Application of ANN technique based on μ-synthesis to load frequency control of interconnected power system

Electrical Power and Energy Systems 28 (2006) 503–511 www.elsevier.com/locate/ijepes

Application of ANN technique based on l-synthesis to load frequency control of interconnected power system H. Shayeghi b

a,*

, H.A. Shayanfar

b

a Technical Engineering Department, The University of Mohaghegh Ardebili, Iran Electrical Engineering Department, Iran University of Science and Technology, Narmak, Tehran 16844, Iran

Received 9 July 2004; received in revised form 15 December 2005; accepted 24 February 2006

Abstract A nonlinear artificial neural networks (ANN) controller based on l-synthesis for solution the load frequency control (LFC) problem is proposed in this paper. Power systems such as other industrial plants subject to some uncertainties and disturbances due to multivariable operating conditions and load changes. In order to take large modeling errors and minimize the effects of area load disturbances, the idea of l-synthesis theory is being used for training ANN based LFC controller. This newly developed design strategy combines advantage of the ANN and l-synthesis control techniques to achieve the desired level of robust performance for all admissible uncertainties and leads to a flexible controller with relatively simple structure, which can be useful in the real world complex power system. A two-area power system is considered as a test system to demonstrate the effectiveness of the proposed method in comparison with the conventional PI and l-based robust controllers under various operating conditions and load changes. The simulation results show that the proposed ANN based controller achieves good robust performance even in the presence of generation rate constraints (GRC) and is superior to the other controllers.  2006 Elsevier Ltd. All rights reserved. Keywords: LFC; Power system control; ANN; l-synthesis; Robust control

1. Introduction Load frequency control (LFC) is one of the most important issues in electric power system design and operation. The objective of the LFC in an interconnected power system is to maintain the frequency of each area and to keep tie-line power near to the scheduled values by adjusting the MW outputs the LFC generators so as to accommodate fluctuating load demands. The load frequency controller design with better performance has received considerable attention during past years and many control strategies have been developed [1–4] for LFC problem. The availability of an accurate model of the system under study plays a crucial role in the development of the *

Corresponding author. Tel.: +98 21 88971630; fax: +98 21 88986563. E-mail addresses: [email protected], [email protected] (H. Shayeghi), [email protected] (H.A. Shayanfar). 0142-0615/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2006.02.012

most control strategies like optimal control. However, an industrial process, such as a power system, contains different kinds of uncertainties due to changes in system parameters and characteristics, loads variation and errors in the modeling. On the other hand, the operating points of a power system may change very much randomly during a daily cycle. Because of this, a fixed controller based on classical theory [3,4] is certainly not suitable for LFC problem. Thus, some authors have suggested variable structure [5–7] and neural networks methods [8,9] for dealing with parameter variations. All the proposed methods are based on state-space approach and require information about the system states, which are not usually known or available. On the other hand, various adaptive techniques [10,11] have been introduced for LFC controller design. Due to requirement of the prefect model, which has to track the state variables and satisfy system constraints, it is rather difficult to apply these adaptive control techniques to

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LFC in practical implementations. Recently, several authors have been applied robust control methodologies [12–15] to the solution of LFC problem. Although via these methods, the uncertainties are directly introduced to the synthesis. But models of large scalar power system have several features that preclude direct application of robust control methodologies. Among these properties, the most prominent are very large (and unknown) model order, uncertain connection between subsystems, broad parameter variation and elaborate organizational structure. In this paper, because of the inherent nonlinearity of power system we address a new nonlinear artificial neural network (ANN) controller based on l-synthesis technique. The motivation of using the l-based robust controller for training of the proposed controller is to take the large parametric uncertainties and modeling error into account. To improve the stability of the overall system and also its good dynamic performance achievement, the ANN controller has been reconstructed with applying the l-based robust controller to power system in the different operating points under different load disturbances by using the learning capability of the neural networks. Moreover, the proposed controller also makes use of a piece of information, which is not used in the conventional and l-based robust controllers (an estimate of the electric load perturbation, i.e. an estimate of the change in electric load when such a change occurs on the bus). The load perturbation estimate could be obtained either by a linear estimator, or by a nonlinear neural network estimator in certain situations. It could also be measured directly from the bus. We will show by simulation that when a load estimator is available, the ANN controller can achieve extremely dynamic response. In the work, a twoarea power system is considered as a test system. Each area

1 R1

B1

(i) conventional integral controller, (ii) l-based robust controller and (iii) ANN controller for different cases of the plant parameter changes under various step load disturbances. The simulation results show that the proposed controller is very effective and gives good dynamic response compared to the conventional PI and l-based robust controllers even in the presence of the plant parameters changes and generation rate constraint (GRC). 2. Plant model A large power system consists of a number of interconnected control areas, which are connected by tie-lines power. There are different complicated nonlinear models for large-scale power systems. However, for the design of LFC a simplified and linearized model is usually used [16]. In advanced control strategies (such as the one considered in this paper) the error caused by simplification and linearization are considered as parametric uncertainties and unmodeled dynamics. A two-area power system is taken as a test system in this study. In each area, all generators are assumed to be coherent group. Fig. 1 shows the block diagram of the system in detail. Each area including steam turbine contains governor, reheater stage of steam turbine. The governor dead-band effects that are important for speed control under small disturbances is considered to

K1

ΔPC1 K I1 s

of power system consists of steam turbines, which include reheater. Therefore, there are the effects of reheater and generating rate boundaries in each area. For comparison, the considered system is controlled by using

1 1+ sTTG 1

ΔX G1

1 1+ sTT 1

ΔPR1

K2

1 1+ sTR 1

ΔPd1

K P1 1+ sTP 1

ΔPT1

ΔF1

u1 ΔPTie

a 12 K I2 s

B2

K1

u2 ΔPC 2

1 1+ sTTG 2 1 R2

ΔX G 2

1 1+ sTT 2

ΔPR 2

a 12

K2

1 1+ sTR 2

2 πTI 2 s

KP2 1+ sTP 2

ΔPT 2 ΔPd 2

Fig. 1. Block diagram of a two-area power system.

ΔF2

H. Shayeghi, H.A. Shayanfar / Electrical Power and Energy Systems 28 (2006) 503–511

be 0.06% [1]. The nomenclature used and the nominal parameter values are given in Appendix A. A state-space model for the given system can be constructed as x_ ¼ Ax þ B1 u þ Fd; y ¼ Cx;

ð1Þ

where x is the state variables vector, u is control signals, d is disturbances vector, y is the measured output vector and

505

where P i is the nominal value of the parameter and e the upper bound of Pi its variation. Therefore, we can separate D that represents the variation element as shown in Fig. 2. Table 1 shows the eight parametric uncertainties of the system described in Section 2 with their nominal, upper and lower bound values where PUi and PLi stand for the upper and lower bound values, respectively. The range of parameter variations are obtained by simultaneously changing of TP, T12 by 50% and all other parameters by 20% of their typical values as given in Appendix A.

u ¼ ½ u1 u2 T ; d ¼ ½ DP d1 DP d2 T ; y ¼ ½ DF 1 DP Tie DF 2 T ; x ¼ ½ DP C1 DX G1 DP R1 DP T1 DF 1 DP Tie DP C2 DX G2 DP R2 DP T2 DF 2 T ; 3 2 K I1 0 0 0 0 0 0 0 0 0 K I1 B1 7 6 7 6 1=T G1 1=T G1 0 0 1=R1 T G1 0 0 0 0 0 0 7 6 7 6 0 1=T T1 1=T T1 0 0 0 0 0 0 0 0 7 6 7 6 7 6 0 K 1 =T T1 a 1=T R1 0 0 0 0 0 0 0 7 6 7 6 7 6 0 0 0 K =T 1=T K =T 0 0 0 0 0 P1 P1 P1 P1 P1 7 6 7 6 0 0 0 0 2pT 12 A¼6 0 0 0 0 0 2pT 12 7; 7 6 6 0 0 0 0 0 K I2 0 0 0 0 K I2 B2 7 7 6 7 6 7 6 0 0 0 0 0 0 1=T 1=T 0 0 1=R T G2 G2 2 G2 7 6 7 6 0 0 0 0 0 0 0 1=T T2 1=T T2 0 0 7 6 7 6 7 6 0 0 0 0 0 0 0 K 2 =T T2 b 1=T R2 0 5 4 0 0 0 0 0 K P2 =T P2 0 0 0 K P2 =T P2 1=T P2 a ¼ ðK 1 þ K 2 Þ=T T1  K 2 =T R1 ; b ¼ ðK 1 þ K 2 Þ=T T2  K 2 =T R2 ; " # 0 1=T G1 0 0 0 0 0 0 0 0 0 B1 ¼ ; 0 0 0 0 0 0 0 1=T G2 0 0 0 " # 0 0 0 0 0 K P1 =T P1 0 0 0 0 0 ; F¼ 0 0 0 0 0 0 0 0 0 0 K P2 =T P2 2 3 0 0 0 0 1 0 0 0 0 0 0 6 7 C ¼ 4 0 0 0 0 0 1 0 0 0 0 0 5. 0 0 0 0 0 0 0 0 0 0 1 2.1. Parametric uncertainty and description Parameters of the linear model described for the LFC problem in Fig. 1 depend on the operating points. On the other hand, because of the inherent characteristics of loads changing and system configuration, the operating points of the power system may change very mach randomly during a daily cycle. Thus, the real power system will contain parametric uncertainty such as other industrial plants. First, we show a way how to consider the change of each parameter. Denoting ith certain parameter by Pi the parameter uncertainty is formulated as   e    P i  e 6 P i 6 P i þ e or P i ¼ P i 1 þ  Di ; jDi j 6 1; Pi ð2Þ

Now let’s define B ¼ ½F

B1 ;

uT1 ¼ ½d T

uT ;

ð3Þ

with these definitions and due to Eq. (2), the state-space model of the system becomes ! ! 8 8 X X d i Ai x þ B0 þ di B i u1 ; x_ ¼ A0 þ ð4Þ i¼1 i¼1 y ¼ Cx. The A0 and B0 matrices are obtained by substituting nominal values of the system parameters into the matrices A and B defined above. The matrices Ai and Bi are obtained by differentiating the matrices of A and B with respect to the ith uncertainty, respectively.

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Table 1 Parametric uncertainties of the system Pi

PLi

P i

PUi

e

Bi 1/TGi 1/RiTGi 1/TTi 1/TRi KPi/TPi 1/TPi T12

0.34 8.33 2.983 2.78 0.0833 4 0.033 0.049

0.43 10.42 4.7 3.4735 0.1042 8 0.0665 0.0707

0.52 12.5 6.51 4.167 0.125 12 0.1 0.093

0.09 2.07 1.81 0.6935 0.0208 4 0.0335 0.0223

zi

ε

Δi

an uncertainty with known structure, bounded value and belonging to the set BD D ¼ fdiagðd1 I r1 ; . . . ; ds I rs ; D1 ; . . . ; DF Þ; di 2 C; Dj 2 C mj mj g; BD ¼ fD 2 Dj rðDÞ 6 1g; ð5Þ ð:Þ denotes the maximum singular value of a matrix where r and two nonnegative integers S and F represent the number of repeated scalar blocks and full blocks, respectively. For a system described in the complex matrix, M 2 Cn·n, the structural singular value (l) is defined as lD ðMÞ ¼

wi

Pi

Pi

Pi

Fig. 2. The parametric uncertainty model.

3. Structure singular value and l-based LFC controller design This section gives a brief over view of structure singular value. Also, the procedure of the l-based LFC controller design is given.

1 . minf rðDÞ : D 2 D; detðI  MDÞ ¼ 0g

ð6Þ

Thus, lD(M) is a measure of the smallest structured l that causes instability of the constant matrix feedback loop shown in Fig. 3(b). Given a desired uncertainty level, the purpose of this design is to look for a control law, which can bring down the closed loop system l level and ensure the stability of the system for all possible uncertainty descriptions. The performance and stability conditions for a system in the presence of structured uncertainty in terms of l are given as 1. Robust stability (RS) F u ðM; DÞ stable 8D 2 BD;

iff

sup lðM 11 ðjxÞÞ 6 1. x

ð7Þ

3.1. Structure singular value and l-synthesis 2. Robust performance (RP) The general framework of l analysis and synthesis [17] shown in Fig. 3, with the generalized plant P, the controller K and the uncertainty block D. Any linear interconnection of inputs, outputs and commands along with perturbations and a controller can be viewed in this context and rearranged to match this diagram. Here, v is the exogenous input vector, e is the error output vector, u is the control input vector and y is the measured output vector. For the purpose of analysis, controller K is obtained into plant P to form the interconnected structure in Fig. 3(b). Given

F u ðM; DÞ stable and kF u ðM; DÞk1 6 1 8D 2 BD;

iff

ð8Þ

sup lðMðjxÞÞ 6 1; x

where Fu(.,.) is upper linear fractional transformation (LFT) and k.k is H1 norm. In other words, the performance and stability of the closed loop system M is a l test, across frequency for the given uncertainty structure D. The synthesis problem is represented by the structure in Fig. 3(c). The control error e 0 can be expressed as the following LFT: 1

e0 ¼ F L ðP ; KÞv0 ¼ ½P 11 þ P 12 KðI  P 22 KÞ P 21 v0 . Δ w

Ideally, the goal is to find a controller K such that

z

v

kF L ðP ; KÞkl 6 1.

e

P

u

y

However, as there is no effective technique to which this K is obtained directly at present, indirectly it is calculated by K and scaling matrix D until Eq. (10) is filled:

K

(a) w

Δ

z

v'{ u

v

M (b)

e

ð9Þ

P

}e' y

min inf kDF L ðP ; KÞD1 k1 6 1; K

D

ð10Þ

K

D ¼ fdiagðd 1 I; d 2 I; . . . ; d n IÞjd i 2 Rþ g.

(c)

In the minimization, fixing either D or K is called especially D–K iteration [18]. It does not become a trouble so much in practice use, and is widely used.

Fig. 3. l Analysis and synthesis structure: (a) general interconnected structure, (b) analysis and (c) synthesis.

H. Shayeghi, H.A. Shayanfar / Electrical Power and Energy Systems 28 (2006) 503–511

3.2. l-Based LFC controller design

4. The ANN controller design for LFC

The objective of the controller design in an interconnected power system is damping of the frequency and tieline power deviations oscillations, stability of the overall system for all admissible uncertainties and load changes. Thus, frequency and tie-line power deviations (DF1, DPtie, DF2) in two-area power system are considered as controller inputs. The first step in designing of the l-based controller is to formulate design problem into l general framework. By using Eq. (4) and following the procedure of Ref. [1], the state-space model along with uncertainties model will be separated as 8 x_ ¼ A0 x þ B0 u1 þ B1 w; > > > > < z ¼ C z x þ Dz u1 ; P0 : ð11Þ > y ¼ Cx; > > > : w ¼ Dz;

4.1. The ANN features

where structured uncertainty block is D ¼ fdiagðd1 I 2 ; . . . ; d7 I 2 ; d8 Þ; di 2 R; kDk 6 1g. The design problem formulation into the l general structure is shown in Fig. 4. In this diagram, P0 is the interconnection of nominal plant and all parametric uncertainties. In order to take modeling error into account, an additional input multiplicative uncertainty is considered by weighting function WC. The WP also indicates the system performance specifications. The weights have been selected based on Ref. [14] as 0:33s þ 1:5 W C ðsÞ ¼ ; 30s þ 0:03 0:7s þ 1:245 W P ðsÞ ¼ : 1:245s þ 0:007

ð12Þ ð13Þ

All the above components are compressed into a generalized plant P to perform the l analysis. A software tool [19] has been used for the design of l-based LFC controller. The control law obtains in the form of state-space notation AK, BK and CK with system order 106. It should be noted, although via l synthesis the uncertainty can be introduced to the controller synthesis, but due to large model order of power systems the order of obtained controller will be very large in general.

Fig. 4. l-Based controller design problem formulation.

507

Recently, computational intelligence systems and among them neural networks, which in fact are model free dynamics, has been used widely for approximation functions and mappings. The main feature of neural networks is their ability to learn from samples and generalizing them and also their ability to adapt themselves to the changes in the environment. In fact, neural networks are very suitable for problems in the real world. These networks with participation of especial kind of parallel processing are able to provide the modeling any kind of nonlinear relations. More accuracy, robustness, generalized capability, parallel processing, learning static and dynamic model of MIMO systems on collected data and its simple implementation are some of the important characteristics of the neural network that caused wide applications of this technique in different branches of sciences and industries, especially in designing of the nonlinear control systems [9,20]. The salient feature of artificial intelligent technique is that they provide a model-free description of control systems and do not require the accurate model of the plant. Thus, they are very suitable for control of the nonlinear plants that their models are unknown or have uncertainty such as a large-scale power system. 4.2. The l-based ANN controller There are some deviations and uncertainties due to changes in system parameters, characteristics and load variations in power systems that for the controller design have to be considered. On the other hand, very large (and unknown) model order, uncertain connection between subsystems, broad parameter variations and elaborate organizational structure of the power system preclude direct application of standard robust control methodologies. In order to overcome this drawback, we propose a new nonlinear artificial neural network (NANN) controller based on l-synthesis technique. Fig. 5 shows the designing procedure of the NANN proposed controller for two-area power system. The multi-layer perceptron (MLP) neural networks for the design of the nonlinear LFC controller in two-area power systems are being used. Since the objective of LFC controller design in an interconnected power system are damping of the frequency and tie-line power deviations in such a way to minimize transient oscillation under the different load conditions. Thus, frequency and tie-line power deviations are chosen as the neural network controller inputs. Moreover, in order to evaluate the control signal (u), the NANN controller is using a piece of information, which is not used in the conventional and modern controller (an estimate of the load perturbation DP^ Di ). In general, the load perturbation of the large system is not directly measurable. Therefore, it must be estimated by a linear estimator or by a nonlinear

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H. Shayeghi, H.A. Shayanfar / Electrical Power and Energy Systems 28 (2006) 503–511

ΔPD1

ΔF1

Area 1

u1 Dem Dem De

NANN

ΔP12

Mux

+ -

Σ µ -Based Robust Controller ΔPD2

second hidden layer and two outputs. The activation function of the networks neurons is hyperbolic tangent. The proposed network has been trained by using back-propagation algorithm [20]. The root mean square (RMS) error criterion is being used to evaluate the learning performance. Learning algorithms cause the adjustment of the weights so that the controlled system gives the desired response. 5. Simulation results

Area 2

ΔF2

u2

Fig. 5. Block diagram of the NANN controller design.

neural network estimator, if the nonlinearities in the system justify it. Such an estimator takes as inputs a series of k samples of the frequency fluctuations at the output of the generator [DF(n)DF(n  1)    DF(n  k + 1)]T and estimates the instantaneous value of the load perturbation based on this input vector. The implementation of such an estimator is beyond the scope of this paper. Here, we assume that the load estimate DP^ Di is available, i.e. DP^ DðnÞ ¼ DPDðnÞ. 4.3. Neural network architecture and training The NANN controller architecture employed here is a MLP neural network, which is shown in Fig. 6. The frequency deviations, tie-line power deviation and load perturbation of each area are chosen as the neural network controller inputs. The outputs of the neural network are the control signals, which are applied to the governors in each area. The data required for the NANN controller training is obtained from the designing and applying the l-based robust controller to power system in different operating points with various load disturbances. After a series of trial and error and modifications, the ANN architecture shown in Fig. 6 provides the best performance. It is a four-layer perceptron with five inputs, 15 neurons in the first hidden layer, seven neurons in the

ΔPD1

u1

ΔF1 ΔP12

The test system for LFC as shown in Fig. 1 consists of two areas control, and its parameters are given in Appendix A. The considered system is controlled by using (1) conventional integral controller; (2) l-based robust controller designed according to the procedure described in Section 3; and (3) the nonlinear ANN controller designed based on the procedure presented in Section 4. These results show that the structure of the ANN controller is simpler than the l-based robust controller whereas the order of the l-based controller is very large that the use of this control strategy is difficult in practical implementations. One of the importance constraints in LFC of the power system is generation rate constraints (GRC), i.e. the practical limit on the rate of change in the generating power. The results in Refs. [21,22] indicated that GRC would influence the dynamic responses of the system significantly and lead to larger overshoot and longer settling time. Furthermore, since the system parameters are unknown, the overall system may become unstable in the presence of a load disturbance. In order to take effect of the GRC into account, in the simulation study, the linear model of a turbine DPR/ DXG in Fig. 1 is replaced by a nonlinear model of Fig. 7. The GRC 0.2 p.u. per minute (d = 0.005) is considered [21]. Also, two limiters, bounded by ±0.005, are used within the above controller to prevent the excessive control action. They limit the following control signals: ju_ i ðtÞj 6 0:005;

jDP_ Ci ðtÞj 6 0:005.

In this section, the performance of ANN controller is compared with the conventional and l-based controllers under four cases of operating conditions. Case 1: We will test the system performance with nominal parameters. We choose the nominal parameters as given in Appendix A and apply load changes of DPd1(t) = 0.02 p.u. MW to one area. The responses of DF1(t) and DPtie(t) are shown in Fig. 8. From these results, it can be seen that the frequency and tie-line power deviations effectively damped to zero with NANN controller. Case 2: We choose the lower bound parameters as given in Table 1 for two areas and apply load changes of

u2

ΔF2 ΔPD2

XGV Input Layer

Hidden Layer

1 TTi

δ

Δ PR −δ

1 s

Δ PR

Output Layer

Fig. 6. The artificial neural network architecture.

Fig. 7. A nonlinear turbine model with GRC.

H. Shayeghi, H.A. Shayanfar / Electrical Power and Energy Systems 28 (2006) 503–511 ΔF1

509

ΔF1 0.4

0.1

0.3 0.2

0.05

0.1 0

0 -0.1

-0.05

-0.2 -0.3

-0.1

-0.4 0

10

20

30

(a)

40

50

60

70

0

80

Time (sec)

20

40

60

80

100

120

80

100

120

Time (sec)

(a)

ΔP12

ΔP12 0.004

0.005

0.002

0

0 -0.005

-0.002 -0.004

-0.01

-0.006 -0.015

-0.008 -0.10 0

10

20

30

(b)

40

50

60

70

80

Time (sec)

20

0

Fig. 8. The performance of the controllers for Case 1: Solid (ANN), Dashed dotted (l-based) and Dotted (PI). The responses of (a) Frequency deviations and (b) Tie-line power deviations.

DPd1(t) = 0.02 and DPd2(t) = 0.005 p.u. MW to one and two areas. The responses of DF1(t) is depicted in Fig. 9. The simulation results indicated that NANN controller has better performance than to other controllers. Case 3: We choose the upper bound parameters for two areas as given in Table 1 and apply load changes of DPd1(t) = 0.02 and DPd2(t) = 0.008 p.u. MW to one and two areas. The responses of DF1(t) and DPtie(t) are depicted in Fig. 10(a) and (b). From these results, it can be seen that NANN controller achieves good dynamic responses.

40

60

Time (sec)

(b)

Fig. 10. The performance of the controllers for the Case 3: Solid (ANN), Dashed dotted (l-based) and Dotted (PI). The responses of (a) Frequency deviations and (b) Tie-line power deviations.

Case 4: We choose the nominal values for all parameters except the plant gain KPi = 180 for two areas and apply load changes of DPd1(t) = 0.015 and DPd2(t) = 0.005 p.u. MW to one and two areas. The response of DF1(t) is depicted in Fig. 11. The simulation results indicated that NANN controller is superior to other controllers. Remark 5.1. The worst case, as seen from all the simulation results, occurs when two areas are using upper bound parameters and having simultaneous load disturbances.

ΔF1

ΔF1 0.15

0.1

0.1 0.05 0.05 0

0 -0.05

-0.05

-0.1 -0.1 -0.15 0

10

20

30

40

50

60

Time (sec)

Fig. 9. The performance of the controllers for the Case 2: Solid (ANN), Dashed dotted (l-based) and Dotted (PI).

0

10

20

30

40

50

60

70

80

Time (sec)

Fig. 11. The performance of the controllers for the Case 4: Solid (ANN), Dashed dotted (l-based) and Dotted (PI).

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Remark 5.2. From the simulation results in Fig. 11, we can see that the responses of overall system are more sensitive to the plant gain KPi than the other parameters. Remark 5.3. We have considered different cases for LFC control of a two-area power system. The simulation results indicated that the proposed ANN controller can guaranty the stability of the overall system and achieves good performance even in the presence of GRC. 6. Conclusion In this paper, a new nonlinear ANN controller based on l-synthesis technique is proposed for the load frequency control of large-scale power system. This control strategy was chosen because of complexity of the actual uncertainty, multivariable operating conditions and large model order of the power system. The motivation of using the lbased robust controller for training of the neural network controller is to take the large parametric uncertainties and modeling error into account. To improve the controller performance, the proposed control strategy makes use of the load perturbation as input control signal, which is not used in the conventional PI and l-based robust controller and leads to a flexible controller with relatively simple structure that is easy to implement and can be useful in the real world complex power system. A two-area power system is used as a test system to demonstrate the effectiveness of the proposed method under various operating conditions for a wide range of parametric uncertainties and area load disturbances. The simulation results show that the proposed ANN controller has better control performance compared to the conventional PI and l-based robust controllers even in the presence of GRC. In addition, it is effective and can ensure the stability of the overall system for all admissible uncertainties and load changes. Appendix A A.1. Nomenclature DFi(t) incremental frequency deviation in Hz DPTi(t) incremental change in the ith subsystem’s output in p.u. MW DPRi(t) incremental change in the output energy of the ith reheat type turbine in p.u. MW DXGi(t) incremental change in the ith governor valve position in p.u. MW DPCi(t) incremental change in the integral controller DPTie(t) incremental change in the tie-line power DPdi(t) load disturbance for the ith area in p.u. MW ui(t) output of the load frequency controller for the ith area TGi ith governor time constant in s TTi ith turbine time constant in s

TRi TPi TTi KPi KIi Bi Ki Ri Tij a12

ith reheat time constant in s ith subsystem-model time constant in s ith reheat time constant in s ith subsystem gain ith subsystem’s integral control gain ith subsystem’s frequency-biasing factor the ratio between output energy of the ith stage of turbine to total output energy speed regulation for ith subsystem due to the ith governor action in Hz/p.u. MW synchronizing coefficient of the tie-line between ith and jth areas the ratio between the base values of two areas

A.2. Nominal parameters of two-area system T G1 ¼ T G2 ¼ 0:1 s; T T1 ¼ T T2 ¼ 0:3 s; T R1 ¼ T R2 ¼ 10 s; T P1 ¼ T P2 ¼ 20 s; K P1 ¼ K P2 ¼ 120 Hz/p.u. MW, a12 ¼ 1; R1 ¼ R2 ¼ 2:4 Hz/p.u. MW, B1 ¼ B2 ¼ 0:425 p.u. MW/Hz, K 1 ¼ K 2 ¼ 0:5; K I1 ¼ K I2 ¼ 0:05; T 12 ¼ 0:0707.

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