Harmonic distortion monitoring for nonlinear loads using neural-network-method

Applied Soft Computing 13 (2013) 475–482

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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc

Harmonic distortion monitoring for nonlinear loads using neural-network-method Claudionor Francisco Nascimento a , Azauri Albano Oliveira Jr. b , Alessandro Goedtel c,∗ , Alvaro Batista Dietrich a a

Federal University of ABC (UFABC), CECS, R. Santa Adelia, 166, 09210-170 Santo Andre, SP, Brazil University of São Paulo (USP), Elect. Eng. Dept., Av. Trab. São-carlense, 400, 13566-590 São Carlos, SP, Brazil c Federal University of Technology (UTFPR), Elect. Eng. Dept., Av. Alberto Carazzai, 1640, 86300-000 Cornélio Procópio, PR, Brazil b

a r t i c l e

i n f o

Article history: Received 14 January 2012 Received in revised form 13 June 2012 Accepted 17 August 2012 Available online 5 September 2012 Keywords: Artificial neural networks Distortion measurement Power quality Total harmonic distortion estimation

a b s t r a c t Nowadays, harmonic distortion in electric power systems is a power quality problem that has been attracting significant attention of engineering and scientific community. In order to evaluate the total harmonic distortion caused by particular nonlinear loads in power systems, the harmonic current components estimation becomes a critical issue. This paper presents an efficient approach to distortion metering, based on artificial neural networks applied to harmonic content estimation of load currents in singlephase systems. The harmonic content is computed using the estimation of amplitudes and phases of the first five odd harmonic components, which are carried out considering the waveform variations of current drained by nonlinear loads, within previously known limits. The proposed online monitoring method requires low computational effort and does not demand a specific number of samples per period at a fixed sampling rate, resulting in a low cost harmonic tracking system. The results from neural networks harmonic identification method are compared to the truncated fast Fourier transform algorithm. Besides, simulation and experimental results are presented to validate the proposed approach. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Power electronic converters are widespread in industrial, commercial, and home applications, increasing the interest in power quality (PQ) issues [1]. Such devices are considered harmonicproducing nonlinear loads, draining nonsinusoidal currents, and can lead to disturbances in AC electric power systems, deteriorating its PQ indices [2–6]. Nonlinear single-phase loads of low power, like ballasts and personal computer power supplies, are used on large-scale in commercial buildings and can cause significant harmonic distortion problems, affecting all loads connected to the point of common coupling (PCC) of a power system. The presence of nonlinear loads in power systems can make the correct quantification of power flows difficult [5]. Harmonic content metering has been used to characterize nonlinear loads behavior, to locate harmonic sources and to quantify the harmonic distortion in power systems [2]. Analysis of waveform distortion in power systems requires a comprehensive and precise

∗ Corresponding author. Tel.: +55 43 35204096; fax: +55 43 35204010. E-mail addresses: [email protected] (C.F. Nascimento), [email protected] (A.A. Oliveira Jr.), [email protected] (A. Goedtel), [email protected] (A.B. Dietrich). 1568-4946/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asoc.2012.08.043

analytical basis in the following subjects: (i) linear loads; (ii) nonlinear loads; (iii) equipment; and (iv) steady-state waveform Fourier analysis. Researches on smart grid aim the evolution of conventional electrical power systems, increasing its controllability and reliability, concomitantly with the ability to connect renewable sources up using power electronic converters. In the implementation of a smart grid, the companies should be able to provide high quality voltage to its consumers, with well-controlled levels of harmonic pollution. Harmonics metering capability for smart meters have been proposed for monitoring and analysis issues in control centers of smart power system [7–9]. Moreover, technologies based on intelligent systems such as artificial neural networks (ANNs), have been integrated in smart meters, in sensors, and in pulse-width modulation (PWM) drivers as a result of the technological progress of online applications in power systems [9–11]. In a smart grid, a PQ online monitoring system can deal with techniques for the determination of harmonic content in distorted voltage power systems [10]. Besides, in some cases, mitigation devices are required to maintain the electric energy quality supplied to end-users [12]. Therefore, even in new concepts of electric power distribution, there is a concern about current total harmonic distortion (THD) and other PQ indices [13]. The THD is the most common harmonic index used to evaluate and meter the conditions of PQ variations under nonsinusoidal conditions.

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In order to compensate or to mitigate harmonic distortion in power systems, some solutions recently presented include the use of active power filters (APFs), whose performance is directly dependent on efficient harmonic content identification algorithms [14–17]. Such identification is performed using two approaches: time domain [15] and frequency domain [17]. The latter method uses traditional techniques for signal spectral analysis as Fourier series and discrete Fourier transform (DFT). The DFT has certain drawbacks for the harmonic analysis of a signal such as spectral leakage, demanding the exact number of AC power supply voltage periods to sample the current signal and the assumption of correct fundamental frequency [18,19]. Usually, the harmonic analysis of electric power system is carried out using fast Fourier transform (FFT) which is a powerful tool, but also subjected to specific restrictions, like phase errors [19–21]. Alternative methods to FFT analysis are found in recent works, which depict ANNs applied to harmonic content determination in three-phase systems and in single-phase systems [14]. In [18] it was reported that ANNs are able to evaluate PQ indices successfully. Nowadays, ANNs are widely used in power electronics and electric power systems issues, including control systems for PQ devices, unified power flow controller (UPFC), and field-programmable gate arrays (FPGA) [22–26]. Harmonic identification using the ANN-based method has some advantages as simplified algorithms and low computational effort [10,18,19]. Such characteristics and the parallel computing architecture of ANN lead to improved software speed and reliability. In commercial and home electric systems, or even in lighting and administrative portions of industrial plants, several kinds of single-phase nonlinear loads connected to balanced three-phase systems are observed [27]. Some of these loads show steady-state current waveforms with a well-known theoretical behavior, e.g. single-phase AC voltage controllers for bulb lamps (dimmers) distributed in four-wire three-phase systems. Nevertheless, variations in controller circuit parameters and in its electrical loads can lead to uncertainties in the resulting current distortion and, consequently, on its harmonic content. Considering a dimmer as an example, such variations are firing angle choice, dependent on human adjustment, and resistance variations of lamp filament, due to its temperature [28,29]. In APF for distributed power harmonic compensation, sampling of current signal with these uncertainties in waveforms leads to increased computation efforts and algorithm complexity, if traditional methods are used [10,18]. In the reviewed literature, harmonic analyses of nonlinear loads are often done without considering steady-state current variations [30]. This work proposes a neural-network-method for current harmonic content identification and THD estimation in single-phase power systems, feeding nonlinear loads that can be analytically modeled on its theoretical behavior, but with uncertainties in model parameters. This ANN method can be used, e.g. in smart meters and APF control system, for selective compensation of some specific and critical harmonics [9,11,20]. The proposed methodology demands previous characterization of nonlinear loads to produce the training patterns for the ANN structure, which is subsequently trained in offline form [1]. Training stage of ANN follows a cross-validation method, hereinafter referred to as double cross-validation method [31,32], which is based on analytical and experimental results to validate the proposed approaches. After the training process, the ANN performs the online identification of harmonic coefficients of load current and the THD computation, for each half-period of voltage source waveform. The neural network converges within waveform variations previously trained. This paper has the following organization: Section 2 describes the nonlinear loads and the single-phase system characteristics;

Fig. 1. Modeled single-phase system which comprises two different nonlinear loads draining iL1 and iL2 currents.

Section 3 discusses the principles of neural approach in harmonic identification; Section 4 shows how the harmonic content identification is carried out; finally, the conclusions drawn from the results of this study are addressed in Section 5. 2. Nonlinear load characterization Characterization and modeling of harmonic sources are important steps in harmonics and THD studies [4,5,12,33–35]. Two kinds of nonlinear loads, which require attention due to its harmonic characteristics, are modeled in this section: the AC controller, applied as dimmers in lighting control, and the full-bridge diode rectifier with output capacitive filter, widely used in AC/DC converters of home appliances and electronics [27,29]. The neural-network-method proposed can be used for nonlinear loads connected to balanced four-wire three-phase systems or connected to single-phase systems. In this work, simulation and tests are carried out considering only the single-phase system shown in Fig. 1, which comprises two nonlinear loads: an AC controller, acting as dimmer for incandescent lamps, and a full-bridge single-phase rectifier, feeding a RC linear load. The resistance RL is composed by three 100 W/220 V incandescent lamps, and the rectifier output has a 470 ␮F capacitor filter and a resistive load of 730 . Both nonlinear loads are fed by a

Fig. 2. Sampling process of load current.

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purely sinusoidal power supply vs (t). The switches S1 and S2 select which loads will contribute for the total current at PCC. Therefore iL (t) = iL1 (t) when only S1 is turned on, iL (t) = iL2 (t) when only S2 is turned on and iL (t) = iL1 (t) + iL2 (t) when both S1 and S2 are turned on. The current signal at PCC is sensed by a hall sensor and is sampled in half-period of voltage source as shown in Fig. 2. The dimmer circuit controls its load power by adjusting the TRIAC A firing angle (˛f ), which is defined as shown in Fig. 3(a). Depending on the dimmer setting point, iL1 waveform varies for firing angle changes and also due to RL resistance variations. Changes in RL occur as a result of lamp (light bulb) filament temperature which varies according to dimmer output RMS voltage. Fig. 3(b) shows the simulated waveform of the input current of the rectifier (iL2 ). All simulation results presented in this work were obtained using MatLab/Simulink software. Analytical harmonic characteristics of system loads are evaluated by Fourier series, by (1)–(4):

 ∞

x(t) =

[An cos(2nf0 t) + Bn sin(2nf0 t)]

(1)

n=1

In this case the Fourier series can be represented as: iL2 (t)

2 =  +



+

k0 k52 + 1

∞  



k4 + 2

∞  

k4 + 2 k3 + 2



k0

n+1

n=3

−



cos(n − 1)˛f − cos(n − 1)

cos nωt

n−1

∞   sin(n + 1)˛f

+

n=3

n+1

−

sin(n − 1)˛f





n−1

sin nωt (5)

The resistance RL of the incandescent lamp set varies in function of filament temperature and, therefore, in function of the dimmer firing angle (˛f ), as described in Fig. 4 [1]. When S2 is turned on and S1 is turned off, the load current at PCC is the rectifier current (iL2 ), as shown in Fig. 3(b), and its Fourier series is given by (6):

[e−k5 ˇ [−k5 cos(nˇ) − n sin(nˇ)] − e−k5 ˛ [−k5 cos(n˛) − n sin(n˛)]]

sin(n − 1)ˇ − sin(n − 1)˛ sin(n + 1)ˇ − sin(n + 1)˛ + n+1 n−1 k0

(6)

 cos(nωt)

cos(n − 1)˛ − cos(n − 1)ˇ cos(n + 1)˛ − cos(n + 1)ˇ + n+1 n−1



[e−k5 ˇ [−k5 sin(nˇ) − n cos(nˇ)] − e−k5 ˛ [−k5 sin(n˛) − n cos(n˛)]]

sin(n − 1)ˇ − sin(n − 1)˛ sin(n + 1)˛ − sin(n + 1)ˇ + n+1 n−1

(2)

where the amplitudes and phase angles are given by: A2n + Bn2

(3)

n

(4)

Bn

∞   cos(n + 1)˛f − cos(n + 1)

+



Cn sin(2nf0 t + n )

n = tan−1

1 [cos 2˛f − 1]cos ωt+ [sin 2˛f + 2 − 2˛f ]sin ωt 2

[e−k5 ˇ [−k5 cos(ˇ) − sin(ˇ)] − e−k5 ˛ [−k5 cos(˛) − sin(˛)]]

n=1

A 

2



k52 + n2

n=3



1

cos(n − 1)ˇ − cos(n − 1)˛ cos(n + 1)˛ − cos(n + 1)ˇ + n+1 n−1

+

Cn =

V RL

=

[e−k5 ˇ [−k5 sin(ˇ) − cos(ˇ)] − e−k5 ˛ [−k5 sin(˛) − cos(˛)]]

k52 + n2

n=3

k3 + 2

x(t) =

iL1 (t)

k3 k4 [cos(2˛) − cos(2ˇ)] + [sin(2˛) − sin(2ˇ) + 2ˇ − 2˛] sin(ωt) 4 4

+

∞ 

to dimmer current (iL1 ), as illustrated in Fig. 3(a). Eq. (5), obtained from (1)–(4), is the Fourier series for the nth order harmonic (n odd) where V is the peak voltage and ω = 2f0 .

k4 k3 [cos(2˛) − cos(2ˇ)] + [sin(2ˇ) − sin(2˛) + 2ˇ − 2˛] cos(ωt) 4 4

 +

k0 k52 + 1

477

where n is a non-zero integer; f0 is the fundamental frequency; x(t) is the Fourier series term; An and Bn are the coefficients for the nth order harmonic; Cn is the amplitude for the nth order harmonic; n is the phase angle for the nth order harmonic. In the single-phase system depicted in Fig. 1, when S1 is turned on and S2 is turned off, the resulting load current at PCC equals





 sin(nωt)

where ˛ is the delay angle and ˇ is the extinction angle of diodes, defined by (7) and (8), being both constant due to the constant RC load. Remaining coefficients in (6) are determined by (9)–(14). The R2 resistance in (10) is the rectifier input resistance, which models conductors and connections between power supply and rectifier input. Such resistances decrease the peak current drained by the rectifier, and thus must be considered in the system input model shown in Fig. 1. The measurement of R2 resulted in 0.47 . Variations in R2 value also lead to uncertainties in the current distortion metering. ˛ = sin−1 (sin(ˇ)e−[(+˛−ˇ)/(RCω)] ) ˇ = tan k0 = e

−1

(−RCω)

−k1 ˛/ω

[k3 sin(˛) + k4 cos(˛)]

(7) (8) (9)

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Fig. 5. Simulated An coefficient for current at PCC. Fig. 3. Simulated currents drained from power supply: (a) AC controller – dimmer and (b) full-bridge rectifier.

1

k1 =

1 C

k2 =

V R2 C

1 k3 = R2 k4 =

1 R2

k5 =

k1 ω

R2

+

1 R

(10)

(11)

V−





k1 k2

(12)

k12 + ω2

ωk2

(13)

k12 + ω2

(14)

Fig. 6. Simulated Bn coefficient for current at PCC.

Analytical An and Bn coefficients of load currents at PCC, when both S1 and S2 switches are turned on, are shown in Figs. 5 and 6. The obtained results are normalized relative to B1 module, when firing angle equals to 0◦ . The amplitude variation of harmonics calculated by (3) is shown in Fig. 7. Fig. 8 shows the THD curves from the load currents at PCC, in function of dimmer firing angle. For firing angles close to 180◦ , THD reaches its upper limit of 251% due to the prevalence of rectifier current over the dimmer current. When only harmonic currents are present in power systems, power factor PF can be defined by the product of harmonic distortion factor (PFdist ) and displacement factor (PFdisp ), given by (15):

 P PF = = PFdist · PFdisp = S

 

1 1 + THD2

[cos(ϕ1 − 1 )]

(15) Fig. 7. Simulated harmonic amplitudes for current at PCC.

Fig. 4. Variation of the resistance R, as a function of the firing angle.

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479

Fig. 8. THD of simulated load current at PCC.

where ϕ1 and  1 are the phase difference between the fundamental harmonic voltage and current. The PF is the ratio of the active power P to the apparent power S. Higher phase difference between voltage and current and higher current distortion lead to limitation for the maximum active power delivered by the utility grid [2], reducing its efficiency. The PF is thus degraded by the presence of harmonics, which is quantified by the THD, given by (16):

∞ THD =

I2 n=2 n

I1

3. ANN metering methodology (16)

where I1 and In are the RMS values for the fundamental current and for its nth order harmonic, respectively. Fig. 9 shows the power factor for each nonlinear load at PCC of the modeled system, in function of dimmer firing angle. Rectifier exhibits low and invariable PF equals to 0.37, due to the invariant values of R2 and RC. The dimmer PF starts from unitary value when the firing angle equals to 0◦ , and then decreases as the firing angle increases. The PF at PCC starts from 0.80 and decreases as the firing angle increases, until it reaches a constant value, due to the reduced impact of dimmer current when firing angle closes to 180◦ . These results demonstrate the influence of THD in PF, which is another important PQ index for utilities and industrial consumers. In this work, several tests and simulations were carried out considering a wide range of dimmer firing angles. Analytical and measurement data were used for the training pattern of the ANN, in order to produce estimation data for current harmonic content and for THD.

Fig. 9. The power factors of simulated load current at PCC.

Fig. 10. Block diagram for ANN-based method proposed.

The use of artificial neural networks has shown success in solving a series of engineering and science problems [22–26] and brings an alternative method to treat the problems related to PQ analysis. ANN was applied to estimate the Fourier coefficients of experimental data in several works [10,18–20], in order to calculate both the harmonic content and the THD of nonlinear load current. In this paper, An and Bn coefficients are estimated from load current of both, the analytical model and the test bench measurements. Phase and amplitude for the fundamental component and its first five odd harmonics are calculated from the current at PCC, during half-period of AC power supply voltage. However, it is possible to determine a larger number of harmonic components, depending on the precision required by THD calculation. After the ANNs were trained and considering that the synaptic weights have been updated, the ANN structure was tested to verify its solution generalization ability. The generalization error is evaluated from the difference between ANN output signal and the desired response. The cross-validation and the denominated double cross-validation method [31,32] are employed in the ANN training process. These methods can be used for the selection of the model to predict An and Bn coefficients. In this work the double cross-validation method was applied to minimize the number of neurons used in each ANN. In order to enhance the performance of the neural estimator, the proposed method modifies the cross-validation process, carrying out ANN training using both, simulation and test data, but without changing its structure. Thus, the system dynamics in which the ANN must operate becomes implicit in the neural structure weights. This new method proposed in this work is denominated double crossvalidation. The block diagram shown in Fig. 10 depicts the proposed methodology. Blocks 1 and 2 represent the Fourier analyzer of the experimental and analytical load currents at PCC. Consequently, in the mathematical model it is possible to shape the desired harmonic content signal and use it as supervised training pattern for the ANN, following the Levenberg–Marquardt algorithm [36]. Blocks 3 and 8 represent the load current reconstructions in time

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C.F. Nascimento et al. / Applied Soft Computing 13 (2013) 475–482 Table 1 ANN training parameters for estimation of Fourier coefficients.

Fig. 11. Neural system structure for the neural-network-method proposed.

domain. Block 7 is the data acquisition system assembled on a test bench. The pre-processing of signal is done in the blocks 4 and 9 where the waveform of current signal is recreated and then sampled in half-period of AC power supply voltage. These samples are put in the training and weight adjusting stage of the ANN, performed in block 5. Test data are used in block 6 for the double cross-validation process of the ANN, which is trained using analytical and experimental data of the previous block. In block 10, the ANN performs the estimation of Fourier coefficients of load current and in the block 11 the THD of the power system is calculated and presented. Fig. 11 shows the neural structure for harmonic identification using ANN. In this figure, a set of 42 samples of current signal is acquired in half-period of the AC power supply voltage, resulting in 5.04 kHz sampling frequency. An ANN Multilayer Perceptron (MLP), also with 42 inputs, reads the current signal samples. Each neural estimator structure comprises 5 neurons in a single hidden layer, and a single output neuron. For the estimation of nth harmonic component, are necessary two neural structures, one for An estimate and another to Bn estimate. Therefore, the neural system is composed of 12 parallel neural structures: 6 with An outputs and 6 with Bn outputs (n = 1, 3, 5, 7, 9, and 11). The computation time depends on the number of operations as well as on the computation platform chosen for the application. The database is composed by 101 vectors, 90 of them obtained from simulation data and 11 from test data. The 90 simulation data vectors were used to carry out the ANN supervised backpropagation training and the 11 test data vectors were used in the validation stage. The choice for the number of neurons depicted in Fig. 11 was based on the cross-validation process [32]. After the convergence

Network architecture

12 Multilayer Perceptrons (MLPs)

Training type Training algorithm Validation Activation function Number of inputs for each MLP Hidden layer (single) for each MLP Output layer for each MLP Learning rate Training epochs Target error

Supervised Levenberg–Marquardt Double cross-validation Hyperbolic tangent 42 5 neurons 1 neuron 1 × 10−3 2000 1 × 10−3

of the cross-validation algorithm using the simulation data and specific objective error, the generalization result was then tested using the experimental test data, which was not used in the training process. Then, as a result of an optimization problem, the best hidden neuron set for the neural network was choose amongst sets which had shown good generalization result under small computational effort. Table 1 presents the parameters details used in the ANN training process. The number of vectors chosen was suitable for the ANN training and test, representing the system dynamics in a satisfactory fashion. Test data used to validate the proposed model was obtained by measurements in a test bench assembly. This structure is able to acquire voltage and current data from a commercial dimmer circuit feeding incandescent lamps, and from a full-bridge diode rectifier with RC load. All data were acquired using a NI-DSQ USB 6009 National Instruments data acquisition system, with Labview software for PC interfacing. The Hall sensors employed to measure currents and the limitations in dimmer firing angle are detailed in [1]. The measurement result from iL (t) = iL1 (t) + iL2 (t), load current at PCC, is shown in Fig. 12. Thus, to carry out the harmonic evaluation using the proposed method, the instantaneous current signal, sampled from dimmer and rectifier, must be introduced to the ANN, which estimates each harmonic coefficient of the load current. Then the THD is calculated using the determined harmonic components. Results shown in Section 4 were obtained by the ANN using An and Bn estimation. Convergence criterion was set to RMS error values under 1 × 10−3 (or 2000 training epochs). This value was chosen so that it allowed the adjustment of the synaptic weights without causing overfitting [32]. 4. Experimental and simulation THD results Results shown in this section compare both, the neural network and the FFT methods, in order to validate the proposed methodology. The FFT method algorithm requires additions and multiplications using complex variables while MLP neural method algorithm performs only real ones. Even considering a larger number of multiplications to estimate the harmonic coefficients [10,37], ANN implementation led to reduction in both complexity and computational demand, when compared to FFT implementation. The average training time (offline) of the ANN was 20 s using a Pentium D CPU (clock 3.0 GHz to 1 GB of RAM). The generalization results performed by the ANNs were obtained almost instantaneously. Tables 2 and 3 show that the proposed ANN-based method for THD metering is able to generalize solutions from FFT coefficient calculations. These results also show the estimated and target output values from the load current at PCC, i.e. the sum of the

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481

Fig. 12. Measurement results for load current at PCC of test bench.

Table 2 Fourier coefficients – dimmer and rectifier – PCC (firing angle, 40◦ ). Harmonic (n)

Estimated

Target

An

Bn

An

Bn

1 3 5 7 9 11

−0.05 −0.11 0.18 −0.14 0.22 −0.23

0.98 −0.37 0.20 −0.22 0.21 −0.09

−0.07 −0.10 0.21 −0.14 0.20 −0.24

0.96 −0.38 0.20 −0.18 0.18 −0.10

Table 3 Fourier coefficients – dimmer and rectifier – PCC (firing angle, 90◦ ). Harmonic (n)

Estimated

Target

An

Bn

An

Bn

1 3 5 7 9 11

−0.21 0.12 0.11 −0.17 0.23 −0.24

0.81 −0.38 0.34 −0.27 0.20 −0.12

−0.22 0.16 0.07 −0.11 0.18 −0.19

0.74 −0.32 0.29 −0.24 0.18 −0.13

dimmer and the rectifier currents at this point. Estimated Fourier coefficients are in reasonably agreement with target ones, validating the proposed online harmonic monitoring method for a known load. Table 4 presents amplitude and phase values for the fundamental component and for its first five odd harmonics of the current at PCC of the dimmer (˛f = 90◦ ) and rectifier. These values were calculated by (3) and (4), and using target and estimates for these components, obtained from the output of ANN as presented in Tables 2 and 3. Target values are obtained from (5) and (6). The waveforms illustrated in Fig. 13 are generated from the harmonic components calculated using amplitudes and phases from Table 4. Fig. 13(a) shows the load current waveform, iL (t), generated using target coefficients obtained from measurements, while Fig. 13(b) shows the waveform generated using the estimated ones. Table 4 Current amplitude and phase – dimmer and rectifier – PCC. Harmonic (n)

1 3 5 7 9 11

Target value

Fig. 13. Current with target (a) and estimated (b) data.

Table 5 shows estimates for FFT coefficients of the load current at PCC (Fig. 1). An and Bn coefficients are obtained by FFT analysis. A1n and B1n coefficients are ANN estimates, trained using simulation data. Coefficients A2n and B2n are also ANN estimates, but trained using both simulation and test data, using the double crossvalidation method. Using this method is possible to reduce the relative error during online Fourier coefficients estimation process of load current. The results obtained for the traditional method, based on truncated FFT in the 6th term, and the proposal of this work, using ANN, where both compared with FFT of 50th order. Both methods shown suitable results for THD evaluation, but computational cost was smaller for the proposed ANN method. The performance of the proposed method for harmonic identification can be verified by THD and PF estimation, using target and estimate values. Target THD is 77% and its estimate is 75%, resulting in 3.3% deviation error. Target PF is 0.75 and its estimate is 0.77, resulting in 2.5% error. Therefore, the ANN-based method proposed is able to predict the system behavior correctly, validating its accuracy.

Table 5 ANN results for different training methods, and FFT results.

ANN estimate

In (A)

 n (◦ )

In (A)

 n (◦ )

Harmonic (n)

An

A1n

A2n

Bn

B1n

B2n

2.07 0.93 0.74 0.66 0.63 0.56

−17.55 150.52 12.80 −154.98 44.97 −121.46

2.34 0.88 0.89 0.75 0.73 0.63

−15.00 155.08 14.33 −152.16 47.79 −119.64

1 3 5 7 9 11

−0.12 −0.05 0.24 −0.19 0.18 −0.19

−0.13 −0.04 0.19 −0.12 0.15 −0.17

−0.11 −0.05 0.22 −0.15 0.17 −0.19

0.92 −0.43 0.26 −0.16 0.13 −0.12

0.93 −0.41 0.26 −0.17 0.20 −0.18

0.93 −0.41 0.24 −0.15 0.17 −0.17

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Fig. 14. Load current with random noise.

Additionally, the neural-network-method robustness was tested under presence of noise on the sampled signal, as shown in Fig. 14. For 10% in amplitude of random noise, the relative average error between the estimated output and target output for the fundamental component was less than 6%. The result obtained for the traditional method using truncated FFT in the 6th term was very similar. Therefore, the ANN method shows good accuracy, even when noise is present on data samples. 5. Conclusion Analytical and experimental evaluation of harmonic content and current total harmonic distortion measurement was carried out in two nonlinear loads widely used in single-phase electronic circuits. FFT coefficient estimation (An and Bn ), for fundamental frequency and its first five odd harmonic components (amplitude and phase), was obtained from ANN methodology in nonlinear and time-varying loads using only half-period of fundamental current signal. The proposed neural-network-method was able to evaluate the harmonic content of the currents efficiently. These currents were drawn by nonlinear load composed by a dimmer and a rectifier, both connected in a single-phase power system. The results obtained by means of the ANN method are in good agreement to those obtained from FFT method, showing its applicability to harmonic monitoring. Generalization results were obtained using the neural proposed method. Acknowledgments The authors acknowledge the financial support received from CNPq under 480352/2010-0, 471825/2009-3 and 474290/2008-5 processes and from Araucaria Foundation under 06/56093-3 process. References [1] C.F. Nascimento, A.A. Oliveira Jr., A. Goedtel, P.J.A. Serni, Harmonic identification using parallel neural networks in single-phase systems, Applied Soft Computing 11 (2) (2011) 2178–2185. [2] L. Sainz, J.J. Mesas, A. Ferrer, Characterization of non-linear load behavior, Electric Power Systems Research 78 (10) (2008) 1773–1783. [3] M.H.J. Bollen, What is power quality? Electric Power Systems Research 66 (1) (2003) 5–14. [4] T. Tarasiuk, Estimator-analyzer of power quality. Part I. Methods and algorithms, Measurement 44 (1) (2011) 238–247. [5] A. Ferrero, Measuring electric power quality: problems and perspectives, Measurement 41 (2) (2008) 121–129.

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