Hybrid NeuroFuzzy B-spline Wavelet Based SSSC Control for Damping Power System Oscillations

Rabiah Badar, Laiq Khan Department of Electrical Engineering

COMSATS Institute of Information Technology, Abbottabad, Pakistan

rabiahbadar@ciit.net.pk

Abstract—Controllable series voltage injection can significantly enhance the damping capability of a power system. Static Synchronous Series Compensator (SSSC) is a well-known FACTS controller used for this purpose. This paper presents a novel adaptive control scheme to damp inter-area oscillations in a multi-machine power system using SSSC. The proposed control paradigm utilizes the local control property of B-spline membership function by hybridizing it with wavelet NNs in the structure of NeuroFuzzy system to design the external control for SSSC. The system parameters are tuned online based on the adaptive NeuroFuzzy rules extracted from rotor speed error and its derivative. The detailed mathematical description of online tuning the control parameters is given. The control scheme utilizes the model free direct control structure which reduces the computational complexity, latency and memory requirements making the control system a good candidate for real time implementation. The robustness of the proposed control system is checked against various faults and operating conditions on the basis of nonlinear time domain simulations. Finally, the results of proposed Hybrid B-spline Wavelet Control (HBsWC) are compared with Adaptive NeuroFuzzy TSK Control (ANeFu-TS).

Keywords-SSSC; B-spline membership function; adaptive NeuroFuzzy control; wavelets; gradient descent; multi-machine power system

I. INTRODUCTION

Bulk power transmission over long distances, weak tie-lines connecting the sub-networks of a large power system and negative damping introduced by any damping control device are few of many reasons of poor damping capability against low frequency oscillations. These low frequency oscillations, ranging from 0.2 to 0.5 Hz, produce due to the mismatch of speed of generating units in one area of the power system or the other. If these oscillations are not damped out using a proper control, the oscillating machines may lose their synchronism leading to a system collapse. Major blackouts due to these low frequency oscillations are reported in literature [1]. An instant and cost effective remedy to this problem was the introduction of exciter, Automatic Voltage Regulators (AVR) or Power System Stabilizer (PSS). The installation of PSS was able to give the “fine adjustment” for electrical speed of synchronous generators keeping the system intact, in contrast to the “coarse adjustment” provided by fast AVR [2-4]. But PSS is a locally controlled device, designed

for a specific operating point and may not incorporate the global behavior of a large power system with time varying dynamics.

Flexible AC Transmission System (FACTS) is a relatively new technology to damp these oscillations effectively by introducing virtual damping in the system over a wide range of operating conditions [5]. SSSC is a second generation FACTS controller which controls the line power flow by injecting a synchronous sinusoidal voltage at fundamental frequency in series with the line voltage. An external damping control can be used along with the primary control of the converter to provide additional damping and to improve the system stability.

Many design techniques have widely been discussed in literature, to damp power system oscillations using external control of SSSC, ranging from conventional to adaptive, linear to nonlinear, heuristic to NeuroFuzzy and their hybrids [6-10]. The design and structural complexity, vulnerability to change in operating condition and sensitivity to system model accuracy are some of the major short-comings in one or the other existing control paradigms.

To overcome these limitations and to effectively incorporate the nonlinearity and dynamicity of nonlinear time varying systems, a class of NeuroFuzzy control systems based on integration of conventional TSK structure and wavelets was introduced. NeuroFuzzy wavelet structures combine the fast learning capability of NNs with computationally efficient fuzzy logic and strong approximation capability of wavelets of analyzing the local details of non-stationary signals. NeuroFuzzy wavelets have found extensive applications in the fields of signal processing and control due to their efficient performance [11-14].

However, these techniques utilize the membership functions like, Gaussian, Triangular, etc. with global tuning approach of their parameters. Another type of NeuroFuzzy systems was evolved to improve the approximation capability of NeuroFuzzy systems from antecedent side to optimize the tuning method of membership functions by introducing the local control capability known as fuzzy B-spline membership function. Fuzzy B-spline based NNs have found extensive applications due to their good approximation capability and computational simplicity [15-18]. However, these systems use

978-1-4673-2252-2/12/$31.00 ©2012 IEEE

singleton fuzzifier in their inference system which cannot create a reasonable rule base for highly nonlinear, dynamic plants like power system.

Therefore, this work proposes a HBsWC based adaptive control scheme to improve the transient and steady-state stability of the power system using SSSC. The proposed control hybridizes the efficient approximation techniques of applied mathematics, nonlinear analysis tools of signal processing and computationally efficient soft-computing techniques. The incorporation of wavelets in the structure of conventional TSK NeuroFuzzy systems evades the inherent drawback of these systems of trapping in local minima and convergence speed when gradient descent based backpropagation technique is used to optimize the control parameters, whereas, the inclusion of B-spline membership functions optimize the approximation capability of a smooth function and its derivatives [19]. The parameters of the proposed control scheme are adapted online and no a-priori information of the system is required.

The main goal of this research is

to model and interface SSSC with AC power system. to present an HBsWC based adaptive control scheme to

damp inter-area modes of oscillations using SSSC. 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 ANeFu-TS based control system.

The rest of the paper is organized as follows; section II gives power system dynamic model installed with SSSC. Section III presents the detailed mathematical description of proposed control strategy. Simulation results are discussed in section IV. Section V concludes the findings of this research.

II. POWER SYSTEM INSTALLED WITH SSSC Dynamic model of a multi-machine power system can be

written in the form of differential algebraic equations as follows;

( )tx = g x, y, (1)

( )t0 = h x, y, (2) Where, x and y are the vectors of state and algebraic

variables, respectively. Differential equations include the machine and control dynamics whereas load flow and other network equations form the algebraic equations.

The nonlinear dynamics of th machine, including controls, using dq reference frame with q-axis leading d-axis is given by,

b

ddt

κκ

δ ω ω= (3)

)1 ( m e D

dP PMdt κ κ κ

κ ωω

−= − (4)

'q' ' '

do q d d d fd

dET = -E - (x - x )I + E

dtκ

κ κ κ κ κ κ (5)

Figure 1. Multi-machine power system

)('d' ' '

qo d q q q

dET = -E + x - x I

dtκ

κ κ κ κ κ (6)

( )''q'' ' '' ' ''

do q q d d d-dE

T = E - E x - x Idt

κ

κ κ κ κ κ κ (7)

( )''d'' '' ''

qo q q q d

dET = x - x I - E

dtκ κ κ κ κ (8)

( )( )fdE E fd E fd RE - K +V

dET = -S E

dtκ

κ κ κ κ κ κ (9)

Stator algebraic equations with zero armature resistance are given as;

'' ''d d q q+ x Iv = E

κ κ κ κ (10)

'' ''q q d dv = E - x I

κ κ κ κ (11)

The generator terminal voltage in machine coordinates is given as;

( )'' '' '' ''tg d q q q d dV = E + x I + j E - x I

κ κ κ κ κ κ κ (12)

Where, 2 2tg d qV v v

κ κ κ= + and κθ is the rotor angle of

κ th machine with 1, 2, , nκ = . n being the total number of machines.

SSSC is modeled as a controllable series voltage source as shown in Fig. 1. The fundamental frequency model, neglecting the harmonics, is sufficient to study the low frequency electromechanical transients [20].

The converter output in phasor form can be written as;

conv DC sV mkV ψ= ∠ (13)

Where, sψ θ ϕ= + such that ϕ is the firing angle, m is the modulation index and k is a constant defining the relationship between converter AC and DC side voltages.

After SSSC installation the resulting equations can be written by applying KCL and KVL to nodes 1 and 2 in Fig. 1 as follows;

'11 1 1 1 0g tg SRY V I+ + =Y V (14)

'22 2 2 2 0g tg SRY V I+ + =Y V (15)

1 1 2 2g g gg tg g=V V ++Y Y Y V I (16)

1 21 2

0

convSR SR

V V VI I

jx

− −= − = (17)

Where, 0 1 2 ssx x xx = + + . Solving (14) to (17) to eliminate

1V and 2V [21], the generator current in network coordinates is given as,

0g Y tg convV= +I C V C (18) Where,

111 2 _12

2 g

gg g gg

Y ϑ−= −

⎛ ⎞⎡ ⎤⎡ ⎤⎜ ⎟⎢ ⎥⎣ ⎦

⎣ ⎦⎝ ⎠C

YY Y Y Y

Y,

1 01 2 _12

0

0

1

1g g

jx

jx

ϑ−=

−

⎡ ⎤⎢ ⎥

⎡ ⎤ ⎢ ⎥⎣ ⎦ ⎢ ⎥⎢ ⎥⎣ ⎦

C Y Y Y and

110 0

_12

220 0

1 1'

1 1'

Yjx jx

Yjx jx

ϑ

+ −

=− +

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

Y

III. CONTROL SYSTEM DESIGN The closed loop structure of the system is shown in Fig. 2.

{ }_ _s dis l disd dτΞ ∈ ∪ is the disturbance applied to the

system, with _s disd representing the set of small disturbances

and _l disd for the set of large disturbances. τ =1, 2, …, p, where, p is the total number of disturbances.

The control of SSSC consists of main control and external control. The primary goal of main control is to keep the injected voltage in quadrature with the line current and maintain it at a constant level in steady-state based on the changes in the reference variables. External control is used to provide the reference injected voltage monitoring the changes in the system variables.

The proposed control scheme is model free direct adaptive control based on controller output matching method. The relative output error and its derivative are taken as inputs to the control block. Plant includes the power system along with SSSC and its internal control.

Figure 2. Closed loop system structure

A. B-spline membership function The tuning approach of fuzzy membership function is of

primary importance and is investigated by many researchers [22-24]. Most of these researchers used the methods of global tuning and off-line preprocessing. However, the local tuning of fuzzy membership functions can significantly enhance their approximation capability. B-spline membership functions are formed using low order polynomial pieces joined together at certain break points known as knot vectors. The order of the B-spline function determines the number of coefficients in the polynomial pieces. The coefficients of the polynomial are known as control points. The B-spline membership function is generally given by,

,0

( ) 1n

ij i i pi

t p nα=

Λ = ℵ ≤ ≤∑ (19)

Where, iα is the ith control point and the total number of control points is n+1. ,i pℵ is the basis function used to define the membership function and is given by,

, , 1 1, 11 1

( ) ( ) ( )

and [ , [

i pii p i p i p

i p i i p i

i i p

t tt tt t t

t t t t

t t t

+− + −

+ − + +

+

⎛ ⎞ ⎛ ⎞−−ℵ = ℵ + ℵ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠

∈

(20)

1,1

1, for [ , [( )

0, otherwisei i

it t t

t +∈⎧ℵ = ⎨

⎩ (21)

Here, ‘t’ is the knot vector, such that 1i it t+ > , defining the support of the B-spline membership function and p is the order of B-spline. The important properties of fuzzy B-spline function like knot vectors, boundary conditions and basis functions are discussed in great detail in [25]. In [26] a NeuroFuzzy control structure, utilizing the locally tuned B-spline membership function, is proposed but the method suffers from overhead of offline tuning. The method is further modified in [27] using online version of B-spline membership function with fixed number of control points. Ideally, the membership function should have infinite number of control points for the applications involving continuous data which is not practical. Therefore, a large number of control points can optimize the solution significantly but at the cost of computational burden and memory requirements.

This research uses eleven control points to tune B-spline membership function with order 2. The number of control points is fixed. The control points are uniformly distributed over the membership function. For B-spline membership function of order two the basis functions in (20) can be simplified to,

,21, for ( )

( )0,

ji j

round xx

otherwise

α=⎧⎪ℵ = ⎨⎪⎩

(22)

The control points are always arranged on the interval

( )[ ]

1f ot t

i−

− . Where, to and tf define the support of the B-

spline membership function and i is the number of control points.

B. HBsWC Power system is a highly dynamic and nonlinear system.

Therefore, the control system applied to it must be able to adapt its nonlinear behavior which cannot be done efficiently, if not impossible, with a controller having linear and fixed parameters architecture. The antecedent part of ANeFu-TS control utilizes the globally tuned Gaussian membership function, whereas, the consequent part of the ANeFu-TS is linear which cannot create a proper knowledge base for complex and uncertain systems.

Wavelets due to their highly nonlinear nature are proven to be very strong and effective tool for non-stationary signal analysis. Therefore, a novel control structure is proposed in this work by replacing the linear part of consequent of conventional TSK with that of wavelets. The incorporation of wavelets enhances the learning capability of the system by analyzing the local details of the signal. The Maxican hat wavelet function is chosen for this purpose and is given by,

( ) ( ) 20.5 0.521 ii i i e ρϕ ρ ς ρ− −= − (23)

Where, i ii

i

x τρς−

= . Here, iς and iτ are the dilation

and translation factors of the wavelet, respectively, and ix is the ith input to the network. Maxican hat wavelet function is a non-orthogonal function of Gaussian probability density function with most of the energy concentrated in the vicinity of origin.

The antecedent part is optimized using B-spline membership function given in (19). T-norm product operator is used to calculate the strength of each rule.

The proposed multiple-input-single-output (MISO) network architecture is based on IF THEN rules of the following generalized form,

1 1 2 2: j j j j j mj mj j jR If x is and x is and x is then y χΛ Λ Λ =

Here, jR denotes the jth fuzzy rule, kjx is the kth input to jth rule such that 1, 2, , k m= where, m is the total number of inputs.

1 2 3 j nTμ = Λ Λ Λ Λ⎡ ⎤⎣ ⎦ (24)

Where, jμ is the firing strength of jth rule and n is the number of membership functions in a rule.

The final output of the network using product inference, centre average and wavelet based fuzzifier, is given by,

1

1

n

i iTi

n

ii

uμ χ

μ

=

=

= =∑

∑ℑχ (25)

where, 1 2 nℑ ℑ ℑ⎡ ⎤⎣ ⎦ℑ = and

1

ii n

ii

μ

μ=

ℑ =

∑known as

fuzzy basis functions (FBFn). 1 2 3 nχ χ χ χ= ⎡ ⎤⎣ ⎦χ is the output of the wavelet network. The wavelet network consists of three layers; the input layer, hidden wavelet layer and the output layer. The kth output of the network is given as,

( )1

m

k ii

wχ ϕ ρ=

= ∑ (26)

Where, w ’s are the weights of the output layer.

The overall network architecture is shown in Fig. 3 and can mainly be divided in two parts i.e. the antecedent and the consequent part. Antecedent part includes layer 1 and layer 2, whereas, the consequent part consists of layer 3, layer 4, layer 5 and layer 6. Layer 1 is the input layer which takes the input and passes it to layer 2. The input vector is fuzzified in layer 2 using (22) and the strength of each rule is computed using (24).

Layer 4 is the consequent layer which contains wavelet NNs in each processing node and basically works in parallel with layer 1. The output of this layer is calculated using (26).

Layer 5 is the product layer which multiplies the outputs of layer 2 and layer 3. Finally, the defuzzification is carried out in layer 6 using (25).

C. Parameters Update Algorithm The parameters of the proposed HBsWC architecture are

updated using the following cost function;

Figure 3. HBsWC architecture

( )2 20.5c rJ y y u⎡ ⎤= − +⎢ ⎥⎣ ⎦ (27)

Where, ry is the reference output value and y is the plant output. The following update rule is used to adapt the control system parameters.

( ) ( )( )1 ( ) ( ) 1cJt t t tγ λ∂

+ = − + − −∂

Ω Ω Ω ΩΩ

(28)

where, ij i i iwς τ[ Λ ]Ω = is the adaptation parameter vector. γ is the learning rate and λ is the momentum term used to speed up the convergence and guarantee the stability of the algorithm.

( )Cc t

J ue

∂ ∂=

∂ ∂Ω Ω (29)

Where, ( )( )c rty

e y y uu

∂= − − −

∂⎛ ⎞⎜ ⎟⎝ ⎠

(30)

The configuration shown in Fig. 2 is based on direct adaptation method to eliminate the need of plant model, therefore, using direct output matching method it can be

assumed that 1yu

∂ =∂

[28]. The parameters are updated using

the following chain rules of calculus;

(31)i

i i i

u u ρχ ϕς χ ϕ ρ ς

∂∂ ∂∂ ∂=

∂ ∂ ∂ ∂ ∂

(32)i

i i i

u u ρχ ϕτ χ ϕ ρ τ

∂∂ ∂∂ ∂=

∂ ∂ ∂ ∂ ∂

(33)j

j j

u u

w w

χ

χ

∂∂ ∂=

∂ ∂ ∂

And

( )1

(34)j i ijij

u uμ χ μ−∂ ⎛ ⎞= − Λ ∑⎜ ⎟∂Λ ⎝ ⎠

using

1 (35)j

jj ij

ij

μμ −

∂∑= Λ

∂Λ

It is to be noted that the adaptation error ce is a scalar quantity and the control system parameters are updated only once during each iteration which minimizes the computational time and makes the controller highly suitable for real time implementation.

IV. SIMULATION RESULTS AND DISCUSSION The performance of the proposed control system is checked

using a two-area, two-machine system with one machine in

Figure 4. Two machines test system

each area. The test system model is shown in Fig. 4. SSSC is installed between buses B2 and B7. The simulation results are generated using SIMULINK SimPowerSystem toolbox.

The parameters detail can be found in [29]. Machine 2 in area 2 is taken as swing bus whereas machine 1 is PV generator bus and L4 is the dynamic load. The relative speed difference of two machines and its derivative are taken as two inputs of the control system. The control system has two rules with two membership functions for each input. The update parameters for HBsWC and ANeFu-TS are 14 and 10, respectively. The robustness of the proposed control system is validated using different case studies as follows;

A. Case 1: Small Disturbance

The performance of the control system against small disturbance is evaluated in this case. The 50 MW load L3 is disconnected at t=1 sec. The system response for this fault is shown in Fig. 5. It is clear from the results that HBsWC quickly damps the oscillations as compared to no control and ANeFu-TS cases and restores the system to a new steady-state. A performance improvement of 93.5 % and 99.75 % is attained in terms of first swing peak amplitude, for rotor speed deviation as compared to ANEFu-TS and no control cases, respectively. The result for power flow on line 3 shown in Fig. 5(c) reveals the excellent power oscillations damping performance of HBsWC.

(a) Rotor speed deviation

(b) Rotor angle deviation

(c) Line power flow

Figure 5. System response for small disturbance

B. Case 2: Large Disturbance In this case, a 10-cycles, 3-φ fault is applied on line 1 near

bus B3, at t=1 sec. The results for this contingency are shown in Fig. 6. It is clear from Fig. 6(a) that system has poorly damped oscillatory behavior when no supplementary control is applied. HBsWC improves the performance with less overshoot and settling time as compared to ANeFu-TS. Performance improvement of 15 % and 29 % as compared to ANeFu-TS and no control cases, respectively, is attained in case of HBsWC for first swing peak amplitude. Figs. 6(b) and 6(c) show the angle deviation and power flow on line 3, respectively. It is evident from the results that HBsWC has better first swing response as compared to ANeFu-TS. Moreover, it can be seen from these results that ANeFu-TS suffers a performance degradation in the steady-state region when SSSC is incorporated in the system at t=0.1 sec.

(a) Rotor speed deviation

(b) Rotor angle deviation

(c) Line power flow

Figure 6. System response for large disturbance

C. Case 3: Series of faults-I

The robustness of the proposed control system is validated by applying series of faults to the system. In this case, a 3-φ fault of duration 10 cycles is applied on line 1 at t=1 sec. followed by another combination of faults of double line outage and load rejection. Lines 1 and 2 are removed from the system along with 100 MW load L2. The simulation results of speed deviation, rotor angle deviation and line power flow are shown in Fig. 7.

Fig. 7 shows that HBsWC has significantly better performance results in terms of overshoot and settling time as compared to ANeFu-TS. The first swing peak amplitude damping improvement for HBsWC is 60.23 % and 66.86 % as compared to ANeFu-TS and no control cases, respectively.

(a) Rotor speed deviation

(b) Rotor angle deviation

(c) Line power flow

Figure 7. System response for series of faults-I

Fig. 7(c) shows that ANeFu-TS has damped oscillations after first swing but HBsWC more effectively damps the oscillation with smaller peak amplitude and less settling time.

D. Case 4: Series of faults-II Finally, the robustness of the proposed control system is

checked against another combination of series of faults. In this case, line 1 and line 2 are removed from the system at t=1 sec. along with load L2. The system is restored to its original condition by reclosing L1, L2 and load L2 at t= 6 sec. This case study is considered to check the adaptive capability of the proposed control system for restoring the system to its original state, returning from any other operating condition. The simulation results for this scenario are shown in Fig. 8.

Fig. 8 shows that HBsWC has significantly better performance results in terms of overshoot and settling time as compared to ANeFU-TS.

(a) Rotor speed deviation

(b) Rotor angle deviation

(c) Line Power flow

Figure 8. System response for series of faults-II

Although, the performance improvement for the proposed HBsWC is clear from given simulation results, different performance indices are used to get further insight for performance improvement in transient and steady-state regions. The performance index is the integral of a function of time and error function of the system parameter. It can be written generally as,

Performance index; ( )( )0

,ts

PI f t e t dt= ∫ (36)

Where, ts is the simulation time and 1 2( , ( )) jif t e t t ω ω= − .

Here, ( ) ( ) ( ) ( ) ( ){ }, 0,1 , 0, 2 , 1,1 , 1, 2i j ∈ for IAE, ISE, ITAE and ITSE, respectively. The performance improvement results are shown in Table I. It is clear from the results that HBSWC performs better in both steady-state and transient regions. Moreover, it can be observed that performance improvement based on ITSE and ISE is greater than that of ITAE and IAE which shows that improvement margin is greater in transient region as compared to steady-state region.

TABLE I. PERFORMANCE IMPROVEMENT FOR HBSWC

Performance Index

Case 1 (%)

Case 2 (%)

Case 3 (%)

Case 4 (%)

ITAE 16.24 16.98 25.20 19.13 ITSE 75.31 37.86 63.06 24.19 IAE 26.71 19.23 41.91 20.80 ISE 77.46 37.41 70.92 25.80

V. CONCLUSION This paper presents a novel adaptive control scheme for

damping inter-area oscillations using SSSC by synergistically integrating the local control asset of B-spline membership function with local nonlinear analysis property of wavelets in the structure of adaptive NeuroFuzzy architecture. The application of B-spline wavelet control scheme significantly improves the damping performance. The proposed control system has better performance in both the steady-state and transient regions for different faults and operating conditions of a multi-machine system. Unlike, the conventional approach of comparing the results with simple, linear control systems, the adaptive NeuroFuzzy control is used for performance comparison. The control without optimization of antecedent and consequent parts can retain its performance for new operating conditions and various faults due to its intelligent and

adaptive structure, however, the proposed control system further improves its performance highlighting the effectiveness of proposed strategy.

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