Forecasting Aggregated Wind Power Production of Multiple WindFarms Using Hybrid Wavelet-PSO-NNs

Paras Mandala,∗, Hamidreza Zareipourb, and William D. Rosehartb

aDepartment of Industrial, Manufacturing and Systems EngineeringUniversity of Texas at El PasoEl Paso, Texas 79968, USA.E-mail: pmandal@utep.edu

bDepartment of Electrical and Computer EngineeringUniversity of Calgary,

Calgary, Alberta T2N 1N4, Canada.E-mails: h.zareipour, rosehart@ucalgary.ca

Abstract

This paper describes the problem of short-term wind power production forecasting based on me-teorological information. Aggregated wind power forecasts are produced for multiple wind farmsusing a hybrid intelligent algorithm that uses a data filtering technique based on wavelet transform(WT) and a soft computing model (SCM) based on neural network (NN), which is optimized byusing particle swarm optimization (PSO) algorithm. To demonstrate the effectiveness of the pro-posed hybrid intelligent WT+NNPSO model, which takes into account the interactions of windpower, wind speed, wind direction, and temperature in the forecast process, the real data of windfarms located in the southern Alberta, Canada are used to train and test the proposed model. Thetest results produced by the proposed hybrid WT+NNPSO model are compared with other SCMsas well as the benchmark persistence method. Simulation results demonstrate that the proposedtechnique is capable of performing effectively with the variability and intermittency of wind powergeneration series in order to produce accurate wind power forecasts.

Keywords: Neural networks, particle swarm optimization, wavelet transform, wind powerforecasting.

∗Corresponding author. Ph: (915) 747-8653, Fax: (915) 747-7184.E-mail: pmandal@utep.edu

Preprint submitted to International Journal of Energy Research January 3, 2014

Nomenclature

ANFIS Adaptive Neuro Fuzzy Inference SystemBPNN Backpropagation Neural NetworkCWT Continuous Wavelet TransformDWT Discrete Wavelet TransformEKF Extended Kalman FilterH High Pass FilterL Low Pass FilterMAE Mean Absolute ErrorMAPE Mean Absolute Percentage ErrorMRA Mallat’s multi-Resolution AnalysisMSE Mean Square ErrorNMAE Normalized Mean Absolute ErrorNN Neural NetworkNRMSE Normalized Root Mean Square ErrorNWP Numerical Weather PredictionPSO Particle Swarm OptimizationRBFNN Radial Basis Function Neural NetworkRES Renewable Energy SourceRMSE Root Mean Square ErrorSCM Soft Computing ModelSOFM Self Organizing Feature MapsWPF Wind Power ForecastingWT Wavelet Transformψ Mother Wavelet

1. Introduction

Wind power is gaining significant attention in the field of renewable energy technology for thepurpose of electricity generation. With increased penetration of this renewable energy source (RES),wind power production is rapidly increasing into a large-scale wind power industry. Electric powergenerated from wind energy will likely play a vital role in balancing future energy supply in manycountries. This implies the need to effectively integrate the power generated from wind energyinto existing power systems. However, power output of wind farm depends on the weather, andunexpected variations of wind power output may increase the operating costs of the power system.Moreover, a major barrier in integrating RES into the grid is their unpredictability, since steadyoutput cannot be guaranteed at any particular time [1, 2]. Since in power systems the supplymust meet the demand at all times, new techniques of balancing supply and demand are necessary.Accurate wind power forecasting (WPF) can play a key role in tackling these challenges. WPF canbe short-term, medium-term and long-term. The short-term (from 30 minutes to 6-hour-ahead) andmedium (from 6-hour to one day-ahead) forecasts are needed in the generation commitment andmarket operation, while the long-term forecast (from 1 day to 1 week-ahead) is required during theunit commitment decision and maintenance planning stage [3, 4]. An accurate forecasting of windpower is thus imperative for the efficient and economical wind power integration and operation.

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The definition of a good forecasting tool in the context of a market must therefore be definedin terms of set-up costs, operational costs, and the revenue improvements that a new methodhas over a more simplified or existing system. Different techniques are available ranging fromsimple heuristics through to full numerical weather modeling systems. All the models howeverhave time frames for which they are best suited, and there are forecasting errors associated withthem all. Several forecasting techniques such as statistical and physical methods are availablein literature to predict wind power. A literature review on forecasting wind power is reportedin [3, 5, 6]. Soft computing models (SCMs) based techniques are gaining attention especially insolving forecasting problems. Various SCMs applied for wind power forecasting works are datamining [1], neural networks (NNs) [7–14], fuzzy logic [15, 16], neuro-fuzzy models [2, 17], andhybrid methods [18–21]. Different types of NNs are backpropagation NN (BPNN), probabilisticNN, radial basis function NN (RBFNN), self-organizing feature maps (SOFM), cascade correlationNN, extended Kalman filter (EKF)-based NN. A case study from Tasmania in Australia for veryshort-term (2.5 minute-ahead) forecasting using adaptive neuro fuzzy inference system (ANFIS) toforecast wind vectors rather than wind speed or power is reported in [2]. A probabilistic methodfor assessing the costs associated with wind generation prediction errors in electricity market isproposed in [22] where case studies showed that the generation prediction error costs could reacharound 10% of the total generation incomes. A short-term probabilistic forecasts of wind poweris presented in [23] where a general method for optimal bidding strategy based on uncertaintyinformation of forecasts is discussed. Pinson and Kariniotakis [16] proposed a fuzzy inferencemodel that permits to integrate expertise on the characteristics of prediction errors for providingconditional interval wind power forecasts. Catalao et al. [19] proposed a novel hybrid wavelettransform (WT)-particle swarm optimization (PSO)-ANFIS approach for short-term wind powerforecasting. Amjady et al. [24] proposed a wind power forecasting strategy composed of a featureselection component and a forecasting engine using the real-life data from wind power producersin Alberta and Oklahoma. Ullah et al. [25] developed the cost model for the possibility of using awind farm’s output for ancillary services, like reactive power support, primary frequency support,power oscillation damping support, etc., in case when wind farm’s output remains unutilized dueto forecast errors. Several works are available in the literature related to wind energy. Mohandeset al. [26] presented an Abductory Induction Mechanis to estimate the mean monthly wind speedbased on wind data. Sahin and Aksakal [27] investigated the wind energy potential in Saudi Arabiaand found that there is a large potential of wind power generation in the eastern part. Al-Garniet al. [28] carried out a regression analysis to model the weather characteristics and wind powerconsidering several weather parameters such as temperature, relative humidity, fog, wind speed,wind power and dust storms. Tina and Gagliano [29] conducted a probabilistic analysis of solarirradiance and wind speed data to evaluate the electric power generation by a photovoltaic systemand a wind system. In [30], discrete Hilbert transform technique was applied to characterize thewind sample data and to estimate wind speed time-series. There are some other works available inthe literature that discuss about energy systems planning [31–33].

In the present context, U.S. electricity markets are utilizing various wind forecasting methods.Independent system operators (ISOs) such as PJM, ERCOT, CAISO, and MISO are currently usingphysical-based models and numerical weather prediction (NWP) [34]. Using these models, PJMand CAISO have shown to receive mean absolute error (MAE) in the range of about 3%−12% for1-hour-ahead wind power forecasting [34, 35]. PJM, ERCOT, CAISO, NYISO, and MISO have alsoexperienced MAE ranging in approximately 8%−17% for day-ahead errors [34–36]. Another methodthat ERCOT and CAISO have shown to apply was statistical models [35, 37]. In addition, ANNs and

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regression models were also applied in the ERCOT and NYISO [37, 38]. It is to be emphasized thatmany wind farms are relatively new and their performances have not been adequately studied [1].Wind power forecasting is a complex problem because of highly volatile nature of wind power time-series. In spite of number of techniques available for wind power forecasting as indicated here, thereis still a need of an algorithm for predicting the wind power generation with higher accuracy. Thedevelopment of such algorithm presents unique challenges that can be effectively solved with theintelligent learning and analysis approaches.

Wind power forecasts could be used by different parties for different purposes and the quantifiedbenefits could significantly vary based on the application. For example, a generation company mayuse wind power forecasts to optimize their bidding strategy, or a system operator may use windpower forecasts to schedule its reserve resources or coordinate facility outages [39–41].

This paper proposes a hybrid intelligent algorithm for aggregated power generation forecasts ofwind farms located in the southern part of Alberta, Canada. The innovative aspect of this paper lieson developing wind power forecast model using the combination of a data filtering technique basedon WT and a SCM based on NN that is optimized by using PSO algorithm, i.e., WT+NNPSOmodel. The proposed hybrid intelligent model considers the interaction of wind power with windspeed, wind direction, and temperature in the forecast process and can effectively capture windpower fluctuations patterns. Following some previous works [19, 42, 43] in the area of wind powerforecasting, this paper mainly focuses on hourly wind power production forecasts with a forecastinghorizon of several-hour-ahead (e.g., 1-hour, 3-hour, 6-hour, 12-hour and 24-hour ahead) for multipleseasons of the year. The effectiveness and efficiency of the proposed wind power forecasting strategyis demonstrated by comparing the results with the benchmark persistence method, other individualSCMs (BPNN, RBFNN, ANFIS, and NNPSO), and hybrid models (WT+BPNN, WT+RBFNN,and WT+ANFIS).

The rest of this paper is organized as follows. Section 2 provides a description of WT andPSO model followed by the forecasting procedure of the proposed hybrid intelligent algorithm inSection 3. Numerical results and discussions are presented in Section 4. Section 5 includes theconclusions of the paper.

2. Description of Wavelet Transform and Particle Swarm Optimization

In this paper, the proposed model uses the combination of WT and NNPSO [44–46]. Detaileddescription of WT and PSO is described in this section.

2.1. Wavelet Transform

WT is a mathematical tool, which has been used in various signal processing purposes in de-scribing a signal at various scales of interest. Wavelet technique has been widely used in forecastingapplications such as wind forecasting [46–48], electricity price forecasting [49]. In this paper, themain reason for using wavelet-based approach for signal processing is that wind power is stochasticin nature and traditional time-series approaches do not tend to obtain a sufficient level of ac-curacy [50]. By using the wavelet technique, time-series data is decomposed into approximatelystationary components and then modeling them separately. No initial assumptions on linearity orstationarity of the time-series have to be made. Finally the aggregate forecasts are obtained as asummation of the separately predicted components.

The regularity of time-series data is an important precondition for forecasting application bythe SCMs. The wind power data series has various fluctuations and spikes, and different types ofnon-stationarities may exist. In order to face the challenge of non-stationarity and stochastic nature

4

S

HL

H

H

L

L

D1

D2

A2

A1 D3

A3

Figure 1: Wavelet decomposition.

of wind power data, WT performs filtering data, which can produce a good local representationof the signal in both time and frequency domains [51]. WTs can be divided into two categories:continuous wavelet transform (CWT) and discrete wavelet transform (DWT). The CWT is definedas

CWTx(a, b) =1√|a|

∫ +∞

−∞ψ∗(t)x(t)dt, a > 0 (1)

where x(t) is the signal to be analyzed, ψa,b(t) is the mother wavelet scaled by a factor a and shiftedby a translated parameter b, and ∗ denotes complex conjugate.

ψa,b(t) =1√aψ

(t− ba

), (2)

where a > 0 and t is finite. The normalization on the right-hand side of (2) was chosen so that‖ψa,b‖ = ‖ψ‖ for all a, b. Low frequencies (large scale) expand the signal and provide non-detailedinformation regarding the signal, whereas high frequencies (low scales) compress the signal andprovide detailed information about the signal. As the CWT is obtained by continuously scalingand translating the mother wavelet, substantial redundant information is generated [49, 51]. So, themother wavelet can be scaled and translated using certain scales and positions known as DWT. TheDWT uses scale and position values based on power of two, called dyadic dilation and translations.

DWTx(m,n) = 2−(m/2)T−1∑t=0

x(t)ψ

(t− n.2m

2m

)(3)

where T is the length of the signal x(t). The scaling and translation parameters are functions ofthe integer variables m and n, where, a = 2m and b = 2n, t is the discrete time index. To imple-ment DWT using filters, Mallat developed an approach called Mallat algorithm or Mallat’s multi-resolution analysis (MRA) [52]. This algorithm has two stages: decomposition and reconstruction.In first stage, the original signal is decomposed into two complementary filters and emerges as twosignals namely approximation (low frequency component) and details (high frequency component).

There are many types of wavelet basis functions that can be used as a mother wavelet for wavelettransforms such as Morlet, Haar, Mexican Hat, and Meyer. Groups of them are called families,among them are Daubechies, Biorthogonals, Coifets, and Symlets [48, 53, 54]. The mother waveletsdetermine the characteristics of the resulting wavelet transform and each one of these functions hasspecial properties. A wavelet function of type Daubechies of order 4 (db4) is used in this work as

5

S

H

L

H

H

L

L

D1

D2

A2

A1D3

A3

Figure 2: Wavelet reconstruction.

the mother wavelet ψ(t) and was selected based on method similar to [19]. Similar wavelets, e.g.,db5 and db2, have been considered in the price and load forecasting works, respectively [49, 55].

2.1.1. Wavelet Decomposition and Reconstruction

Wavelet decomposition defines a signal to break down into many lower resolution components.The decomposition can proceed only until the individual detail consists of a single sample. Asuitable number of decomposition levels can be selected based on the nature of signal [53]. Twotypes of filters are used to decompose the original signal. The high pass filter (H) and low passfilter (L) are used to downsample the original signal as shown in Fig. 1.

The decomposed components of the wavelet signal can be assembled back into the originalsignal without loss of any information. This is called wavelet reconstruction as shown in Fig. 2.Like wavelet decomposition, high pass filter (H) and low pass filter (L) are used and the decomposedsignal are upsampled to get the original signal, S. The reconstructed original signal S can be foundin the following way:

S = A1 +D1

= A2 +D2 +D1

= A3 +D3 +D2 +D1 (4)

The approximation and detail signals A1 and D1 are obtained by down sampling and are onlyhalf the length of the original signal. So, before reproducing the original signal, it is necessary toreconstruct the approximation and detail coefficients.

If the WT is applied to an ill-behaved time-series (non constant mean and variance, outliers), theresulting constitutive series behave usually better than the original time-series. The approximationseries consistently presents a more stable variance than the original time-series and no outliers [49].Thus, the future values of the constitutive series can be forecasted accurately. The inverse WT to thepredictions of the constitutive series allows producing accurate forecast to the original time-series.Therefore, the application of WT to wind power time-series helps enhance forecasting capability.

2.2. Particle Swarm Optimization

The PSO is an efficient evolutionary computational stochastic optimization technique developedby Kennedy and Eberhart [56, 57]. Unlike the other population based search algorithm, PSO tracksthe optimal solution not by survival of the fittest but by a process motivated by the personal andsocial behavior of a flock of birds. PSO has been successfully applied to solve complex global op-timization problems [57–59] as well as forecasting problems [19, 45, 60]. Its solution methodology

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current motion

influence

swarm influence

particle memory

influence

xik

pgk

xik+1 pi

vik

vik+1

Figure 3: Velocity and position update in PSO.

is hardly affected by the dimension, complexity and nonlinearity of the problem. PSO is there-fore a smart alternative to solve problems associated with large scale power system optimization,power systems planning and operation as well as power forecasting application. PSO performsthe search procedure by a population of particles called a swarm. The particle is characterized byD-dimensional vector representing the position of the particle in the search space. The positionvector represents a potential solution to an optimization problem. The particles traverse the entiresolution space with a certain velocity while algorithm runs. Each particle of the swarm is associ-ated with a fitness value evaluated using the fitness function or objective function at the particlescurrent position. Each particle memorizes its individual best position and the swarm remembersthe position of the best performer among the population. The particles update their position byadding a certain velocity at each iteration. The velocity of each particle is determined by its pre-vious velocity, the distance from its individual best position (cognitive) and the distance from thebest particle in the swarm (social). A weighted combination of these three parameters gives thenew velocity [57, 59].

The basic PSO algorithm consists of three steps, i.e., generating particles’ positions and veloc-ities, velocity update, and position update. A particle refers to a point in the design space thatchanges its position from one iteration to another based on its velocity updates. The initial positionxik and velocity vik are randomly generated using the upper and lower bound of wind power, whichare shown in (5) and (6). The initialization process allows the swarm particles to be randomlydistributed across the design space.

xi0 = xmin + rand(xmax − xmin) (5)

vi0 =xmin + rand(xmax − xmin)

∆t(6)

The second step is to update the velocities of all particles at time k+1 using the particles fitness orobjective values which are functions of the particles current positions in the design space at time k.The fitness function determines which particle has the best global value in the current swarm, pgk,and also determines the best position of each particle over time. Equation (7) updates the velocityincluding uniformly distributed rand variables to ensure good coverage of design space and avoid

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Table 1: BPNN parameters for wind power forecast.

Parameters Value

Number of neurons in hidden layer 7

Learning coefficient (η) 0.9

Momentum (α) 0.2

Activation functions in hidden layer TANSIG

Activation functions in output layer PURELIN

Length of training data 30 days

Training function TRAINLM

Number of epochs 1000

Table 2: PSO network parameters for wind power forecast.

Parameters Value

Swarm size 50

Initial weight w1 0.9

Final weight w2 0.4

φ1, φ2 1.4

Max. number of iterations 100

entrapment in local optima.

vik+1 = wvik + φ1 rand1pi − xik

∆t+ φ2 rand2

pgk − xik∆t

, (7)

where vik+1 is the velocity of particle i at time k + 1, pi is the local best position, w is the inertiafactor, φ1 self confidence parameter, φ2 is the swarm confidence parameter.

A variable inertia weight was used to balance the inertia weight w between the current velocityand local/global bests. This ceases the oscillation of the particle around the optimum point [58, 61].The inertia weight w is defines as:

w = (w1 − w2)itermax − iter

itermax+ w2 (8)

where iter is the current iteration, itermax is the maximum number of iterations. Position updateis the last step in each iteration. The position of each particle as shown in Fig. 3 is updated usingvelocity vector, which is given by

xik+1 = xik + vik+1∆t (9)

3. Overview of the Proposed Forecasting System

The proposed wind power prediction strategy considers the combination of WT and NN, whichis optimized using PSO. Development of NNPSO model and the procedure for forecasting wind

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power using the proposed WT+NNPSO model is described below.

3.1. NNPSO Forecasting Model

The application of NNs technology for power system forecasting problems has received muchattention because of their capability in dealing with non-linear systems. In wind power forecastingapplications, the main function of NN is to predict wind power for the next several hours. In general,the main variable that drives the wind power is the wind speed, and their relationship is highlynonlinear. For the development of NNPSO model in this paper, we employ BPNN and its initialvalues are determined by the PSO algorithm, which is used to establish a reasonable particle modeland search space. The BPNN parameters used during training stage are presented in Table 1. Sincethe PSO searching process adjusts the velocity and position of its different dimensions, the initialweights and thresholds of BPNN should correspond with particle position. If the initial values ofNN contain weights, D-dimensional X vectors are formed while each X vector represents a particlein PSO. The PSO network parameters are given in Table 2. Following are the steps considered inthis paper to model NNPSO.

• Step−1: For initialization of particles, assign randomly m groups of initial weights andthresholds within [-1, 1] under fixed network architecture and form initial set, where m equalsthe number of particles. The position vectors and velocity vectors are separately Xk =(xk1, xk2, . . . , xkD) and Vk = (vk1, vk2, . . . , vkD), where k = 1, 2, ...,m.

• Step−2: We choose the reciprocal of mean square error (MSE) as the objective function ofthe NN in the training stage. The objective function fobj is denoted as

fobj =N

N∑i=0

(WP truei −WP forecast

i )2

, (10)

where WP truei is the actual wind power, WP forecast

i is the forecasted wind power at trainingstage, and N is the total number of data points.

• Step−3: Compare with current actual values; if fobj > pi, then pi = fobj , and if fobj > pgk,then pgk = fobj .

• Step−4: Adjust velocity and position using (7) and (9).

• Step−5: If satisfying the stopping criteria, then stop. Otherwise, return to Step 2. Thestopping criterion is given as follows.

|f (pgk)− f (pgk −m)| ≤ ε m = 1, 2, . . . , itermax, (11)

where ε is specified as 10−6.

• Step−6: Select the particle with the best fitness value from particle swarm and its posi-tion just represents the optimal initial weights and thresholds of NN under fixed networkarchitecture.

• Step−7: Completion of NN training. Input that optimal initial weights and thresholds intoa NN with fixed network architecture and complete training for wind power forecast.

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Wind power

Wavelet decomposition

Neural Network

Wavelet reconstruction

Wind power

production forecasts

D2

WPt

A3D3D1

D3 A3D2D1

WStWind speed

Wind direction

TemperatureTt

WDtParticle Swarm

Optimization

Reciprocal of MSE as

fitness function to PSO

Optimized neuron weights

of NN by using PSO

Figure 4: Flow of wind power forecasting process using hybrid WT+NNPSO model.

3.2. Proposed Wind Power Forecasting Procedure

Detailed description of WT decomposition and reconstruction process, PSO algorithm, andformulation of NNPSO model applied in this paper was described in previous section. Fig. 4 showsschematic diagram of the proposed hybrid intelligent method based on the combination of WT andNN, which is optimized using PSO algorithm. The step by step wind power forecasting procedureis explained below.

Step-1: Only the wind power data series is passed through WT and it then undergoes throughthe wavelet decomposition process. The wind power data series is decomposed into four componentsby WT. The decomposed approximation signal, A3 (low frequency component) and detail signals,D1, D2, D3 (high frequency components) are obtained from the wavelet decomposition process (seeFigs. 1 and 4).

Step-2: The decomposed wind power signals (A3, D1, D2, and D3) along with the WS, WD,and T data are then fed into the NN that is optimized using PSO algorithm. The training andoptimization process using the approximation signal (A3) in the NNPSO network is described inSection 3.1. Other detail (D1, D2, and D3) signals follow a similar training procedure. The NNPSOnetwork is trained by using the past 30 days data before the forecast day.

Step-3: The output components of the NNPSO network are individual forecast of the decom-posed approximation (A3) and detail (D1, D2, and D3) signals. Then, the process of waveletreconstruction follows (see Figs. 2 and 4) in order to produce the final wind power forecasts.

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4. Numerical Results and Discussion

This paper presents a short-term wind power forecasting methodology based on a data filteringtechnique using WT and a soft computing model based on NN that is optimized by using PSO algo-rithm. This paper uses the data of wind farms located in Pincher Creek, southern part of Alberta,Canada, as the testing system for our forecasting models. There are five wind farms in PincherCreek, namely − Castler River (44MW), Cowley Ridge (21MW), Kettles Hill (63MW), Summerview(70MW), Summerview−2 (66MW), making the total capacity of Pincher Creek as 264MW (Pmax).This paper uses the aggregated wind power of Pincher Creek. The following data are collected toestablish and verify the robustness of the model: i) aggregated hourly wind power (MW) data fromthe wind farms in Pincher Creek under consideration and ii) hourly weather data from Pincher Creekweather station that includes wind speed (in km/h), wind direction (in degree), and temperature (inCelsius). The collected data is for the period of January to December 2009. For a fair comparison,the same data and weather parameters are taken into consideration in all the forecasting modelsadopted in this paper, i.e., BPNN, RBFNN, ANFIS, WT+BPNN, WT+RBFNN, WT+ANFIS, andWT+NNPSO. In a short form, WT+SCMs represents WT+BPNN, WT+RBFNN, WT+ANFIS,and WT+NNPSO. The system operator requires hourly wind power production forecast in advancein order to carry out planning operations such as unit commitment and economic dispatch. In thispaper, wind power forecasting has been carried out using two major cases:

• Case−I: 1-hour-ahead wind power forecasts.

• Case−II: Wind power forecasts for longer forecasting horizon (3-hour, 6-hour, 12-hour and24-hour ahead).

4.1. Accuracy Measures

The principal statistics used to evaluate the performance of the proposed model is mainlymeasured by using MAPE [19]:

MAPE =1

N

N∑t=1

|WP truet −WP forecast

t |WPt

true,N× 100%, (12)

where N=24 for wind power forecasts over a 24 hour period, WP truet is the actual wind power

in hour t, WP forecastt is the predicted wind power for that hour, and WPt

true,Nis the average of

actual wind power for all the N test hours, which is given by

WPttrue,N

=1

N

N∑t=1

WP truet (13)

The reason for considering average of actual wind power in the denominator of equation (12) isthat if the actual value of wind power is small, this will contribute a large term in MAPE evenif the difference between actual and forecasted values is small. In addition, if the predicted valueis small and actual value is large, then absolute percentage error will be close to 100%. Also, inorder to avoid the adverse effect of wind power close to zero, the MAPE definition as mentioned in

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equation (12) is used in this paper. In addition, normalized root mean square error (NRMSE) isalso calculated as [24]

NRMSE(%) =

√√√√ 1

N

N∑t=1

(WP true

t −WP forecastt

WPN

)2

× 100, (14)

where WPN is the nameplate capacity of wind farm. Note that the total nameplate capacity ofPincher Creek wind farm is 264MW.

Furthermore, in order to assess the prediction capacity of the proposed hybrid intelligent model,the normalized mean absolute error (NMAE) criterion, which provides an indication of error range,is calculated as [24]

NMAE(%) =1

N

N∑t=1

|WP truet −WP forecast

t |WPN

× 100 (15)

4.2. 1-Hour-Ahead Wind Power Forecasting Results

Table 3 presents case-I results obtained from the proposed hybrid WT+NNPSO model when theforecasting look-ahead time is 24 hours, and the results are then compared with other individual(BPNN, RBFNN, NNPSO) and combined (WT+BPNN, WT+RBFNN, WT+ANFIS) SCMs aswell as compared with the persistence method, which is used as a benchmark for comparing othermethods for short-term wind power forecasting. Utilizing the MAPE definition as described by (12),it is seen from Table 3 that the MAPE values obtained from the persistence method are 10.13%,11.31%, 21.58%, and 14.79% for winter, spring, summer and fall, respectively. Note that December3 (Thursday), May 4 (Monday), July 7 (Tuesday) and October 15 (Thursday) of the year 2009have been chosen from the seasons winter, spring, summer and fall, respectively. The test resultsindicate that the performance of individual and combined SCMs (except the proposed WT+NNPSOmodel) as well as that of the persistence method are found to be inconsistent with varying MAPEsfor the multiple seasons. Especially in winter and spring, it can be observed that the forecastingperformance of the persistence method is found to be comparatively better than that of BPNN,RBFNN, ANFIS, NNPSO, WT+BPNN, WT+RBFNN, and WT+ANFIS. Particularly in winter,the MAPEs obtained from the ANFIS (14.81%) and BPNN (13.62%) models show a low degree ofperformance when compared with the persistence method. In contrary, the prediction capability ofthe persistence method is found to be worse than other individual and combines SCMs in summerand fall. Particularly in summer, the persistence method generated a very large MAPE valueof 21.56% showing its inconsistency in forecasting performance. These observations depict thatthe performances of the considered SCMs as well as that of the persistence method are seasonalsensitive and their forecasting performances are inconsistent. However, in all the simulated cases,the forecasting capability of the proposed hybrid WT+NNPSO model is not only superior than thepersistence method but also it outperforms all the considered SCMs with or without a combinationof WT. This demonstrates an effective predicting performance of the proposed hybrid model formultiple seasons with higher accuracy.

In order to have a better comparison of the proposed WT+NNPSO model with other modelsas shown in Table 3, an average value of MAPEs of the four seasons has been calculated for eachmodel, and histogram is generated for the comparison purpose as shown in Fig. 5 where we canobserve the lower value of average MAPE (8.19%) for the proposed WT+NNPSO model when

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Table 3: Comparison of errors of 1-hour-ahead forecasts over a 24 hour period.

Season ErrorModels

a∗ b∗ c∗ d∗ e∗ f∗ g∗ h∗ i∗p

Winter

MAPE 10.03 13.62 10.41 14.81 9.54 11.26 8.26 11.08 7.28

NMAE 4.18 4.32 4.61 4.75 4.09 4.29 4.19 4.55 3.87

NRMSE 5.41 5.73 5.84 6.13 5.38 5.44 5.43 5.91 5.07

Spring

MAPE 11.31 12.42 11.07 13.51 11.41 11.73 9.22 11.76 8.73

NMAE 4.58 4.88 4.61 4.71 4.51 4.67 4.18 4.39 4.11

NRMSE 6.11 6.38 5.89 6.43 6.20 6.09 5.41 6.22 5.83

Summer

MAPE 21.58 17.44 16.72 19.30 12.26 14.11 12.39 16.38 11.27

NMAE 8.48 7.46 7.21 7.75 5.94 6.97 7.04 7.18 5.29

NRMSE 11.25 9.23 8.88 10.16 7.33 8.96 8.36 9.63 7.02

Fall

MAPE 14.79 13.93 12.73 12.04 12.82 11.67 14.86 11.08 5.48

NMAE 7.48 7.26 7.31 7.76 6.85 7.14 7.08 7.32 6.17

NRMSE 9.19 8.79 8.99 9.38 7.43 8.22 8.60 8.93 7.21

a∗:persistence; b∗:BPNN; c∗:RBFNN; d∗:ANFIS; e∗:NNPSO; f∗:WT+BPNN; g∗:WT+RBFNN;h∗:WT+ANFIS; i∗p:WT+NNPSO (proposed hybrid model)

Table 4: Comparison of errors of 1-hour-ahead forecasts over a 72 hour period.

Season ErrorModels

a∗ b∗ c∗ d∗ e∗ f∗ g∗ h∗ i∗p

Winter

MAPE 13.27 14.82 11.51 16.29 10.11 13.16 10.09 13.83 8.66

NMAE 5.86 4.72 4.76 4.93 4.65 4.54 4.33 4.42 4.11

NRMSE 8.11 5.80 5.93 6.24 5.39 5.56 5.62 5.98 5.21

Spring

MAPE 9.77 12.33 10.89 14.67 11.07 11.34 9.14 12.17 9.32

NMAE 5.00 4.74 4.64 4.63 4.28 4.52 4.10 4.42 4.13

NRMSE 6.71 6.40 5.82 6.46 6.15 5.94 5.53 6.16 5.64

Summer

MAPE 17.46 19.87 17.93 20.17 14.79 17.29 15.90 18.63 12.38

NMAE 5.36 7.21 7.33 8.16 5.99 7.08 7.10 7.56 5.17

NRMSE 8.12 9.82 9.07 9.56 7.83 9.19 8.46 9.12 7.18

Fall

MAPE 18.29 14.67 12.92 17.94 13.77 13.40 11.85 15.79 12.61

NMAE 7.61 7.63 7.83 7.96 7.11 7.38 7.23 7.68 6.55

NRMSE 9.90 9.14 9.20 9.42 7.86 8.43 8.74 9.15 7.32

a∗:persistence; b∗:BPNN; c∗:RBFNN; d∗:ANFIS; e∗:NNPSO; f∗:WT+BPNN; g∗:WT+RBFNN;h∗:WT+ANFIS; i∗p:WT+NNPSO (proposed hybrid model)

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0

5

10

15

20

Models

MA

PE

(%)

: Persistence

: BPNN : RBFNN

: ANFIS : NNPSO

: WT+NNPSO(proposed)

: WT+BPNN

: WT+RBFNN

: WT+ANFIS

Figure 5: Average MAPE histogram comparing 1-hour-ahead forecasting performance of the pro-posed model with tested alternatives over a 24 hour period.

0

5

10

15

20

Models

MA

PE

(%)

: Persistence

: BPNN : RBFNN

: ANFIS : NNPSO

: WT+NNPSO(proposed)

: WT+BPNN

: WT+RBFNN

: WT+ANFIS

Figure 6: Average MAPE histogram comparing 1-hour-ahead forecasting performance of the pro-posed model with tested alternatives over a 72 hour period.

compared with the persistence (14.42%), BPNN (14.35%), RBFNN (12.73%), ANFIS (14.91%),NNPSO (11.50%), WT+BPNN (12.19%), WT+RBFNN (11.18%), and WT+ANFIS (12.57%). Inother words, the percentage error improvement due to the proposed hybrid WT+NNPSO modelover the tested alternatives are in the range of around 26-45%. Table 3 also presents NRMSE and

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NMAE results obtained for daily wind power forecast. The average NRMSE (6.28%) of the proposedWT+NNPSO shows the best prediction capability over other models. With the application ofWT+SCM, the NMAEs are greatly improved as it can be observed in Table 3 that when WTis combined with NNPSO, the average NMAE value is found to be lower (4.86%) than that of asingle use of NNPSO (5.34%). The test results demonstrate that optimizing the NN by using PSOalgorithm and combining with WT, i.e., WT+NNPSO model certainly produces a higher degree ofwind power forecasting accuracy when compared not only with an individual SCM(s) but also withother hybrid SCMs as well as the persistence method. The improved forecasting accuracy is becauseof a significant impact of the WT on volatile wind power time-series and the NNPSO model capturesthe non-linear wind power fluctuation in a better way as PSO is a robust stochastic optimizationalgorithm. These attributes make the proposed hybrid intelligent forecasting algorithm robust andhighly efficient, and as a result, this forecasting system has the advantages of higher accuracy.

The predicting performance of the proposed model is further validated by carrying out hour-ahead forecasting considering the forecasting look-ahead time up to three days. The seasonalaverage MAPEs obtained from the persistence method is 14.69%, whereas this value is compara-tively lower (10.74%) using the proposed WT+NNPSO method. In other words, the benchmarkpersistence method is outperformed by our proposed model in three-day forecasts as well. As statedearlier, the wind power data series contain a various fluctuations, spikes, and different types of non-stationarities, and the WT is used as filtering those spikes. In Table 4, we can see that the averageseasonal MAPEs obtained from the BPNN, RBFNN, ANFIS, and NNPSO models are 15.42%,13.31%, 17.26%, and 12.43%, respectively. However, when these individual SCMs are combinedwith WT, the MAPEs are found to be reduced by around 10%, 11%, 12%, and 13%, respectively.This demonstrates the effectiveness of using WT. The forecasting performances of these models arefurther compared by calculating NRMSE and NMAE as shown in Table 4. Note that in all thesimulated cases, the test results demonstrate the superior predicting performance of the proposedWT+NNPO model over the persistence method, individual SCMs, and other WT+SCMs. Fig. 6shows the average three-day seasonal MAPE comparison for all wind power forecasting models. Itis clear from the Fig. 6 that the proposed hybrid WT+NNPSO forecasting model outperforms allother forecasting models. It is evident that the combination of the data filtering effect due to WTon stochastic wind power time-series and the PSO based optimization technique to enhance theforecasting performance of the NN is able to predict wind power efficiently.

4.3. Wind Power Forecasting Results for Longer Forecasting Horizons

As there is a need to forecast wind power accurately at various time spans, it is also important tostudy the performance of the forecasting models for different forecasting horizons. Hence, to showan effectiveness of the proposed hybrid WT+NNPSO model, this paper further reports resultsfor longer forecasting horizons, i.e., k-step-ahead wind power forecasts, where k=3, 6, 12, and 24,and the results are compared with other WT+SCMs as presented in Table 5. As expected, theMAPE values start deteriorating when the forecasting horizon or the value of k increases. This isdue to propagation of the error. It is found that the 24-hour-ahead (or day-ahead) MAPE valuesare more than 20% in all the seasons. As Table 5 presents the errors comparison among varioushybrid forecasting models, the proposed hybrid WT+NNPSO model outperforms all other hybridmodels for longer forecasting horizons. The proposed forecasting model utilizes the actual dataup to time “t” only and produces the k-step-ahead wind power forecasts. Figs. 7 and 8 comparethe forecasting performance of the proposed model with other hybrid models in terms of MAPEfor winter and spring, respectively. From Figs. 7 and 8, it is confirmed that the proposed hybrid

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Table 5: Errors comparison of the proposed model with other hybrid models in the case of longerforecasting horizons.

Season Error3-hour-ahead 6-hour-ahead 12-hour-ahead 24-hour-ahead

a∗ b∗ c∗ d∗ a∗ b∗ c∗ d∗ a∗ b∗ c∗ d∗ a∗ b∗ c∗ d∗

Winter

MAPE 13.52 10.25 12.31 9.79 16.18 14.29 16.98 13.81 21.06 18.73 22.64 18.16 25.11 23.59 27.44 23.19

NMAE 4.87 4.77 4.83 4.21 5.34 5.15 5.67 4.91 6.17 5.70 6.61 5.72 7.25 6.57 7.19 6.42

NRMSE 5.91 5.83 6.46 5.64 6.50 6.33 7.32 5.97 7.82 7.34 7.84 6.83 8.73 8.62 9.28 8.14

Spring

MAPE 12.95 11.83 14.64 10.09 17.13 14.23 18.16 14.20 21.69 17.91 23.11 17.88 27.92 24.67 29.72 23.81

NMAE 4.93 4.44 5.02 4.32 5.61 4.83 5.74 4.73 6.28 5.33 6.27 5.09 7.04 6.27 7.38 5.96

NRMSE 6.72 5.90 6.84 5.97 7.17 6.45 7.45 6.53 7.62 6.86 7.92 6.94 8.62 7.99 9.15 7.90

Summer

MAPE 16.43 15.75 19.46 13.28 19.53 17.72 22.80 16.04 23.76 20.38 25.96 20.39 27.57 24.37 31.71 25.82

NMAE 7.68 7.57 7.74 5.86 7.98 7.94 7.83 6.29 8.43 8.17 8.51 6.73 8.72 8.52 9.32 7.41

NRMSE 9.48 8.86 9.85 7.64 9.73 9.33 10.28 8.15 9.97 9.58 10.75 8.81 10.44 9.94 11.07 9.82

Fall

MAPE 14.12 14.70 17.68 13.82 17.28 18.22 20.77 16.54 22.61 21.69 24.71 19.72 26.84 26.93 28.67 24.73

NMAE 7.65 7.66 8.10 6.72 7.74 7.95 8.43 7.04 8.09 8.36 8.86 7.64 8.58 9.24 9.22 7.95

NRMSE 8.73 8.94 9.37 7.75 9.08 9.45 9.82 7.97 9.55 9.72 10.53 8.89 9.83 10.46 10.84 9.33

a∗: WT+BPNN; b∗: WT+RBFNN; c∗: WT+ANFIS; d∗: WT+NNPSO (proposed hybrid model)

5

10

15

20

25

30

MA

PE

(%)

3-hrahead

6-hrahead

12-hrahead

24-hrahead

WT+BPNNWT+RBFNNWT+ANFIS

WT+NNPSO(proposed)

Figure 7: Comparison of hybrid models for longer forecasting horizons in winter.

intelligent forecasting model performs better even in longer horizon cases. However, it can alsobe observed in Figs. 7 and 8 that MAPE deteriorates as the hour-ahead increases and such trendis expected in time-series modeling. Though the curves apparently look linear trend due to scaleconstraint, the curves are non-linear in nature. In this paper, cases−I and −II are carried out byrandomly choosing the days of the year 2009. It is emphasized that presented test results are onlyrepresentative and several other results were obtained, which have not been reported in this paper.The average computation time required by the proposed hybrid WT+NNPSO model for short-term(hour-ahead) daily wind power forecasts is around 1−4 minute using MATLAB on a PC with 4 GBof RAM and a 2.7-GHz-based processor.

Forecasting short-term wind power with a higher rate of accuracy is extremely important for the

16

5

10

15

20

25

30

WT+BPNNWT+RBFNNWT+ANFIS

WT+NNPSO(proposed)

MA

PE

(%)

3-hrahead

6-hrahead

12-hrahead

24-hrahead

Figure 8: Comparison of hybrid models for longer forecasting horizons in spring.

power system operators as they face challenges associated with fluctuating wind power productionwith the increasing installed wind power capacity. In this paper, the developed forecasting strategyis composed of an efficient data filtering technique based on WT and a SCM based on NN, which isoptimized by PSO. The presented data filtering technique is capable of filtering out non-stationarywind power data series effectively. The forecast engine is comprised of NN and by hybridizingit with stochastic search technique, i.e., PSO, its forecasting performance is greatly enhanced,thus achieving a higher degree of prediction accuracy as demonstrated by the test results with asignificant improvement over the tested alternatives.

The model developed in this paper is considered for short-term forecasting (1-hour ahead).Traditionally, computational fluid dynamics predictions are not considered for short-term predictiondue to difficulties in model parameter data acquisition and computation. Therefore, in future work,the models developed in this paper can be used as a complement to the numerical weather predictionmodel (NWP) models to compensate for their ineffectiveness for short-term predictions of less than6 hours. It can be beneficial to examine the benefit of combining the NWP model with the proposedtechniques in this study. It is expected that, this hybrid forecasting system will take advantage ofthe higher accuracy of the proposed SCMs in shorter forecast horizons and the advantage from theNWP model for longer forecast horizons.

5. Conclusions

This paper presented a hybrid intelligent approach for predicting short-term wind power pro-duction. The NNPSO based prediction model combined with the WT technique was proposed toforecast aggregated wind power production of multiple wind farms. The proposed WT+NNPSOmodel was rigorously compared with other models such as BPNN, RBFNN, ANFIS, NNPSO,WT+BPNN, WT+RBFNN, WT+ANFIS including the benchmark persistence method. In all thetest cases, the proposed model showed superior performance over the tested alternatives. The pre-sented work contributed to alleviate an important problem of wind power forecasting as the testresults obtained through the simulation demonstrate that the proposed hybrid intelligent algorithm

17

is more accurate, efficient, capable of transforming the numerical weather data into the power out-put of wind farms, and it also performs well in multiple seasons. The possible future work wouldbe to consider other weather parameters such as humidity and pressure in the forecast process andcarry out uncertainty associated with wind power forecasting.

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References

[1] A. Kusiak, H. Zheng, Z. Song, Short-term prediction of wind farm power: A data miningapproach, Energy Conversion, IEEE Transactions on 24 (1) (2009) 125 – 136.

[2] C. Potter, M. Negnevitsky, Very short-term wind forecasting for Tasmanian power generation,IEEE Transactions on Power Systems 21 (2) (2006) 965 – 972.

[3] M. Negnevitsky, P. Mandal, A. Srivastava, An overview of forecasting problems and techniquesin power systems, in: Power Energy Society General Meeting, 2009. PES ’09. IEEE, 2009, pp.1 – 4.

[4] S. Soman, H. Zareipour, O. Malik, P. Mandal, A review of wind power and wind speed fore-casting methods with different time horizons, in: North American Power Symposium (NAPS),2010, pp. 1 – 8.

[5] G. Giebel, R. Brownsword, G. Kariniotakis, M. Denhard, C. Draxl, The State-Of-The-Art inShort-Term Prediction of Wind Power: A Literature Overview, ANEMOS.plus, 2011.

[6] M. Lei, L. Shiyan, J. Chuanwen, L. Hongling, Z. Yan, A review on the forecasting of windspeed and generated power, Renewable and Sustainable Energy Reviews 13 (4) (2009) 915 –920.

[7] S. Li, D. Wunsch, E. O’Hair, M. Glesselmann, Using neural networks to estimate wind turbinepower generation, in: Power Engineering Society Winter Meeting, IEEE, Vol. 3, 2001, p. 977vol.3.

[8] T. Barbounis, J. Theocharis, M. Alexiadis, P. Dokopoulos, Long-term wind speed and powerforecasting using local recurrent neural network models, Energy Conversion, IEEE Transactionson 21 (1) (2006) 273 – 284.

[9] M. Mabel, E. Fernandez, Estimation of energy yield from wind farms using artificial neuralnetworks, Energy Conversion, IEEE Transactions on 24 (2) (2009) 459 – 464.

[10] S. Kelouwani, K. Agbossou, Nonlinear model identification of wind turbine with a neuralnetwork, Energy Conversion, IEEE Transactions on 19 (3) (2004) 607 – 612.

[11] K. Methaprayoon, C. Yingvivatanapong, W.-J. Lee, J. Liao, An integration of ANN windpower estimation into unit commitment considering the forecasting uncertainty, Industry Ap-plications, IEEE Transactions on 43 (6) (2007) 1441 – 1448.

[12] R. Bessa, V. Miranda, J. Gama, Entropy and correntropy against minimum square error inoffline and online three-day ahead wind power forecasting, Power Systems, IEEE Transactionson 24 (4) (2009) 1657 – 1666.

[13] M. Alexiadis, P. Dokopoulos, H. Sahsamanoglou, Wind speed and power forecasting based onspatial correlation models, Energy Conversion, IEEE Transactions on 14 (3) (1999) 836 – 842.

[14] G. Kariniotakis, G. Stavrakakis, E. Nogaret, Wind power forecasting using advanced neuralnetwork models, IEEE Transactions on Energy Conversion 11 (4) (1996) 762 – 767.

19

[15] I. Damousis, M. Alexiadis, J. Theocharis, P. Dokopoulos, A fuzzy model for wind speed predic-tion and power generation in wind parks using spatial correlation, Energy Conversion, IEEETransactions on 19 (2) (2004) 352 – 361.

[16] P. Pinson, G. Kariniotakis, Conditional prediction intervals of wind power generation, PowerSystems, IEEE Transactions on 25 (4) (2010) 1845 – 1856.

[17] G. Sideratos, N. Hatziargyriou, An advanced statistical method for wind power forecasting,Power Systems, IEEE Transactions on 22 (1) (2007) 258 – 265.

[18] T. El-Fouly, E. El-Saadany, M. Salama, Improved grey predictor rolling models for wind powerprediction, Generation, Transmission Distribution, IET 1 (6) (2007) 928 – 937.

[19] J. Catalao, H. Pousinho, V. Mendes, Hybrid wavelet-PSO-ANFIS approach for short-termwind power forecasting in Portugal, IEEE Transactions on Sustainable Energy 2 (1) (2011) 50– 59.

[20] S. Fan, J. Liao, R. Yokoyama, L. Chen, W.-J. Lee, Forecasting the wind generation using a two-stage network based on meteorological information, Energy Conversion, IEEE Transactions on24 (2) (2009) 474 – 482.

[21] N. Amjady, F. Keynia, H. Zareipour, A new hybrid iterative method for short-term wind speedforecasting, European Transactions on Electrical Power 21 (1) (2011) 581 – 595.

[22] A. Fabbri, T. Roman, J. Abbad, V. Quezada, Assessment of the cost associated with wind gen-eration prediction errors in a liberalized electricity market, Power Systems, IEEE Transactionson 20 (3) (2005) 1440 – 1446.

[23] P. Pinson, C. Chevallier, G. Kariniotakis, Trading wind generation from short-term proba-bilistic forecasts of wind power, Power Systems, IEEE Transactions on 22 (3) (2007) 1148 –1156.

[24] N. Amjady, F. Keynia, H. Zareipour, Wind power prediction by a new forecast engine composedof modified hybrid neural network and enhanced particle swarm optimization, SustainableEnergy, IEEE Transactions on 2 (3) (2011) 265 – 276.

[25] N. Ullah, K. Bhattacharya, T. Thiringer, Wind farms as reactive power ancillary serviceproviders–technical and economic issues, Energy Conversion, IEEE Transactions on 24 (3)(2009) 661 – 672.

[26] M. A. Mohandes, S. Rehman, S. M. Rahman, Spatial estimation of wind speed, InternationalJournal of Energy Research 36 (4) (2012) 545 – 552.

[27] A. Z. Sahin, A. Aksakal, A statistical analysis of wind energy potential at the eastern regionof saudi arabia, International Journal of Energy Research 23 (10) (1999) 909 – 917.

[28] A. Z. Al-Garni, A. Z. Sahin, A. Al-Farayedhi, Modelling of weather characteristics and windpower in the eastern part of saudi arabia, International Journal of Energy Research 23 (9)(1999) 805 – 812.

[29] G. Tina, S. Gagliano, Probabilistic analysis of weather data for a hybrid solar/wind energysystem, International Journal of Energy Research 35 (3) (2011) 221 – 232.

20

[30] S. Alpay, L. Bilir, S. Ozdemir, B. Ozerdem, Wind speed time series characterization by hilberttransform, International Journal of Energy Research 30 (5) (2006) 359 – 364.

[31] Y. P. Cai, G. H. Huang, Z. F. Yang, Q. G. Lin, B. Bass, Q. Tan, Development of an optimizationmodel for energy systems planning in the region of waterloo, International Journal of EnergyResearch 32 (11) (2008) 988 – 1005.

[32] A. Mackillop, Canada’s energy options, International Journal of Energy Research 2 (4) (1978)307 – 336.

[33] Y.-S. Jang, D.-J. Lee, H.-S. Oh, Evaluation of new and renewable energy technologies in koreausing real options, International Journal of Energy Research.

[34] J. Rogers, K. Porter., Central wind power forecasting programs in North America by regionaltransmission organizations and electric utilities (2009, Dec.).URL http://www.uwig.org/WindForecastingbyTSOs.pdf

[35] J. Rogers, K. Porter., Wind power production forecasting for CA ISO PIRP (2005, Mar.).URL http://www.caiso.com/docs/2005/03/22/2005032215434015268.pdf

[36] D.Eldson., Wind forecasting and curtailment experience at NYISO (2009, Oct.).URL http://www.uwig.org/iowawork/Edelson.pdf

[37] J. Zack., AWSTs implementation of wind power production forecasting for ERCOT (2008,Mar.).URL http://www.ercot.com

[38] New york independent system operator (NYISO) (2010).URL http://www.ferc.gov/industries/electric/indus-act/rto/metrics/nyiso-rto-metrics.pdf

[39] A. Botterud, Z. Zhou, J. Wang, R. Bessa, H. Keko, J. Sumaili, V. Miranda, Wind power tradingunder uncertainty in LMP markets, Power Systems, IEEE Transactions on 27 (2) (2012) 894– 903.

[40] P. Pinson, C. Chevallier, G. Kariniotakis, Trading wind generation from short-term proba-bilistic forecasts of wind power, Power Systems, IEEE Transactions on 22 (3) (2007) 1148 –1156.

[41] C. Wang, Z. Lu, Y. Qiao, A consideration of the wind power benefits in day-ahead schedulingof wind-coal intensive power systems, Power Systems, IEEE Transactions on 28 (1) (2013) 236– 245.

[42] M. Mabel, E. Fernandez, Estimation of energy yield from wind farms using artificial neuralnetworks, Energy Conversion, IEEE Transactions on 24 (2) (2009) 459 – 464.

[43] J. Catalao, H. Pousinho, V. Mendes, Hybrid intelligent approach for short-term wind powerforecasting in portugal, Renewable Power Generation, IET 5 (3) (2011) 251 – 257.

[44] R. Welch, S. Ruffing, G. Venayagamoorthy, Comparison of feedforward and feedbackneural network architectures for short term wind speed prediction, in: Neural Net-works, 2009. IJCNN 2009. International Joint Conference on, 2009, pp. 3335 – 3340.doi:10.1109/IJCNN.2009.5179034.

21

[45] J. Catalao, H. Pousinho, V. Mendes, Hybrid Wavelet-PSO-ANFIS approach for short-termelectricity prices forecasting, Power Systems, IEEE Transactions on 26 (1) (2011) 137 – 144.

[46] D. Faria, R. Castro, C. Philippart, A. Gusmao, Wavelets pre-filtering in wind speed predic-tion, in: Power Engineering, Energy and Electrical Drives, POWERENG ’09. InternationalConference on, 2009, pp. 168 – 173.

[47] A. Khan, M. Shahidehpour, One day ahead wind speed forecasting using wavelets, in: PowerSystems Conference and Exposition, PSCE ’09. IEEE/PES, 2009, pp. 1 – 5.

[48] C. Lei, L. Ran, Short-term wind speed forecasting model for wind farm based on waveletdecomposition, in: Electric Utility Deregulation and Restructuring and Power Technologies,DRPT. Third International Conference on, 2008, pp. 2525 – 2529.

[49] A. Conejo, M. Plazas, R. Espinola, A. Molina, Day-ahead electricity price forecasting using thewavelet transform and ARIMA models, Power Systems, IEEE Transactions on 20 (2) (2005)1035 – 1042.

[50] P. Pinson, H. Madsen, Adaptive modelling and forecasting of offshore wind power fluctuationswith Markov-switching autoregressive models, Journal of Forecasting 31 (4).

[51] N. Amjady, F. Keynia, Day ahead price forecasting of electricity markets by a mixed data modeland hybrid forecast method, International Journal of Electrical Power & Energy Systems 30 (9)(2008) 533 – 546.

[52] S. Mallat, A theory for multiresolution signal decomposition: the wavelet representation, Pat-tern Analysis and Machine Intelligence, IEEE Transactions on 11 (7) (1989) 674 – 693.

[53] A. Pandey, D. Singh, S. Sinha, Intelligent hybrid wavelet models for short-term load forecasting,Power Systems, IEEE Transactions on 25 (3) (2010) 1266 – 1273.

[54] Y. Chuanan, Y. Yongchang, A hybrid model to forecast wind speed based on wavelet andneural network, in: Control, Automation and Systems Engineering (CASE), 2011 InternationalConference on, 2011, pp. 1 – 4.

[55] A. Reis, A. da Silva, Feature extraction via multiresolution analysis for short-term load fore-casting, Power Systems, IEEE Transactions on 20 (1) (2005) 189 – 198.

[56] R. Eberhart, J. Kennedy, A new optimizer using particle swarm theory, in: Micro Machineand Human Science, MHS ’95., Proceedings of the Sixth International Symposium on, 1995,pp. 39 – 43.

[57] J. Kennedy, R. Eberhart, Particle swarm optimization, in: Neural Networks, 1995. Proceed-ings., IEEE International Conference on, Vol. 4, 1995, pp. 1942 – 1948.

[58] Y. Shi, R. Eberhart, Empirical study of particle swarm optimization, in: Evolutionary Com-putation, 1999. CEC 99. Proceedings of the 1999 Congress on, Vol. 3, 1999, pp. 3 vol.(xxxvii+2348).

[59] D. Boeringer, D. Werner, Particle swarm optimization versus genetic algorithms for phasedarray synthesis, Antennas and Propagation, IEEE Transactions on 52 (3) (2004) 771 – 779.

22

[60] Z. Bashir, M. El-Hawary, Applying wavelets to short-term load forecasting using PSO-basedneural networks, Power Systems, IEEE Transactions on 24 (1) (2009) 20 – 27.

[61] Y. del Valle, G. Venayagamoorthy, S. Mohagheghi, J.-C. Hernandez, R. Harley, Particle swarmoptimization: Basic concepts, variants and applications in power systems, Evolutionary Com-putation, IEEE Transactions on 12 (2) (2008) 171 – 195.

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