Robust decentralized neural networks based LFC in a deregulated power system

Electric Power Systems Research 77 (2007) 241–251

Robust decentralized neural networks based LFC in a deregulated power system H. Shayeghi a,∗ , H.A. Shayanfar b , O.P. Malik c a

Technical Engineering Department, The University of Mohaghegh Ardebili, Ardebil, Iran Electrical Engineering Department, Iran University of Science and Technology, Tehran, Iran c Electrical and Computer Engineering Department, The University of Calgary, Calgary, Canada b

Received 10 July 2005; received in revised form 25 December 2005; accepted 2 March 2006 Available online 19 April 2006

Abstract In this paper, a decentralized radial basis function neural network (RBFNN) based controller for load frequency control (LFC) in a deregulated power system is presented using the generalized model for LFC scheme according to the possible contracts. To achieve decentralization, the connections between each control area with the rest of system and effects of possible contracted scenarios are treated as a set of input disturbance signals. The idea of mixed H2 /H∞ control technique is used for the training of the proposed controller. The motivation for using this control strategy for training the RBFNN based controller is to take large modeling uncertainties into account, cover physical constraints on control action and minimize the effects of area load disturbances. This newly developed design strategy combines the advantage of the neural networks and mixed H2 /H∞ control techniques to provide robust performance and leads to a flexible controller with simple structure that is easy to implement. The effectiveness of the proposed method is demonstrated on a three-area restructured power system. The results of the proposed controllers are compared with the mixed H2 /H∞ controllers for three scenarios of the possible contracts under large load demands and disturbances. The resulting controller is shown to minimize the effects of area load disturbances and maintain robust performance in the presence of plant parameter changes and system nonlinearities. © 2006 Elsevier B.V. All rights reserved. Keywords: Load frequency control; RBF neural network; Decentralized control; Deregulated power system; Mixed H2 /H∞ control

1. Introduction Global analysis of the power system markets shows that the frequency control is one of the most profitable ancillary services at these systems. This service is related to the short-term balance of energy and frequency of the power systems. The most common methods used to accomplish frequency control are generator governor response (primary frequency regulation) and load frequency control (LFC). The goal of LFC is to reestablish primary frequency regulation capacity, return the frequency to its nominal value and minimize unscheduled tieline power flows between neighboring control areas. From the mechanisms used to manage the provision this service in ancil-

∗

Corresponding author at: Narmak, Tehran 16844, Iran. Tel.: +98 21 780822; fax: +98 21 7454055. E-mail address: [email protected] (H. Shayeghi). 0378-7796/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2006.03.002

lary markets, the bilateral contracts or competitive offers stand out [1]. During the past decade several proposed LFC scenarios have attempted to adapt traditional LFC schemes to the change of environment in the power systems under deregulation [2–4]. In a deregulated power system, each control area contains different kinds of uncertainties and various disturbances due to increased complexity, system modeling errors and changing power system structure. As a result, a fixed controller based on classical theory is not very suitable for the LFC problem. It is desirable that a flexible controller be developed. Efforts have been made to design automatic generation controllers with better performance to cope with parameter changes, using various decentralized robust, optimal control and neural network methods during the past two decades [5–9]. These approaches are based on state-feedback and the state-vector for the entire system should be made available for the generating of local feedback control signals. This requirement may be met if the system

242

H. Shayeghi et al. / Electric Power Systems Research 77 (2007) 241–251

state-vector is observable from area measurements. However, even if the observability condition is satisfied, the resulted controllers with appropriately designed observers are normally quite complicated and therefore these approaches are not suitable for a large-scale power system where the total number of the state variables is large. Moreover, these methods applied to the traditional power systems and the effects of possible load following contracts were not considered on system dynamics. Also, some of them [8,9] have a centralized scheme, which is not feasible for a large-scale power system because of computational and economical difficulties in implementing this scheme. Recently, several optimal and robust control strategies have been developed for LFC synthesis according to change of environment in power system operation under deregulation [10–13]. These methods show good dynamic response, but robustness in the presence of model dynamic uncertainties and system nonlinearities is not considered. Also, some complex state feedback or high order dynamic controllers have been suggested, but they are not practical for industry practices. In this paper, a new decentralized radial basis function neural network (RBFNN) controller for LFC in a deregulated power system is proposed using the generalized model for LFC scheme according to the possible contracted scenarios. To achieve decentralization, the effects of the possible contracted scenarios and connections between each area with the rest of system are treated as a set of new input disturbance signals in each control area. LFC goals, i.e. frequency regulation and tracking the load demand, maintaining the tie-line power interchanges to specified values in the presence of model uncertainties, system nonlinearities and area load disturbances, determine the LFC synthesis as a multi-objective optimization problem. For this reason, the idea of mixed H2 /H∞ control technique is being used for training the proposed RBFNN based controller. The motivation for using this control technique for training the RBFNN is to take large modeling uncertainties into account, cover physical constraints on control and minimize the effects of area load disturbances. To achieve the desired level of robust performance, the training data is obtained by designing mixed H2 /H∞ controllers for various operating conditions and applying them to the restructured power system in the presence of plant parameter changes and generation rate constraints (GRC). The proposed controller is then reconstructed using the learning capability of neural networks. The salient feature of the RBFNN based controller is that it provides a non-model based control system and does not require an accurate model of the plant. Moreover, the proposed control strategy has simple structure and is a decentralized LFC scheme that requires only the area control error (ACE). Thus, its implementation is fairly easy and can be used in the real world complex power systems. The proposed control strategy is tested on a three-area power system for three scenarios in the presence of model uncertainties and GRC under various load changes. The results show that the proposed method provides robust performance for a wide range of operating conditions and is superior to the mixed H2 /H∞ controller.

2. Generalized LFC scheme model In the deregulated power systems, the vertically integrated utility no longer exists. However, the common LFC objectives, i.e. restoring the frequency and the net interchanges to their desired values for each control area, still remain. The deregulated power system consists of GENCOs, TRANSCOs and DISCOs with an open access policy. In the new structure, GENCOs may or may not participate in the LFC task and DISCOs have the liberty to contract with any available GENCOs in their own or other areas. Thus, various combinations of possible contracted scenarios between DISCOs and GENCOs are possible. All the transactions have to be cleared by the independent system operator (ISO) or other responsible organizations. In this new environment, it is desirable that a new model for LFC scheme be developed to account for the effects of possible load following contracts on system dynamics. Based on the idea presented in [14], the concept of an ‘Augmented Generation Participation Matrix’ (AGPM) to express the possible contracts following is presented here. The AGPM shows the participation factor of a GENCO in the load following contract with a DISCO. The rows and columns of AGPM matrix equal the total number of GENCOs and DISCOs in the overall power system, respectively. Consider the number of GENCOs and DISCOs in area i be ni and mi in a large-scale power system with N control areas. The structure of AGPM is given by: ⎡ ⎤ AGPM11 . . . AGPM1N ⎢ ⎥ .. .. .. ⎥ AGPM = ⎢ (1) . . . ⎣ ⎦ AGPMN1 . . . AGPMNN where,

⎡

gpf(si +1)(zj +1) ⎢ .. AGPMij = ⎢ . ⎣ gpf(si +ni )(zj +1)

⎤ gpf(si +1)(zj +mj ) ⎥ .. ⎥ . ⎦ . . . gpf(si +ni )(zj +mj ) ... .. .

For i, j = 1, · · · , N and si =

i−1  k=1

ni , z j =

j−1 

mj ,

and

s1 = z1 = 0

k=1

In the above, gpfij refers to ‘generation participation factor’ and shows the participation factor of GENCO i in total load following requirement of DISCO j based on the contract. Sum of all entries in each column of AGPM is unity. The diagonal sub-matrices of AGPM correspond to local demands and offdiagonal sub-matrices correspond to demands of DISCOs in one area on GENCOs in another area. Block diagram of the generalized LFC scheme for control area i in a restructured system is shown in Fig. 1. Dashed-dot lines show interfaces between areas and the demand signals based on the possible contracts. These new information signals are absent in the traditional LFC scheme. As there are many GENCOs in each area, ACE signal has to be distributed among them ni due to their ACE participation factor in the LFC task and j=1 apfji = 1.

H. Shayeghi et al. / Electric Power Systems Research 77 (2007) 241–251

243

Fig. 1. Generalized LFC scheme for area i in the deregulated system.

It can be seen from Fig. 1 that four input disturbance channels, di , ζ i , ηi and ρi are considered for the decentralized LFC design. They are defined as bellow: mi  di = PLoc,i + Pdi , PLoc,i = PLj−i ,

for area i can be obtained as: x˙ i = Ai xi + Biu ui + Biw wi  yi = Ci xi + Diw wi 

(9)

where,

j=1

Pdi =

mi 

xiT = [xai x1i . . . xki . . . xni i ], (2)

PULj−i

j=1

ηi =

N 

N 

k = 1, . . . , ni ,

wi = [ di

yi = ACEi ,

ηi

ρi = [ρ1i . . . ρki . . . ρni i ], (3)

Tij fj

j=1 j=i

ζi =

xki = [PTki PVki ],

xai = [fi Ptie,i ],

Ai =

(4)

Ptie,ik,schdueled

k=1 k=i

A11i A21i



A12i = 

A12i



A22i

KPi /TPi 0

−1/TPi

⎢ , A11i = ⎢ ⎣ 0 0

ρi ] ,

ξi ⎡

ui = Pci ,

T

N 

−KPi /TPi 0

Tji

j=1 and j=i



... 

KPi /TPi 0

0 0

⎤ ⎥ ⎥ ⎦

 , 

ni blocks

Ptie,ik,schdueled =

mk ni  

T A21i = [DP1i . . . DPkiT . . . DPnTi i ],

apf(si +j)(zk +t) PLt−k

j=1 t=1

−

nk  mi 

apf(sk +t)(zi +j) PLj−i

(5)

t=1 j=1

Ptie,i−error = Ptie,i−actual − ζi

(6)

ρi = [ρ1i . . . ρki . . . ρni i ] ,

(7)

T

Pm,k−i =

mj N   j=1

ρki = Pm,k−i 

gpf(si +k)(zj +t) PLt−j

i=1

+ apfki

mi 

PULj−i

(8)

j=1

Pm,k-i is the desired total power generation of a GENCO k in area i and must track the demand of the DISCOs in contract with it in the steady state. Considering Fig. 1, the state space model

A22i = diag(TG1i , . . . , TGki , . . . , TGni i ),

 0 0 DPki = , −1/(Rki THki ) 0 

1/TTki −1/TTki TGki = , 0 −1/THki T T T T Biu = [02×1 B1iu . . . Bkiu . . . BnTi iu ], Bkiu = [0 apfki /THki ]T , T T T T Biw = [Baiw B1iw . . . Bkiw . . . BnTi iw ], 

−KPi /TPi 0 01×(ni +1) Baiw = , 0 −1 01×(ni +1) 

0...0...0 01×3 Bkiw = , 01×3 b1i . . . bki . . . bni i  −1/THki j = k bji = , Ci = [ Cai 0 j = i

Cai = [ Bi

1 ], Diw = [ 01×2

−1

01×ni ]

01×2ni ],

244

H. Shayeghi et al. / Electric Power Systems Research 77 (2007) 241–251

Fig. 2. The Proposed mixed H2 /H∞ synthesis framework. Fig. 3. Formulation of mixed H2 /H∞ based control design problem.

3. LFC problem formulation via H2 /H∞ mixed control In the real world LFC problem, multi objectives such as stability, disturbance attenuation and reference tracking under model uncertainties and practical constraints are followed simultaneously. Meeting all LFC design objectives by a single control approach, particularly with increasing complexity and change of the power system structure, is difficult. It is shown that the mixed H2 /H∞ control synthesis gives a powerful multiobjectives design addressed by the linear matrix inequalities (LMI) techniques [15]. The regulation against random disturbances is more naturally expressed by the H2 synthesis, while the H∞ approach is more useful for closed-loop stability and formulation of physical control constraints. Thus, to take advantage of two methods the mixed H2 /H∞ control technique is being used to solve the LFC problem. The main synthesis framework for the formulation of the LFC problem as a mixed H2 /H∞ control design in a given control area (Fig. 1) is shown in Fig. 2. In the restructured power systems, each control area contains different kinds of uncertainties because of plant parameter variations, load changes and system modeling errors due to some approximations in model linearization and un-modeled dynamics. Usually, the uncertainties in a power system can be modeled as multiplicative and/or additive uncertainties [16]. In Fig. 2 the ui block models the unstructured uncertainties as a multiplicative type and Wui is the associated weighting function. The output channels z∞i,1 and z∞i,2 are associated with the H∞ performance. The first channel is used to meet robustness against uncertainties and reduce their impact on the closed-loop system performance. In the second channel (z∞i,2 ), WCi , sets a limit on the allowed control signal to penalize fast changes and large overshoot in the control signal with regard to practical constraints. The output channel z2i is associated with the H2 performance and Wpi sets the performance goal i.e. zero tracking error and minimizing the effects of disturbances on the area control error (ACEi ). Fig. 2 can be drawn as a mixed H2 /H∞ general framework synthesis as shown in Fig. 3, where Poi (s) and Ki (s) denote the nominal area model as given by (9) and controller, respectively. Also, yi is the measured output (performed by ACEi ), ui is the control output and wi includes the perturbed, disturbance and reference signals in the control area. In Fig. 3, Pi (s) is the augmented plant (AP) that includes nominal model of control area i and associated weighting functions. The state-space model of the AP can be obtained

as: x˙ APi = AAPi xAPi + B1i wi + B2i ui , z∞i = C∞i xAPi + D∞1i wi + D∞2i ui , z2i = C2i xAPi + D21i wi + D22i ui ,

yi = Cyi xAPi + Dy1i wi (10)

where, wTi = [ vi

di

ηi

ζi

ρi

yref ] , zT∞i = [ z∞1i

z∞2i ]

Based on Fig. 3 the synthesis problem can be expressed by the following optimization problem: design a controller Ki (s) that minimizes a trade off criterion of the form: γ1 ||T∞i (s)||2∞ + γ2 ||T2i (s)||22

(γ1 and γ2 ≥ 0)

(11)

Where T∞i (s) and T2i (s) are the transfer functions from wi to z∞i and z2i . An efficient algorithm for solving this problem is available in the LMI control toolbox of MATLAB [17]. 4. RBFNN based controller design In parallel with the development of the technology and complexity of the systems, such as power systems, their modeling and control by using the conventional mathematical-modelbased analytical techniques have become more difficult or in rare cases are impossible to apply. On the other hand, human abilities for control of complex plants have encouraged researchers to pattern on the human neural network systems. Artificial neural networks (ANN) with their massive parallelism and ability to learn any type of nonlinearity are now being used in different branches of science and industry, especially in designing practical controllers [18,19]. An ANN-based controller with adjustable weights can be designed based on adaptive control techniques. The main advantages of such a controller are: parallel architecture, mapping of nonlinearities, training for various operating conditions and adapting for any desired situation, providing a model free description of control system and not requiring an accurate model of the plant. Thus, it is very suitable for the control of nonlinear plants whose models are unknown or have uncertainties. A power system has several features such as: very large (unknown) model order, uncertain connection between subsystems and elaborate organizational structure that precludes direct application of robust methodologies. In order to overcome these drawbacks, a RBFNN controller based on the mixed

H. Shayeghi et al. / Electric Power Systems Research 77 (2007) 241–251

245

Where M is the number of neurons in the hidden layer, and can be less than or equal to the number of training samples. The output of this network is an approximation of the desired output and is obtained as follows: ∗

F (x) = b +

M 

wi ϕi (x)

(13)

i=1

Fig. 4. RBFNN control design problem based on the mixed H2 /H∞ technique.

H2 /H∞ control technique is proposed. This hybrid structure is being used to take advantage of both the neural networks and mixed H2 /H∞ control techniques. Due to uncertain connection between sub-systems (control areas), the interconnection among some areas is considered as the channels of disturbances in the proposed LFC scheme dynamical model. The main framework of the RBFNN control design problem based on the mixed H2 /H∞ technique for a given control area is shown in Fig. 4. For training the proposed controller, the LFC problem is formulated first as a decentralized multi-objective optimization control problem via a mixed H2 /H∞ technique and designed for different operating conditions. Then, the training and test data are obtained by applying these controllers to a power system in the presence of GRC in different operating conditions under various load changes. Moreover, in order to account the effects of organizational change of power system under deregulation, different ACE participation factors (apfij ) are being used for GENCOs of each control area under various possible contracts. The results of previous work [20] in the traditional power systems for LFC design show that the performance of the RBFNN controller is better than a MLP neural network controller. Thus, in this study the RBF neural network is being used for the LFC problem in a restructured power system. The architecture of the RBF neural network employed here is shown in Fig. 5. This network consists of three layers; the input layer with two source nodes (ACEi and the rate of change of ACEi ), the hidden layer, which has enough number of neurons, and the output layer that defines the response of the network with regard to the applied inputs and applies to the governor load set point in each area (Fig. 1). In this network, the activation functions between input layer and hidden layer are nonlinear and between hidden layer and output layer are linear. Also, the radial basis functions are Green functions of the form: Φi (x) = G(||x − ti ||) = exp(−||x − ti ||2 ),

The weights in the output layer wi and the center of the Green functions ti is obtained during the training of the network [18]. In summary, the design procedure for the RBFNN based controller has the following steps: Step 1: Formulation of LFC problem as a decentralized mixed H2 /H∞ optimization control problem. Step 2: Obtaining mixed H2 /H∞ controllers in different operating conditions using the LMI approach. Step 3: Applying mixed H2 /H∞ controllers to a power system for obtaining training and test data in the presence of GRC in different operating points under the possible contracted scenarios and various load changes. Step 4: Training the neural network according to Fig. 4 and testing it. The design strategy includes enough flexibility to set the desired robust performance and gives a flexible controller with simple structure. Due to its practical merit, the proposed method is a decentralized LFC scheme and requires only the area control error. Thus, its implementation is fairly easy and can be used in the real world power systems. 5. Case study A three control area power system, shown in Fig. 6, is considered as a test system to illustrate the effectiveness of the proposed control strategy. It is assumed that each control area includes two GENCOs and DISCOs. The power system parameters are given in Tables 1 and 2.

i = 1, . . . , M (12)

Fig. 5. Architecture of the RBF network.

Fig. 6. Three area power system.

246

H. Shayeghi et al. / Electric Power Systems Research 77 (2007) 241–251

Table 1 GENCOs parameter GENCOs (k in area i)

MVAbase (1000 MW) parameter

1–1

2–1

1–2

2–2

1–3

2–3

Rate (MW) TT (s) TH (s) R (Hz/pu) apf

1000 0.32 0.06 2.4 0.5

800 0.30 0.08 2.5 0.5

1100 0.30 0.06 2.5 0.5

900 0.32 0.07 2.7 0.5

1000 0.31 0.08 2.8 0.6

1020 0.34 0.06 2.4 0.4

Table 2 Control area parameters Parameter

Area 1

Area 2

Area 3

KP (Hz/pu) TP (s) B (pu/Hz) Tij (pu/Hz)

120 20 0.4250 T12 = 0.245 , T13 = 0.212

120 25 0.3966

125 20 0.3522

5.1. Uncertainty and performance weights selection Simulation results and eigenvalue analysis show that the open loop system performance is affected more significantly by changes in the values of Kpi , Tpi , Bi and Tij than changes of other parameters. Thus, it is assumed that these parameters have uncertain values in each area and their variation range is considered as ±50%. These uncertainties are modeled as an unstructured multiplicative uncertainty (Wui ) block (Fig. 2). Let Pi (s) denote the transfer function from the control input ui to control output yi at operating points other than the nominal point. This transfer function can be represented as: |ui (s)Wi (s)| = |(Pˆ i (s) − Poi (s))/Poi (s)|, ||Δui (s)||∞ = sup|ui (s)| ≤ 1

Poi (s) = 0, (14)

Where, ui (s) shows the uncertainty block corresponding to the uncertain parameters and Poi (s) is the nominal transfer function model. Thus, Wui (s) is such that its magnitude Bode plot covers the Bode plot of all possible plants. Using (14) some sample uncertainties corresponding to different values of Kpi , Tpi , Bi and Tij , in the range ±50% from nominal values are shown in Fig. 7 for one area. Based on this figure, the following multiplicative uncertainty weight was chosen for the control design: Wu1 =

5.35s2 + 36.32s + 2.87 s2 + 0.57s + 45.40

3.28s2 + 18.79s + 24.76 , = s2 + 1.01s + 36.09

Wu3 =

4.37s2 + 21.91s + 28.69 s2 + 0.58s + 43.45

5.2. Performance weights selection The selection of performance weights WCi and WPi entails a trade off among different performance requirements, particularly good area control error minimization versus peak control action. The weight on the control input, WCi , must be chosen close to a differentiator to penalize fast change and large overshoot in the control input due to corresponding practical constraints. The weight on the output, WPi , must be chosen close to an integrator at low frequency in order to get disturbance rejection and zero steady state error. Based on the above discussion, a suitable set of performance weighting functions for one control area is chosen as: WC1 =

0.3s + 1 , s + 10

WP1 =

0.03s + 0.75 250s + 1

(17)

Using the same procedure and setting similar objectives as discussed above the set of suitable weighting function for the other control area synthesizes are given in Table 3. 5.3. Mixed H2 /H∞ based control design Based on the synthesis methodologies as given in Secion 4, a set of three decentralized robust controllers is designed using the hinfmix function in the LMI control toolbox. This function gives an optimal controller through the mentioned optimization problem, (11), with γ 1 and γ 2 fixed at unity. The resulting controllers are dynamic type, whose orders are the same as the size of the augmented plant model (here 10). It should be noted that, due to the complexity of actual uncertainties and

(15)

Using the same method the uncertainty weighting functions for areas 2 and 3 are calculated as follow: Wu2

Fig. 7. Uncertainty plot due to change of Kp1 , Tp1 , B1 and T1j (dashed) and Wui (solid).

(16)

Table 3 Performance weighting functions Weight

Area 2

Area 3

WPi

0.02s + 0.65 100s + 1

0.03s + 0.7 200s + 1

WCi

0.3s + 1 s + 10

0.2s + 1.2 s + 15

H. Shayeghi et al. / Electric Power Systems Research 77 (2007) 241–251

247

large model order of the real world power systems this method yield complex controllers whose size will be very large in general. 5.4. RBFNN based control design A decentralized RBFNN controller is designed for each of the three areas according to the synthesis procedure described in Section 5. This control strategy, that has an adaptive control structure, is simple and suitable for the LFC applications. The controller has enough neurons in the hidden layer. Training and test data was obtained from the decentralized H2 /H∞ controllers designed for each of the three areas and applying them to the power system with setting different apfij for GENCOs for three possible contracts under large load demand and GRC according to the following sets of parameters: • Nominal parameters. • Uncertain parameters of the system are reduced by 50% from nominal value. • Uncertain parameters are increased by 50% from nominal value. 6. Simulation results In the simulation study, the linear model of a turbine PVKi /PTKi in Fig. 1 is replaced by a non-linear model of Fig. 8 (with ±0.1 limit). This is to take GRC into account, i.e. the practical limit on the rate of change in the generating power of each GENCO. Simulations are carried out for different scenarios of the possible contracts under large load demands and disturbances. The performance of the proposed RBFNN based controllers is compared with the mixed H2 /H∞ controllers in the presence of plant parameters changes and GRC.

Fig. 8. Nonlinear turbine model with GRC.

6.1. Scenario 1: Poolco based transactions In this scenario GENCOs participate only in load following control of their areas. It is assumed that a large step load is demanded by DISCOs of areas 1 and 2 as follow: PL1 − 1 = 100,

PL2 − 1 = 80,

PL1−2 = 100,

PL2−2 = 100 MW A case of Poolco based contracts between DISCOs and available GENCOs is simulated based on the following AGPM. It is noted that the GENCOs of area 3 do not participate in the LFC task. ⎡ ⎤ 0.6 0.5 0 0 0 0 ⎢ 0.4 0.5 0 0 0 0⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 0.5 0.5 0 0 ⎥ ⎥ AGPM = ⎢ ⎢ 0 0 0.5 0.5 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ 0 0 0 0 0 0⎦ 0

0

0

0

0

0

Power system responses with 50% increase in uncertain parameters KPi , TPi , Bi and Tij are depicted in Fig. 9. Using the proposed method, the frequency deviation of all areas and the tie-line power flows are quickly driven back to zero and have small overshoots. Since there are no contracts between areas, the scheduled steady state power flow, Eq. (5), over the tie-lines is zero. The

Fig. 9. Power system response to scenario 1: Solid (RBFNN), Dashed (H2 /H∞ ). (a) Area 1 (b) Area 2 (c) Tie lines powers.

248

H. Shayeghi et al. / Electric Power Systems Research 77 (2007) 241–251

Fig. 10. Power system response to scenario 2: Solid (RBFNN), Dashed (H2 /H∞ ). (a) Frequency deviation (b) Tie line powers deviations.

actual generated powers of GENCOs, according to Eq. (8), properly converge to the desired values in the steady state. i.e.: Pm, 1 − 1,

Pm,2−1 = 0.08,

Pm,1−2 = Pm,2−2 = 0.1 pu MW

6.2. Scenario 2: combination of Poolco and bilateral based transactions In this case, DISCOs have the freedom to have a contract with any GENCO in their and other areas. Consider that all the DISCOs contract with the available GENCOs for power as the

following AGPM: ⎡ 0.25 0 ⎢ 0.5 0.25 ⎢ ⎢ ⎢ 0 0.5 AGPM = ⎢ ⎢ 0.25 0 ⎢ ⎢ ⎣ 0 0.25 0 0

0.25 0 0 0.25 0.25 0 0.5 0.75 0 0

0 0

⎤ 0.5 0 0 0⎥ ⎥ ⎥ 0 0⎥ ⎥ 0 0⎥ ⎥ ⎥ 0.5 0 ⎦ 0

1

All GENCOs participate in the LFC task. The GENCO 1 in area 2 and GENCO 2 in area 3 only participate for performing the LFC task in their areas, while other GENCO track the load demand in their areas and/or others. It is assumed that a large

Fig. 11. GENCOs powers changes: Solid (RBFNN), Dashed (H2 /H∞ ). (a) Area 1 (b) Area 2 (c) Area 3.

H. Shayeghi et al. / Electric Power Systems Research 77 (2007) 241–251

249

Fig. 12. Power system response to scenario 3: Solid (RBFNN), Dashed (H2 /H∞ ). (a) Frequency deviation (b) Tie line powers deviations.

step load demand is requested by all DISCOs as follow:

6.3. Scenario 3: contract violation

PLl − 1 = 80,

PL2 − l = 70, PLl − 2 = 60,

PL2 − 2 = 80,

PLl − 3 = 50 and PL2 − 3=100 MW

In this case, DISCOs may violate a contract by demanding more power than that specified in the contract. This excess power is reflected as a local load of the area (un-contracted demand). Consider scenario 2 again. It is assumed that in addition to specified contracted load demands and 50% increase in uncertain parameters, the DISCO 1 in area 2 and DISCO 2 in area 3 demand 0.1 and 0.05 pu MW as a large un-contracted load, respectively. Using the Eq. (2), the total local load in all areas is obtained as:

Also the ACE participation factor of each GENCO in the LFC is defined as bellow: apf11 = 0.75, apf22 = 0.5,

apf21 = 0.25,

apf12 = 0.5,

apf13 = 0.6 and apf23 = 0.4

Power system responses with 50% increase in uncertain parameters KPi , TPi , Bi and Tij are depicted in Figs. 10 and 11. Using the proposed method, the frequency deviation of all areas is quickly driven back to zero and the tie-line power flows properly converges to the specified value of Eq. (5) in the steady state. i.e.: Ptie,21,sch = 0.02,

Ptie,31,sch = −0.0075 pu MW

As shown in Fig. 11, the actual generated powers of GENCOs properly reach the desired values in the steady state as given by Eq. (13). i.e.: Pm,1−1 = 0.06,

Pm,2−1 = 0.0775,

Pm,2−2 = 0.11,

Pm,1−3 = 0.0425,

Pm,2−3 = 0.1 pu MW

Pm,1−2 = 0.05,

PLoc,1 = 0.15,

PLoc,2 = 0.24,

PLoc,3 = 0.2 pu MW

The purpose of this scenario is to test the effectiveness of the proposed controller against uncertainties and large load disturbances in the presence of GRC. The power system responses for this scenario are shown in Figs. 12 and 13. As AGPM is the same as in scenario 2 and the un-contracted load in each area is taken up by the GENCOs in the same area, the tie-line powers are the same as in scenario 2 in the steady state (Fig. 12b). The un-contracted load of DISCO1–2 and DISCO2–3 is taken up by GENCOs of their areas according to ACE participation factors in the steady state. As shown in Fig. 13, the actual generated powers of GENCOs properly reach the desired values using the proposed strategy. Using the Eq. (8) the actual generated power of GENCOs in area 2 and 3 in the steady state is given by: Pm,1−2 = 0.1,

Pm,2−2 = 0.16,

Pm,2−3 = 0.12 pu MW

Pm,1−3 = 0.0725,

250

H. Shayeghi et al. / Electric Power Systems Research 77 (2007) 241–251

Fig. 13. GENCOs power changes: Solid (RBFNN), Dashed (H2 /H∞ ). (a) Area 2 (b) Area 3. Table 4 Performance index of ITAE Method

Scenario 1 Case A

RBFNN H2 /H∞

175.7 344.8

Scenario 2 Case B

Case C

199.0 398.3

217.7 278.7

Case A 73.0 192.5

Scenario 3 Case B

Case C

104.2 242.7

88.6 121.6

Case A 231.3 579. 5

Case B

Case C

358.3 905.0

245.5 438.5

Table 5 Performance index of FD Method

Scenario 1 Case A

RBFNN H2 /H∞

578.2 989. 6

Scenario 2 Case B

Case C

Case A

503.2 898.7

776.5 1242.0

370.4 946.0

The simulation results show that the proposed RBFNN based controllers track the load changes and achieve good robust performance than the mixed H2 /H∞ controllers for a wide range of load disturbances and the possible contracted scenarios in the presence of plant parameters changes and system nonlinearities. To demonstrate robustness of the proposed control strategy, the performance indexes of integration time absolute error (ITAE) based on the ACE and figure of demerit (FD) based on the system performance characteristics (suitably weighted) are being used as:  20 ITAE = 100 t|ACE1 |dt 0

FD = (OS × 100)2 + (US × 40)2 + (Ts × 2)2 Overshoot (OS), undershoot (US) and settling time for 4% band of the total step load demand in area (1) of frequency deviation of area 1 are considered for the evaluation of FD. Numerical results of the performance robustness for the above three scenarios by the following sets of system parameters are listed in Tables 4 and 5.

Scenario 3 Case B

Case C

Case A

Case B

Case C

312.3 920.0

412.5 1065.1

403.2 1551.1

1048.3 2956.0

391.0 1429.7

Case A: Nominal values. Case B: Uncertain parameters of the system are reduced by 50% from nominal values. Case C: Uncertain parameters of the system are increased by 50% from nominal values. Examination of these Tables reveal that the performance of RBFNN based controllers is better than the mixed H2 /H∞ controllers. 7. Conclusions In this paper, a new decentralized RBFNN based controller for LFC is proposed using the generalized LFC scheme model in the deregulated environments. Since, each control area in the deregulated power systems contains different kinds of uncertainties and disturbances, the LFC problem first is formulated as a multi-objective optimization control problem via a mixed H2 /H∞ control technique and the proposed controller is then trained based on samples obtained from applying the mixed H2 /H∞ controllers to the power system in different operating

H. Shayeghi et al. / Electric Power Systems Research 77 (2007) 241–251

conditions under large load demands and contracts variations. This control strategy includes enough flexibility to set the desired level of robust performance and has been chosen because of large model order and complexity of the real world power systems. The effectiveness of the proposed method is tested on a threearea restructured power system for a wide range of load demands and disturbances under uncertainties and contract variations. The following conclusions can be drawn about the proposed method. 1. Due to non-model base, it can be used to control a wide range of complex and nonlinear systems. 2. It uses both the learning capability of RBFNN and advantage of mixed H2 /H∞ control technique for achieving the desired level of robust performance. 3. It is effective and ensures robust performance such as frequency regulation, tracking of load demand and disturbance attenuation for a wide range of operating conditions and area load disturbances. 4. It dose not require an accurate model of the LFC problem and has relatively simple structure. Thus its construction and implementation are fairly easy. 5. The system performance characteristics in terms of ‘ITAE’ and ‘figure of demerit’ indexes reveal that this control strategy is a promising control scheme for the LFC problem in the deregulated power systems. Appendix A. List of symbols F PTie PT PV PC ACE apf  KP TP TT TH R B Tij Pd PLj-i PULj-i Pm,j-i PLoc η ζ

area frequency net tie-line power flow turbine power governor valve position governor set point area control error ACE participation factor deviation from nominal value subsystem equivalent gain subsystem equivalent time constant turbine time constant governor time constant droop characteristic frequency bias tie line synchronizing coefficient between areas i and j area load disturbance contracted demand of Disco j in area i un-contracted demand of Disco j in area i power generation of GENCO j in area i total local demand area interface scheduled power tie line power flow deviation (Ptie,sch. )

251

References [1] R. Raineri, S. Rios, D. Schiele, Technical and economic aspects of ancillary services markets in the electric power industry: an international comparison, Energy Policy, in press. [2] R.D. Christie, A. Bose, Load frequency control issues in power system operations after deregulation, IEEE Trans. Power Syst. 11 (3) (1996) 1191–1200. [3] J. Kumar, N.G. Hoe, G. Sheble, LFC simulator for price-based operation part I: modeling, IEEE Trans. Power Syst. 12 (2) (1997) 527– 532. [4] V. Donde, A. Pai, I.A. Hiskens, Simulation and optimization in a LFC system after deregulation, IEEE Trans. Power Syst. 16 (3) (2001) 481– 489. [5] K.Y. Lim, Y. Wang, R. Zhou, Robust decentralized load frequency control of multi-area power systems, IEE Proc. Gener. Transm. Distrib. 143 (5) (1996) 377–386. [6] M.H. Kazemi, M. Karrari, M.B. Menhaj, Decentralized robust adaptiveoutput feedback controller for power system load frequency control, Electr. Eng. 84 (2) (2002) 75–83. [7] A. Feliachi, Optimal decentralized load frequency control, IEEE Trans. Power Syst. 2 (1987) 379–384. [8] C.M. Liaw, K.H. Chao, On the design of an optimal automatic generation controller for interconnected power systems, Int. J. Control 58 (1993) 113–127. [9] H.L. Zeynelgil, A. Demiroren, N.S. Sengor, The Application of ANN technique to automatic generation control for multi-area power system, Elect. Power Energy Syst. 24 (2002) 354–545. [10] A. Feliachi, On load frequency control in a deregulated environment, in: Proceedings of the IEEE International Conference on Control Applications, 15–18 September, 1996, pp. 437–441. [11] D. Rerkpreedapong, A. Hasanovic, A. Feliachi, Robust load frequency control using genetic algorithms and linear matrix inequalities, IEEE Trans. Power Syst. 18 (2) (2003) 855–861. [12] F. Liu, Y.H. Song, J. Ma, Q. Lu, Optimal load frequency control in restructured power systems, IEE Proc. Gener. Transm. Distrib. 150 (1) (2003) 87–95. [13] B. Tyagi, S.C. Srivastava, A fuzzy logic based load frequency controller in a competitive electricity, 2, in: Proceedings of the IEEE-PES General Meeting, Canada, 2003, pp. 560–565. [14] H. Bevrani, Y. Mitani, K. Tsuji, Robust decentralized AGC in a restructured power system, Energy Convers. Manage. 45 (15/16) (2004) 2297– 2312. [15] C. Scherer, P. Gahinet, M. Chilali, Multi-objective output-feedback control via LMI optimization, IEEE Trans. Autom. Control 42 (7) (1997) 896– 911. [16] M.B. Djukanovic, M.H. Khammash, V. Vittal, Sensitivity based structured singular value approach to stability robustness of power systems, IEEE Trans. Power Syst. 15 (2) (2000) 825–830. [17] P. Gahinet, A. Nemirovski, A.J. Laub, M. Chilali, LMI Control Toolbox, The MathWorks Inc., South Natick, 1995. [18] S. Haykin, Neural Networks–A Comprehensive Foundation, 2nd ed., Prentice Hall, New Jercy, 1999. [19] H. Shayeghi, H.A. Shayanfar, Application of ANN technique for interconnected power system load frequency control, Int. J. Eng. Natl. Center Sci. Res. 16 (3) (2003) 247–254. [20] H. Shayeghi, H.A. Shayanfar, Power system load frequency control using RBF neural networks based on ␮-synthesis theory, in: Proceedings of the IEEE International Conference on Cybernetics and Intelligent Systems, 1–3 Dec. 2004, Singapore, 2004, pp. 93–98.