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Electric Power Systems Research 104 (2013) 42– 51

Contents lists available at ScienceDirect

Electric Power Systems Research

jou rn al hom epage: www.elsev ier .com/ locate /epsr

odel of discharge lamps with saturated magnetic ballast andon-square arc voltage

ulio Molinaa, Luis Sainzb,∗, Lluis Monjob

Department of Power, School of Electrical Engineering-UCV, Los Chaguaramos 1040, Caracas, VenezuelaDepartment of Electrical Engineering, ETSEIB-UPC, Av. Diagonal 647, 08028 Barcelona, Spain

r t i c l e i n f o

rticle history:eceived 24 November 2012eceived in revised form 1 May 2013ccepted 8 June 2013

a b s t r a c t

Harmonic modeling of high intensity discharge lamps with magnetic ballast has been extensively studiedbecause they are energy-efficient lighting devices commonly used in industrial and public installations,and can be an important source of harmonics. This kind of modeling usually considers the linear sat-

vailable online 18 July 2013

eywords:ID lampsarmonic modelingower system harmonics

uration curve of the magnetic ballast and represents the arc voltage by a square waveform. However,both assumptions can be far from describing actual lamp behavior, affecting the accuracy of the model.This paper proposes a novel characterization of high intensity discharge lamps considering both the non-linear behavior of the magnetic ballast and the non-square waveform of the arc voltage. The accuracyof the new model is validated with experimental measurements and compared to that of the traditionalmodels.

. Introduction

Lighting installations currently represent approximately 20%f worldwide electric energy consumption, producing 1900 MtO2 emissions per year. For this reason, the use of new lightingechnologies for energy efficiency improvement has become anmportant issue, allowing energy savings of 122-133 TWh/year andO2 emission reductions of 15–20%. The energy-efficient lightingevices commonly used in residential and commercial installationsre compact fluorescent lamps (CFLs) and light-emitting diodesLEDs) while industrial and public lighting installations use highntensity discharge (HID) lamps [1,2]. HID lamps have a power con-umption range of approximately 70–500 W and can be classifiedccording to the gas and pressure inside the bulb [low pressureodium (LPS), high pressure sodium (HPS), high pressure mer-ury (HPM), metal halide (MH) and ceramic metal halide (CMH)amps] and the use of magnetic or electronic ballast [3]. Amongll previous lamps, HPS and MH lamps are the most commonlysed due to their high luminous flux (7000–40000 lm) and effi-iency (40–140 lm/W), long useful life 5000-22000 h) and higholor-rendering index (15–62) [1]. In the last decade, electronicallasts have been promoted as substitutes for magnetic ballasts in

hese lamps (particularly in the European Union) because of theirigher energy efficiency and ability to deliver a constant power tohe lamp throughout its useful life. Nevertheless, magnetic ballast

∗ Corresponding author. Tel.: +34 93 4011759; fax: +34 93 4017433.E-mail address: sainz@ee.upc.edu (L. Sainz).

378-7796/$ – see front matter © 2013 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.epsr.2013.06.003

© 2013 Elsevier B.V. All rights reserved.

lamps are still widely used because of their lower cost, longer lifeand higher robustness [3]. Moreover, the power range of electronicballast lamps is limited to 150 W due to the acoustic resonancephenomenon at very high frequencies [4].

One of the main concerns about magnetic and electronic ballastHID lamps is their harmonic current emission [2,3]. Although theharmonic emissions of magnetic ballast lamps are lower than thoseof electronic ballast lamps, they are also considered in power qual-ity studies because a large number of magnetic ballast lamps areusually connected at the same bus. For this reason, many model-ing works on these lamps try to determine the harmonic currentsinjected into installations by such nonlinear loads [5–11]. Thesestudies usually assume the linear saturation curve of the magneticballast and square waveform of the arc voltage. However, bothassumptions can be far from describing actual magnetic ballastbehavior and arc voltage waveforms, affecting the accuracy of theHID lamp model [10,11]. In the knowledge of the authors, no workin the literature has examined the non-linear saturation curve ofthe magnetic ballast in HID lamp modeling, and the non-squarewaveform of the arc voltage has only been solved in [11].

This paper extends the HID lamp model in [11] by consideringthe non-linear saturation curve of the magnetic ballast togetherwith the non-square waveform of the arc voltage. An HID lampmodel based on the piecewise linear modeling of the ballast satu-ration curve and the arc voltage model in [11] is developed and the

analytical expressions of the lamp harmonic emissions are deter-mined. The lamp model also includes harmonic supply voltages andthe magnetic ballast resistance. The study is validated with exper-imental measurements, and the limitations of several models in

J. Molina et al. / Electric Power Syste

Fs

tm

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watsitHmwti

ig. 1. HID lamp modeling: (a) HID lamp equivalent circuit. (b) Magnetic ballastaturation curve. (c) Arc voltage waveform. (d) Ac current and voltage waveforms.

he literature are discussed and compared to those of the proposedodel.

. HID lamp modeling

Fig. 1(a) illustrates the typical circuit of magnetic DLs [5,6,10,11],hich is formed by the magnetic ballast and the lamp bulb. Fig. 1(b)

nd (c) shows the magnetic ballast saturation curve, b = Lb(i)·i, andhe arc voltage vA respectively, measured in an actual lamp. Fig. 1(d)hows the ac current i and the supply voltage v, which has a typ-cal flat-topped waveform with a total harmonic distortion equalo 5.62% (i.e., THDv = 5.62%). According to the previous figures, theID lamp ballast and the arc voltage phenomenon in the bulb are

odeled in Fig. 1(a) with its non-linear inductance Lb(i) (togetherith its associated resistance Rb) and the voltage source vA, respec-

ively. Moreover, a non-sinuosidal supply voltage v is also includedn the model.

ms Research 104 (2013) 42– 51 43

The HID lamp consumed current i and its harmonic spectrumare determined by analyzing the circuit of Fig. 1(a), [10,11], wherehalf-wave symmetry is commonly assumed [i.e., i(�) = −i(� + �)],and the following hypotheses are considered in the proposed HIDlamp model:

• Non-sinusoidal supply voltage

v(�) =√

2∑k≥1

Vk cos(k · � + �Vk) =√

2∑k≥1

Re{Vk · ejk·�

}

=√

2∑k≥1

Re{Vk · ejk·�

}, (1)

where � = ω1·t = 2�f1·t and f1 is the fundamental frequency of thesupply system.

• Non-linear inductive ballast with its associated resistance

vb(�) = Rb · i(�) + ω1d( b(�))d�

= Rb · i(�) + ω1d(Lb(i) · i(�))

d�. (2)

• Arc voltage represented with its Fourier series

vA(�) =√

2∑k≥1

VAk cos(k · � + �VAk) =√

2∑k≥1

Re{VAk · ejk·�

}

=√

2∑k≥1

Re{VAk · ejk·�

}. (3)

In the literature, the ballast inductance is considered linear (i.e.,Lb(i) = Lb) [5–11] and the arc voltage is commonly modeled as asquare voltage waveform (vsqA in Fig. 1(c)) [5–10]. However, fromFig. 1(b)–(d), it is concluded that both hypotheses might be wrongin actual DLs [10,11]. Fig. 1(b) and (d) shows that the ac currentconsumed by DLs could be higher than the current correspond-ing to saturation knee point of the DL ballast (i.e., |ikp| ≈ 4.5 A inFig. 1(b)). Therefore, the ballast inductance exhibits a non-linearbehavior. In Fig. 1(c), it is observed that the actual arc voltage andthe square voltage waveforms could be significantly different, andthereby Fourier series expressions must be used to fit any shape ofthese actual voltages.

Thus, the current waveform i(�) can be obtained from the cir-cuit in Fig. 1(a) by solving the equation characterizing HID lampbehavior under the previous hypothesis:

Rb · i(�) + ω1d(Lb(i)i(�))

d�+ vA(�) = v(�) (0 ≤ � ≤ �), (4)

and assuming half-wave symmetry, i(�) = −i(� + �).

2.1. Non-linear ballast model

The literature usually considers a linear behavior of the HIDlamp ballast inductance, which is modeled as a constant inductancevalue (i.e., Lb(i) = Lb) [5–11]. Nevertheless, this model might not fitaccurately actual HID lamp ballasts because of possible saturation,Fig. 1(b) and (d). In this case, the ballast saturation curve b = Lb(i)·ishould be considered in HID lamp modeling. On the other hand, itis difficult to characterize this curve analytically, and its analyti-cal functions are too complicated to be used in frequency domainmodeling of HID lamps. To avoid this problem, the ballast satu-ration curve is approximated by the piecewise linear method, i.e.the curve is broken down into several pieces and each piece p is

characterized by a linear polynomial function (see Fig. 2(a)), i.e.,

b,p = p + Lpi (p = 1, 2 . . . pm), (5)

44 J. Molina et al. / Electric Power Systems Research 104 (2013) 42– 51

Fw

w

ac

2

tdic[

•

•

v

wv

h(tate

Fig. 3. Measured arc voltage waveforms of the 400 W MH lamp (up) and the 400 WHPS lamp (down): (a) supplied with sinusoidal voltage and (b) supplied with dis-

ig. 2. HDI lamp characterization: (a) piecewise linear method. (b) Lamp currentaveform study.

here

p, Lp =

⎧⎪⎪⎨⎪⎪⎩

1, L1 Piecewise #1 : I1 = 0 < |i| ≤ I2

... Piecewise #p : Ip < |i| ≤ Ip+1

pm, Lpm Piecewise #pm : Ipm < |i|

(6)

nd pm is the maximum number of pieces into which the saturationurve is divided.

.2. Arc voltage model

To study arc voltage modeling, two public lighting lamps wereested in the laboratory (details of the power supply and nominalata and circuit parameters are given in Appendix A). Accord-

ng to the arc voltage measurements for different supply voltageonditions, the arc voltage model is based on two assumptions11]:

Assumption #1: Arc voltage stable performance. It allows arc volt-age characterization from the pattern arc voltage, vp

AAssumption #2: Arc voltage linear relation with respect to thesupply voltage rms value in the ±10% range of the HID lamp ratedvoltage. It serves to define the relation between the arc voltage vAand the pattern arc voltage vp

A at any supply voltage (distorted ornon-distorted) in the ±10% range of the HID lamp rated voltage,as follows:

A(�) = �p · vpA(�) (�p = �1 · V + �2), (7)

here V is the rms value of the supply voltage.

As an example, Fig. 3 shows the arc voltage of a 400 W metalalide (MH) lamp and a 400 W high-pressure sodium (HPS) lampsee DL details in Appendix A) for different supply voltage condi-

ions (210, 230 and 250 V sinusoidal supply voltages in Fig. 3(a)nd distorted supply voltages in Fig. 3(b)). It can be observed thathe arc voltage waveform patterns do not change significantlyxcept for a small difference in amplitude depending on supply

torted voltage.

voltage rms values. From this, it is proved in [11] that, within a±10% DL rated voltage range, there is a linear relation betweenamplitudes and the supply voltage rms values. As a result, thearc voltage can be characterized from (7). The expression of �p isobtained considering that (7) can be extended to the arc voltage rmsvalues

VA =(

12�

∫2�

(vA(�))2 · d�

)1/2

= �p

(1

2�

∫2�

(vpA(�))

2 · d�

)1/2

= �p · VpA , (8)

and observing experimentally that �p = VA/VpA can be expressed

in function of the supply voltage rms value as �p = �1·V + �2 in the±10% range of the HID lamp rated voltage (see �p determination inthe experimental tests of Section 4).

Thus, considering the previous assumptions, HID lamp arc volt-age modeling involves two steps (see Section 4) [11].

The first step is the experimental characterization of the patternarc voltage harmonic spectrum (i.e., the pattern arc voltage vp

A) at

Systems Research 104 (2013) 42– 51 45

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v

b

tsttVr

eamv

V

wai

2

cth

i

w(a

mi

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as

i

wt−

J. Molina et al. / Electric Power

he sinusoidal supply voltage of the HID lamp in rated conditions:

pA(�) =

√2 ·∑k≥1

VpAk

cos(k · � + �pVAk

) =√

2∑k≥1

Re{VpAk

· ejk·�}

=√

2∑k≥1

Re{VpAk

· ejk·�}. (9)

The zero crossing of the pattern waveform is located at � toetter develop the current analytical expressions below [11].

The second step is the experimental determination of the func-ion �p to obtain the arc voltage vA for any supply voltage v from thecaling of the pattern arc voltage vp

A according to the rms value V ofhe supply voltage (7). Considering (8), the �p expression is charac-erized experimentally by relating the ratio VA/V

pA to the rms value

of the sinusoidal supply voltage and fitting the results to the linearelation in (7).

In the literature, the stable behavior of the arc voltage isxperimentally validated and assumed [2,5] but this voltage ispproximated by a fixed square waveform with a well-known har-onic spectrum [7–11]. Thus, considering the square arc voltage

psqA associated with the pattern arc voltage vp

A, their spectrum is

psqAh = � · V

psqA

h, �psq

VAh = −�2, (10)

here � = 2(√

2)/�, VpsqA is the average value of the pattern arc volt-

ge in half cycle at the HID lamp rated voltage and the �psqVAh

values fixed by the zero crossing of the pattern voltage at �.

.3. Proposed discharge lamp model

According to the previous assumptions, the HID lamp half periodurrent (0 ≤ � ≤ �) is divided in sm = 2pm − 1 segments defined byhe commutation angles �0 to �sm (with �sm = �0 + � due to thealf-wave symmetry hypothesis) as follows, Fig. 2(b):

(�) =sm∑s=1

is(�)

⇒

⎧⎨⎩IL,s−1 = Is ≤ is(�) < IL,s = Is+1 (s = 1, . . . pm − 1)

IL,s−1 = IL,s = Is ≤ is(�) (s = pm)

IL,s−1 = I2pm−s+1 ≥ is(�) > IL,s = I2pm−s (s = pm + 1, . . . sm)

, (11)

here IL,s (s = 1, . . . sm) are the segment current limits and Iss = 1, . . . pm) are the piecewise current limits in (6) [see Fig. 2(a)nd (b), respectively].

The equations characterizing the HID lamp current of each seg-ent s can be derived from (4) and the piecewise linear curve

nductances (5):

b · is(�) + Lbsω1dis(�)d�

+ vA(�) = v(�) (0 ≤ � ≤ �)

⇒{Lbs = Ls (s = 1, . . . pm)

Lbs = L2pm−s (s = pm + 1, . . . sm). (12)

The commutation angles characterizing the HID lamp currentre obtained by solving the equations in (12) under their respectiveegment change and continuity conditions, Fig. 2(b),

s(�s) = IL,s is(�s) = is+1(�s) = IL,s (s = 1, . . . sm), (13)

here, assuming the half-wave symmetry, the continuity condi-ion of the last current segment can be rewritten as ism+1(�sm ) =i1(�0 + �).

Fig. 4. Ballast saturation curve measurements.

Thus, the expression of the HID lamp current is

i(�) =sm∑s=1

is(�) =sm∑s=1

(i(I)s (�) + i(II)s (�)) (0 ≤ � ≤ �)

i(I)s (�) = Ks · e−�/�s , i(II)s (�) =√

2∑k≥1

Re

{Vk − VAkZsk

· ejk�}

(s = 1, . . . sm),

(14)

where

�s = Lbsω1

RbZsk = Rb + jLbskω1 (s = 1, . . . sm)

VAk = �pVPAk · ejk(�−�sm) = �pVp

Ak· ejk(�−(�0+�)) = �pVp

Ak· e−jk�0

(15)

The commutation angles are obtained by solving the error func-tion nonlinear system F(x) = 0 derived from the segment changeconditions in (13):

F(x) = (f1(x), . . . fsm (x)) = 0, x = (�0, . . . �sm−1) ⇒ fs(x)

= is(�s) = Ks · e−�s/�s +√

2∑k≥1

Re

{Vk − VAkZsk

· ejk�s}

− IL,s

= 0 (s = 1, . . . sm), (16)

and considering that the commutation angle �sm verifies the rela-tion �sm = �0 + � due to the half-wave symmetry hypothesis.

The expressions of the constants Ks (s = 1, . . . sm) are obtainedfrom the continuity conditions in (13):

Ks =(IL,s−1 −

√2∑k≥1

Re

{Vk − VAkZsk

· ejk�s−1

})· e�s−1/�s

× (s = 1, . . . sm). (17)

The complex (or exponential) Fourier series of the HID lamp cur-rent i, (14), provides its fundamental and harmonic phasors, whichare easily handled with complex toolboxes of commercial software

1 2∫ �sm

(sm∑ )

1 1sm∑∫ �s

(I)

Ih = √2 � �0 s=1

is(�) · e−jh� d� = √2 �

s=1 �s−1

(is (�)

+ i(II)s (�)) · e−jh� d� = I(I)h

+ I(II)h, (18)

4 Syste

w

ejkn�s−

k

a

ˇ

2

f

agtlAoa

i

act

(

wf

f

acc

I

6 J. Molina et al. / Electric Power

here

I(I)h

=√

2�

sm∑s=1

Ksˇs(jhˇs − 1)

1 + h2ˇ2s

(e−�s − e−�s−1 )(j(hˇs−1)ω1)/ˇs

I(II)h

= 1�

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝j∑k≥1

V∗Dk

kp

sm∑s=1

e−jkp�s − e−jkp�s−1

Z∗sk

− j∑k ≥ 1

k /= h

VDkkn

sm∑s=1

ejkn�s −

Zs

nd

s = ω1�s VDk = Vk − VAk kp = k + h kn = k − h. (20)

.4. Simplified discharge lamp models

The HID lamp models in the literature [10,11] can be derivedrom the general model in the previous section.

If a linear behavior of the HID lamp ballast inductance isssumed, the ballast saturation curve can be characterized by a sin-le piece, pm = 1, with a constant inductance (i.e., Lb1(i) = Lb = L1). Inhis way, the HID lamp model in [11], which is called linear bal-ast and non-square arc voltage model (LB-NS model), is obtained.ccording to that, the HID lamp half period current (0 ≤ � ≤ �) isnly divided into one segment, sm = 1, defined by the commutationngles �0 and �1 (with �1 = �0 + �),

(�) = i1(�) (0 ≤ � ≤ �) ⇒ IL,0 = IL,1 = I1(= 0) ≤ i1(�), (21)

nd the HID lamp current equation and the segment change andontinuity conditions can be derived from (12) and (13), respec-ively:

Rb · i1(�) + Lbω1di1(�)d�

+ vA(�) = v(�) (0 ≤ � ≤ �)

i1(�1) = I1 = 0 i1(�1) = i2(�1) = −i1(�0 + �) = I1 = 0.(22)

Thus, the expression of the HID current can be rewritten from14):

i(�) =√

2∑k≥1

Re

{Vk − VAkZk

· ejk�}

(0 ≤ � ≤ �)

Zk = Rb + jLbkω1 VAk = �pVpAk

· e−jk�0 ,

(23)

here the commutation angle �0 is obtained by solving the errorunction derived from the segment change condition in (22):

1(�0) = i1(�1 = �0 + �) =∑k≥1

Re

{Vk − VAkZk

· ejk(�0+�)

}

=∑k≥1

Re

{Vk − �pVp

Ake−jk�0

Zk· ejk(�0+�)

}

=∑k≥1

Re

{VkZkejk�0

}−∑k≥1

Re

{�pVp

Ak

Zk

}= 0, (24)

nd the complex (or exponential) Fourier series of the HID lampurrent (23) provides its fundamental and harmonic phasors, whichan also be derived from (18)∫ �

h = 1√2

2�

1

�0

i1(�) · e−jh� d� = 1�VDh

�1 − �0

Zh= VDhZh

= Vh − VAhZh

.

(25)

ms Research 104 (2013) 42– 51

1 + VDh

sm∑s=1

�s − �s−1

Zsh

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,

(19)

If the arc voltage is also modeled as a square voltage wave-form (10), the HID lamp model in [8], which is called linear ballastand square arc voltage model (LB-S model), is obtained. Thus, theexpression of the HID lamp current (23) is

i(�)=√

2∑k≥1

Re

{1Zk

(Vk + j

� · VpsqA

k· e−jk�0

)· ejk�

}(0 ≤ � ≤ �),

(26)

where the commutation angle �0 is obtained by solving the errorfunction derived from the segment change condition in (24):

f1(�0) = i1(�1 = �0 + �)

=∑k≥1

Re

{1Zk

(Vk − � · V

psqA

k· e−jk(�0+�/2)

)· ejk(�0+�)

}

=∑k≥1

Re

{VkZk

· ejk�0

}−∑k≥1

Re

{1Zk

� · VpsqA

k· e−jk�/2

}

=∑k≥1

Re

{VkZk

· ejk�0

}+ � · Vpsq

A Lbω1

∑k≥1

1

Z2k

= 0, (27)

and the complex (or exponential) Fourier series of the HID lampcurrent (23) provides its fundamental and harmonic phasors, whichcan also be derived from (25):

Ih = 1Zh

(Vh + j

� · VpsqA

h· e−jh�0

). (28)

The model in [5,9] can be obtained from the previous oneneglecting the ac resistance, and the model in [6,7] can be derivedby considering the sinusoidal voltage as well.

3. Numerical resolution of the HID lamp model

The non-linear error functions of the previous HID lamp models[(16), (24) or (27)] must be solved numerically in order to char-acterize the commutation angles and lamp consumed currents. Forthis end, these functions are solved satisfactorily by using the New-ton method with the adequate initial values for the commutationangles and calculating the terms of the Jacobian matrix by the finitedifference approach. In the simplified HID lamp models, the initialvalue of �0 is easily obtained from (24) or (27) considering the sinu-soidal supply voltage. In the proposed HID lamp model, the initialvalues of �s (s = 0 to sm − 1) are obtained from the LB-NS model, asfollows:

• The commutation angle �0 is obtained from (24) considering thesinusoidal supply voltage.

• The others (i.e., �s with s = 1 to sm − 1) are determined by imposingthe segment current limits (11) in (23) and obtaining a set of sm

independent equations which can be sequentially (and easily)solved by the Newton method. In this resolution, the non-linear

J. Molina et al. / Electric Power Systems Research 104 (2013) 42– 51 47

Table 1Piecewise linear model parameters of the ballast saturation curve.

Piecewise # p p (Wb) Ip (A) Lp (H)

MH lamp (|ikp| ≈ 3.8 A) 1 0 0 0.16552 0.0564 3.0 0.14803 0.2350 4.7 0.1100

HPS lamp (|i | ≈ 5.2 A) 1 0 0 0.1371

f

w

4

tp

•

•

wdt

aTtr

aasctapttdMftatvtcll

a

Fig. 5. Arc voltage measurements: (a) pattern arc voltage at HID lamp rated voltage.

kp

2 0.0339 4.2 0.12903 0.2943 6.2 0.0870

equations to determine the commutation angles �s (s = 2 to sm − 1)are initialized with the angles �s−1, i.e.:

s(�s) =√

2∑k≥1

Re

{Vk − �pVp

Ak· e−jk�0

Zk· ejk�s

}− IL,s = 0;

�s,i.v. = �s−1 (s = 1, sm − 1), (29)

here �s,i.v. are the initial values of the commutation angles.

. Experimental tests

To determine the accuracy of the HID lamp model, six laboratoryests were conducted with the power supply and the MH and HPSublic lighting lamps described in Appendix A.

Tests #1, #2 and #3: the lamps were fed with 220, 230 and 240 Vrms typical flat-topped voltages with a total harmonic distortionof 5.62% (i.e., THDv ≈ 5.62%).Tests #4, #5 and #6: the lamps were fed with 220, 230 and 240 Vrms non-sinusoidal voltages with a total harmonic distortion of7.03% (i.e., THDv ≈ 7.03%).

The magnetic ballast saturation curve and the arc voltageaveform of these lamps had previously been experimentallyetermined and numerically characterized from the models in Sec-ions 2.1 and 2.2, respectively.

The magnetic ballast saturation curves of the MH and HPS lampsre accurately modeled by the piecewise linear method in Fig. 4.able 1 contains data of the three pieces that correctly characterizehese curves and their saturation knee point (|ikp| = 3.8 and 5.2 A,espectively).

The pattern arc voltages vpA of the MH and HPS lamps measured

t a 230 V sinusoidal supply voltage are illustrated in Fig. 5(a), whichlso contains the rms value Vp

A of the patterns and the patternquare arc voltages vpsq

A derived from the measurements. As dis-ussed in Section 2.2, it can be observed that the zero crossing ofhe patterns is placed at � to better develop the further currentnalytical expressions. Table 2 shows the harmonic spectra of theattern arc voltages. The errors between the proposed model andhe square waveform are also presented in Table 2. It is worth notinghat, although the square waveform model is generally adopted, itoes not fit actual arc voltages correctly, especially in the case of theH lamp [10,11]. Once the pattern arc voltage is characterized, the

unction �p in (7) is obtained experimentally in Fig. 5(b) by relatinghe ratio VA/V

pA (8) to the rms value V of the sinusoidal supply volt-

ge for which the arc voltage vA is measured. For example, whenhe MH lamp is fed with a 210 V sinusoidal supply voltage, the arcoltage rms value is VA = 120 V [point P in Fig. 5(b)]. Consideringhat Vp

A = 122.6 V [Fig. 5(a)], then �p = 120/122.6 = 0.979 (8), whichorresponds to point Q in Fig. 5(b). It must be noted that �p is a

inear function coinciding with those calculated from (7) for bothamps, Fig. 5(b).

The results of the 230 and 240 V experimental tests, which show higher degree of magnetic ballast saturation, are summarized in

(b) Supply voltage influence on arc voltage.

Fig. 6 (tests #2 and #3) and Fig. 7 (tests #5 and #6), correspond-ing to THDv ≈ 5.62 and 7.03%, respectively. The lamp ac currentswere measured and compared with the waveforms obtained withthe HID lamp models in Sections 2.3 and 2.4. Figs. 6(a) and 7(a)plots measured and simulated waveforms of the ac currents andalso labeling the saturation knee points of the HID lamp magneticballasts in Fig. 4. The measured supply voltage is also plotted as areference. The harmonic content of the measured and simulatedac currents (Ih,M∠�Ih,M and Ih∠�Ih, respectively), their numericaldifferences (shown at the top of the model bars)

ıI,h = 100 ·∣∣Ih,M − Ih

∣∣Ih,M

ı�,h =∣∣�h,M − �h

∣∣ , (30)

and the rms value of the measured fundamental currents (given inparentheses at the top of the fundamental current bar) are shownin Figs. 6(b) and 7(b). The current phase angles (�Ih,M and �Ih) arereferred to the phase angles of the fundamental supply voltage.Note that the results obtained with the novel HID lamp modelbased on the proposed ballast and arc voltage characterizationagree closely with the experimental measurements. However, this

is not true of the results of the simplified models as illustrated inFig. 8(a) and (b), which shows the average value of the differences

48 J. Molina et al. / Electric Power Systems Research 104 (2013) 42– 51

Table 2Pattern arc voltage harmonic spectrum of the MH (up) and HPS (bottom) lamps.

h 1 3 5 7 9 11 13 15

VpAh

(V) 116.5 34.9 13.5 5.77 3.22 2.09 1.50 1.15�pVAh

(o) −83.1 −94.9 −110.3 −112.2 −108.8 −106.0 −104.4 −103.5

VpsqAh

(V) 106.6 35.5 21.3 15.2 11.8 9.69 8.20 7.11�psqVAh

(o) −90

εV (%) 8.5 1.7 57.8 163.4 266.5 363.6 446.7 518.2ε� (%) 8.3 5.2 18.4 19.8 17.3 15.1 13.8 13.0

VpAh

(V) 73.3 26.2 17.2 12.6 9.66 7.55 5.93 4.68�pVAh

(o) −84.4 −80.1 −82.4 −86.7 −92.0 −97.4 −102.8 −107.8

VpsqAh

(V) 72.4 24.1 14.5 10.3 8.04 6.58 5.57 4.82�psqVAh

(o) −90

εV (%) 1.2 8.0 15.7 18.2 16.8 12.8 6.1 3.0.8

N

bi

ı

Fc

ε� (%) 6.6 12.3 9.2 3

ote: εX (%) = 100|XpAh

− XpsqAh/|Xp

Ah(X = V, �).

etween the measured and simulated HID lamp harmonic currentsn Figs. 6 and 7 for the six experimental tests, i.e.,

¯x,h = 1

(H + 1)/2

H∑h=1,3...

ıx,h (x = I, �; H = 7). (31)

ig. 6. Experimental tests #2 and #3: (a) voltage and current waveforms. (b) Magnitude

urrents.

2.2 7.6 12.4 16.5

Furthermore, the contribution of the proposed model to HIDlamp current fitting for magnetic ballasts operating beyond their

saturation knee point is verified in Fig. 8(c). In this figure, the accu-racy of the HID lamp models in Sections 2.3 and 2.4 is comparedfrom the average value of the square error between the measured

(left) and phase angle (right) of the fundamental and harmonics of the HID lamp ac

J. Molina et al. / Electric Power Systems Research 104 (2013) 42– 51 49

F itude (c

ai

ε

wf

aa

•

ig. 7. Experimental tests #5 and #6: (a) voltage and current waveforms. (b) Magnurrents.

nd simulated test current waveforms (iM and i, respectively),.e.,

¯ sq = 1Nsεsq = 1

Ns

Ns∑n=1

(iM(�n) − i(�n))2,(�n = 2�

Ns(n − 1)

), (32)

here Ns = 104 is the number of samples of the test current wave-orms.

Fig. 8 sows that the discrepancy between current measurementsnd simplified model results may reach unacceptable values thatre improved by the proposed HID lamp model when

The supply voltage is greater than the lamp rated voltage andthe ac current exceeds the saturation knee point of the magneticsaturation curve (e.g., in the 240 V tests). This can be verifiedby comparing the LB-NS and the proposed model ıx,h and εsq

curves for the different supply voltage rms values (i.e., for tests#1, #2 and #3 and for tests #4, #5 and #6). Higher supply voltagerms values lead to greater difference between the LB-NS and theproposed model curves.

left) and phase angle (right) of the fundamental and harmonics of the HID lamp ac

• The magnetic ballast of the HID lamp is saturated due to small val-ues of the ballast inductance, which do not limit the ac currentbelow the saturation knee point of the magnetic ballast, or due tosmall values of the saturation knee point of the magnetic ballast.This can be verified by comparing the LB-NS and the proposedmodel ıx,h and εsq curves for the MH and HPS lamp. The differ-ence between the LB-NS and the proposed model curves is higherfor the HPS lamp because of its smaller inductance (L1 = 0.1371and 0.1655 H, respectively). It results in higher ac currents, whichreach values above the saturation knee-point of the lamp ballast(Figs. 6(a) and 7(a)).

Fig. 8 also allows comparing the LB-S and LB-NS models and ver-ifying the inaccuracy of the first model for the MH lamp becauseof the difference between arc and square voltage waveforms.However, the MH lamp tests show that the square error of theLB-S model becomes smaller for higher supply voltage rms val-

ues (i.e., for higher magnetic ballast saturation values). A possibleexplanation for this could be that the magnetic ballast saturationphenomenon compensates for the inaccuracy of the arc voltagemodel.

50 J. Molina et al. / Electric Power Systems Research 104 (2013) 42– 51

menta

5

pblaFwMmoTptcssbmi

A

CC

Fig. 8. Comparison of experi

. Conclusions

A novel model of magnetic ballast HID lamps is presented in theaper. The model considers the non-linear saturation curve of theallast and the non-square waveform of the arc voltage. The bal-

ast saturation curve is modeled by the piecewise linear methodnd the arc voltage is characterized from the waveform patternourier series of the HID lamp arc voltage and its linear relationith the supply voltage rms value. Experimental tests with anH and an HPS lamp were performed to validate the proposedodel and the limitations of traditional HID lamp models based

n the ballast linear saturation curve and the square arc voltage.he results reveal that the harmonic currents obtained with theroposed model are more accurate than those obtained with tradi-ional models, which could provide inaccurate results under certainircumstances. The proposed model should be particularly used forupply voltage rms values greater than lamp rated voltages andmall ballast inductance and/or the saturation knee point valuesecause these conditions could lead to saturation of the HID lampagnetic ballast. Moreover, the new model can be incorporated

nto the harmonic load flow program in a similar way.

cknowledgment

This research was carried out with the financial support of theonsejo de Desarrollo Científico y Humanístico of the Universidadentral de Venezuela, which the authors gratefully acknowledge.

l and simulated test results.

Appendix A. HID lamp laboratory measurements

The following DLs were tested in the laboratory and their ac cur-rent and arc voltage were measured under different supply voltageconditions:

• Metal halide (MH) lamp: a SYLVANIA 400 W Tubular (HSI-THX400W/4K model) with a LOYJE magnetic ballast 230 V (4.340-I model). The measured inductance of the ballast is shown inTable 1; the measured resistance was R = 2 .

• High-pressure sodium (HPS) lamp: a SYLVANIA 400 W Tubular(SHP-T 400 W model) with a LOYJE magnetic ballast 230 V (4.240-I model). The measured inductance of the ballast is shown inTable 1; the measured resistance was R = 1.8 .

The lamps were fed with a power source AC ELGAR SmartwaveSwitching Amplifier of 4.5 kVA that can generate distorted wave-forms, and measurements were made with a YOKOGAWA DL 708Edigital scope. The test scheme consists of the power supply feedingthe discharge lamp and the oscilloscope suitably connected for themeasurement.

References

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Julio Molina was born in Caracas, Venezuela, in 1965. He received his B.S. and M.Sc.degrees in Electrical Engineering from the UCV, Venezuela, in 1993 and 2000, respec-tively. Since 1995, he has been a Professor at the Department of Power in the Schoolof Electrical Engineering of the UCV. He is currently a Ph.D. student at the Depart-ment of Electrical Engineering of the UPC. His research interest lies in the areas ofpower quality and power systems.

Luis Sainz was born in Barcelona, Spain, in 1965. He received his B.S. degree in Indus-trial Engineering and his Ph.D. degree in Industrial Engineering from UPC-Barcelona,Spain, in 1990 and 1995, respectively. Since 1991, he has been a Professor at theDepartment of Electrical Engineering of the UPC. His research interest lies in theareas of power quality. He is the head of the Electrical Supply Quality Group (QSE)at the UPC, recognized by the government of Catalonia.

Lluís Monjo was born in Tremp (Spain) in 1983. He received his B.S. degree in Elec-

Catalunya, Barcelona, Spain, in 2007 and 2009, respectively. He is currently pursuingthe Ph.D. degree. He is with the Electrical Engineering Department of the UniversitatPolitècnica de Catalunya since 2007. His area of interest includes electric machines,renewable generation and power system quality.