978-1-4799-3043-2/13/$31.00 ©2013 IEEE 7

Online Adaptive Legendre Wavelet Embedded NeuroFuzzy Damping Control Algorithm

Rabiah Badar, Laiq Khan

Department of Electrical Engineering COMSATS Institute of Information Technology, Abbottabad, Pakistan

rabiahbadar@ciit.net.pk

Abstract-Power system stability can significantly be enhanced

by installing a Static Synchronous Series Compensator (SSSC) using appropriate damping control scheme. In this work, a new Online Adaptive Legendre Wavelet (OALeW) based NeuroFuzzy control of SSSC is proposed. The proposed control strategy tunes the rule base using the current estimate of plant model, based on the online sensitivity measure, provided by the identification block. The parameters of the controller are updated online using gradient descent based backpropagation algorithm. The robustness of the proposed control strategy is validated using nonlinear time domain simulations and different performance indices for Single Machine Infinite Bus (SMIB) and multi-machine test systems. A comparative analysis with singleton Takagi-Sugeno-Kang (TSK) reveals that the proposed OALeW control performs better in both the transient and steady-state regions for different operating conditions and faults with improvement in control effort smoothness.

Keywords-SSSC; indirect adaptive control; Legendre wavelets; NeuroFuzzy; power system stability.

I. INTRODUCTION

Power system stability is a matter of concern for secure and reliable operation. In stable operating condition, all the interconnected machines, in a large power system, operate at a synchronous speed. However, this synchronism may be lost in the event of a fault or perturbation resulting in low frequency oscillations. These oscillations not only cause the wear and tear losses in mechanical parts of the system but may also lead to major blackouts if sustained for long duration [1, 2].

It has long been realized that power system oscillations can effectively be damped by using Flexible AC Transmission Systems (FACTS) controllers via efficient control of different network variables. Moreover, due to their flexibility and versatility, FACTS controllers will be an important part of smart grids in future [3]. SSSC is a series FACTS controller, proved to be very effective in controlling line power flow, by injecting a controllable voltage in series with line voltage. SSSC is superior to capacitive series compensation, because, its operation does not depend on magnitude of line current and it can inherently mitigate the phenomena of SubSynchronous Resonance (SSR) by injecting a purely sinusoidal voltage at system fundamental frequency. However, low frequency oscillations damping is not the primary objective of SSSC, this additional feature can be achieved by installing an auxiliary damping control. Many control techniques proposed in literature for SSSC are based on linearized model of power

system, losing the global dynamics of the system, by reducing the complex nonlinear system model to a first order state space model [4,5].

These controllers may perform well in the event of small faults when there is no large variation in the system variables from their equilibrium position for which the controller is tuned. But for major disturbances, the performance of these controllers suffers degradation. Therefore, the control must be nonlinear, adaptive and fast enough to respond to the critical faults, occurring in highly nonlinear and non-stationary plants like power system.

In recent years, NeuroFuzzy based techniques have been evolved as an effective tool for control design of dynamic plants [6,7]. However, most of these control techniques are based on conventional TSK structure with linear consequent part and thus cannot efficiently translate the nonlinearities of the system in its rule base which led to the emergence of wavelets based NeuroFuzzy systems. Wavelet based NeuroFuzzy systems have good approximation and control capabilities due to time-frequency localization property of wavelets [8,9]. The primary goal of introduction of NeuroFuzzy wavelet systems is to improve the function approximation capability and elimination of 'curse of dimensionality' arising from number of wavelet basis. Moreover, wavelet based Neural Networks (NNs) are highly suitable to use with adaptive control schemes as the problem of learning the plant dynamics can simply be replaced by adapting the wavelet coefficients. Other advantages of wavelet NNs include computational efficiency, compression abilities, border effects and Multi-Resolution Analysis (MRA) [10]. Legendre wavelets are piecewise polynomials based wavelets with low computational complexity. Due to this property, Legendre wavelet based neural networks have found extensive applications in function approximation and in solution of differential and integral equations [11-13].

This work proposes an online adaptive NeuroFuzzy control scheme based on Legendre wavelets for damping local and inter-area modes of oscillations using SSSC. Since, power system is highly complex, dynamic and nonlinear plant, therefore, identification block is used to provide the online sensitivity measure of the plant. The proposed control scheme involves online adaptation of the parameters of the controller based on the current estimate of plant model. Therefore, it does not suffer from drawbacks of heavy computation and latency

8

due to offline training. A comparative analysis is made with online adaptive TSK controller.

The main goal of this research is to present an OALeWC based SSSC to damp local and

inter-area modes of oscillations. to study the performance of the proposed algorithm for

different operating conditions and contingencies. to compare the results of the proposed algorithm with those

of Online Adaptive TSK Control (OATSC). The rest of the paper is organized as follows; section II

presents the detailed mathematical description of the proposed control strategy and the update parameters learning algorithm are given in section III. Simulation results are discussed in section IV. Section V concludes the findings of this research.

II. ONLINE ADAPTIVE CONTROL

The closed-loop structure of proposed control strategy is shown in Fig. 1. The plant includes power system installed with SSSC. The nonlinear differentio-algebraic model of power system including all the machines, loads and control dynamics is given as;

( )( )

g t

h t

⎧⎪⎨⎪⎩

x = x, y,

0 = x, y, (1)

Where, g and h are unknown nonlinear functions, x and y are the vectors of state and algebraic variables, respectively.

Differential equations include the machine and control dynamics, whereas, algebraic equations comprise load flow and other network equations. The control block contains online adaptive neural network based on Legendre wavelets. The parameters of the control block are updated using rotor speed deviation and its derivative. The output of the control block is compared with the measured injected voltage and the relative error is used to calculate the Pulse Width Modulation (PWM) parameters to regulate the voltage injected by SSSC.

A. OALeW NN LeWN is a three layered network with Legendre wavelets as

activation functions in its hidden layer. The extended view of network is shown in Fig. 2. Legendre wavelets are Legendre polynomial based, orthonormal, compactly supported wavelets.

Definition 1: The Legendre wavelets are defined as;

( ) ( )1 12 2 1 2 0.5 2 2

0 otherwise

n nkk n n

nm

m mk x m xxϕ

+ +⎧ + − − ∀ ≤ <⎪= ⎨⎪⎩

L (2)

Where, n and m are the decomposition level and integer translation, respectively, such that 1,2, ,n m= and

10,1, ,2nm −= . kL denotes the Legendre polynomials. The Legendre polynomial is defined using Rodrigues

formula [14];

( )2

1( ) 12 !

n

n

n

n n

dL x xn dx

= − (3)

Where, [ ]0, 1 1n x≥ ∈ − .

Remark 1. Since, ( )nL x is the nth derivative of a

polynomial of degree 2n , therefore, ( )nL x is a polynomial of degree n .

In this work, six Legendre wavelet basis functions corresponding to 1n = and 0, 1, 2k = defined on [ [0 1 are used.

Theorem 1: A function ( ) ( )2f x L R∈ defined on [0 1],

with bounded second derivative, say ( )''f x M≤ , can be

expanded as infinite sum of Legendre wavelets and the series converges uniformly to the function ( )f x , i.e.,

( ) ( ),1 0

j nm n mn m

f x xρ ϕ∞ ∞

= ==∑∑ (4)

Where, ( ) ,,nm n mp f x ϕ=< > and .,.< > is the inner

product of nmρ and ,n mϕ . Proof of this theorem is given in [14]. Corollary 1. If the infinite series in (4) is truncated then it

can be written as;

( )2 1

2 ,1 0 0

ˆ,nN K

k kj nm n m

n m kxζ ρ ϕ

−

= = == =∑∑ ∑θ ρϕ (5)

ρ represents the Legendre wavelet coefficients, whereas,

ρ and ϕ are matrices of order ( )2 1 1n K + × such that

10 1 2, , , nn n nρ ρ ρ −⎡ ⎤⎣ ⎦ρ = and 10 1 2, , , nn n nϕ ϕ ϕ −⎡ ⎤

⎣ ⎦ϕ = with

0,1, , K= .

B. NeuroFuzzy OALeWC The proposed NeuroFuzzy architecture is shown in Fig. 2.

The NeuroFuzzy structure is based on the following form of IF-THEN rules;

1 1: j i i m m j jR IF x is and x is and x is THEN fμ μ μ ζ =

Where, jR is the jth fuzzy rule,

{ }1 2, , , nmx x x x U∈ ∈ ⊂ is the crisp input and

j Vζ ∈ ⊂ is the output of jth nonlinear function of

consequent part. ijμ are fuzzy sets with Gaussian membership

functions given as;

Fig. 1. Closed-loop system structure

9

( )2

1,i ij

ij

x

ij ix e

υςμ χ

⎛ ⎞−−⎜ ⎟⎜ ⎟⎝ ⎠= =θ (6)

Where, ( )1 1ˆ , [ ]f c ij ijeβ υ ς= =θ θ is the adaptation

parameters vector for antecedent part. The NeuroFuzzy rules are realized in a layered fashion as

follows; Layer 1 is the input layer. No processing is done in this

layer and the output of this layer is same as the input pattern. Layer 2 is the fuzzification layer. This layer fuzzifies the

inputs using (6). Layer 3: The firing strength of each rule is calculated in

layer 3 using T-norm product operator, given as;

1

m

j iji

μ=

℘ = ∏ (7)

Layer 4 is basically the first layer of consequent part and contains the sub-layers of underlying network. Each node of this layer contains the nonlinear function ( )2,j xζ θ .

Where, jζ is the final output of jth LeWN and

( )2 2ˆ ˆ , k

g c nm je wβ ρ⎡ ⎤= = ⎣ ⎦θ θ is the adaptation parameters

vector for consequent part. Remark 2. The definition of Legendre wavelets restricts the

input data to be fed directly to LeWNN. Therefore, some transformation needs to be applied to the input data to map it within the defined limits, such that;

[ ]( ) | ( ) 0 1T x x and T xτ = ∈ℜ ∈ (8) This constraint leads to an optimization of the network

structure by exploiting the output of second layer of the network, instead of preprocessing the input data, which improves the efficiency of the algorithm also.

Layers 5 & 6 are used to perform defuzzification using

center of gravity (COG) method. Therefore, the final output of the network is given as;

( ) 1

1

n

j jj

n

jj

u xζ

=

=

℘=

℘

∑

∑ (9)

III. PARAMETER UPDATE LEARNING ALGORITHM

The parameters of the control and identification block are updated using gradient descent backpropagation algorithm. In what follows is the detail of parameters update for each block.

A. Online adaptation of identifier parameters The identification block contains Gaussian membership

functions in the antecedent part to fuzzify the inputs and weights in the consequent part to form the rule base. The parameters of the identification block are updated by minimizing the following cost function;

( )221 1 ˆ2 2IJ e ω ω= = Δ − Δ (10)

The generalized adaptive update law for parameters adaptation is given as;

( ) ˆ( 1) II I I

I

Jk k η ∂

+ = −∂

(11)

Where, I ij ij jwυ ς⎡ ⎤= ⎣ ⎦ and ˆIη is the learning rate. The

differential in (11) can be simplified using the following chain rules of calculus for each update parameter;

ˆˆ

j ijI I

ij j ij ij

J J μωυ ω μ υ

∂℘ ∂∂ ∂ ∂Δ=∂ ∂Δ ∂℘ ∂ ∂

(12)

ˆˆ

j ijI I

ij j ij ij

J J μως ω μ ς

∂℘ ∂∂ ∂ ∂Δ=∂ ∂Δ ∂℘ ∂ ∂

(13)

ˆˆ

I I

ij j

J Jw w

ωω

∂ ∂ ∂Δ=∂ ∂Δ ∂

(14)

B. Online adaptation of control parameters The cost function used to optimize the control parameters is

defined as;

( )22 2 21 12 2 2 2c rJ e u uω ω= + = Δ − Δ + (15)

The generalized adaptive update law for parameters adaptation can be written as;

( ) ( ) ( )( )ˆˆ( 1) 1cc c c c c c

c

Jk k k kη λ∂

+ = − + − −∂

(16)

Where, ˆcη and cλ are the learning rate and momentum terms, respectively, adjusted in the range of ]0 1[. The inclusion of momentum term improves the stability and convergence of the algorithm. c is the update parameters

vector. Here, c is defined as; kc ij ij mnυ ς ρ⎡ ⎤= ⎣ ⎦ .

The gradient of the cost function is given as;

Fig. 2. Structure of online adaptive NeuroFuzzy network

10

c

c c

J ue uuω∂ ∂Δ ∂⎛ ⎞= − −⎜ ⎟∂∂ ∂⎝ ⎠

(17)

Here, is the gain factor. After simplifying the partials in (17) and using (16) and

(17), the following update relations can be found for parameters update of OALeWC .

( )( )

2

1

ˆˆ1 ( ) .2j i ij ij

ij ij c jnij

jj

u xk t e υ

ζ υυ υ η λγ

ς

=

− ⎛ ⎞−⎜ ⎟+ = + ℘ +⎜ ⎟⎝ ⎠℘∑

(18)

( )( ) ( )2

3

1

ˆˆ1 ( ) .2j i ij ij

ij ij c jnij

jj

u xk t e ς

ζ υς ς η λγ

ς

=

⎛ ⎞− −⎜ ⎟+ = + ℘ +⎜ ⎟⎜ ⎟℘ ⎝ ⎠∑ (19)

( )

1

ˆˆ1 ( ) .j k knm ij c nmn

jj

k t e ρρ ς η ϕ λγ

=

℘+ = + +

℘∑ (20)

Where, ce e uuω∂Δ⎛ ⎞= −⎜ ⎟∂⎝ ⎠

and ( ) ( )1ijij ijk kγ = − − .

The plant output sensitivity measure provided by the identification block is given as;

( )2

2( ). j ijj j

j j ijj

uu

μ υζ μωμ ς

⎛ ⎞−− −∂Δ ⎜ ⎟= ∑ ⎜ ⎟∂ ∑⎜ ⎟⎝ ⎠

(21)

IV. SIMULATION RESULTS AND DISCUSSION

The performance of OALeWC is validated for local and inter-area modes of oscillation using SMIB and multi-machine test systems. Load flow is performed using ode23tb solver with phasor simulation method. The simulations were carried out for different operating conditions and various faults using three scenarios; 1) when SSSC is equipped with no supplementary damping control, 2) SSSC installed with OATSC, 3) SSSC installed with OALeWC. The details of each are given as follows;

A. Case-1: Nominal Loading ( )00 00.75 . ., 20.17eP p u δ= =

Fig. 3 shows the SMIB test system for simulation results generation with SSSC installed near the generating unit. The detail of system parameters can be found in [8]. The performance of the proposed control strategy is validated for nominal loading condition by applying a 5-cycles, 3-ϕ , self-cleared fault. The fault is applied at the middle of line 2, at t=1 sec. The performance results for this contingency are shown in Fig. 4.

Fig. 4 shows that OALeWC quickly damps the oscillations

and converges in less iterations as compared to OATSC. There is a performance improvement of 31.12% and 28.81%, in terms of settling time for rotor speed deviation and line power flow, respectively.

B. Case-2: Heavy Loading ( )00 01 . ., 25eP p u δ= =

In this case, the performance of OALeWC is compared with OATSC by changing the loading condition from nominal to heavy loading for the same fault as discussed in previous case. The system response for this scenario is shown in Fig. 5. It can be seen that the application of fault, in this case, makes the system to lose its synchronism when no supplementary control is applied. The application of OATSC restores the system stability with poorly damped oscillations, whereas, OALeWC effectively restores the system equilibrium with improved performance. The settling time performance improvement of 21.62% and 22.32% is achieved for rotor speed deviation and line power flow, respectively.

Fig. 3. SMIB test system (b)

Fig. 5. SMIB case-2; (a) Rotor speed deviation (b) Power flow on line 1

(a)

(b) Fig. 4. SMIB case-1; (a) Rotor speed deviation (b) Power flow on line 1

(a)

11

C. Case-3: Multiple Faults In order to compare the robustness and online stability of

proposed control scheme, the system is subjected to a series of fault events. A 3-ϕ self-cleared fault of duration 5-cycles is applied near bus B2 at t=1 sec. followed by another fault of permanent line outage by removing line 2 from the system at t=5 sec. The performance results for this scenario are shown in Fig. 6. The performance improvement in terms of settling time is 12.92% and 15.72% for speed deviation and line power flow, respectively.

D. Case-4: Multi-machine System with Large Disturbance The performance of the proposed control is further examined

in a more complex and practical scenario of multi-machine system, shown in Fig. 7. The parameters detail of the system can be found in [15]. Machine 1 and 2 are in area 1, whereas, machine 3 and 4 lie in area 2 with machine 2 taken as swing bus. No PSS is installed in the system and whole damping is provided by SSSC, placed at the middle of the system.

The performance of the OALeWC is validated against large disturbances by applying a 3-ϕ fault of duration 8-cycles on line 4 near area 2, at 1 sec.t = The system performance is shown in Fig. 8. Figs. 8(a) and 8(b) show the results of local and inter-area modes of oscillations, respectively. Fig. 8(c) shows the results of tie-line power flow from bus B-3 to B-4. It is clear from these results that the application of 3-ϕ fault leaves the system in unstable state when no supplementary control is applied. The application of OALeWC not only restores the system equilibrium but also significantly improves the damping performance as compared to OATSC.

E. Case-5: Robustness The online stability and robustness of OALeWC is checked

by exposing the system to a series of fault events. Firstly, line 2 is temporarily disconnected at t=1 sec. for a duration of 3 secs. by reclosing the line at t=4 sec. Then, a step increase in mechanical input power of machine 1 is made at t=8 sec. The simulation results are shown in Figs. 9(a)-9(c) for local and inter-area modes of oscillations and tie-line power flow, respectively. The results reveal that in absence of supplementary damping control, the oscillations grow indefinitely, eventually causing the simulation to stop at t=8 secs.

Fig. 7. Multi-machine test system

(b)

(a)

(b) Fig. 6. SMIB case-3; (a) Rotor speed deviation (b) Power flow on line 1

(a)

(c)

Fig. 8. Multi-machine system; (a) Local mode of oscillation (b) Inter-areamode of oscillation (c) Tie-line power flow

(b)

(a)

12

The results show that OALeWC performs better as

compared to OATSC for normal operating condition and also maintains its performance at a different operating point. The performance improvement of OALeWC as compared to OATSC is further justified on the basis of quantitative analysis using different performance indices. The performance index is an integral of the function of time and error and is given as;

0

sthPI t dtω= Δ∫ (22)

Where, and h are fixed numbers, such that

( ) ( ) ( ) ( ) ( ){ }, 0,1 , 0,2 , 1,1 , 1,2h ∈ for Integral Time Absolute

Error (ITAE), Integral Absolute Error (IAE), Integral Time Square Error (ITSE) and Integral Square Error (ISE), respectively. Another important measure of good control is the smoothness of control effort to avoid switching losses. The control effort smoothness is measured using the following relation;

( )22

1

1 L

Mi

S u iL =

⎡ ⎤= Λ⎣ ⎦∑ (23)

Where, L is the total length of the control signal and 2Λ is the discrete second order derivative. The smaller value of this smoothness measure means the smoother output. Table 1 presents the statistics based on these performance measures. The results confirm that OALeWC performs better in both the transient and steady-state regions as compared to OATSC with smooth control effort.

TABLE I COMPARATIVE ANALYSIS

Perf

orm

ance

im

prov

emen

t for

O

ALe

WC

[%]

Test cases Case-1 Case-2 Case-3 Case-4 Case-5

ITAE 24.58 24.67 18.38 31.30 14.93

ITSE 30.93 22.77 19.96 39.37 14.14 IAE 21.97 18.28 15.40 25.29 17.63 ISE 27.39 18.84 15.35 28.21 22.16

Con

trol

effo

rt sm

ooth

ness

OALeWC 205.94 104.04 204.27 455.91 176.78

OATSC 3798.8 123.76 3278.4 478.17 283.70

It can be observed that although OALeWC maintains its performance improvement at different operating conditions

other than normal operating point, however, maximum performance improvement is achieved at normal operating conditions for large disturbances.

V. CONCLUSION

This article presents a novel online adaptive NeuroFuzzy supplementary damping control based on Legendre wavelets using SSSC. The parameters of the proposed control scheme are updated online based on the current estimate of plant model using identification block. The nonlinear time domain simulation results and different performance indices reveal that the proposed control strategy effectively damps the local and inter-area modes of oscillations with optimal control effort. It has been observed that in case of more stressed conditions like heavy loading and multi-machine system, only the installation of SSSC is not sufficient to maintain system stability. The elimination of PSS, in case of multi-machine system, to avoid any destabilizing action and smoothness of converter output to reduce the megavar ratings of the converter, eventually, make the system cost-effective.

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(c) Fig. 9. Multi-machine system; (a) Local mode of oscillation (b) Inter-areamode of oscillation (c) Tie-line power flow