Designing an adaptive type-2 fuzzy logic system load frequency control for a nonlinear time-delay power system

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Designing an adaptive type-2 fuzzy logic system load frequency control for a nonlinear time-delay power system

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Kamel Sabahi, Sehraneh Ghaemi ∗ , Mohammadali Badamchizadeh Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran

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a r t i c l e

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i n f o

a b s t r a c t

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Article history: Received 28 July 2015 Received in revised form 9 February 2016 Accepted 9 February 2016 Available online xxx

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Keywords: Load frequency control Type-2 fuzzy logic system Time-delay and restructured power system

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1. Introduction

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In this paper, a combination of type-2 fuzzy logic system (T2FLS) and a conventional feedback controller (CFC) has been designed for the load frequency control (LFC) of a nonlinear time-delay power system. In this approach, the T2FLS controller which is designed to overcome the uncertainties and nonlinearites of the controlled system is in the feedforward path and the CFC which plays an important role in the transient state is in the feedback path. A Lyapunov–Krasovskii functional has been used to ensure the stability of the system and the parameter adjustment laws for the T2FLS controller are derived using this functional. In this training method, the effect of delay has been considered in tuning the T2FLS controller parameters and thus the performance of the system has been improved. The T2FLS controller is used due to its ability to effectively model uncertainties, which may exist in the rules and data measured by the sensors. To illustrate the effectiveness of the proposed method, a two-area nonlinear time-delay power system has been used and compared with the controller that uses the gradient-descend (GD) algorithm to tune the T2FLS controller parameters. © 2016 Elsevier B.V. All rights reserved.

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The objective of load frequency control (LFC) in an interconnected power system is to maintain the frequency of each area within limits and keep tie-line power flows within some prespecified tolerances [1]. In LFC, the tie-lines are utilities for the contracted energy exchange between areas and they provide interarea support in abnormal conditions. Changes in the area loads and abnormal conditions lead to mismatches in frequency and the scheduled power interchanges between areas. These mismatches have to be corrected by LFC, which is defined as the regulation of the power output of generators within a prescribed area [2–4]. Various controllers have been proposed for LFC, including classical controllers, such as adaptive [5–8] and robust controllers [9–13], and intelligent controllers, such as fuzzy logic system (FLS) [14–20] and artificial neural network (ANN) [21–24] based controllers. Although these controllers have led to the promising results, some of the undeniable power system characteristics such as delays and nonlinearities have not been taken into account. It is clear that ignoring these issues in designing of the LFC not only degrades its dynamic

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∗ Corresponding author. Tel.: +98 4133393740. E-mail addresses: [email protected] (K. Sabahi), [email protected] (S. Ghaemi), [email protected] (M. Badamchizadeh).

performance, but may also cause system instability. Any signal processing, filtering, and breakdown in the communication channel can introduce delays, and speed governor dead band can be a source of nonlinearity in the power system. Dedicated communication channels in conventional LFC schemes and open communication networks in the restructured power system will introduce constant and time-varying delays, respectively. Hence, time-delays have been considered in designing LFC in the traditional and restructured power systems [25–32]. Most of these works have employed a robust control design method in which delay was dealt with as a part of uncertainties. For example, in [27] a robust PI based controller, which uses the static output feedback control law in a linear matrix inequality (LMI) framework, has been designed for LFC. In [26], an H∞ controller has been proposed for a two-area power system LFC with multiple state delays. In [30], a robust PID-type LFC scheme has been proposed for a power system with constant and time-varying delays. Also, authors of [31,32] have proposed a delay margin, which is a new performance index, as guidance for designing a robust controller for LFC. The main drawback of the robust control schemes is that they lead to high-order controllers and thus cause difficulty when implement in large scale systems such as LFC. Moreover, in many cases, all of the state variables of the system are needed for designing the controller. Therefore, it is necessary to have a controller which can overcome these problems. Based on artificial intelligence approaches, the FLS and ANN

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Please cite this article in press as: K. Sabahi, et al., Designing an adaptive type-2 fuzzy logic system load frequency control for a nonlinear time-delay power system, Appl. Soft Comput. J. (2016), http://dx.doi.org/10.1016/j.asoc.2016.02.012

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Fig. 1. Configuration of the ith control area.

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controllers are more suitable in this respect. The most salient feature of these techniques is that they provide a model-free description of control systems and do not require model identification or an exact model of the system. Also, type-2 FLSs (T2FLSs), which are an extension of the type-1 FLSs (T1FLSs), have been considered as a suitable solution to deal with measurement noise and parametric uncertainties of a system [33–40]. In type-2 fuzzy sets, membership functions (MFs) themselves are fuzzy and, therefore, they are a favourite solution for handling uncertainties [41,42]. Following the aforementioned properties of T2FLSs, in [43,50], type-2 fuzzy controllers have been designed for LFC in a typical nonlinear power system with system parametric uncertainty. These controllers have a fixed structure and the effect of delay has not been considered in the synthesis of the controllers. Besides, an adaptive T2FLS controller for a delay-free power system has been proposed in [44]. This control strategy is based on feedback error learning (FEL) approach. In this approach, the T2FLS controller is in the feedforward path and a conventional feedback controller (CFC) (i.e. proportional-derivative (PD)) is in the feedback path. The classic controller is used for stabilization and the intelligent part (the T2FLS controller) is designed to overcome the variations and nonlinearities in the controlled system. In that work, a simple gradient descent (GD) algorithm has been used to tune the T2FLS parameters and the effect of delay has been ignored in the synthesis of the controller. Motivated by the above-mentioned observations, in this paper an adaptive T2FLS controller based on FEL approach has been designed for LFC in a nonlinear time-delay power system. The most important contributions of this paper, and improvements to the work [44], can be considered as follows:

(i) The delayed power system with speed governor dead band nonlinearity has been considered in this study. Whereas, in [44] the delay-free power system was considered and in the dynamic model of the power system the linear model was used for the governor unit. (ii) The Lyapunov–Krasovskii functional has been utilized to evaluate the stability of the system; and the adaptation laws for the parameters of the T2FLS controller have been derived using this functional. The adaptation laws involve relationships between the controller parameters and the maximum values of the timevarying delays. So, the proposed strategy can deal with the time-varying delays and increase the performance of the power system. It is worth pointing out that the in the previous work, there was not any stability analysis for the proposed controller

and the effect of delay has been ignored in the designing of the controllers. Since only the training error signal and its derivative have been used in the construction of the Lyapunov–Krasovskii functional, the achieved adaptation laws for the T2FLS parameters are very simple. Unlike the other Lyapunov based control approaches such as robust controller, in the synthesis of the proposed scheme, there is no need for any prior knowledge about the mathematical model or its parameters. A two-area nonlinear time-delay power system is assumed to show the effectiveness of proposed controller. The proposed Lyapunov–Krasovskii based controller has been compared with the controller proposed in [44]. The remaining part of the paper is organized as follows: in Section 2, the dynamic model of a two-area nonlinear time-delay power system is presented. The proposed adaptive controller is derived for LFC in Section 3. Section 4 includes the simulation results for a two-area power system. Finally, the conclusions are given in Section 5.

2. Model description In a traditional power system structure, a single entity called a Vertically Integrated Utility (VIU) generates, transmits and distributes power to customers at the regulated rates and tie-lines connect all control areas. The load disturbance in a given area results in a transient change in the areas. Through a feedback mechanism, the turbine tries to modify the generation corresponding to the load. In the steady state, there is a match between the generation and load and then the tie-line power and frequency deviations are driven to zero. In the restructured power systems, there are some generation companies (Gencos) and distribution companies (Discos) in each area. A block diagram of the ith control area in the restructured power system for LFC is shown in Fig. 1 [3,30,32]. In this system, in an open energy market, Gencos may or may not participate in the LFC task and they can sell power to various Discos at competitive price [3]. Also, Discos have the liberty to choose the Gencos for contracts. In this power system, the concept of “generation participation matrix” (GPM) is used to facilitate visualization of the contracts. GPM shows the participation factors of each Genco in the considered control area and each control area is determined by a Disco. The rows of a GPM correspond to Genco and the columns correspond to the control areas that contract power. For example, for a large scale power system with m control areas (Discos) and n

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Fig. 2. Nonlinear model of governor and turbine [4,29].

147

Gencos, GPM will have the following structure [3,27]:

⎡

148

gpf11

⎢ gpf ⎢ 21 ⎢ ⎢ GPM = ⎢ .. ⎢. ⎢ ⎣ gpf(n−1)1 gpfn1

gpf12

...

gpf1(m−1)

gpf22

...

gpf2(m−1)

gpf1m

Fig. 3. Proposed controller for the ith area.

⎤

gpf(n−1)2

...

gpf(n−1)(m−1)

⎥ ⎥ ⎥ ⎥ .. ⎥ . ⎥ ⎥ gpf(n−1)(m) ⎦

gpfn2

...

gpfn(m−1)

gpfnm

.. .

.. .

Based on Refs. [3,31,32], we can write the scheduled Ptie i (w3i ) for an N control area power system as follows:

gpf2m

w3i

=

(1)

149

=



151 152 153

where gpfij refers to “generation participation factor” and shows the participation factor of the ith Genco in load flowing of the jth area (based on a specified bilateral contract). The sum of all the entries in a column is equal to unity, i.e.

⎛

n N  

n  154

 gpfkj

k=1

According to Fig. 1, it can be written that: i,error

gpfij = 1

(2)

= Ptie

i,actual

− w3i

 N

w4i−1 =

157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177

178

179

The dashed lines in Fig. 1 show the demand signals based on the likely contracts between Gencos and Discos, indicating that a Genco has to meet the load demands of the Disco. Any time-delay in LFC occurs chiefly between the control centres and the operating stations on the communication channels. Because of dedicated communication channels in the traditional LFC scheme, delay can be considered as a constant value [31,32]. On the other hand, due to the open communication channels in the restructured LFC scheme, constant and time-varying delays will be introduced. It should be noted that packet dropout and disordering, updating of area control error (ACE) signal, and faults of communication channels are other issues that lead to delays in the power system. To simplify the analysis, the mentioned delays are combined as one single delay and represented in the input of the controlled system by an exponential block e−s , where  is the amount of delay in the ith area and can be considered as a time-varying signal. In the power system, the turbine with generation rate constraint (GRC) and the speed governor dead band are the common sources of nonlinearities. These constraints can be modelled in several ways, among them the block diagram shown in Fig. 2 is adopted in this study [4,29]. In Fig. 1 we have:

183

(6)

w1i = PLoc i + Pdi

w2i =

N  j=1

Tij fj ,

j= / i

(3)

(4)



184 185

N

gpf1j PLj , . . ., w4i−n =

gpfnj PLj

(7)

186

j=1

The desired total power generation of the ith Genco in terms of GPM entries can be calculated as:



182

/ i j=

j=1

156

(5)

and the elements of w4i vector can be expressed as:

i=1

155

⎞

⎜ ⎟ ⎜ ⎟ ⎜ ⎟ n N  ⎜ ⎟ ⎜ P Lj − gpfjk ⎟ ⎜ ⎟ P Li ⎟ k=1 ⎜ j = 1 ⎜ ⎟ ⎝ ⎠

/ i j=

Ptie

181

(Total export power − Total import power)

j=1 150

180

187 188

N

Pmi =

gpfij PLj

(8)

189

j=1

The input signal for controller Ki (s) (where s indicates Laplace transformation) is ACE which can be defined from Fig. 1 as below: ACEi = Ptie

i,error

+ Bi fi

(9)

where Ptie i,error is the tie-line power deviation in the ith area. For more details about the considered power system, the interested reader can refer to Ref. [3,44]. 3. Adaptive controller design for LFC The proposed controller structure for the ith area of the nonlinear time-delay power system is illustrated in Fig. 3. This scheme, renowned as the FEL approach, was originally introduced by Kawato [45]. According to this strategy, the PD controller in the feedback and the intelligent controller (T2FLS controller) in the feedforward path simultaneously control the system. The final output of proposed controller, ui , consists of uffi and ufbi where uffi is the output of the feedforward controller and ufbi is the output of the PD controller. In this structure, the PD controller plays an important role in the transient state and is essential to guarantee the stability of the system. Moreover, the output of this controller (ufbi ) is used as the training signal to tune the T2FLS parameters such as standard deviations, centres, and consequent coefficients. On the other hand, by driving ufbi to zero, type-2 fuzzy controller in the feedforward path

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The lower and upper firing strengths of the Rj rule are obtained as follows: j1

Fig. 4. Type-2 membership function with an uncertain standard deviation.

j2

213 214 215 216 217 218 219 220 221

222

223 224 225 226 227 228 229 230 231 232 233 234 235 236 237

238

239 240 241 242 243

takes over the control in the steady state. It is obvious that if ufbi is driven to zero, the tracking and regulation problems of the closed loop system will be achieved. In order to take into consideration the existence of uncertain parameters, measurement noise, and unmodeled dynamics of the power system in the controller design process, T2FLS is a better choice than the ANN [46] and T1FLS controllers [15]. When a system has a large amount of uncertainties, the T1FLS and ANN controllers may not be able to achieve the desired level of performance with a reasonable complexity of the structure [40]. 3.1. Proposed T2FLS controller

(11)

246

f¯j =  ¯ j1 (x1 ) ∗  ¯ j2 (x2 )

(12)

247

where * shows the t-norm operation (which is considered the product operator in this study). The lower ((x)) and upper ((x)) ¯ MFs, using the Gaussian MFs in the above expression, are of the following forms:

j

R :

if x1

˜ j1 and x2 is A ˜ j2 , is A



1 jn (xn ) = exp ⎝ − 2

⎛



2 ⎞ ⎠ , n = 1, 2

(14)

253

xn − mjn jn



then uffi = cj

(10)

˜ jk are type-2 MFs for the jth rule and the kth input and cj is where, A the consequent parameters of the jth rule which are updated during training of the T2FLS controller. In this study, for the T2FLS controller, type-2 Gaussian MF with an uncertain standard deviation is used for the antecedent parts (see Fig. 4).

Fig. 5. Structure of the proposed T2FLS controller.



M

f j cj -

fj + -

j=1

M 

+ f¯j

f¯j cj

j=1 M 

j=1

j=1

M 

254 255 256 257

M

fj + -

M  j=1

= f¯j

cj (f j + f¯j ) -

j=1 M 

(15)

258

(f j + f¯ j ) -

j=1

where M is the number of rules. After normalization of (15), the output signal of the type-2 fuzzy controller (for the ith control area) acquires the following form: uffi (t) =

251

252

2

j=1 M 

250

(13)

where  1 ,  2 , and m are the tuning parameters of the T2FLS controller. The output of the T2FLS controller as proposed in [48], can be considered as follows:

uffi (t) =

249

1

1  ¯ jn (xn ) = exp ⎝ − 2



248

2 ⎞ xn − mjn ⎠ , n = 1, 2 jn

M

The T2FLS controller considered here benefits from type-2 MFs in the antecedent part (see Fig. 4) and crisp numbers in the consequent part [41,42]. In type-2 fuzzy sets, membership grades can be any subset in [0,1] which is called primary membership. Additionally, there is a secondary membership value corresponding to each primary membership value that defines the possibility of the primary memberships. It should be noted that a type-2 fuzzy set appears as two main cases: Interval type-2 fuzzy sets and generalized type-2 fuzzy sets. The mathematics that is needed for the interval type-2 fuzzy sets is much simpler than the mathematics that is needed for the general type-2 fuzzy sets. The structure of the proposed interval T2FLS controller for the ith control area in the power system is shown in Fig. 5. This controller has two inputs (x1 = ACEi and x2 = ACEi ) and one output (uffi ). The fuzzy if-then Rj rule is defined as follows:

245

f j =  (x1 ) ∗  (x2 ) -

⎛

212

244

cj j

(16)

259 260 261

262

j=1

where j =

263

f j + f¯j -

 M

j=1

(f j + f¯ j ) -

=

fj M 

(17)

264

fj

j=1

3.2. Stability analysis and adaptation law for the proposed controller By considering the nonlinear time-delay system illustrated in Fig. 1 and the proposed controller in Fig. 3, the control objective is to synthesize and implement a T2FLS controller to force the frequency and tie-line power deviations to zero. The gradient descent (GD) method (such as backpropagation algorithm) is a common method for online tuning of the T2FLS parameters [44,47]. In this method, to get the adaptation law, one needs to calculate the derivative of the squared error function with respect to the T2FLS parameters. In this algorithm, the learning rate plays an important role in the stability of the system, with a large value leading to unstable learning, and a small value resulting in a slow learning speed. Moreover, since the backpropagation algorithm uses only the derivative of the squared error function and ignores the effect of delay in designing the controller parameters (i.e. T2FLS controller), it can degrade the performance of the time-delay system. The Lyapunov–Krasovskii theorem is the main time-domain approach in the stability analysis as well as in the controller synthesis of such systems [49]. In this method, the main purpose is to obtain the sufficient conditions

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for the stability of the time-delay system by constructing an appropriate Lyapunov–Krasovskii functional. According to the definition of this functional, two classes of sufficient conditions, i.e. delayindependent and delay-dependent ones, have achieved a great deal of attention [49]. Therefore, in order to consider the effect of delay in the designing procedure of the proposed controller, the following Lyapunov–Krasovskii functional has been introduced: 1 Vi (t) = ei (t)2 + 2





t

0



t 2

2

ei (s) ds +

ei () dds −max

t−(t)

(18)

318

where

⎛ (fj (t))

⎝



M 

⎞ ⎛ fj ⎠ − ⎝

j=1

˙ j (t) =

⎛

M 

319

⎞ fj ⎠ fj (t)

j=1

M 

⎞2

320

fj ⎠

j=1

where ei (t) = ufbi (t) = ui (t) − uffi (t)



(19)

299

Assumption a.

297

(28)

j=1

⎝

298

296

(c˙ j j + ˙ j cj )

t+

As aforesaid, if ufbi in each control area is driven to zero, the regulation purposes of the closed loop system can be achieved. Moreover, in the power system dynamic model, the following assumptions are considered:

295

M 

u˙ ffi (t) =

5

f˙ j + f¯˙ j -

⎛



⎝

M  j=1

=

⎞ ⎛ fj ⎠ − ⎝

⎛ ⎝

M  

f˙ j + f¯˙ j -

301

where and ˇ are the known positive constants.

|u˙ i (t)| < ˇ

⎠fj

j=1

M 

⎞2

(29)

(20)

321

fj ⎠

j=1 322



|ui (t)| < ,

⎞

and

ui (t) and u˙ i (t) are bounded signals, i.e.

300





f˙ j = (j1 (x1 )) j2 (x2 ) + j1 (x1 )(j2 (x2 )) f¯˙ j = ( ¯ j1 (x1 ))  ¯ j2 (x2 ) +  ¯ j1 (x1 )( ¯ j2 (x2 ))

(30)

and

323

324 325

303

Assumption b. Delay and its derivative are bounded in each control area

304

(t) ≤ max ,

302

˙ |(t)| <1

(jn (xn )) -

jn



jn

jn

∂ (xn ) ∂ (xn ) ∂xn ∂ (xn ) ∂m = + ∂t ∂xn ∂t ∂t ∂mjn

=



jn

(21)

jn

˙ jn − x˙ n )1 + (xn − mjn )˙ 1 (m



 jn 2

=

305 306

+

∂ (xn ) jn ∂t ∂1

xn − mjn



326

n = 1, 2

jn (xn ), -

jn

1

1

jn ∂1

jn

jn

(31)

Theorem 1. If the adaptation laws for the parameters of the proposed T2FLS controller are chosen as:

327 328

jn

307

˙ jn = x˙ n (t), m

n = 1, 2

(22)

( ¯ jn (xn ))

∂ ¯ jn (xn ) ¯ jn (xn ) ∂xn ¯ jn (xn ) ∂mjn ¯ jn (xn ) ∂2 ∂ ∂ ∂ = = + + jn ∂t ∂xn ∂t ∂t ∂t ∂mjn ∂2





jn

jn 3

308

jn ˙ 1

= −˛

˙ 2 = −˛

310

c˙ j =

312

314 315 316

317

jn 3 (2 )

(xn − mjn )

j ˚˚T

e (t), 2 i

n = 1, 2

j =

M 

(f j + f¯ j ) -



 jn 2

xn − mjn



jn 2

2

329

∂ ¯ jn (xn ),

n = 1, 2 330 331

e (t), 2 i

n = 1, 2

(24)

(25)

So, by substituting (31) and (32) into (30) we have:

⎛

,

˚ = [1 , 2 , . . ., M ]

n=1

1

jn

1

⎛

⎜ ⎜ f¯˙ j =f¯j ⎝ ⎝ 2

jn ˙ jn − x˙ n )2 (m



(26)

⎞

jn + (xn − mjn )˙ 2 2 jn 2



⎟ ⎠



⎞  (33) xn − mjn ⎟ ⎠ jn

˙ j (t) = 0

j=1

(27)

then V˙ i (t) < 0 is satisfied. It should be noted that for the purpose of simplicity, the time variable is omitted from the arguments of some functions. Proof. The time derivative of uffi (the output of the T2FLS controller in the ith area) can be considered as:

334

(34)

Using Assumption b, the time derivative of Vi (t) in (18) can be considered as follows: 2

˙ V˙ i (t) = ei (t)e˙ i (t) + ei (t) − (1 − (t))e i (t − (t))



333

2

Substituting Eqs. (22)–(24) into (29) results in:

fj

Ei (t) = ˇ + (1 + max + ˛)ei (t)

332

⎞

n=1

fj M 

⎛

⎞

 2 jn jn jn jn  jn ⎜ ⎜ (m˙ − x˙ n )1 + (xn − m )˙ 1 ⎟ xn − m ⎟ f˙ j =f j ⎝ ⎝ ⎠ ⎠  2 jn - ⎛



jn

˙ jn − x˙ n )2 + (xn − mjn )˙ 2 (m

(32)

Ei (t)

f j + f¯j -

=

(23)

where ˛ is a positive designing parameter and

j=1

313

(xn − mjn )

jn

309

311

(1 )

335 336 337

2

t

+max ei (t)2 −

ei (s)2 ds t+

≤ ei (t)e˙ i (t) + ei (t)2 + max ei (t)2

(35)

= ei (t)(u˙ i (t) − u˙ ffi (t)) + (1 + max )ei (t)2

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Fig. 6. Two control area power system (4 Gencos and 4 Discos).

339 340

Using Assumption a and by considering (34), the inequality below can be obtained: V˙ i (t)

≤ ei (t)(u˙ i (t) −

M 

2

(c˙ j j )) + (1 + max )ei (t)

j=1

341

< ei (t)(ˇ −

M 

(c˙ j j )) + (1 + max )ei (t)

Genco i (for i = 1:4)

343 344 345 346

Area i (for i = 1,2)

Remark 1. Because of the limit in valve position on the governor units, Assumption a is reasonable. This limit can be described by the upper and lower bounds.

350

351

4. Simulation results

348 349

352 353 354 355 356 357 358 359 360 361 362

Tgi

˛i

2.4

0.36

0.06

0.5

K pi (Hz/pu)

Tpi (s)

Bi (pu/Hz)

T12 (pu/Hz)

120

20

0.5

0.545

(37)

Remark 2. From the adaptation laws, it is clear that only the output of the PD controller (training signal), the upper bound of delay, ˙ and u(t) are required for training of the T2FLS controller parameters.

347

Tti (s)

Table 2 Applied control area parameters.

Finally, substituting (25) into (36) gives: V˙ i (t) < −˛ei (t)2

Ri (Hz/pu)

(36) 2

j=1 342

Table 1 Applied data for Gencos.

For demonstrating the efficiency of the proposed controller, some simulations are performed. A two control area power system, shown in Fig. 6, is considered as a test system. The power system considered here is the dynamic model that was considered in works [3,14,26,31,32]. It should be noted that in some of mentioned works, the linear delay-free model [3], nonlinear delay-free [43] and linear delayed one [31,32,43] have been used. In the this paper, a nonlinear delayed restructured power system has been used to demonstrate the effectiveness of the proposed Lyapunov–Krasovskii stable T2FLS controller in the form of FEL approach.

It is assumed that each control area includes two Gencos, which use the same ACEi participation factor, and two Discos. In these simulations, the PD controller coefficients, which are tuned using trial-and-error methods, are Kpi = 4.3 and KDi = 5.1. For T2FLS controller, the antecedent part is composed of three Gaussian type-2 MFs with uncertain standard deviations for each input. Hence, the type-2 fuzzy controller benefits from nine rules (M = 9). The MFs for x1 and x2 (centres) are initially distributed in the closed interval of [−1,1] with random values for standard deviations  1 ,  2 . Also, random value has been considered for consequent parameters cj . After initialization, these parameters are evolved during training, according to Eqs. (22)–(25), to find their optimal positions. The nominal parameters of the two-area interconnected power system used in the simulations are given in Tables 1 and 2 [4–5,26,32]. Furthermore, to consider the uncertainties, it is assumed that the measured output of the system (ACEi ) in each control area is distorted by the random noise with a normal distribution. For the ith area, we have: ACEi = ACEi + 0.1 × n(t)

(38)

Please cite this article in press as: K. Sabahi, et al., Designing an adaptive type-2 fuzzy logic system load frequency control for a nonlinear time-delay power system, Appl. Soft Comput. J. (2016), http://dx.doi.org/10.1016/j.asoc.2016.02.012

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0.1 Proposed T2FLS controller GD based method [44]

Proposed T2FLS controller GD based method [44]

0.1

0.08

0.06

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0.04

Δ Ptie

0

Δf1

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Fig. 8. The tie-line power deviation (Ptie ) ( max = 2 s).

Proposed T2FLS controller GD based method [44]

0.2

392

0.1

The nominal values for parameters of the power system are introduced in Tables 1 and 2. Furthermore, the Discos contract with the available Gencos according to the following GPM:

Δf2

0

-0.1

⎡

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10

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25

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40

Fig. 7. The frequency deviations for the first (a) and second (b) areas for case 1 ( max = 2 s).

383 384

where n(t) is a random number with the Gaussian distribution. The proposed controller has been compared with the controller proposed in [44] through some simulations.

386

Case 1. The communication delays in each control area are assumed to have a small value as follows:

387

(t) = 1.5 + 0.5 sin(t)

385

(39)

389

in which  max is 2 s. And a large step load is demanded by Discos of areas 1 and 2 as follows:

390

PL1 = 0.1 pu, PL2 = 0.2 pu, PL3 = 0.2 pu, and PL4 = 0.1 pu

0.0

0.5

0.2 0.15

Δpg2

0.2

0.1 0.05

0.5

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0

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40

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Δpg4

0.25

0.1 0.05 0

396

From GPM, it is clear that there are no contracts between areas. Therefore, the scheduled steady state power flow through the tieline is zero (Eq. (5)): Fig. 7 illustrates the frequency deviations (f1 and f2 ) of the closed loop system when the power system is equipped with the proposed controller and the one proposed in [44]. It can be seen that as the effect of delays has been considered in the tuning procedure of the proposed adaptive controller, the performance of the proposed controller surpasses that of the other controller. On the other hand, in the controller proposed in [44], with a small amount of increase in delays, the frequency deviations start to oscillate slowly. Then, at around t = 25 s, the frequency deviations are damped to zero. This is because the effect of delays has been ignored in the tuning of T2FLS parameters. It is clear that if these increases continue, the oscillation of the frequency deviations increase. The tie-line power (Ptie ) and the actual generated

0.15

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394 395

⎤

Proposed T2FLS controller GD based method [440.25 ]

0.25

Δpg1

391

Δpg3

388

0.0

⎢ 0.5 0.5 0.0 0.0 ⎥ ⎥ ⎣ 0.0 0.0 0.5 0.5 ⎦

Time(sec)

382

0.5

GPM = ⎢

-0.3

-0.4 0

0.5

393

20

Time(sec)

Fig. 9. The generated power of each Genco (Pgi ) ( max = 2 s).

Please cite this article in press as: K. Sabahi, et al., Designing an adaptive type-2 fuzzy logic system load frequency control for a nonlinear time-delay power system, Appl. Soft Comput. J. (2016), http://dx.doi.org/10.1016/j.asoc.2016.02.012

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ufb1 uff1

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Proposed T2FLS controller GD based method [44]

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ufb2 & uff2

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ufb2 uff2

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Time(sec)

Fig. 10. The frequency deviations for the first (a) and second (b) areas for case 2 ( max = 3 s).

413 414 415 416

417

Fig. 12. Output of PD and T2FLS controllers for the first (a) and second (b) areas for case 2.

power of Gencos, Pgi , for the controllers under consideration are shown in Figs. 8 and 9, respectively. It should be noted that the desired value for generated powers, according to Eq. (8), can be calculated as follows:

a

0.2

Pm1 = gpf11 PL1 + gpf12 PL2 + gpf13 PL3 + gpf14 PL4 = 0.5 × 0.1 + 0.5 × 0.2 + 0.0 × 0.2 + 0.0 × 0.1 = 0.15 pu

0.1

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Pm2 = 0.15 pu, Pm3 = 0.15 pu, Pm4 = 0.15 pu

419 420

Using the proposed method, the tie-line power is quickly driven back to zero and the generated powers properly converge to specified values.

424

Case 2. In this case we want to see how the two mentioned controllers respond to the 50% increasing of the maximum delays in Case 1. For this purpose, consider the following delay equation ( max is 3 s):

425

(t) = 2.5 + 0.5 sin(t)

423

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(40)

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Proposed T2FLS controller GD based method [44]

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422

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ΔPtie

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Δ f1

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418

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20

Time(sec)

5

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35

40

Fig. 13. The frequency deviations for the first (a) and second (b) areas for case 3 with the proposed strategy.

Time(sec)

Fig. 11. The tie-line power deviation (Ptie ) ( max = 3 s).

Please cite this article in press as: K. Sabahi, et al., Designing an adaptive type-2 fuzzy logic system load frequency control for a nonlinear time-delay power system, Appl. Soft Comput. J. (2016), http://dx.doi.org/10.1016/j.asoc.2016.02.012

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Time(s)

Fig. 14. Training errors (the output of PD controllers) for two areas in the proposed controller.

427

It is assumed that a large step load is demanded by Discos of areas 1 and 2 as follows (contracted demands):

428

PL1 = 0.2 pu, PL2 = 0.2 pu, PL3 = 0.2 pu,

426

429

and PL4 = 0.2 pu.

430 431 432

Besides, according to the following GPM the Discos contract with the available Gencos:

⎡

433

0.25

435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453

0.25

⎢ 0.25 0.25 0.25 GPM = ⎢ ⎣ 0.0 0.5 0.5 0.5

434

0.0

0.25

0.0

0.25

⎤

⎥ ⎥ 0.5 ⎦ 0.0

0.25

In addition to the specified contracted load demands, the Discos in areas 1 and 2 demand 0.1 and 0.2 pu (Pd1 = 0.2 and Pd2 = 0.1) as a large un-contracted load, respectively. The amount of these violated demands distributed among the Gencos according to the values of the ACE participation factors ˛ij . The un-contracted load of Discos is taken up by Gencos of those areas according to the ACE participation factors in the steady state. Moreover, we consider the operating point to be Tp1,2 = 15, T12 = 0.34, and B1,2 = 0.425. The responses of f1 and f2 are shown in Fig. 10. It can be seen that increase in delays can properly be dealt with by the proposed controller. Due to the fact that the effect of delays has been considered in designing of the proposed strategy, this controller acts to maintain the area frequency close to the scheduled values. On the other hand, by increasing delays, the FEL strategy in [44] starts to give an oscillatory output. The simulations performed for  max > 3 s reveal that the controller proposed in [44] is not capable of maintaining the stability of the closed loop system, and therefore, the delays will lead to instability in the power system. By considering the GPM, the scheduled steady state power flow through the tie-line is as follows: Ptie,1−2 = [(gpf13 + gpf23 )PL3 + (gpf14 + gpf24 )PL4 ] − [(gpf31 + gpf41 )PL1 + (gpf32 + gpf42 )PL2 ]

454

455 456 457 458 459 460 461 462

= [(0.25 + 0.25)0.2 + (0.25 + 0)0.2] −[(0 + 0.5)0.2 + (0.5 + 0.25)0.2] = −0.1 pu The actual tie-line powers are depicted in Fig. 11. Fig. 12 illustrates the output of the PD controller (ufbi ), considered as the training error, and the output of the T2FLS controller (uffi ) for the proposed controller (for each control area). As mentioned before, the PD controller plays an important role in the transient state. The output of this controller is used to train the feedforward T2FLS controller. After deriving ufb to zero, uff becomes the leading signal and T2FLS obviously takes over the control of the

system in the steady state. These figures show that after about 10 s, the proposed T2FLS controller takes the responsibility of controlling the system. It should be noted that if again a change occurs in the system parameters and load demand, the PD controller will definitely be activated for a short time. But once the feedforward controller is trained for a new condition, the control of the system is again assigned to the T2FLS controller. Case 3. Consider the conditions stated in Case 2 for Gencos and Discos with the un-contracted loads Pd1 = 0.0 and Pd2 = 0.1. In Case 3, a long delay has been considered in the communication links in the power system: (t) = 6.5 + 0.5 sin(t)

(41)

The responses of f1 and f2 for the proposed adaptive controller are shown in Fig. 13. Despite the long delay in the communication links, the T2FLS controller with the proposed learning algorithms result in a satisfactory performance. It should be noted that the controller strategy proposed in [44] becomes unstable and incapable of dealing with the long delay in the power system. The training errors for the proposed controller are illustrated in Fig. 14. It is evident from this figure that within a short time, the T2FLS controllers take the responsibility of controlling the power system in the areas. 5. Conclusion An adaptive T2FLS controller in combination with a classical PD controller has been designed for LFC in a nonlinear time-delay power system. In this controller, T2FLS in the feedforward path and the PD controller in the feedback path control the system simultaneously. The PD controller has a crucial role in the transient state while T2FLS has been trained to take over the control of the system in the steady state and also deal with nonlinearities, uncertainties, and delay of the controlled system. In order to analysis the stability of the proposed strategy, a suitable Lyapunov–Krasovskii functional has been introduced. The adaptation laws for the consequent and antecedent parameters of the T2FLS controller have been derived using this functional. Compared with the existing adaptive structures, the achieved equations are very simple and only the training error and its derivative are required for tuning the T2FLS controller parameters. Simulation results indicate the efficiency of the proposed strategy even in presence of the long time-delays. References [1] H. Saadat, Power System Analysis, McGraw-Hill, 2002. [2] P. Kundur, Power System Stability and Control, McGraw Hill, New York, 1994. [3] D. Vaibhav, M.A. Pai, I.A. Hiskens, simulation and optimization in an AGC system after deregulation, IEEE Trans. Power Syst. 16 (3) (2001).

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