función real base ps feb 07

96 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 1, FEBRUARY 2007

An RBF Network With OLS and EPSO Algorithmsfor Real-Time Power Dispatch

Chao-Ming Huang and Fu-Lu Wang

AbstractThis paper proposes a novel technique that combinesorthogonal least-squares (OLS) and enhanced particle swarmoptimization (EPSO) algorithms to construct the radial basisfunction (RBF) network for real-time power dispatch (RTPD).The goals considered are fuel cost, power wheeling cost, andNOx

/CO2

emissions. The RBF network is composed of three-layerstructures, which contain the input, hidden, and output layer. Tosimplify the network, the OLS algorithm is used first to determinethe number of centers in the hidden layer. With an appropriatenetwork structure, the EPSO algorithm is then used to tune theparameters in the network, including the dilation and translationof RBF centers and the weights between the hidden and outputlayer. The proposed approach has been tested on the IEEE 30-bussix-generator and practical Taiwan Power Company (Taipower)systems. Testing results indicate that the proposed approach canmake a quick response and yield accurate RTPD solutions as soonas the inputs are given. Comparisons of learning performance aremade to the existing artificial neural network (ANN), conventionalRBF network, and basic particle swarm optimization (PSO)methods.

Index TermsEnhanced particle swarm optimization (EPSO),orthogonal least-squares (OLS), radial basis function (RBF), real-time power dispatch (RTPD).

I. INTRODUCTION

MUCH research has been devoted to coping with thetechniques of power dispatch. Conventionally, electricpower plants are operated based on minimizing operationalcosts while meeting diverse system constraints. As a result ofincreasing public concern for the environmental protection,the operating dispatch strategies, considering both economicfactors and emission reduction, deserve greater attention. Thesestrategies include emission constrained economic dispatch [1],minimum emission dispatch [2], and economic emission powerdispatch (EEPD) methods [3][5]. Although power dispatchproblems have been effectively solved by several excellenttechniques, the related dispatch programs need to be rerunwhen the system load changes and thus is unsatisfactory for theRTPD environment.

The Hopfield neural networks have been applied to deal withthe classical economic dispatch for nonsmooth cost functions

Manuscript received April 11, 2006; revised September 29, 2006. This workwas supported by the National Science Council, Taiwan, R.O.C., under GrantNSC 93-2213-e-168-014. Paper no. TPWRS-00231-2006.

C.-M. Huang is with the Department of Electrical Engineering, Kun ShanUniversity, Tainan 710, Taiwan, R.O.C. (e-mail: [email protected]).

F.-L. Wang is with the Fenshan District Office, Taiwan Power Company,Kaohsiung 830, Taiwan, R.O.C. (e-mail: [email protected]).

Color versions of Figs. 38 are available online at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TPWRS.2006.889133

[6] and the EEPD problem [7]. To speed up the computationeffect, an adaptive learning rate approach has been incorpo-rated into Hopfield neural networks to improve the convergencecharacteristics and oscillations in solving the economic dispatchproblem [8].

Artificial neural network (ANN) methods [9], [10] have beenpresented to deal with the real-time power dispatch problem.These methods can accurately and efficiently capture complexinput-output relations, have well interpolated and extrapolatedcapabilities based on their experiences, and can provide real-time response in practical applications. However, ANN still hassome unsolved problems, including the local and slow conver-gence during training and the fact that the network structure andparameters are problem dependent.

The radial basis function (RBF) networks [11][13] providean alternative method to accomplish the same work as ANN.However, in contrast to ANN, the RBF network has a more com-pact topology and less training time for learning. A commonlearning strategy for an RBF network is to randomly select someinput data sets as the RBF centers in the hidden layer. Theweights between hidden and output layer can then be estimatedby using the stochastic gradient approach. However, it is clearlyunsatisfactory to use such a mechanism to build RBF networks.

This paper employs the orthogonal least-squares (OLS)learning algorithm [14] to select a suitable set of centers fromthe input data. The OLS is a systematic method that employsthe forward regression procedure to reduce the size of RBFnetworks. Furthermore, to make the network more suitableto fit the multidimensional space, the EPSO approach is usedto tune the parameters in the network, including the positionof RBF centers, the width of RBFs, and the weighting valuesbetween hidden and output layer.

The PSO was first introduced by Kennedy and Eberhart in1995 [15]. Through the simulation of a simplified social system,the behavior of PSO can be treated as an optimization process.Since the PSO requires short computer time and less memory, ithas successfully been applied to solve the power dispatch prob-lems [16][18]. To enhance the capability of particle swarm op-timization (PSO) in converging performance, a hybrid (or en-hanced) PSO method has been employed to deal with the var-ious problems of power system [19][21]. In this paper, an en-hanced PSO algorithm with the addition of a mutation mech-anism from evolutionary computation is presented in order togenerate a high-quality solution with shorter execution time.

The rest of this paper is organized as follows. In Section II, theproblem formulation of RTPD is briefly reviewed. Section IIIdescribes the proposed optimization algorithms. In Section IV,practical test results of the proposed method by employing

0885-8950/$25.00 2007 IEEE

HUANG AND WANG: RBF NETWORK WITH OLS AND EPSO ALGORITHMS 97

IEEE and Taipower systems are illustrated. Comparisons withthe other methods are also included in this section. Conclusionsare given in Section V.

II. PROBLEM FORMULATION

In a deregulated and competitive environment, the rapid re-sponse of the RTPD problem will be more and more importantfor the power utilities than ever before from the viewpoints ofoperating cost, environmental protection, and system security.Therefore, the RTPD has become an important topic in compet-itive power markets. The objectives and constraints of the EEPDand RTPD problems solved in this paper are summarized below.

A. Power Wheeling CostThe techniques for evaluating power wheeling cost can

roughly be divided into four methods: the postage stampmethod, contract path method, megawatt mile method, andmarginal cost method [22]. The megawatt mile method isadopted in this paper. For this

(1)

where is the power wheeling cost of the th exchange,is the weighted value of the th transmission line, is theabsolute value of real power flow for the th transmission lineon the th exchange, and is the length of the th transmissionline.

The real power flow in (1) can be calculated using thedc power flow method [23] as follows:

(2)

where is the reactance of the th transmission line, anddenotes the variation of voltage phase angle for the th

transmission line on the th exchange.

B. Fuel CostThe fuel cost function of the system can be expressed as a

quadratic function of generator active power output as follows:

$ (3)

where is the total fuel cost of the system; is thepower output of the th unit; indicates the number of gen-erators; and , and are the cost coefficients, which aregenerally obtained by curve-fitting technique [1].

Aggregating (1) and (3), the total cost can be defined as fol-lows:

$ (4)

where are the weights of power wheeling cost.

C. EmissionThe emission function of the system can be expressed as

the polynomial function of generator active power output asfollows:

(5)where , and are emission coefficients. The emissionfunction of (5) may be for any type of pollutants, such as NO ,CO , SO , particulates, or thermal pollutants. The emissions ofNO and CO are separately considered as two different casestudies in this paper.

D. Economic Emission Power DispatchReducing costs and reducing emissions are conflicting ob-

jectives and cannot be minimized simultaneously. To solve theEEPD problem, the two conflicting functions can be convertedinto a single objective optimization problem by giving relativeweights

(6)

where the objectives and correspond to (4) and(5), respectively, is the weight of the total cost, is theweight of emissions, and . The optimization of(6) must be subjected to power balance constraints, generationcapacity constraints, and line overload prevention constraints,as defined in

(7)

(8)

(9)

where is the total system load demand; is the trans-mission loss of the system; and are the lowerand upper bounds of the th power generation, which meansthat if , andif ; is the active power flow of the th line;

is the maximal power flow capacity of the th line; and isthe number of transmission lines.

To solve the EEPD problem, it is necessary to minimize theobjective function in (6) for the successive adjustmentof from zero (one) to one (zero). In this paper, the goal-attainment (GA) method [5] is employed to obtain the non-in-ferior solutions.

E. Real-Time Power DispatchTraditionally, the EEPD methods, including the GA model,

can serve as a useful tool to acquire non-inferior solutions fora specified load demand by using the successive adjustmentof weighting value as given in (6). However, the related dis-patch programs of the EEPD methods need to be rerun when


Fig. 1. Schematic diagram of the proposed method.

the system load changes and thus unsuitable for the RTPDenvironment.

To overcome the limitations of existing EEPD methods, thispaper proposes a combination of OLS and EPSO algorithmsto construct the best RBF network. Once this network is con-structed, the desired outputs of the power generated by eachgenerator can be determined quickly as soon as the inputs ofsystem load with different weights are given. Details of the pro-posed optimization algorithms are described in the followingsection.

III. PROPOSED OPTIMIZATION ALGORITHMS

To construct the optimal RBF network, historical records ofEEPD training data for different power demands with variousweights are set up by the GA method. Note that the collection ofhistorical training data must cover all of the normal load profiles.Details of the proposed method are depicted in Fig. 1, and themain steps are stated in the subsections that follow.

A. RBF Network

Fig. 2 shows a schematic diagram of the RBF network. Thenetwork is comprised of three layers: input layer, hidden layer,and output layer. The input and output layers are presented withtraining pairs, each consisting of a vector from an input spaceand a desired network response. Through a defined learning al-gorithm, the error between the actual and desired response isminimized relative to some optimization criterion.

As depicted in Fig. 2, the th output node of the RBF networkcan be expressed as follows:

(10)

where is an input vector; is the numberof input node; is the th center node in the hidden layer,

, in which is the number of hidden nodes;denotes Euclidean distance [14] between and

vector; is a nonlinear transfer function of the th center;is the weighting value between the th center and the th

output node; and is the number of output nodes.

Fig. 2. Schematic diagram of RBF network.

Fig. 3. RBF centers with different width and position.

Equation (10) reveals that the output of the network is com-puted as a weighted sum of the hidden layer outputs. The non-linear output of the hidden layer is radially symmetrical.In this paper, the most widely used Gaussian function is chosenas follows:

(11)

where and are the parameters that control the width andposition of the RBF centers, respectively. Fig. 3 presents theRBF centers with different width and position.

It follows from (10) and (11) that there are four sets of pa-rameters governing the mapping properties of the network: thenumber of centers in the hidden layer, the position of RBF cen-ters, the width of RBFs, and the weights . In general, a suffi-cient number of centers are randomly chosen as a subset of theinput space according to the probability density function of thetraining data. Then the stochastic gradient approach is used totune the other three sets of parameters (that is, the position ofRBF centers, the width of RBFs, and the weights). The maindisadvantage of this method is that it is very difficult to quantifyhow many numbers of center should be adequate to cover theinput vector space. Furthermore, the training algorithm is proneto getting stuck in local minimum.

To overcome these limitations, this paper employs the OLSalgorithm to determine the number of centers, and the remain-ders are tuned using the EPSO approach. The OLS and EPSOalgorithms are further described below.


B. OLS Algorithm

As shown in (10), the OLS algorithm can be implemented byintroducing an error term as follows:

(12)

Using matrix form, (12) can be expressed as

(13)

where is the vector of the desired network outputs;can be regarded as a linear regression matrix with

each column vector ; is a vector ofweights; and is the vector of errors between thedesired and actual network outputs.

By using the GramSchmidt orthogonalization [14], the re-gression matrix can be decomposed into a set of orthogonalbasis vectors as follows:

.

.

.

.

.

.

.

.

.

.

.

.

(14)

where is an upper triangular matrix, andis a matrix with mutually orthogonal vector such that

(15)

where is a diagonal element of matrix.

Aggregating (13) to (15), the desired network outputs can berewritten as

(16)

where . Since the GramSchmidt orthogonalizationensures the orthogonality between and DG in (16), we have

(17)

Therefore, the error reduction ratio (ERR) due to the inclusionof the th center can be defined as

(18)

The ERR in (18) provides an effective criterion for selectingthe RBF centers in a forward regression manner. At every stepof the forward regression, an adequate RBF center is selected

so that the value of ERR is the maximum. The regression isterminated at the th step when

(19)

where is a tolerance value selected by the operators.Once the number of centers is selected by the OLS algorithm,

the EPSO approach is employed to tune the parameters in thenetwork. The EPSO algorithm is described below.

C. EPSO Algorithm

The PSO simulates the behavior of a swarm as a simplifiedsocial system. In a PSO system, each particle adjusts its positionin light of its own experience and the experiences of neighbors,including the current velocity, position, and the best previousposition experienced by itself and its neighbors as follows:

(20)

where is the current velocity of the th particle,, in which is the population size; is the

dimension of population; is the best previous positionof the th particle; is the best previous position amongall the particles in the swarm; is the current position ofthe th particle; is an acceleration factor; represents theuniform random number between 0 and 1; and denotes theinertial weight, which is set according to the following equation[15]:

(21)

where is the maximum number of iterations, and is thecurrent number of iterations. Equation (21) restricts the valueto the range .

In the above procedures, the PSO utilizes Pbest and Gbest tomodify the current search point to avoid the particle flying inthe same direction. With model (20), the next position of the thparticle can be modified by

(22)

Furthermore, model (20) reveals that the search procedure byPSO depends heavily on Pbest and Gbest. If the initial pop-ulation of particles cannot effectively cover the whole region,the particles will converge usually on a suboptimal solution. Toavoid these limitations, a mutation mechanism of evolutionarycomputation is then incorporated into PSO algorithm in thispaper. In general, the particles with poor individuals are selectedfor mutation from a subset of the swarm. The next position ofthe th selected particle can be modified by

(23)


where represents a vector of Gaussian random vari-ables with mean zero and standard deviation ,in which is the number of particles selected for mutation. Gen-erally, is set at 10% of population. In addition, in (23) isgiven according to the following equation:

(24)

where denotes the Euclidean distance be-tween and is a scaling factor, and representsan offset.

As given in (23) and (24), the next position of the th particleselected for mutation is attained according to the distance be-tween and . If the distance is relatively large, thenext position will be searched within a wider range.

D. Parameters Adjustment Using EPSOThe EPSO algorithm for tuning the position of RBF centers,

the width of RBFs, and the weights between hidden and outputlayer is described as follows.

1) Initialization: Randomly generate the initial trial vec-tors ( is the population size,is the dimension of population), which indicate the possiblesolutions for the translation and dilation of RBF centers orthe connection weights. This step is accomplished by setting

, where , and represent the de-sired values of the position of RBF centers, the width of RBFs,and the weighting vector, respectively. The element in vector

is randomly generated as follows:

(25)where designates the outcome of a uniformlydistributed random variable ranging over the given lower- andupper-bounded values and of the weighting factorsor the parameters of translation and dilation.

2) Determination of Fitness Function: For each trial vector, a fitness value should be assigned and evaluated. In this

paper, the criterion of least-squared fitting error (LSFE) functiondefined below is adopted to stand for the fitness value of the RBFnetwork

(26)

where is the th computed output of the RBF network by using(10), is the corresponding actual output, and is the numberof network output nodes.

3) Selection and Memorization: Each particle memorizesits own fitness value and chooses the minimum one that hasbeen better so far as . We then have

in the population. On the other hand, eachparticle also memorizes another particles fitness values to knowtheir experiences. The particle with the best fitness value amongPbest is denoted as Gbest. Note that in the first iteration, eachparticle is set directly to , and the particle with thebest fitness value among Pbest is set to Gbest.

TABLE ICOEFFICIENTS OF FUEL COST AND NO EMISSION FUNCTIONS

4) Modification of Velocity and Position: Modify the velocityof each particle according to (20). An offspring vector is thencreated using (22) within the range of , whereand are real values.

5) Mutation: In step 4), the particles are ranked in de-scending order of their corresponding fitness value. Theparticles with poor individuals are selected for mutation fromswarm subset by using (23) and (24). In this step, the populationsize remains unchanged.

6) Stopping Rule: Repeat steps 2) to 5) until the best fitnessvalue (Gbest) is not obviously improved or the given count oftotal generations is reached. The solution with the lowest fitnessvalue is chosen as the best parameter in the RBF network thatshall further be applied to real-time power dispatch.

IV. NUMERICAL RESULTS

The proposed approach has been verified on two diversesystems under various power demands. For comparison, theANN approach [10], conventional RBF network method [11],and basic PSO method [16] implemented by the commer-cial MATLAB package [24] were also tested using the samedatabase.

A. Case I: IEEE 30-Bus Six-Generator System [4], [5]For the IEEE 30-bus six-generator system, the objectives of

fuel cost, power wheeling cost, and NO emission are convertedinto a single objective optimization problem as given in (6).Table I shows the coefficients of fuel cost and NO emissionfunctions. The power losses are ignored, and the length of trans-mission line defined in (1) is assumed to be proportional to thereactance of the line. Table II reveals the plan of training datacreated by the GA method. These load levels are defined in sucha way that they cover the whole range of system load within thenormal condition. As shown in this table, a total of 315 trainingsamples are created.

Table III shows the parameter settings for diverse methods.Note that the conventional RBF network has 315 centers dis-tributed over the defined input space, while the proposed methodemploys an OLS algorithm to perform the reduction of the net-work size. Furthermore, the same numbers of input and outputnodes are given to the existing and proposed methods, while


TABLE IIPLAN OF TRAINING DATA CREATED BY GA METHOD

TABLE IIIPARAMETER SETTINGS FOR DIFFERENT METHODS

Fig. 4. Trends of LSFE and number of centers versus ERR value.

the number of intermediate layers remains to be determinedindependently.

To determine an adequate number of RBF centers using OLS,Fig. 4 exhibits the typical relationships of LSFE and the numberof centers to the ERR. As noted, with the value of ERR [given in(18)] increased from 0.05 to 0.55, the number of centers neededto construct the network rises, while the LSFE value decreases

Fig. 5. Distribution of initial (315) and retained (65) RBF centers.

Fig. 6. Optimization of the existing and proposed methods in case I.

progressively. In this case, 65 centers werechosen to perform the exact mapping of all 315 centers, as givenin Fig. 5.

Fig. 6 depicts the optimization processes of the best fitnessvalue obtained by the diverse methods. As shown in this figure,the conventional RBF network converges toward the best valueafter 312 iterations (341 s), the basic PSO method need about47 iterations (15.73 s), and the proposed method converges afterabout 31 iterations (10.73 s). The ANN method, learned throughthe well-known back-propagation algorithm, takes about 14 748iterations (6060 s) to construct the network, which is unsuitablefor presenting in this figure.

Table IV summarizes the comparisons of test results for thediverse load levels. Notably, the cases of and2.6065 (p.u.) are in the range of trained historical data (interpo-lation cases), while the cases of , 2.2334, 2.8964,and 2.9380 (p.u.) are beyond the trained range (extrapolationcases). Results of this table indicate that the proposed methodis more accurate than the existing methods for both interpola-tion and extrapolation cases. The associated values of fuel cost,power wheeling cost, and NO emission are also listed for ref-erence. Note that the values of fuel cost and NO emission are


TABLE IVTEST RESULTS FOR DIFFERENT LOAD LEVELS IN CASE I

obtained from Table I, while the power wheeling cost is calcu-lated by aggregating (1) and dc power flow method.

To compare the accuracy of different methods, the criterionof average percentage absolute error (APAE), defined in (27) atthe bottom of the page, is adopted in this paper, where is thenumber of generators.

B. Case II: Taipower 388-Bus 27-Generator System [25]In case II, the objectives considered are fuel cost, power

wheeling cost, and CO emission. The fuel cost functionadopted in this case was a cube curve, and the CO emission iscalculated as follows [25]:

(28)

where is a constant. is the heat rate of the th unit, whichis obtained from fuel cost function. represents the carbonemission coefficient of the th unit, and is the oxidizingrate of carbon for the th unit.

The inputs of the training networks are power demand andoperators preferences ( and ), while the outputs arethe power generated by 27 thermal generators. To create thetraining data by GA, the load demands are varied from 8815 to9265 (MW) with a step size of 50 (MW). The variation ofand are the same as case I. As described above, a total of210 training samples are created in this case.

The architecture of ANN contains three input nodes, ninehidden nodes, and 27 output nodes. With the same numbers ofinput and output nodes, the conventional RBF network has 210hidden nodes, while the proposed method selects an adequatenumber of hidden nodes by OLS algorithm. The other param-eter settings for case II are the same as case I.

Fig. 7 reveals that a total of 66 RBF centers are selected by theOLS algorithm. Fig. 8 depicts the optimization processes of theexisting and proposed methods. The conventional RBF networktakes about 208 iterations (195 s) to construct the network, thebasic PSO method needs about 41 iterations (13.25 s), while theproposed method converges after about 26 iterations (8.32 s).The ANN method takes about 11 211 iterations (4322 s) to trainthe network and so is not presented in the figure.

% % (27)


Fig. 7. Distribution of initial (210) and retained (66) RBF centers.

Fig. 8. Optimization of the existing and proposed methods in case II.

Table V summarizes the comparisons of existing and pro-posed methods for the different load levels. As shown in thistable, the cases of , and (MW) arethe extrapolation cases, while and 9185 (MW) arethe interpolation cases. Test results from this table indicate thatthe proposed method is more accurate than the existing methodswhen the GA results are used as a benchmark.

V. DISCUSSIONS

Considering the results obtained from the two cases, the fol-lowing observations are in order.

1) As shown in Figs. 5 and 7, the OLS can effectively reducethe size of RBF network that provides some of the contri-butions in decreasing the computation time.

2) Test results from Tables IV and V indicate that the pro-posed method can provide better accuracy for diverse loadlevels. Even in the case of extrapolation, where a higherestimating error was obtained by the existing methods, the

TABLE VTEST RESULTS FOR DIFFERENT LOAD LEVELS IN CASE II

TABLE VIDETAILED RESULTS OF ANN AND PROPOSED METHODSCASE OFP = 2:9380 (P.U.) IN TABLE IV

proposed method still provides an accurate mapping be-tween inputs and outputs.

3) As defined in (27), the criterion to assess the performanceof the proposed method is related to the estimating accu-racy of active power output of each generator. Since theestimated value of each generator may be higher or lowerthan the actual one, the solution with lower APAE valueseems not to ensure that its objective values including fuelcost, power wheeling cost, and No /CO emissions arecloser to the actual objective value than the other ones withhigher APAE value. Table VI exhibits the detailed resultsof case (p.u.) in Table IV for both ANNand proposed methods. This table reveals that the proposedmethod has better estimating accuracy than ANN, but thefuel cost of ANN is closer to the actual one than the pro-posed method.


VI. CONCLUSION

A new technique that combines OLS and EPSO algorithms toconstruct the optimal RBF network has been presented to solvethe RTPD problem in this paper. Compared to the existing tech-niques, the superiority of the proposed method is summarizedas follows.

1) Compared with the conventional RBF network method, theproposed approach provides an effective method to sim-plify the network structure.

2) Based on the same network structure, the proposed EPSOalgorithm provides a more efficient search scheme to deter-mine the related parameters of the RBF network than thePSO method.

3) Testing on two different cases has shown that the pro-posed approach is superior to the existing methods in con-structing the network and estimating the outputs of the gen-erating units.

4) After the network is constructed, the proposed approachcan make a quick response and yield accurate RTPD solu-tions as soon as the inputs of system load with the weightof cost are given.

The proposed method, however, still has some unsolved prob-lems that should further be studied in future work.

1) The EPSO has been verified to be a fast optimizationtechnique. However, the convergence characteristic ofEPSO is sensitive to the tuning of acceleration factorand weighting values of and . Generally, it isnot easy for the dispatchers to obtain an adequate value forthese parameters. This may be a weakness in comparingwith ANN and classic RBF methods. Therefore, a sensi-tive study to determine the optimal parameters in EPSOfor various power system problems will be valuable.

2) Although this paper focuses on the power dispatch of satis-fying the limits of the units, extension of the proposed ap-proach to consideration of the case of violating the limits ofthe units is feasible by appropriately formulating the RTPDproblem.

REFERENCES[1] R. Ramanathan, Emission constrained economic dispatch, IEEE

Trans. Power Syst., vol. 9, no. 4, pp. 19942000, Nov. 1994.[2] J. H. Talaq, F. El-Hawary, and M. E. El-Hawary, Minimum emission

power flow, IEEE Trans. Power Syst., vol. 9, no. 1, pp. 429435, Feb.1994.

[3] J. Nanda, D. P. Kothari, and K. S. Lingamurthy, Economic emissionload dispatch through goal programming technique, IEEE Trans. En-ergy Convers., vol. 3, no. 1, pp. 2632, Mar. 1988.

[4] R. Yokoyama, S. H. Bae, T. Morita, and H. Sasaki, Multiobjective op-timal generation dispatch based on probability security criteria, IEEETrans. Power Syst., vol. 3, no. 1, pp. 317324, Feb. 1988.

[5] P. C. Chen and C. M. Huang, Bi-objective power dispatch using goal-attainment method and adaptive polynomial networks, IEEE Trans.Energy Convers., vol. 19, no. 4, pp. 741747, Dec. 2004.

[6] J. H. Park, Y. S. Kim, I. K. Eom, and K. Y. Lee, Economic load dis-patch for piecewise quadratic cost function using Hopfield neural net-work, IEEE Trans. Power Syst., vol. 8, no. 3, pp. 10301038, Aug.1993.

[7] T. D. King, M. E. El-Hawary, and F. El-Hawary, Optimal environ-mental dispatching of electric power systems via an improved Hop-field neural network model, IEEE Trans. Power Syst., vol. 10, no. 3,pp. 15591565, Aug. 1995.

[8] K. Y. Lee, A. Sode-Yome, and J. H. Park, Adaptive Hopfield neuralnetworks for economic load dispatch, IEEE Trans. Power Syst., vol.13, no. 2, pp. 519526, May 1998.

[9] M. Djukanovic, M. Calcvic, B. Milosevic, and D. J. Sobajic,Neural-net based real-time economic dispatch for thermal powerplants, IEEE Trans. Energy Convers., vol. 11, no. 4, pp. 755761,Dec. 1996.

[10] P. S. Kulkarni, A. G. Kothari, and D. P. Kothari, Combined eco-nomic and emission dispatch using improved backpropagation neuralnetwork, Elect. Mach. Power Syst., vol. 28, no. 1, pp. 3144, Jan. 2000.

[11] P. K. Hota and S. K. Dash, Multiobjective generation dispatch througha neuro-fuzzy technique, Elect. Power Compon. Syst., vol. 32, no. 11,pp. 11911206, Nov. 2004.

[12] P. S. Kulkarni, A. G. Kothari, and D. P. Kothari, Application of ra-dial basis function neural network for economic dispatch, J. Inst. Eng.(India): Elect. Eng. Div., vol. 83, pp. 8186, Sep. 2002.

[13] P. Aravindhababu and K. R. Nayar, Economic dispatch based on op-timal lambda using radial basic function network, Int. J. Elect. PowerEnergy Syst., vol. 24, no. 7, pp. 551556, Sep. 2002.

[14] S. Chen, C. F. N. Cowan, and P. M. Grant, Orthogonal least squareslearning algorithm for radial basis function networks, IEEE Trans.Neural Netw., vol. 2, no. 2, pp. 302309, Mar. 1991.

[15] J. Kennedy and R. Eberhart, Particle swarm optimization, inProc. IEEE Int. Conf. Neural Networks, Nov./Dec. 1995, vol. 4, pp.19421948.

[16] A. I. S. Kumar, K. Dhanushkodi, J. J. Kumar, and C. K. C. Paul, Par-ticle swarm optimization solution to emission and economic dispatchproblem, in Proc. IEEE Int. Conf. TENCON, Oct. 2003, pp. 435439.

[17] J. B. Park, K. S. Lee, J. R. Shin, and K. Y. Lee, A particle swarmoptimization for economic dispatch with nonsmooth cost functions,IEEE Trans. Power Syst., vol. 20, no. 1, pp. 3442, Feb. 2005.

[18] D. N. Jeyakumar, T. Jayabarathi, and T. Raghunathan, Particle swarmoptimization for various types of economic dispatch problems, Int. J.Elect. Power Energy Syst., vol. 28, no. 1, pp. 3642, Jan. 2006.

[19] S. Naka, T. Genji, T. Yura, and Y. Fukuyama, A hybrid particleswarm optimization for distribution state estimation, IEEE Trans.Power Syst., vol. 18, no. 1, pp. 6068, Feb. 2003.

[20] A. A. A. Esmin, G. Lambert-Torres, and A. C. Zambroni de Souza, Ahybrid particle swarm optimization applied to loss power minimiza-tion, IEEE Trans. Power Syst., vol. 20, no. 2, pp. 859866, May 2005.

[21] J. S. Heo, K. Y. Lee, and R. Garduno-Ramirez, Multiobjective controlof power plans using particle swarm optimization technique, IEEETrans. Energy Convers., vol. 21, no. 2, pp. 552561, Jun. 2006.

[22] Y. R. Sood, N. P. Padhy, and H. O. Gupta, Wheeling of power underderegulated environment of power systemA bibliographical survey,IEEE Trans. Power Syst., vol. 17, no. 3, pp. 870878, Aug. 2002.

[23] A. J. Wood and B. F. Wollenberg, Power Generation Operation andControl, 2nd ed. New York: Wiley, 1996.

[24] The Math Work Inc., Optimization Toolbox Users Guide, Dec. 1992.[25] C. M. Huang, H. T. Yang, Y. Y. Hong, S. P. Hong, and K. P. Liou,

Power dispatching considering fuel cost and CO Emission, MonthlyJ. Taipowers Eng., vol. 610, pp. 3248, Jun. 1999.

Chao-Ming Huang was born in Kaohsiung, Taiwan, R.O.C., on December 6,1962. He received the M.S. and Ph.D. degrees from National Cheng Kung Uni-versity, Tainan, Taiwan, in 1992 and 1997, respectively, both in electrical engi-neering.

Since 2003, he has been a Full Professor in the Department of Electrical En-gineering, Kun Shan University, Tainan. His research interests are in power dis-patch and load forecasting of power systems.

Fu-Lu Wang was born in Kaohsiung, Taiwan, R.O.C., on September 27, 1964.He received the M.S. degree from Kun Shan University, Tainan, Taiwan, in 2003.

Since 1990, he has been a Senior Engineer at Taiwan Power Company, Kaoh-siung. His major interests are in power dispatch and SCADA systems.

función real base ps feb 07

Documents