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Parameter estimation procedure for the equivalent circuit model of compact fluorescent lamps
Electric Power Systems Research 116 (2014) 128–135
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Parameter estimation procedure for the equivalent circuit model of compact fluorescent lamps Julio Molina a , Juan José Mesas b , Luis Sainz b,∗ a b
Department of Power, School of Electrical Engineering – UCV, Los Chaguaramos 1040, Caracas, Venezuela Department of Electrical Engineering, ETSEIB-UPC, Av. Diagonal 647, 08028 Barcelona, Spain
a r t i c l e
i n f o
Article history: Received 13 November 2013 Received in revised form 26 March 2014 Accepted 17 May 2014 Keywords: Compact fluorescent lamps Estimation procedures Non-linear load modeling Power system harmonics
a b s t r a c t The spreading use of compact fluorescent lamps (CFLs) in utility distribution systems is leading to increased concerns over power quality because CFLs consume highly distorted currents, which may account for significant power consumption of distribution feeders. For this reason, CFL models and estimation procedures of CFL model parameters must be studied in order to predict CFL harmonic current emissions into networks. This paper describes estimation procedures of CFL model parameters and presents estimation algorithms based on least-square techniques and actual measurements. The estimation procedures are validated with extensive laboratory measurements. © 2014 Elsevier B.V. All rights reserved.
1. Introduction CFLs are small-power, energy-efficient lighting devices increasingly used in residential and commercial installations due to their low energy consumption and long useful life in comparison with incandescent lamps. CFLs consume highly distorted current waveforms, which can pose a harmonic issue because CFL power consumption of all residential and commercial customers in a power distribution system may be of the order of mW, causing unacceptable voltage distortion in distribution feeders [1–3]. For the above reason, CFL modeling is currently studied in the literature in order to assess CFL harmonic current injection and predict its impact on power quality [1,2,4–6]. In [1], supply voltage harmonic interaction in CFL harmonic currents is modeled using the concept of tensor analysis with phase dependence. In [2], CFL harmonic currents are introduced in power flow calculation with Norton equivalent circuits. In [4], CFL study is based on the CFL equivalent circuit without considering its ac equivalent resistance because it can lead to unrealistic infinite slopes in the ac current rising edge. In [5], the CFL equivalent circuit is improved by considering the ac equivalent resistance because it enhances the accuracy of the model at the expense of a slight increase in model complexity. In [6], CFL external behavior is modeled using a double-exponential function to characterize ac current waveform dependence on the supply voltage without considering the internal electric circuit. CFL
∗ Corresponding author. Tel.: +34 93 4011759; fax: +34 93 4017433. E-mail address: [email protected] (L. Sainz). http://dx.doi.org/10.1016/j.epsr.2014.05.009 0378-7796/© 2014 Elsevier B.V. All rights reserved.
parameter estimation procedures are also necessary to allow use of the previous models in harmonic studies but they have not been studied as extensively as CFL modeling. Little detailed information about CFL parameter values is available and only a few works deal with parameter estimation of the CFL equivalent circuit model [1,4,5,7]. In [1], typical dc capacitor values of either 4.7 F or 10 F are suggested for CFL modeling. In [4], a simple procedure for determining the parameters of the 120 V, 60 Hz CFL equivalent circuit model from limited information is described. In [5], a straightforward method for CFL parameter estimation using experimental measurements of the CFL supply voltage and ac consumed current is proposed. In [7], a range of typical values for estimation of CFL equivalent circuit components is presented. Among these works, only the study in [5] provides an accurate estimation of CFL equivalent circuit parameters, although these results can be improved with least-square algorithms. These algorithms are used in [8] for estimating the parameters of the CFL double-exponential model in [6]. Other studies deal with the estimation of other non-linear loads using least-square algorithms [9–11]. In particular, [9,10] investigate parameter estimation of single-phase rectifiers by analyzing several non-linear sets of equations. This paper examines CFL parameter estimation by non-linear least-square procedures based on actual measurements and the CFL equivalent circuit model in [5]. These procedures are experimentally validated with two laboratory tests performed on 12 CFLs of different power ratings and trade names. From this study, a nonlinear least-square procedure based on the minimization of the square error between the temporal samples of the CFL measured and simulated ac currents is finally proposed.
J. Molina et al. / Electric Power Systems Research 116 (2014) 128–135
129
Table 1 Rated values and tolerances of 450 V CFL capacitors. C (F) ...
1
1.5
2.2
3.3
4.7
6.8
10
...
Tol. (%)
±1
±2
±3
±5
±10
±20
F
G
H
J
K
M
+80 −20 Z
±100
Code
+50 −20 S
Tube Inverter
i
which must be below 20% at low frequencies and capacitor rated voltages above 63 V [12]. The contribution to R of the rectifier diode dynamic resistance (RDiode ≈ 0.6 ) and the equivalent parallel resistance of the dc electrolytic capacitor (RP ∼ M) can be neglected. The CFL electrolytic capacitors directly influence the dc voltage ripple and, together with the load resistance RD , determine the CFL harmonic current emissions [7]. These capacitors have standardized rated values and tolerances (permissible relative deviation from the rated value), which are identified with a letter code (see Table 1) [12]. The usual CFL capacitor size for 230 V systems is 4.7 or 10 F [1,5]. According to Fig. 1(b), the ac current waveform starts with a time delay from the zero crossing of the supply voltage and has a very pronounced peak with a steep and a gentle slope of the ac current rising and falling edges, respectively. Two working modes can be distinguished in the CFL equivalent circuit operation [4,5]:
Rectifier DC busbar
AC input R
vC
v
XC = 1/(Cω)
RD
a) v, i, vC
i
vC I 0 θ1 i(θp)
II
θ f1 θ2 θp π θ 3
III
IV
θ4 v
2π θ =ω·t (rad) θ1 + 2π
• Segments I ( 1 < < 2 ) and III ( 3 < < 4 ): The diodes are off and the capacitor discharges through the equivalent resistance RD . • Segments II ( 2 < < 3 ) and IV ( 4 < < 1 + 2): The ac current i flows through the rectifier diodes, charging the capacitor and feeding the tube inverter.
b)
Fig. 1. CFL modeling: (a) CFL equivalent circuit. (b) Supply voltage v, ac current i and dc voltage vC waveforms.
The commutation angles ( 1 – 4 ), which define the CFL ac current and dc voltage in Fig. 1(b), must be determined by analyzing the circuit topologies of the above segments to characterize CFL behavior. Nevertheless, assuming half-wave symmetry, these angles verify that j+2 = j + (j = 1, 2), and therefore only segments I and II must be studied. Thus, considering a non-sinusoidal supply voltage with K harmonics
2. CFL modeling The estimation procedures in this paper are based on the CFL equivalent circuit model in [5], which characterizes CFL behavior in “poor-average” harmonic spectrum category [1,2]. This CFL category covers most of the low-Watt (<25 W) CFL market share because the minimum power factor requirement of several Standards (e.g. ANSI C82.77-2002 and ENERGY STAR program requirements for CFLs) is 0.5 only [5]. CFLs with input power above 25 W are generally better because they must comply with the requirements for class C equipment of Standard IEC 61000-32. Fig. 1(a) illustrates the equivalent circuit of this category and Fig. 1(b) shows the typical ac current i and dc voltage vC , which characterize CFL behavior where ω = 2·f and f is the fundamental frequency of the supply voltage v. According to Fig. 1(a), the equivalent circuit is supplied from a non-sinusoidal “stiff” system and is composed of a diode bridge with an ac equivalent resistance R and a dc electrolytic capacitor C that feeds the inverter and the tube modeled as an equivalent resistance RD [5]. The “stiff” supply system can be considered because CFL consumed power is much smaller than the short-circuit power of the supply system at the point of common coupling of the lamps. The ac equivalent resistance R represents the CFL input resistance Rin plus the contribution of the equivalent series resistance (ESR) of the dc electrolytic capacitor. ESR represents the capacitor ohmic losses and is characterized by the dissipation factor (DF = ESR/XC ), K1 =
√
Ik = −
XC
2 · e RD
1
1
·
k +j RD XC
K k=1
√
K
v() = 2
Re Vk · ejk·
=
√
2
K
k=1
Re V k · ejk· ,
(1)
k=1
and the previous half-wave symmetry assumption, the analysis of Fig. 1(a) reveals that the equations characterizing the ac current and dc voltage waveforms of segments I and II are [5]
⎧ (I) ⎪ ⎨ i () = 0
(I) :
⎪ ⎩ v(I) () = K · e− RXDC 1 ⎧C 1 1
K √ ⎪ K ⎪ ⎪ i(II) () = 2 e−XC RD + R + 2 Re(I k · ejk ) ⎪ ⎨ R k=1 , (II) : 1 1
K ⎪ √ ⎪ −X + (II) C R jk ⎪ R D ⎪ vC () = K2 · e + 2 Re(V Ck · e ) ⎩
(2)
k=1
where
√ X Re(V k · ejk1 ) K2 = − 2 · e C
· V Ck
P
1 RD
+
1 R 2
·
K
Re (1 + ˇ )V k · ejk2
k=1
V Ck = ˇ · V k = k
jXC k · R − jXC
R RD
k
· V k.
+1
(3)
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J. Molina et al. / Electric Power Systems Research 116 (2014) 128–135
The commutation angles x = ( 1 , 2 ) are obtained by solving the error function non-linear system of the form F(x) = (f1 , f2 ) = 0: (I)
f1 (x) = vC (2 ) + v(2 ) =
K
Re(V k · ejk1 ) · e
k=1
f2 (x) = i(II) (1 + ) =
K
Re(I k · ejk1 ) +
k=1
K
X − C RD
2 − 1
+
K
Re(V k · ejk2 ) = 0
k=1
Re(I k · ejk2 ) · e
−XC
1 RD
+
3. CFL parameter estimation procedures Any study of CFL impact on harmonic power quality from the lamp model in the previous section involves the determination of the CFL equivalent circuit parameters R, C (or XC = 1/(C·ω)) and RD in Fig. 1(a). Several procedures in the literature determine these parameters from the supply voltage and ac current measurements [vM (t) and iM (t), respectively]: • Estimation procedure #1 (EP1 ) [4,5]: The ac resistance R is neglected (i.e. R = 0), the dc capacitor is assumed equal to 15 F for the 120 V, 5–30 W CFLs and 4.7 or 10 F for the 230 V, 5–30 W CFLs. The dc equivalent resistance RD is approximated as (V ) D
·
V1, M I1, M
(120, 230 V)
(KR
D
= 3.927, 1.676 pu).
(5)
where V1,M and I1,M are the measured rms values of the CFL supply voltage and ac current at fundamental frequency. These values can also be approximated with the CFL rated voltage and current, VRate and IRate , as V1,M = VRate and I1,M = 0.85·IRate if no measurements are available. • Estimation procedure #2 (EP2 ) [5]: The dc resistance RD is estimated as RD =
PM 2 Idc, M
,
(6)
where PM and Idc,M are the CFL active power and dc current calculated from the CFL sinusoidal supply voltage and ac current measurements. In (6), the dc resistance RD represents the CFL power consumption because the ohmic losses in the ac resistance, the dynamic resistance of the diodes and the equivalent parallel resistance of the capacitor are below 3% compared with the CFL power consumption. The dc capacitor C and the ac resistance R are calculated from the ratios XCN = XC /RD and RN = R/RD , which can be approximately related to the commutation angle 2,M and the rising slope mi,M , of the normalized measured current iN,M () = (1/IRef )·iM () = (RD /VM )·iM () as follows: 2 XCN (pu) = xC1 · 2,M − xC2 · 2,M + xC3
RN (pu) =
r1 m2i,M
r2 + + r3 mi,M
(4) (1 −2 +)
= 0.
k=1
This system is derived from the symmetry and segment change conditions of the ac current and dc voltage in Fig. 1(b) [5], and can be solved satisfactorily by the Newton method calculating the terms of the Jacobian matrix by finite difference approach. The initial values of the commutation angles can be obtained from (4) considering sinusoidal supply voltage and zero ac input resistance (R = 0) [5]. Under distorted supply voltage conditions, it may be useful to solve the problem using lower distortion values first and use the results as initialization of the original problem. Once the current and voltage waveforms are known, the fundamental and harmonic currents consumed by the CFL can be determined from the complex Fourier series of the ac current (2) [5].
RD = KR
1 R
(xC1 = 0.8990 rad−2 ,
(r1 = 2.1094 rad
−2
,
where the current rising slope is calculated as −1
mi,M (rad
)=
iN,M (p,M ) p,M − 2,M
(8)
,
with VM being the rms value of the measured sinusoidal supply voltage and p,M the angle at which the measured current peak value iM ( p,M ) occurs, see i( p ) in Fig. 1(b). According to experience with different CFLs, in practical applications, the XCN ratio usually ranges from 5% to 20% and the RN ratio from 0.05% to 1%. As discussed in [5], the best of the above estimation procedures is EP2 , which is a simple calculation method that provides acceptable results. Nevertheless, these results can be improved using non-linear least-square algorithms. The next subsections propose two procedures to enhance the previous parameter estimation methods. In the procedures, the CFL parameters are estimated by supplying the CFL with a sinusoidal supply voltage and comparing the measured ac current iM () with the simulated ac current i() (2). For the comparison, a non-linear least-square problem formulated as follows is solved: min(S(y)) = min(r(y)T r(y)), y
(9)
y
where y = (R, XC , RD ) and r(y) is the residual vector, which depends on the procedure. The above non-linear least-square problem is solved by the MATLAB function lsqnonlin (·) [13] and considering the R, XC and RD values obtained with EP2 as initial values. The MATLAB function is based on the Levenberg–Marquardt algorithm and the finite difference approach Jacobian matrix [14,15]. Each residual of the proposed estimation procedures is generally scaled to achieve a uniformly-weighted residual vector. Scaling is important in problems where the difference in the magnitude of individual variables and/or residuals in the iterative process can be significant. In the analyzed problem, this is not necessary because all the residuals are related to the consumed ac current and only range from 0 to 0.5 A. Although CFL parameters are estimated from sinusoidal supply voltage tests, once the parameters are obtained, CFL behavior can be simulated with the model in Section 2 under any supply voltage condition. 3.1. Estimation procedure #3 (EP3 ) The residual vector is given by
⎡
iM (0) − i(0)
⎢ .. ⎢ ⎢ . ⎢ r(y) = ⎢ () − i() i M ⎢ ⎢ ⎢ .. ⎣ .
⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎦
(10)
iM (2) − i(2)
xC2 = −4.7769 rad−1 ,
r2 = 0.2767 rad
⎤
−1
,
xC3 = 6.4138 pu)
r3 = −5.487 · 10−4 pu),
(7)
J. Molina et al. / Electric Power Systems Research 116 (2014) 128–135
200
Actual Data Smoothed Data
i (mA)
b) a)
a)
0 -50 0
π/5
2π/5 θ (rad)
3π/5
4π/5
Fig. 2. Smoothing details of an actual CFL ac current waveform: (a) zero crossings, (b) peak value.
This estimation procedure minimizes the square error between the temporal samples of the CFL measured and simulated ac currents evaluated at all sampled time instants in one current waveform period. Although this estimation procedure provides very good results, the use of more equations than unknowns to define the estimation procedure does not generally ensure better results and can lead to convergence problems [10]. For this reason, sometimes it is better to choose the smallest number of equations defining the problem correctly as residuals. Following this criterion, the next subsection presents another residual vector based on the framework in [8,9]. 3.2. Estimation procedure #4 (EP4 ) The residual vector is given by
⎡
i(2,M )
⎤
⎢ ⎥ i(3,M ) ⎢ ⎥ ⎢ ⎥. r(y) = ⎢ ⎥ ⎣ iM (p,M ) − i(p,M ) ⎦
(11)
di(p,M )/dt This estimation procedure minimizes the error between the temporal samples of the CFL measured and simulated ac current evaluated at the angles 2,M , 3,M and p,M , and the current derivative at the current peak angle p,M (see Fig. 1(b)). Other residuals could be chosen in (11) but those proposed characterize the ac current pulse correctly and provide good results. The commutation and the peak value conditions [i.e. i( 2 ) = i( 3 ) = 0, i( p ) and di( p )/dt = 0] are the most characteristic points of the current pulse, and therefore the chosen set is the one that best characterizes the ac current pulse. Sometimes, the proper choice of current falling edge temporal samples to be added to the residuals in (11) can be useful in fitting the different shapes of this falling edge. It is advisable to choose the samples at the middle of the current falling edge, i.e. at the angle f1 = ( p + 3 )/2, or even between the previous angle and the commutation angles, i.e. at the angles f2 = ( p + f1 )/2 and f3 = ( f1 + 3 )/2, Fig. 1(b). 3.3. On measured ac current uncertainties In general, smoothing of measured ac current is necessary as a preliminary step to parameter estimation in order to reduce the influence of measurement uncertainty on this estimation. The aim of smoothing is to improve the location of current commutation and peak angles by avoiding multiple zero crossings and peak oscillations of measured ac current. Fig. 2 illustrates a measured CFL current and the way smoothing eliminates its ripple and correctly locates the current zero crossing and peak point. This data pre-processing step is performed by the MATLAB function
131
smooth (·), which smoothes curve data by fitting a data moving window with a second-degree polynomial model and a weighted linear least-square method [13]. To analyze the effects of measurement uncertainties on the accuracy of CFL parameter estimation, a study of this estimation was performed from simulated ac currents iSim (t) with a certain degree of normally-distributed noise ei (t), i.e. i(t) = iSim (t) + ei (t) where ei (t) ∼ N( = 0, 2 ). The results confirm that data pre-processing by the function smooth (·) provides robustness to the estimation procedure. The errors obtained are acceptable, even for very noisy ac currents (e.g. ac currents with 3 = 25%ISim , where ISim is the rms value of the simulated ac current). 4. Experimental tests To determine the accuracy of the estimation procedures, the following two laboratory tests were conducted on 12 CFLs of different power ratings and trade names (Table 2). • Test #1: The CFLs were fed with a 230 V/50 Hz sinusoidal voltage. • Test #2: The CFLs were fed with a 230 V/50 Hz flat-topped voltage with a total harmonic distortion of 5.83% (i.e. THDV ≈ 5.83%). The lamps were supplied with a power source AC ELGAR Smartwave Switching Amplifier of 4.5 kV A that can generate distorted waveforms, and measurements were made with a YOKOGAWA DL 708E digital scope. The test set-up consisted of the power supply feeding the CFLs and the oscilloscope suitably connected for measurements. The CFL parameters were estimated from Test #1 measurements and according to procedures EP2 , EP3 and EP4 in Section 3. The results are presented in Table 3. The obtained C/Pi ratio and the time constant C = RD ·C [5,7], are in the ranges C/Pi = 0.21–0.56 F/W and C = 15.6–55.8 ms, which agree with those proposed in [7] (C/Pi = 0.2–0.6 F/W and C = 10–70 ms). Because the results for procedure EP1 are less accurate, they are not given. Using the estimated parameters in Table 3, the ac current waveforms of the 12 CFLs were obtained from simulations and compared with the measured ac current of Tests #1 and #2 in Figs. 3 and 4, respectively. The measured supply voltage is plotted as a reference. The harmonic content of the measured and simulated ac currents (Ih,M and Ih , respectively) and their numerical differences (shown at the top of the model bars) are also plotted. Note that the simulation results obtained with the parameters of the three estimation procedures agree closely with the experimental measurements (errors of the harmonic current magnitudes are approximately below 8%). Procedure EP3 provides the best results, followed by procedure EP4 , whose results are actually quite similar to those obtained by EP2 . This is also true for the harmonic phase angles (not shown due to lack of space). The above conclusions are corroborated with the plots in Figs. 5 and 6, which show the mean square relative error between measured and simulated current waveforms and the mean value of the relative errors in the magnitude determination of the odd harmonic currents below 39th order:
εc =
εI, H
1 IM
Ns
(i ( ) − i(n )) n=1 M n
100 = (H + 1)/2
Ns
2
Ih,M − Ih
n =
H
h=1, 3,...
Ih,M
2 (n − 1) Ns
(12)
(H = 39),
where Ns = 104 and IM are the number of samples and the rms value of the measured test current waveforms, respectively. The three procedures give acceptable results but EP3 has the best accuracy and EP4 yields smaller relative errors in the harmonic magnitudes than EP2 . Specifically, procedure EP3 has mean square
132
J. Molina et al. / Electric Power Systems Research 116 (2014) 128–135
Fig. 3. Experimental test and estimation procedures of CFLs fed with 230 V/50 Hz sinusoidal voltage: (a) voltage and current waveforms, (b) magnitude of the fundamental and harmonics of CFL currents.
J. Molina et al. / Electric Power Systems Research 116 (2014) 128–135
133
Fig. 4. Experimental test and estimation procedures of CFLs fed with 230 V/50 Hz flat-topped voltage with THDV = 5.83%: (a) voltage and current waveforms, (b) magnitude of the fundamental and harmonics of CFL currents.
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J. Molina et al. / Electric Power Systems Research 116 (2014) 128–135
Table 2 Manufacturer CFL technical data. CFLs
Trade name
Made in
P (W)
I (mA)
˚V (lm)
L1P L2P L3P
Philips Genie Philips Ecotone Economy Philips SL-E Pro Philips Genie Rapid Start Philips SL-Electronic General Electric Biax Electronic Laes Mini Carrefour Economic Eut-15 Vidal Sylvania Mini Lynx Sylvania Mini Lynx Ikea
China Europe Poland
11 14 20
80 115 145
600 900 1200
China
11
80
600
China
20
141
1280
Hungary Spain China Spain India India China
23 20 15 9 20 15 11
180 130 108 70 150 110 80
1500 827 800 400 1200 900 600
L4P L5P L6G L7L L8C L9V L10S L11S L12Ik
a) 102
a) 102
EP2
EP3
EP2
EP3 EP4
εc (%)
101
100
101
100 b)
b) 1
101
εH,I (%)
εH,I (%)
10
L11S 15W
L10S 20W
L9V 9W
L8C 15W
L12Ik 11W
CFLs
L7L 20W
L6G 23W
L5P 20W
L4P 11W
L3P 20W
L1P 11W
L12Ik 11W
L11S 15W
L10S 20W
L9V 9W
L8C 15W
L7L 20W
L6G 23W
L5P 20W
L4P 11W
L3P 20W
100 L2P 14W
L1P 11W
100
L2P 14W
εc (%)
EP4
CFLs
Fig. 5. Errors of the estimation procedures for CFLs fed with 230 V/50 Hz sinusoidal voltage: (a) mean square relative error, (b) mean error of the odd harmonic magnitudes up to H = 39.
Fig. 6. Errors of the estimation procedures for CFLs fed with 230 V/50 Hz flat-topped voltage (THDV = 5.83%): (a) mean square relative error, (b) mean error of the odd harmonic magnitudes up to H = 39.
Table 3 CFL estimated electrical parameters. CFLs
L1P L2P L3P L4P L5P L6G L7L L8C L9V L10S L11S L12Ik
EP2
EP3
EP4
R ()
C (F)
RD ()
R ()
C (F)
RD ()
R ()
C (F)
RD ()
38.36 50.19 14.57 14.87 16.98 3.21 1.96 18.63 8.48 13.46 21.97 20.56
2.94 3.61 5.78 3.03 4.81 6.64 11.26 5.06 2.61 4.57 3.18 3.04
7878 5980 4787 7183 4494 3998 3993 6359 9991 4008 4933 7253
37.88 53.10 14.34 15.65 14.83 6.16 3.39 17.47 15.34 12.76 15.18 19.86
2.95 3.71 5.92 2.97 4.75 6.60 11.03 5.17 2.46 4.77 3.36 3.04
7872 5853 4767 7359 4575 4038 5056 6385 10658 3936 4837 7330
40.44 51.36 14.78 16.94 17.20 6.22 3.18 17.46 14.23 12.40 16.86 24.88
2.94 3.66 5.79 2.97 4.76 6.74 10.96 4.99 2.49 4.64 3.40 3.08
7715 5823 4671 7180 4463 3840 4907 6356 10232 3912 4599 7041
J. Molina et al. / Electric Power Systems Research 116 (2014) 128–135
relative errors between measured and simulated current waveforms approximately below 8% and mean relative errors in the harmonic current magnitude below 6% while the other procedures (in particular EP2 ) can have errors above 10% for the same cases. The accuracy of the estimations in Table 3 was also verified opening several lamps in Table 2. • The dc capacitors of these lamps had one of the standardized rated values in Table 1 and their tolerances were always 20% (capacitors with code M). Considering this, the estimated values in Table 3 are in agreement with the following standardized rated values: • • • • •
2.2 F ± 20%: lamp L9 V. 3.3 F ± 20%: lamps L1P, L2P, L3P, L4P, L11S and L12Ik. 4.7 F ± 20%: lamps L5P, L8C, and L10S. 6.8 F ± 20%: lamp L6G. 10 F ± 20%: lamp L7L.
• The ac equivalent resistance values in Table 3 correspond to the contribution of the CFL ac input resistance Rin and the ESR of the dc electrolytic capacitor. The ac input resistance of the opened lamps ranges from 0 to 20 and, considering a dc electrolytic capacitor dissipation factor below 20% [12], the ESR maximum values would approximately be 120 for lamp L9 V, 80 for lamps L1P, L2P, L3P, L4P, L11S and L12Ik, 56 for lamps L5P, L8C, and L10S, 39 for lamp L6G, and 26.4 for lamp L7L.
5. Conclusions This paper proposes a non-linear least-square algorithm for CFL parameter estimation based on the minimization of the square error between the temporal samples of the CFL measured and simulated ac currents (procedure EP3 ). This procedure is compared with the simple estimation procedure in [5] and the non-linear leastsquare algorithm based on the framework in [8] (procedures EP2 and EP4 , respectively). Although the three methods give acceptable errors, procedure EP3 performs better than the other procedures. There are also other simple CFL estimation methods in the literature but they provide worse results. The CFL model, together with the proposed parameter estimation procedure, makes it possible to include CFLs in further harmonic load flow studies to
135
accurately analyze their impact on the power quality in electrical installations. Acknowledgment This research was carried out with the financial support of the Consejo de Desarrollo Científico y Humanístico of the Universidad Central de Venezuela (B-08-4302-2008), which the authors gratefully acknowledge. References [1] Z. Wei, N.R. Watson, L.P. Frater, Modelling of compact fluorescent lamps, in: Proceedings of the 13th IEEE Int. Conf. on Harmonics and Quality of Power, 2008, pp. 1–6. [2] N.R. Watson, T.L. Scott, S.J.J. Hirsch, Implications for distribution networks of high penetration of compact fluorescent lamps, IEEE Trans. Power Deliv. 24 (July (3)) (2009) 1521–1528. [3] G.A. Vokas, I.F. Gonos, F.N. Korovesis, F.V. Topalis, Influence of compact fluorescent lamps on the power quality of weak low-voltage networks supplied by autonomous photovoltaic stations, in: Proceedings of the IEEE Porto Power Tech, 2001, pp. 1–5. [4] J. Yong, L. Chen, A.B. Nassif, W. Xu, A frequency-domain harmonic model for compact fluorescent lamps, IEEE Trans. Power Deliv. 25 (April (2)) (2010) 1182–1189. [5] J. Molina, L. Sainz, Model of electronic ballast compact fluorescent lamps, IEEE Trans. Power Deliv. 29 (June (3)) (2014) 1363–1371. [6] J. Cunill-Solà, M. Salichs, Study and characterization of waveforms from lowWatt (<25 W) compact fluorescent lamps with electronic ballasts, IEEE Trans. Power Deliv. 22 (October (4)) (2007) 2305–2311. [7] J. Slezingr, J. Drapela, R. Langella, A. Testa, A new simplified model of compact fluorescent lamps in the scenario of smart grids, in: Proceedings of the 15th IEEE Int. Conf. on Harmonics and Quality of Power, 2012, pp. 835–841. [8] L. Sainz, J. Cunill, J.J. Mesas, Parameter estimation procedures for compact fluorescent lamps with electronic ballasts, Elect. Power Syst. Res. 95 (February) (2013) 77–84. [9] J.J. Mesas, L. Sainz, J. Molina, Parameter estimation procedure for models of single-phase uncontrolled rectifiers, IEEE Trans. Power Deliv. 26 (July (3)) (2011) 1911–1919. [10] R.N.N. Souza, Nonlinear loads parameter estimation and modeling, in: Proceedings of the IEEE Int. Symp. on Industrial Electronics, 2007, pp. 937–942. [11] J.M. Maza, A. Gómez, J.L. Trigo, M. Burgos, Parameter estimation of harmonic polluting industrial loads, Int. J. Elect. Power Energy Syst. 27 (November–December (9–10)) (2005) 635–640. [12] IEC 60384-4, Fixed capacitors for use in electronic equipment – Part 4: Sectional specifications – aluminum electrolytic capacitors with solid (MnO2 ) and nonsolid electrolyte, 2007. [13] Matlab 7.9 (R2009b), The MathWorks, Inc., Natick, MA, 2009. [14] K. Levenberg, A method for the solution of certain non-linear problems in leastsquares, Quart. Appl. Math. 2 (July (2)) (1994) 164–168. [15] D.W. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, SIAM J. Appl. Math. 11 (June (2)) (1963) 431–441.
Contents lists available at ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Parameter estimation procedure for the equivalent circuit model of compact fluorescent lamps Julio Molina a , Juan José Mesas b , Luis Sainz b,∗ a b
Department of Power, School of Electrical Engineering – UCV, Los Chaguaramos 1040, Caracas, Venezuela Department of Electrical Engineering, ETSEIB-UPC, Av. Diagonal 647, 08028 Barcelona, Spain
a r t i c l e
i n f o
Article history: Received 13 November 2013 Received in revised form 26 March 2014 Accepted 17 May 2014 Keywords: Compact fluorescent lamps Estimation procedures Non-linear load modeling Power system harmonics
a b s t r a c t The spreading use of compact fluorescent lamps (CFLs) in utility distribution systems is leading to increased concerns over power quality because CFLs consume highly distorted currents, which may account for significant power consumption of distribution feeders. For this reason, CFL models and estimation procedures of CFL model parameters must be studied in order to predict CFL harmonic current emissions into networks. This paper describes estimation procedures of CFL model parameters and presents estimation algorithms based on least-square techniques and actual measurements. The estimation procedures are validated with extensive laboratory measurements. © 2014 Elsevier B.V. All rights reserved.
1. Introduction CFLs are small-power, energy-efficient lighting devices increasingly used in residential and commercial installations due to their low energy consumption and long useful life in comparison with incandescent lamps. CFLs consume highly distorted current waveforms, which can pose a harmonic issue because CFL power consumption of all residential and commercial customers in a power distribution system may be of the order of mW, causing unacceptable voltage distortion in distribution feeders [1–3]. For the above reason, CFL modeling is currently studied in the literature in order to assess CFL harmonic current injection and predict its impact on power quality [1,2,4–6]. In [1], supply voltage harmonic interaction in CFL harmonic currents is modeled using the concept of tensor analysis with phase dependence. In [2], CFL harmonic currents are introduced in power flow calculation with Norton equivalent circuits. In [4], CFL study is based on the CFL equivalent circuit without considering its ac equivalent resistance because it can lead to unrealistic infinite slopes in the ac current rising edge. In [5], the CFL equivalent circuit is improved by considering the ac equivalent resistance because it enhances the accuracy of the model at the expense of a slight increase in model complexity. In [6], CFL external behavior is modeled using a double-exponential function to characterize ac current waveform dependence on the supply voltage without considering the internal electric circuit. CFL
∗ Corresponding author. Tel.: +34 93 4011759; fax: +34 93 4017433. E-mail address: [email protected] (L. Sainz). http://dx.doi.org/10.1016/j.epsr.2014.05.009 0378-7796/© 2014 Elsevier B.V. All rights reserved.
parameter estimation procedures are also necessary to allow use of the previous models in harmonic studies but they have not been studied as extensively as CFL modeling. Little detailed information about CFL parameter values is available and only a few works deal with parameter estimation of the CFL equivalent circuit model [1,4,5,7]. In [1], typical dc capacitor values of either 4.7 F or 10 F are suggested for CFL modeling. In [4], a simple procedure for determining the parameters of the 120 V, 60 Hz CFL equivalent circuit model from limited information is described. In [5], a straightforward method for CFL parameter estimation using experimental measurements of the CFL supply voltage and ac consumed current is proposed. In [7], a range of typical values for estimation of CFL equivalent circuit components is presented. Among these works, only the study in [5] provides an accurate estimation of CFL equivalent circuit parameters, although these results can be improved with least-square algorithms. These algorithms are used in [8] for estimating the parameters of the CFL double-exponential model in [6]. Other studies deal with the estimation of other non-linear loads using least-square algorithms [9–11]. In particular, [9,10] investigate parameter estimation of single-phase rectifiers by analyzing several non-linear sets of equations. This paper examines CFL parameter estimation by non-linear least-square procedures based on actual measurements and the CFL equivalent circuit model in [5]. These procedures are experimentally validated with two laboratory tests performed on 12 CFLs of different power ratings and trade names. From this study, a nonlinear least-square procedure based on the minimization of the square error between the temporal samples of the CFL measured and simulated ac currents is finally proposed.
J. Molina et al. / Electric Power Systems Research 116 (2014) 128–135
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Table 1 Rated values and tolerances of 450 V CFL capacitors. C (F) ...
1
1.5
2.2
3.3
4.7
6.8
10
...
Tol. (%)
±1
±2
±3
±5
±10
±20
F
G
H
J
K
M
+80 −20 Z
±100
Code
+50 −20 S
Tube Inverter
i
which must be below 20% at low frequencies and capacitor rated voltages above 63 V [12]. The contribution to R of the rectifier diode dynamic resistance (RDiode ≈ 0.6 ) and the equivalent parallel resistance of the dc electrolytic capacitor (RP ∼ M) can be neglected. The CFL electrolytic capacitors directly influence the dc voltage ripple and, together with the load resistance RD , determine the CFL harmonic current emissions [7]. These capacitors have standardized rated values and tolerances (permissible relative deviation from the rated value), which are identified with a letter code (see Table 1) [12]. The usual CFL capacitor size for 230 V systems is 4.7 or 10 F [1,5]. According to Fig. 1(b), the ac current waveform starts with a time delay from the zero crossing of the supply voltage and has a very pronounced peak with a steep and a gentle slope of the ac current rising and falling edges, respectively. Two working modes can be distinguished in the CFL equivalent circuit operation [4,5]:
Rectifier DC busbar
AC input R
vC
v
XC = 1/(Cω)
RD
a) v, i, vC
i
vC I 0 θ1 i(θp)
II
θ f1 θ2 θp π θ 3
III
IV
θ4 v
2π θ =ω·t (rad) θ1 + 2π
• Segments I ( 1 < < 2 ) and III ( 3 < < 4 ): The diodes are off and the capacitor discharges through the equivalent resistance RD . • Segments II ( 2 < < 3 ) and IV ( 4 < < 1 + 2): The ac current i flows through the rectifier diodes, charging the capacitor and feeding the tube inverter.
b)
Fig. 1. CFL modeling: (a) CFL equivalent circuit. (b) Supply voltage v, ac current i and dc voltage vC waveforms.
The commutation angles ( 1 – 4 ), which define the CFL ac current and dc voltage in Fig. 1(b), must be determined by analyzing the circuit topologies of the above segments to characterize CFL behavior. Nevertheless, assuming half-wave symmetry, these angles verify that j+2 = j + (j = 1, 2), and therefore only segments I and II must be studied. Thus, considering a non-sinusoidal supply voltage with K harmonics
2. CFL modeling The estimation procedures in this paper are based on the CFL equivalent circuit model in [5], which characterizes CFL behavior in “poor-average” harmonic spectrum category [1,2]. This CFL category covers most of the low-Watt (<25 W) CFL market share because the minimum power factor requirement of several Standards (e.g. ANSI C82.77-2002 and ENERGY STAR program requirements for CFLs) is 0.5 only [5]. CFLs with input power above 25 W are generally better because they must comply with the requirements for class C equipment of Standard IEC 61000-32. Fig. 1(a) illustrates the equivalent circuit of this category and Fig. 1(b) shows the typical ac current i and dc voltage vC , which characterize CFL behavior where ω = 2·f and f is the fundamental frequency of the supply voltage v. According to Fig. 1(a), the equivalent circuit is supplied from a non-sinusoidal “stiff” system and is composed of a diode bridge with an ac equivalent resistance R and a dc electrolytic capacitor C that feeds the inverter and the tube modeled as an equivalent resistance RD [5]. The “stiff” supply system can be considered because CFL consumed power is much smaller than the short-circuit power of the supply system at the point of common coupling of the lamps. The ac equivalent resistance R represents the CFL input resistance Rin plus the contribution of the equivalent series resistance (ESR) of the dc electrolytic capacitor. ESR represents the capacitor ohmic losses and is characterized by the dissipation factor (DF = ESR/XC ), K1 =
√
Ik = −
XC
2 · e RD
1
1
·
k +j RD XC
K k=1
√
K
v() = 2
Re Vk · ejk·
=
√
2
K
k=1
Re V k · ejk· ,
(1)
k=1
and the previous half-wave symmetry assumption, the analysis of Fig. 1(a) reveals that the equations characterizing the ac current and dc voltage waveforms of segments I and II are [5]
⎧ (I) ⎪ ⎨ i () = 0
(I) :
⎪ ⎩ v(I) () = K · e− RXDC 1 ⎧C 1 1
K √ ⎪ K ⎪ ⎪ i(II) () = 2 e−XC RD + R + 2 Re(I k · ejk ) ⎪ ⎨ R k=1 , (II) : 1 1
K ⎪ √ ⎪ −X + (II) C R jk ⎪ R D ⎪ vC () = K2 · e + 2 Re(V Ck · e ) ⎩
(2)
k=1
where
√ X Re(V k · ejk1 ) K2 = − 2 · e C
· V Ck
P
1 RD
+
1 R 2
·
K
Re (1 + ˇ )V k · ejk2
k=1
V Ck = ˇ · V k = k
jXC k · R − jXC
R RD
k
· V k.
+1
(3)
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The commutation angles x = ( 1 , 2 ) are obtained by solving the error function non-linear system of the form F(x) = (f1 , f2 ) = 0: (I)
f1 (x) = vC (2 ) + v(2 ) =
K
Re(V k · ejk1 ) · e
k=1
f2 (x) = i(II) (1 + ) =
K
Re(I k · ejk1 ) +
k=1
K
X − C RD
2 − 1
+
K
Re(V k · ejk2 ) = 0
k=1
Re(I k · ejk2 ) · e
−XC
1 RD
+
3. CFL parameter estimation procedures Any study of CFL impact on harmonic power quality from the lamp model in the previous section involves the determination of the CFL equivalent circuit parameters R, C (or XC = 1/(C·ω)) and RD in Fig. 1(a). Several procedures in the literature determine these parameters from the supply voltage and ac current measurements [vM (t) and iM (t), respectively]: • Estimation procedure #1 (EP1 ) [4,5]: The ac resistance R is neglected (i.e. R = 0), the dc capacitor is assumed equal to 15 F for the 120 V, 5–30 W CFLs and 4.7 or 10 F for the 230 V, 5–30 W CFLs. The dc equivalent resistance RD is approximated as (V ) D
·
V1, M I1, M
(120, 230 V)
(KR
D
= 3.927, 1.676 pu).
(5)
where V1,M and I1,M are the measured rms values of the CFL supply voltage and ac current at fundamental frequency. These values can also be approximated with the CFL rated voltage and current, VRate and IRate , as V1,M = VRate and I1,M = 0.85·IRate if no measurements are available. • Estimation procedure #2 (EP2 ) [5]: The dc resistance RD is estimated as RD =
PM 2 Idc, M
,
(6)
where PM and Idc,M are the CFL active power and dc current calculated from the CFL sinusoidal supply voltage and ac current measurements. In (6), the dc resistance RD represents the CFL power consumption because the ohmic losses in the ac resistance, the dynamic resistance of the diodes and the equivalent parallel resistance of the capacitor are below 3% compared with the CFL power consumption. The dc capacitor C and the ac resistance R are calculated from the ratios XCN = XC /RD and RN = R/RD , which can be approximately related to the commutation angle 2,M and the rising slope mi,M , of the normalized measured current iN,M () = (1/IRef )·iM () = (RD /VM )·iM () as follows: 2 XCN (pu) = xC1 · 2,M − xC2 · 2,M + xC3
RN (pu) =
r1 m2i,M
r2 + + r3 mi,M
(4) (1 −2 +)
= 0.
k=1
This system is derived from the symmetry and segment change conditions of the ac current and dc voltage in Fig. 1(b) [5], and can be solved satisfactorily by the Newton method calculating the terms of the Jacobian matrix by finite difference approach. The initial values of the commutation angles can be obtained from (4) considering sinusoidal supply voltage and zero ac input resistance (R = 0) [5]. Under distorted supply voltage conditions, it may be useful to solve the problem using lower distortion values first and use the results as initialization of the original problem. Once the current and voltage waveforms are known, the fundamental and harmonic currents consumed by the CFL can be determined from the complex Fourier series of the ac current (2) [5].
RD = KR
1 R
(xC1 = 0.8990 rad−2 ,
(r1 = 2.1094 rad
−2
,
where the current rising slope is calculated as −1
mi,M (rad
)=
iN,M (p,M ) p,M − 2,M
(8)
,
with VM being the rms value of the measured sinusoidal supply voltage and p,M the angle at which the measured current peak value iM ( p,M ) occurs, see i( p ) in Fig. 1(b). According to experience with different CFLs, in practical applications, the XCN ratio usually ranges from 5% to 20% and the RN ratio from 0.05% to 1%. As discussed in [5], the best of the above estimation procedures is EP2 , which is a simple calculation method that provides acceptable results. Nevertheless, these results can be improved using non-linear least-square algorithms. The next subsections propose two procedures to enhance the previous parameter estimation methods. In the procedures, the CFL parameters are estimated by supplying the CFL with a sinusoidal supply voltage and comparing the measured ac current iM () with the simulated ac current i() (2). For the comparison, a non-linear least-square problem formulated as follows is solved: min(S(y)) = min(r(y)T r(y)), y
(9)
y
where y = (R, XC , RD ) and r(y) is the residual vector, which depends on the procedure. The above non-linear least-square problem is solved by the MATLAB function lsqnonlin (·) [13] and considering the R, XC and RD values obtained with EP2 as initial values. The MATLAB function is based on the Levenberg–Marquardt algorithm and the finite difference approach Jacobian matrix [14,15]. Each residual of the proposed estimation procedures is generally scaled to achieve a uniformly-weighted residual vector. Scaling is important in problems where the difference in the magnitude of individual variables and/or residuals in the iterative process can be significant. In the analyzed problem, this is not necessary because all the residuals are related to the consumed ac current and only range from 0 to 0.5 A. Although CFL parameters are estimated from sinusoidal supply voltage tests, once the parameters are obtained, CFL behavior can be simulated with the model in Section 2 under any supply voltage condition. 3.1. Estimation procedure #3 (EP3 ) The residual vector is given by
⎡
iM (0) − i(0)
⎢ .. ⎢ ⎢ . ⎢ r(y) = ⎢ () − i() i M ⎢ ⎢ ⎢ .. ⎣ .
⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎦
(10)
iM (2) − i(2)
xC2 = −4.7769 rad−1 ,
r2 = 0.2767 rad
⎤
−1
,
xC3 = 6.4138 pu)
r3 = −5.487 · 10−4 pu),
(7)
J. Molina et al. / Electric Power Systems Research 116 (2014) 128–135
200
Actual Data Smoothed Data
i (mA)
b) a)
a)
0 -50 0
π/5
2π/5 θ (rad)
3π/5
4π/5
Fig. 2. Smoothing details of an actual CFL ac current waveform: (a) zero crossings, (b) peak value.
This estimation procedure minimizes the square error between the temporal samples of the CFL measured and simulated ac currents evaluated at all sampled time instants in one current waveform period. Although this estimation procedure provides very good results, the use of more equations than unknowns to define the estimation procedure does not generally ensure better results and can lead to convergence problems [10]. For this reason, sometimes it is better to choose the smallest number of equations defining the problem correctly as residuals. Following this criterion, the next subsection presents another residual vector based on the framework in [8,9]. 3.2. Estimation procedure #4 (EP4 ) The residual vector is given by
⎡
i(2,M )
⎤
⎢ ⎥ i(3,M ) ⎢ ⎥ ⎢ ⎥. r(y) = ⎢ ⎥ ⎣ iM (p,M ) − i(p,M ) ⎦
(11)
di(p,M )/dt This estimation procedure minimizes the error between the temporal samples of the CFL measured and simulated ac current evaluated at the angles 2,M , 3,M and p,M , and the current derivative at the current peak angle p,M (see Fig. 1(b)). Other residuals could be chosen in (11) but those proposed characterize the ac current pulse correctly and provide good results. The commutation and the peak value conditions [i.e. i( 2 ) = i( 3 ) = 0, i( p ) and di( p )/dt = 0] are the most characteristic points of the current pulse, and therefore the chosen set is the one that best characterizes the ac current pulse. Sometimes, the proper choice of current falling edge temporal samples to be added to the residuals in (11) can be useful in fitting the different shapes of this falling edge. It is advisable to choose the samples at the middle of the current falling edge, i.e. at the angle f1 = ( p + 3 )/2, or even between the previous angle and the commutation angles, i.e. at the angles f2 = ( p + f1 )/2 and f3 = ( f1 + 3 )/2, Fig. 1(b). 3.3. On measured ac current uncertainties In general, smoothing of measured ac current is necessary as a preliminary step to parameter estimation in order to reduce the influence of measurement uncertainty on this estimation. The aim of smoothing is to improve the location of current commutation and peak angles by avoiding multiple zero crossings and peak oscillations of measured ac current. Fig. 2 illustrates a measured CFL current and the way smoothing eliminates its ripple and correctly locates the current zero crossing and peak point. This data pre-processing step is performed by the MATLAB function
131
smooth (·), which smoothes curve data by fitting a data moving window with a second-degree polynomial model and a weighted linear least-square method [13]. To analyze the effects of measurement uncertainties on the accuracy of CFL parameter estimation, a study of this estimation was performed from simulated ac currents iSim (t) with a certain degree of normally-distributed noise ei (t), i.e. i(t) = iSim (t) + ei (t) where ei (t) ∼ N( = 0, 2 ). The results confirm that data pre-processing by the function smooth (·) provides robustness to the estimation procedure. The errors obtained are acceptable, even for very noisy ac currents (e.g. ac currents with 3 = 25%ISim , where ISim is the rms value of the simulated ac current). 4. Experimental tests To determine the accuracy of the estimation procedures, the following two laboratory tests were conducted on 12 CFLs of different power ratings and trade names (Table 2). • Test #1: The CFLs were fed with a 230 V/50 Hz sinusoidal voltage. • Test #2: The CFLs were fed with a 230 V/50 Hz flat-topped voltage with a total harmonic distortion of 5.83% (i.e. THDV ≈ 5.83%). The lamps were supplied with a power source AC ELGAR Smartwave Switching Amplifier of 4.5 kV A that can generate distorted waveforms, and measurements were made with a YOKOGAWA DL 708E digital scope. The test set-up consisted of the power supply feeding the CFLs and the oscilloscope suitably connected for measurements. The CFL parameters were estimated from Test #1 measurements and according to procedures EP2 , EP3 and EP4 in Section 3. The results are presented in Table 3. The obtained C/Pi ratio and the time constant C = RD ·C [5,7], are in the ranges C/Pi = 0.21–0.56 F/W and C = 15.6–55.8 ms, which agree with those proposed in [7] (C/Pi = 0.2–0.6 F/W and C = 10–70 ms). Because the results for procedure EP1 are less accurate, they are not given. Using the estimated parameters in Table 3, the ac current waveforms of the 12 CFLs were obtained from simulations and compared with the measured ac current of Tests #1 and #2 in Figs. 3 and 4, respectively. The measured supply voltage is plotted as a reference. The harmonic content of the measured and simulated ac currents (Ih,M and Ih , respectively) and their numerical differences (shown at the top of the model bars) are also plotted. Note that the simulation results obtained with the parameters of the three estimation procedures agree closely with the experimental measurements (errors of the harmonic current magnitudes are approximately below 8%). Procedure EP3 provides the best results, followed by procedure EP4 , whose results are actually quite similar to those obtained by EP2 . This is also true for the harmonic phase angles (not shown due to lack of space). The above conclusions are corroborated with the plots in Figs. 5 and 6, which show the mean square relative error between measured and simulated current waveforms and the mean value of the relative errors in the magnitude determination of the odd harmonic currents below 39th order:
εc =
εI, H
1 IM
Ns
(i ( ) − i(n )) n=1 M n
100 = (H + 1)/2
Ns
2
Ih,M − Ih
n =
H
h=1, 3,...
Ih,M
2 (n − 1) Ns
(12)
(H = 39),
where Ns = 104 and IM are the number of samples and the rms value of the measured test current waveforms, respectively. The three procedures give acceptable results but EP3 has the best accuracy and EP4 yields smaller relative errors in the harmonic magnitudes than EP2 . Specifically, procedure EP3 has mean square
132
J. Molina et al. / Electric Power Systems Research 116 (2014) 128–135
Fig. 3. Experimental test and estimation procedures of CFLs fed with 230 V/50 Hz sinusoidal voltage: (a) voltage and current waveforms, (b) magnitude of the fundamental and harmonics of CFL currents.
J. Molina et al. / Electric Power Systems Research 116 (2014) 128–135
133
Fig. 4. Experimental test and estimation procedures of CFLs fed with 230 V/50 Hz flat-topped voltage with THDV = 5.83%: (a) voltage and current waveforms, (b) magnitude of the fundamental and harmonics of CFL currents.
134
J. Molina et al. / Electric Power Systems Research 116 (2014) 128–135
Table 2 Manufacturer CFL technical data. CFLs
Trade name
Made in
P (W)
I (mA)
˚V (lm)
L1P L2P L3P
Philips Genie Philips Ecotone Economy Philips SL-E Pro Philips Genie Rapid Start Philips SL-Electronic General Electric Biax Electronic Laes Mini Carrefour Economic Eut-15 Vidal Sylvania Mini Lynx Sylvania Mini Lynx Ikea
China Europe Poland
11 14 20
80 115 145
600 900 1200
China
11
80
600
China
20
141
1280
Hungary Spain China Spain India India China
23 20 15 9 20 15 11
180 130 108 70 150 110 80
1500 827 800 400 1200 900 600
L4P L5P L6G L7L L8C L9V L10S L11S L12Ik
a) 102
a) 102
EP2
EP3
EP2
EP3 EP4
εc (%)
101
100
101
100 b)
b) 1
101
εH,I (%)
εH,I (%)
10
L11S 15W
L10S 20W
L9V 9W
L8C 15W
L12Ik 11W
CFLs
L7L 20W
L6G 23W
L5P 20W
L4P 11W
L3P 20W
L1P 11W
L12Ik 11W
L11S 15W
L10S 20W
L9V 9W
L8C 15W
L7L 20W
L6G 23W
L5P 20W
L4P 11W
L3P 20W
100 L2P 14W
L1P 11W
100
L2P 14W
εc (%)
EP4
CFLs
Fig. 5. Errors of the estimation procedures for CFLs fed with 230 V/50 Hz sinusoidal voltage: (a) mean square relative error, (b) mean error of the odd harmonic magnitudes up to H = 39.
Fig. 6. Errors of the estimation procedures for CFLs fed with 230 V/50 Hz flat-topped voltage (THDV = 5.83%): (a) mean square relative error, (b) mean error of the odd harmonic magnitudes up to H = 39.
Table 3 CFL estimated electrical parameters. CFLs
L1P L2P L3P L4P L5P L6G L7L L8C L9V L10S L11S L12Ik
EP2
EP3
EP4
R ()
C (F)
RD ()
R ()
C (F)
RD ()
R ()
C (F)
RD ()
38.36 50.19 14.57 14.87 16.98 3.21 1.96 18.63 8.48 13.46 21.97 20.56
2.94 3.61 5.78 3.03 4.81 6.64 11.26 5.06 2.61 4.57 3.18 3.04
7878 5980 4787 7183 4494 3998 3993 6359 9991 4008 4933 7253
37.88 53.10 14.34 15.65 14.83 6.16 3.39 17.47 15.34 12.76 15.18 19.86
2.95 3.71 5.92 2.97 4.75 6.60 11.03 5.17 2.46 4.77 3.36 3.04
7872 5853 4767 7359 4575 4038 5056 6385 10658 3936 4837 7330
40.44 51.36 14.78 16.94 17.20 6.22 3.18 17.46 14.23 12.40 16.86 24.88
2.94 3.66 5.79 2.97 4.76 6.74 10.96 4.99 2.49 4.64 3.40 3.08
7715 5823 4671 7180 4463 3840 4907 6356 10232 3912 4599 7041
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relative errors between measured and simulated current waveforms approximately below 8% and mean relative errors in the harmonic current magnitude below 6% while the other procedures (in particular EP2 ) can have errors above 10% for the same cases. The accuracy of the estimations in Table 3 was also verified opening several lamps in Table 2. • The dc capacitors of these lamps had one of the standardized rated values in Table 1 and their tolerances were always 20% (capacitors with code M). Considering this, the estimated values in Table 3 are in agreement with the following standardized rated values: • • • • •
2.2 F ± 20%: lamp L9 V. 3.3 F ± 20%: lamps L1P, L2P, L3P, L4P, L11S and L12Ik. 4.7 F ± 20%: lamps L5P, L8C, and L10S. 6.8 F ± 20%: lamp L6G. 10 F ± 20%: lamp L7L.
• The ac equivalent resistance values in Table 3 correspond to the contribution of the CFL ac input resistance Rin and the ESR of the dc electrolytic capacitor. The ac input resistance of the opened lamps ranges from 0 to 20 and, considering a dc electrolytic capacitor dissipation factor below 20% [12], the ESR maximum values would approximately be 120 for lamp L9 V, 80 for lamps L1P, L2P, L3P, L4P, L11S and L12Ik, 56 for lamps L5P, L8C, and L10S, 39 for lamp L6G, and 26.4 for lamp L7L.
5. Conclusions This paper proposes a non-linear least-square algorithm for CFL parameter estimation based on the minimization of the square error between the temporal samples of the CFL measured and simulated ac currents (procedure EP3 ). This procedure is compared with the simple estimation procedure in [5] and the non-linear leastsquare algorithm based on the framework in [8] (procedures EP2 and EP4 , respectively). Although the three methods give acceptable errors, procedure EP3 performs better than the other procedures. There are also other simple CFL estimation methods in the literature but they provide worse results. The CFL model, together with the proposed parameter estimation procedure, makes it possible to include CFLs in further harmonic load flow studies to
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accurately analyze their impact on the power quality in electrical installations. Acknowledgment This research was carried out with the financial support of the Consejo de Desarrollo Científico y Humanístico of the Universidad Central de Venezuela (B-08-4302-2008), which the authors gratefully acknowledge. References [1] Z. Wei, N.R. Watson, L.P. Frater, Modelling of compact fluorescent lamps, in: Proceedings of the 13th IEEE Int. Conf. on Harmonics and Quality of Power, 2008, pp. 1–6. [2] N.R. Watson, T.L. Scott, S.J.J. Hirsch, Implications for distribution networks of high penetration of compact fluorescent lamps, IEEE Trans. Power Deliv. 24 (July (3)) (2009) 1521–1528. [3] G.A. Vokas, I.F. Gonos, F.N. Korovesis, F.V. Topalis, Influence of compact fluorescent lamps on the power quality of weak low-voltage networks supplied by autonomous photovoltaic stations, in: Proceedings of the IEEE Porto Power Tech, 2001, pp. 1–5. [4] J. Yong, L. Chen, A.B. Nassif, W. Xu, A frequency-domain harmonic model for compact fluorescent lamps, IEEE Trans. Power Deliv. 25 (April (2)) (2010) 1182–1189. [5] J. Molina, L. Sainz, Model of electronic ballast compact fluorescent lamps, IEEE Trans. Power Deliv. 29 (June (3)) (2014) 1363–1371. [6] J. Cunill-Solà, M. Salichs, Study and characterization of waveforms from lowWatt (<25 W) compact fluorescent lamps with electronic ballasts, IEEE Trans. Power Deliv. 22 (October (4)) (2007) 2305–2311. [7] J. Slezingr, J. Drapela, R. Langella, A. Testa, A new simplified model of compact fluorescent lamps in the scenario of smart grids, in: Proceedings of the 15th IEEE Int. Conf. on Harmonics and Quality of Power, 2012, pp. 835–841. [8] L. Sainz, J. Cunill, J.J. Mesas, Parameter estimation procedures for compact fluorescent lamps with electronic ballasts, Elect. Power Syst. Res. 95 (February) (2013) 77–84. [9] J.J. Mesas, L. Sainz, J. Molina, Parameter estimation procedure for models of single-phase uncontrolled rectifiers, IEEE Trans. Power Deliv. 26 (July (3)) (2011) 1911–1919. [10] R.N.N. Souza, Nonlinear loads parameter estimation and modeling, in: Proceedings of the IEEE Int. Symp. on Industrial Electronics, 2007, pp. 937–942. [11] J.M. Maza, A. Gómez, J.L. Trigo, M. Burgos, Parameter estimation of harmonic polluting industrial loads, Int. J. Elect. Power Energy Syst. 27 (November–December (9–10)) (2005) 635–640. [12] IEC 60384-4, Fixed capacitors for use in electronic equipment – Part 4: Sectional specifications – aluminum electrolytic capacitors with solid (MnO2 ) and nonsolid electrolyte, 2007. [13] Matlab 7.9 (R2009b), The MathWorks, Inc., Natick, MA, 2009. [14] K. Levenberg, A method for the solution of certain non-linear problems in leastsquares, Quart. Appl. Math. 2 (July (2)) (1994) 164–168. [15] D.W. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, SIAM J. Appl. Math. 11 (June (2)) (1963) 431–441.