20th Iranian Conference on Electrical Engineering, (ICEE2012), May 15-17,2012, Tehran, Iran

Intelligent Control of SSSC Via an Online Self­Tuning PID to Damp the Subsynchronous

Oscillations Mohsen Farahani, Sohei1 Ganjefar and Mojtaba A1izadeh

Department of Electrical Engineering Bu-Ali Sina University

Hamedan, Iran m.farahani@basu.ac.ir

Abstract- This paper proposes an intelligent control system

which is an online self-tuning PID for controlling a static

synchronous series compensator (SSSC) to suppress

subsynchronous resonance (SSR). By considering the PID

controller similar to a single layer neural network, its parameters

can be updated in online mode. To train the PID controller, the

gradient descent method is employed where the learning rate is

adapted in every iteration in order to accelerate the speed of

convergence. In the proposed controller design, the parameters of

PID are intelligently adjusted according to the design objectives.

A wavelet neural network (WNN) is also used to identify the

controlled system dynamic. To update the parameters of WNN, the gradient descent (GD) along with the adaptive learning rates

derived by the Lyapunov method is used. To show the

performance of proposed controller, the computer simulations

using MATLAB are carried out on the IEEE second benchmark

model.

Keywords- Subsynchronous resonance; Self-tuning PID; Static

synchronous series compensator (SSSC); Wavelet neural network;

Adaptive learning rates.

I. INTRODUCTION

With the rapid growth of power systems in recent decades, the stability in these systems has become a critical issue such that the stability is the most important factor to determine the capability of power transfer in transmission lines and the loading of generator. On the other hand, in order to increase the capability of power transfer and transient stability, series capacitors are widely used in transmission lines as an effective tool to decrease the reactance of lines. However, these capacitors can cause torsional oscillations between the electrical system and turbine-generator due to electrical resonance.

Flexible AC transmission system (FACTS) provides a powerful mechanism in order to control power flow in transmission lines and regulate the voltage at buses. Although, FACTS devices can improve transient stability but they are unable to damp the low frequency oscillation and sub synchronous resonance [I]. As a result, a supplementary sub synchronous damping controller (SSDC) must be designed and added to these devices [1]. Many FACTS devices with

978-1-4673-1148-9/12/$31.00©2012IEEE 336

different SSDCs in order to damp the SSR have been suggested in the literatures [1-11]. ThampaUy et al. [2] proposed a TCSC based on an adaptive neurocontroller in order to damp the SSR. Two neural networks (NNs) are used to provide an effective damping. One of the NNs identifies the dynamic of system and another generates the firing angle of TCSC. Ganjefar and Farahani [12] developed a self-tuning PID controller for damping the sub synchronous resonance. This self-tuning PID is designed and added to the automatic voltage regulator (A VR) system. These two methodologies are combined and implemented in the present work.

Most of the controllers proposed for the oscillation damping are designed based on a linearized model. Such controllers cannot guarantee their performance in a real and highly nonlinear power system. Also, the performance of these linear controllers will not be satisfactory in other operating conditions and in presence of large disturbances. To accommodate the controller in a wide range of operating conditions, the parameters of controller must periodically be retuned, so the controller can always maintain its good performance. So, an intelligent controller with adaptive learning capability is required to overcome unknown disturbances and unmodelled dynamics. The self-tuning PID controller can be used as an intelligent controller to control systems with complex dynamic. Since accurate modeling of the system in the design of self-tuning PID is not required, it can be used to control nonlinear models.

In this paper, a self-tuning PID is designed and added to a SSSC installed in the transmission line in order to damp the torsional oscillations as well as the swing mode. The proposed self-tuning PID controller with the adaptive learning rate is implemented in real time and its parameters are update in online mode. To identify the dynamic of system, a WNN trained by the gradient decent method is used and to guarantee its convergence, adaptive learning rates (ALRs) are extracted using the Lyapunov method. The simulation results show that the proposed controller is so attractive as far as the practical implementation is concerned and can stabilize the unstable modes. The system we considered in this paper is the IEEE second benchmark model (SBM).

II. MODELLING OF SYSTEM UNDER STUDY

A. Power system model

To evaluate the risk of SSR in power systems, the IEEE SBM shown in Fig. 1 is used [13]. A 600 MVA synchronous generator as shown in Fig. 1 via two 500 kV transmission lines is connected to a large grid that is approximated by an infinite bus. One of the transmission lines is compensated by the series capacitor for increasing the capability of transfer power and improving the transient stability. The SSSC is also installed at line-l to damp the SSR. In order to model the shaft of turbine­generator system, four masses shown in Fig. 2 are considered. These masses are: exciter mass, generator mass, low pressure turbine mass and high pressure turbine mass.

600MAV 22kV

sssc

Infinite

Bus

Figure I. IEEE second benchmark model along with SSSc.

( EXCC¢:::( Gen (¢=( LP (¢=( HP ( ) Figure 2. Modeling of the turbine-generator system.

B. Overview of SSSC

The Static Synchronous Series Compensator (SSSC) is a series device of the Flexible AC Transmission Systems (FACTS) family using power electronics to control power flow and improve power oscillation damping on power grids [14]. The SSSC injects a voltage V, in series with the transmission line where it is connected. The control system block diagram is shown in Fig. 3. As the SSSC does not use any active power source, the injected voltage must stay in quadrature with line current. By varying the magnitude Vq of the injected voltage in quadrature with current, the SSSC performs the function of a variable reactance compensator, either capacitive or inductive. The variation of injected voltage is performed by means of a V oltage-Sourced Converter (VSC) connected on the secondary side of a coupling transformer. The VSC uses forced­commutated power electronic devices (GTOs, IGBTs or IGCTs) to synthesize a voltage Vconv from a DC voltage source. A capacitor connected on the DC side of the VSC acts as a DC voltage source. A small active power is drawn from the line to keep the capacitor charged and to provide transformer and VSC losses, so that the injected voltage V, is practically 90 degrees out of phase with current I. In the control system block diagram Vd_conv and Vq_conv designate the components of converter voltage Vconv which are respectively

337

in phase and in quadrature with current. The output of the proposed controller u as shown in Fig. 3 is also added to Vqref in order to modulate the output voltage of SSSC in the torsional modes.

A.

VSC pulses

Figure 3. Bock diagram of control of the SSSc.

III. SELF-TUNING PID CONTROLLER

Design of Self-tuning

PID controllers are widely used in industry processes, because of simple structure, easy design, proper performance and simplicity of implementation. In design of PID controller, it is so important to tune the PID gains, if the tuning is not good, not only the control provides good performance but also the control system becomes inefficient.

The general structure of the proposed controller is shown in Fig. 4. To tune the parameters of PID controller, we consider the PID controller similar to a single layer neural network that its activation function is the purelin [13]. Fig. 5 depicts the PID controller based on the neural network. Therefore, the parameters of PID controller are the weights of the neural network that can be updated.

x

ec y Vd , ----�+-r-----------�------�

Figure 4. Bock diagram of the proposed controller.

---------------PiD-: I I I

Wd I ----------------- Acti vation function (Pure lin)

u

Figure 5. The PID controller based on neural network.

According to Fig. 5, control input u can be calculated from the following equation [2]:

(1)

Where x, T., and z are the input of PID controller, the sampling time and the operator of z-transform. Here, the self­tuning mechanism is explained in detail in order to train the PID controller. Let us define the following cost function:

(2)

Where yin) is the desired output for the plant. Next, the parameters of PID controller are tuned using the GD method such that the cost function given in Eq. (4) is minimized. The GD method is defined as follows:

( oEc (n) J W(n+I)=W(n)+t-w(n)=w(n)+ar oW(n) (3)

Where W=[WP,Wi,W(tJ and (J. are the weight vector and learning rate of PID, respectively. The partial derivative of the cost function with respect to W is:

oEc(n) =-ec(n) oy(n) ou(n) oW(n) . ou(n) oW(n) (4)

By applying the chain rule, the error term is first calculated and then the PID parameters are adjusted. The derivative of PID input with respect to the components of the weighting vector by assuming B=8u(n)/8W(n) are:

� I ou ( n ) OU ( n ) OU ( n) l [ Ts B= lowp(n) owdn) oWd(n)f x z-1 x

-1 ] l-z --

x Ts (5)

In (4), the term 8y(n)/8u(n) is the system sensitivity. By using the WNN which is a real-time plant tracker, the system sensitivity is computed in each time step n.

B. Convergence analyses of the Self-tuning PID

The strategy of training includes on-line tuning of the PID weights such that the control error converges to zero. The principle of the proof is obtained from [15] where the stability

338

of controller is based on proper selection of learning rate. Finding the optimal interval for the learning rate is done by using Lyapunov function approach. Let us consider a discrete Lyapunov function given by:

(6)

The change in the discrete Lyapunov function at the nth step is given by:

1 [ 2 2 ] t- V (n ) = V ( n + 1) - V ( n ) = '2 ee (n + 1) -ee (n) (7)

Where the control error in the next sampling period is approximated as:

By substituting LlW(n) from (3), we have:

ec (n + 1) = ee (n) + Me (n)

Then

l JT oy n � oy n � =ec(n)+ -�( ))11 aec(n)�( )

)11 OU n OU n

h('''l)ll� "(+-"[�:i:ir wn

sh(n)11 1-a l:�i:jr liTiJ

If (J. is selected as follows:

(8)

(9)

(10)

(11)

Where j3 > I . Then, the Lyapunov stability of V>O and

LI V<O is guaranteed. Choice of fJ coefficient is a result of trade­off between the speed and stability of training algorithm, for each specific case. A lower value of fJ guarantees more stable adaptation of weights.

C. WNN identifier

The WNN structure shown in Fig. 6 has Ni inputs, one output, and NiXNw mother wavelets. The WNN structure consists of four layers.

The layer I as an input layer accepts the inputs and transmits the inputs to the next layer directly. The layer 2 is a mother wavelet layer. Each node of this layer has a mother wavelet. In this paper, we select the first derivative of a

Gaussian function ¢( r) = -rexp(-O.5r2

), as a mother wavelet

function. A wavelet 9ij of each node is derived from its mother

wavelet 9 as follows:

(12)

Figure 6. The WNN structure.

Where, mij and dij are the translation factor and the dilation factor of the wavelets, respectively. The subscript ij indicates the jth input variable of the ith wavelet. In the layer 3 which is a product layer, the nodes in this layer are given by the product of the mother wavelets as follows:

N +d 0 ) = IT ¢i} ( zi) ) j=l

(13)

The layer 4 is an output layer. The node output is a linear combination of consequences obtained from the output of the layer 3. Therefore, the output of the WNN is obtained as follows:

N y(n)= L w/I\(0) (14) i=l

Where Wi is the connection weight between product node and output node. The GD method is used to train the WNN. Our purpose is to update the parameters of WNN such that the following cost function is minimized:

1 2 1 A 2 El(n)=-(eJ(n)) =-( y(n)- y(n)) 2 2 (15)

339

The parameters of the WNN can be updated using the GD method as follows:

(16)

Where rr, r( and rl

are learning rates of the WNN parameters. By calculating the partial derivatives in term of the cost function, we have:

dEl (n) del (n) =-el (n) del (n) dy(n) dy(n) = +d 0) dy( n) = _2 dJ.d 0) dwd n ) , dmi} ( n ) di} dZi} , dy(n) Wi d+i(c")) Mi} (n) = - di} zi} a;;;-

Where

The WNN can be represented as:

(17)

(18)

y(n)=I[ t-y(n-I),t-y(n- m ),u(n),u(n-I), ... ,u(n-l)] (19)

Where y(n) is the predicted speed deviation at the time step

n. In this paper, the series-parallel method is used to identify the dynamic of system. In this study, the current output of the

WNN is obtained as follows:

y( n) = I [y(n -I), y( n - 2 ),u (n ),u (n -I)] (20)

Important assumption: If difference of the WNN output and the plant output approaches zero, then the self-tuning PID controller can minimize the difference between the desired output and the plant output, i.e., as ej = (y(n)- y(n)) approaches

zero, then the PID controller minimizesec=( Yd(n)- Y(n)) . So,

the term dy(n)/du(n) in (4) is equal to dy(n)fdu(n) and written as:

dy(n) N. Wi d<I\ (0) -= L ---du(n) i=l di} dZi} (21)

The subscript j is related to the control signal u or the PID controller output.

D. Convergence analyses of the WNN

Selection of the values for the learning rates has a significant effect on the WNN performance. In order to effectively train the WNN, adaptive learning rates which guarantee the convergence of tracking error based on the analyses of a discrete-type Lyapunov function, are derived in this section. The convergence analyses in this study are to derive specific learning rates for specific types of WNN parameters to assure convergence of the tracking error. Here, the method proposed for calculating the ARLs in [15] are applied to the parameters ofWNN.

Theorem 1. Let r( be the learning rate of the WNN weights and let p".max be defined as P" max=maxNIIP".(n)lI, where

P" (n) = ay(n )/aw, (n) and 11.11 is the Euclidean norm in Rn. The

convergence is guaranteed if 1Jw is chosen as 1Jw=),jp2".max =J,.INw, in which J,. is a positive constant gain; Nw is the number of nodes in the product layer of the WNN.

Theorem 2. Let 1Jm and 1Jd be the learning rates of the

translation and dilation of the mother wavelet for the WNN; let Pm, min be defined as Pm.mm:=maxNIIPm(n)ll, where Pm(n) = a:; (n)/amij ( 11 ) ; let Pd,mm be defined as Pilmax =maxNIIPin)lI, where 1',1 (n) = a:; (n)/adij ( 11 ) ; The convergence is

guaranteed if 1Jm and 1Jd are, respectively, chosen as 1Jm=

1JW[IWilmax(2e -o5/Idijlmin)r2 and 1Jd=1JW[lwilmax (2e05/Idijlmin)r2, in which 1Jw=AlNw; J,. is a posItIve constant gain; IWilmax=maXNlwi(n)l; Idijlmin= minNldij(n)l; 1.1 is the absolute value,

The proof of theorems 1 and 2 are completely explained in [15] .

The overall performance of the proposed controller IS summarized as follows:

• Initial the WNN (offline training) and PID gains.

• Calculate u form (1).

• If ec<acceptable error or iteration>iterationmax go the step 5; else go to the step 4,

• Calculate learning rate from (11) and Update the gains of PID from (3).

• Apply u to the plant

• Calculate the output ofWNN form (14),

• If e,<acceptable error or iteration>iterationmax go the step 9; else go to the step 8.

• Calculate learning rates and Update the parameters of WNN from (16).

• 00 to the step 2

IV. CONTROL OF SSSC VIA THE SELF-TUNING PID

Here, the self-tuning PID controller designed in the previous section is implemented as a powerful supplementary

340

sub synchronous resonance controller. The input of PID controller as shown in Fig. 7 is selected the accelerating energy of generator, i.e. P=(Pm-Peo)' This signal contains components of all the torsional modes. Although local control signal can easily be obtained, but may not consist of the oscillation modes. By considering the recent advances in optical fiber communication and global positioning system (O PS), this difference can be measured and deliver to the control centre [9]. In Fig. 7, P d is the set-point, which is identically zero in a regulatory setup. By adding the output of PID controller to the reference voltage, the SSSC tracks the signal u so that in addition to controlling the voltage, it can also suppress the torsional oscillations.

Vq Voltage

Regulator

-------------�---------------. u '-----;,--__ ---1

Selt�tuning mechanism

Figure 7. Block diagram of implementation of self-tuning PID for SSe.

� -���.-:------------------------� o 2 3 4 5 t o.o� P\F-----------------l

-0.01 L __ "----�� __ � __ � ___ � __ ---' o 1 2 3 4 5

2 3 4 5 " 1 .

8:= - O··O�

1/III11/1W ... ,,---------------1

E-� _

o 1 2 3 4 5 Time (s)

Figure 8. Responses of the power system to a three-phase to ground fault.

I. SIMULATION

To demonstrate the ability of designed controller, the computer simulation results on the SBM are presented in this section. It is assumed that a three-phase to ground fault occurs at the end of the line-l during 80ms. The system has faced such large disturbance to prove the performance of the proposed controller. All the simulations are carried out in MATLAB/SIMULINK environment for 55% compensation ratio of line-l and the operating conditions P=0.9pu and Vr=lpu. The difference output active power, rotor speed

deviation and torsional torque of Gen-LP and LP-HP sections are shown Fig. 8, respectively. Clearly, the self-tuning PID controller with the adaptive learning rate is able to eliminate the oscillations during the short time. Also, it is clear from Fig. 3 that there are the severe vibrations on the shaft of turbine­generator system. If these vibrations are not suppressed, the shaft will warp and if this condition continues, the shaft will eventually break. The SSSC with the self-tuning PID can control the vibrations on the shaft so that the torsional torques on the shaft will be minimal. The online performance of the proposed controller and the WNN are shown in Figs. 9 and 10.

3 ��: -8 " -5

0 1 4000 t l ti 200g "j, : ��l� ;::::

0 0 1

: 2

: 2

: 2

: : I 3 4 5

: : j 3 4 5

: : 1 3 4 5

S: �I : : :� -5 0.1

0 1 2 3 4

��O.O�� : : : 0 1 2 Time(s) 3 4

Figure 9. The online performance of self-tuning PID.

5 r-----�----�----�----�----�

IJluh.u.lI. I\ --- WNN --- P

� 0 Yl�����====�==========�==�� � 10'nl'""" V�I =

� -5 0 r V V '-/ 'V � 1 _1 L-------------------� -10 1 1.05 1.1 1.15 1.2

� >="

::: >="

� >="

20 Of -2

� 10-5 1

��"':" Ck 10-4 1

��''''''�, 0 1

2 3 : : 2 3

2 3

2 . 3 TIme (s)

4 : 4

4

4 Figure 10. The online performance of the WNN.

II. CONCLUSION

5

, 5

5

5

5

] 5

This paper presents an online adaptive self-tuning PID controller for SSSC in order to suppress the sub synchronous oscillations and the swing mode. By assuming similarity of PID controller to a single layer neural network, the weights of

341

this PID controller are updated. To guarantee the convergence of the proposed controller, the learning rate is computed using the Lyapunov stability method. A wavelet neural network is used to identify the dynamic of controlled plant. To guarantee the convergence of WNN, the adaptive learning rates derived by the Lyapunov method are employed. In the proposed controller design, the parameters of PID are adjusted in online mode according to the objectives of design and system. The simulation results show the suitable performance of SSSC as well as the proposed controller in the damping of torsional modes

REFERENCES

[1] K. R. Padiyar and Nagesh Prabhu, "Design and Performance Evaluation of Subsynchronous Damping Controller With STATCOM", IEEE Transactions on Power Delivery, vol. 21, pp. 1398-1405,2006.

[2] K.C. Sindhu Thampatty, M.P. Nandakumar, Elizabeth P. Cheriyan, "Adaptive RTRL based neurocontroller for damping subsynchronous oscillations using TCSC", Engineering Application of Artificial Intelligence, vol. 24, pp. 60-76, 2011.

[3] Mohammad S. Widyan, "On the effect of A VR gain on bifurcations of subsynchronous resonance in power systems". Electrical Power and Energy Systems, vol. 32, pp. 656-663, 2010.

[4] Fernando Cattan Jusan , Sergio Gomes Jr., Glauco Nery Taranto, "SSR results obtained with a dynamic phasor model of SVC using modal analysis", Electrical Power and Energy Systems, vol. 32, pp. 571-582, 2010.

[5] M. R. A. Pahlavani, H. A. Mohammadpour, " Damping of sub­synchronous resonance and low-frequency power oscillation in a series­compensated transmission line using gate-controlled series capacitor", Electric Power Systems Research, vol. 81, pp. 308-317, 2011.

[6] Majdi M. Alomari, Jian Guo Zhu, "Bifurcation control of sub synchronous resonance using TCSC", Commun Nonlinear Sci Numer Simulat, vol. 16, pp. 2363-2370, 2011.

[7] Sidhartha Panda, "Multi-objective evolutionary algorithm for SSSC­based controller design", Electric Power Systems Research, vol. 79, pp. 937-944,2009.

[8] G. N. Pillai, Arindam Ghosh, Avinash Joshi, "Torsional Interaction Studies on a Power System Compensated by SSSC and Fixed Capacitor", IEEE Transaction on Power Delivery, vol. 18, pp. 988-993, 2003.

[9]

[10]

[11]

[12]

Mohammed El Moursi, A.M. Sharaf, Khalil EI-Arroudi, "Optimal control schemes for SSSC for dynamic series compensation", Electric Power Systems Research, vol. 78, pp. 646-656, 2008. Jianhong Chen, Tjing T. Lie, D.M. Vilathgamuwa, "Damping of power system oscillations using SSSC in real-time implementation", Electrical Power and Energy Systems, vol. 26, pp. 357-364, 2004. Ghosh, S. V. J. Kumar, S. Sachchidanand, "Sub synchronous resonance analysis using a discrete time model of thyristor controlled series compensator", Electrical Power & Energy Systems, vol. 21, pp. 571-578, 1999. Soheil Ganjefar, Mohsen Farahani, "Damping of subsynchronous resonance using self-tuning PID and wavelet neural network", The International Journal for Computation and Mathematics in Electrical and Electronic Engineering (COMPEL), in press.

[13] IEEE Subsynchronous resonance working group. "Second benchmark model for computer simulation of subsynchronous resonance". IEEE Trans Power Apparatus Syst, vol. 104, pp. 1057-1066, 1985.

[14] Narain G. Hingorani and Laszlo Gyugyi, Understanding FACTS, Concepts and technology for Flexible AC transmission Systems, IEEE press, 2000.

[15] Rong-Jong Wai, Jia-Ming Chang, "Intelligent control of induction servo motor drive via wavelet neural network", Electric Power Systems Research, vol. 61, pp. 67-76, 2002.