads: Load1 ¼ 250 MW, Load2 ¼100 MW, Load3 ¼50 MW. Transformers: SBT1 ¼ 2100 MVA, SBT2 ¼1400 MVA, 13.8/500 kV, f ¼60 Hz, R1 ¼R2 ¼0.002, L1 ¼0, L2 ¼0.12, D1/Yg connection, Rm ¼500, Lm ¼500. Transmission lines: 3-Ph, 60 Hz, L1 ¼280 km, L2 ¼ 300 km, L3 ¼ 50 km, R1 ¼0.02546 Ω/km, R0 ¼0.3864 Ω/km, L1 ¼0.9337e  3 H/km, L0 ¼4.1264e  3 H/km, C1 ¼ 12.74e  9 F/km, C0 ¼ 7.751e  9 F/km. SSSC: Snom ¼ 100 MVA, Vnom ¼500 kV, f ¼60 Hz, maximum rate of change of reference voltage (Vqref) ¼3 pu/s; R¼0.00533, L ¼0.16; VDC ¼40 kV; CDC ¼375e 6 F; KP ¼0.00375, Ki ¼0.1875; KP ¼0.1e 3, Ki ¼20e  3; Vq ¼ 70.2. PSSs: Ts ¼15e  3, Tw ¼1, Kp ¼0.25, T1 ¼ 0.06, T2 ¼ 1, T3 ¼ T4 ¼0.

So we have pffiffiffiffiffiffiffiffiffi 2 exp ð0:5Þ Smax ¼ N I;w wI jmax jdI jmin

Appendix B. Training cycle A. If n ¼ 1, then go to step B, else: (a) Calculate ALRs for SRWNN-PU having the “Data I-b” from previous time step, using (48)–(51) (b) Calculate ∂PE(n)/∂Wci(n) using (8)–(11) (c) Train the SRWNN-PU having the “Data I-a” and u(n) from previous time step, using (16) and (17) B. Calculate u(n) using (12), and apply it to both the plant and the SRWNNI, and also save it for using in the next time step. C. Calculate following items using SRWNNI ^ 12(n+1) using (19) and then ec(n+1) (also save Δω ^ 12(n (a) Δω +1) for using in the next time step as Δω ^ 12(n)) (b) The system sensitivity ∂Δω12(n+1)/∂u(n) using (22) and save it as “Data I-a” (c) The system sensitivity S using (52) and save it as “Data I-b” (d) Calculate ALRs for SRWNNI using (33)–(36) and save them as “Data I-c” ^ 12(n)/∂Wci(n) using (8)–(11) and save them as (e) Calculate ∂Δω “Data I-d”

15

Appendix D. The PID parameters Kp ¼40, Ki ¼ 3.9687, Kd ¼0.001. References Alizadeh, M., Tofighi, M., 2013. Full-adaptive THEN-part equipped fuzzy wavelet neural controller design of FACTS devices to suppress inter-area oscillations. Neurocomputing 118, 157–170. Ashraf, M.H., 2005. Wavelet neural network load frequency controller. Energy Convers. Manage. 46, 1613–1630. Acha, E., Agelidis, V., 2001. Power electronic control in electrical system. Butterworth-Heinemann Press, London. Astrom, K.J., Hagglund, T., 1995. PID Controllers: Theory Design and Tuning. Instrument Society of America, Research Triangle Park, NC. Banaei, M.R., Kami, A.R., 2011. Interline power flow controller (IPFC) based damping recurrent neural network controllers for enhancing stability. Energy Convers. Manage. 52, 2629–2636. Banaei, M.R., Kami, A.R., 2010. Improvement of dynamical stability using interline power flow controller. Adv. Electric Comput. Eng. 10 (1), 42–49. Coelho, L.D.S., 2009. Tuning of PID controller for an automatic regulator voltage system using chaotic optimization approach. Chaos Solitons Fractals 39 (4), 1504–1514. Ganjefar, S., Alizadeh, M., 2012. Inter-area oscillations damping by multi-objective wavelet neural inverse controlled SSSC. Int. Rev. Electr. Eng. (IREE) 7 (2), 4000–4012. Hong, W.C., 2010. Application of chaotic ant swarm optimization in electric load forecasting. Energy Policy 38, 5830–5839. Hingorani, N.G., Gyugyi, L., 2000. Understanding FACTS Concepts and Technology of Flexible AC Transmission System. IEEE Press, New York. Kao, C.C., Chuang, C.W., Fung, R.F., 2006. The self-tuning PID control in a slidercrank mechanism system by applying particle swarm optimization approach. Mechatronics 16 (8), 513–522. Kazemi, A., Sohrforouzani, M.V., 2006. Power system damping using fuzzy controlled facts devices’. Electr. Power Energy Syst. 2006 (28), 349–357. Lekutai, G., 1997. Adaptive Self-Tuning Neuro Wavelet Network Controllers. PhD Thesis. State University Blacksburg. Narendra, K.S., Parthasarathy, K., 1990. Identification and control of dynamic system using neural network. IEEE Trans. Neural Networks 1 (1), 4–27. Oussar, Y., Dreyfus, G., 2000. Initialization by selection for wavelet network training. Neurocomputing 34, 131–143.

Please cite this article as: Alizadeh, M., et al., Wavelet neural adaptive proportional plus conventional integral-derivative controller design of SSSC for transient stability improvement. Eng. Appl. Artif. Intel. (2013), http://dx.doi.org/10.1016/j.engappai.2013.06.018i

16

M. Alizadeh et al. / Engineering Applications of Artificial Intelligence ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Oussar, Y., Rivals, I., Personnaz, L., Dreyfus, G., 1998. Training wavelet networks for nonlinear dynamic input–output modeling. Neurocomputing 20 (1–3), 173–188. Panda, S., 2011. Robust coordinated design of multiple and multi-type damping controller using differential evolution algorithm. Electr. Power Energy Syst. 33, 1018–1030. Panda, S., 2009. Multi-objective evolutionary algorithm for SSSC-based controller design. Electr. Power Syst. Res. 79, 937–944. Shayeghi, H., Safari, A., Shayanfar, H.A., 2010a. PSS and TCSC damping controller coordinated design using PSO in multi-machine power system. Energy Convers. Manage. 51, 2930–2937. Shayeghi, H., Shayanfar, H.A., Jalilzadeh, S., Safari, A., 2010b. Tuning of damping controller for UPFC using quantum particle swarm optimizer. Energy Convers. Manage. 51, 2299–2306. Shayeghi, H., Shayanfar, H.A., Jalilzadeh, S., Safari, A., 2009. Design of output feedback UPFC controller for damping of electromechanical oscillations using PSO. Energy Convers. Manage. 50, 2554–2561. Sauer, P.W., Pai, M.A., 1998. Power System Dynamic and Stability. Englewood Cliffs, NJ, Prentice-Hall. Vlachos, C., Williams, D., Gomm, J.B., 2002. Solution to the Shell standard control problem using genetically tuned PID controllers. Contr. Eng. Pract. 10 (2), 151–163.

Venayagamoorthy, G.K., Kalyani, R.P., 2005. Two separate continually onlinetrained neurocontrollers for a unified power flow controller. IEEE Trans. Indust. Appl. 41 (4), 906–916. Vilathgamuwa, M., Zhu, X., Choi, S.S., 2000. A robust control method to improve the performance of a unified power flow controller. Electr. Power Syst. Res. 55, 103–111. Wai, R.J., Chang, J.M., 2002. Intelligent control of induction servo motor drive via wavelet neural network. Electr. Power Syst. Res. 61, 67–76. Yoo, S.J., Park, J.B., Choi, Y.H., 2007. Indirect adaptive control of nonlinear dynamic systems using self-recurrent wavelet neural networks via adaptive learning rates. Inf. Sci. 177, 3074–3098. Yoo, S.J., Park, J.B., Choi, Y.H., 2005. Stable predictive control of chaotic systems using self-recurrent wavelet neural network. Int. J. Contr. Automat. Syst. 3 (1), 43–55. Zhang, J., Walter, G., Miao, Y., Lee, W.N.W., 1995. Wavelet neural networks for function learning. IEEE Trans. Signal Process. 43 (6), 1485–1497. Zhang, Q., Benveniste, A., 1992. Wavelet networks. IEEE Trans. Neural Networks 3 (6), 889–898. Ziegler, J.G., Nichols, N.B., 1942. Optimum setting for automatic controllers. Trans. ASME, 64; , pp. 759–765.

Please cite this article as: Alizadeh, M., et al., Wavelet neural adaptive proportional plus conventional integral-derivative controller design of SSSC for transient stability improvement. Eng. Appl. Artif. Intel. (2013), http://dx.doi.org/10.1016/j.engappai.2013.06.018i