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Deviations between the commonly-used model and measurements of harmonic distortion in low-voltage installations
Electric Power Systems Research 180 (2020) 106166
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Deviations between the commonly-used model and measurements of harmonic distortion in low-voltage installations
T
Tatiano Busattoa,*, Vineetha Ravindrana, Anders Larssona, Sarah K. Rönnberga, Math H.J. Bollena, Jan Meyerb a b
Electric Power Engineering, Luleå University of Technology, Skellefteå, Sweden Electric Power Engineering, Technische Universität Dresden, Dresden, Germany
A R T I C LE I N FO
A B S T R A C T
Keywords: Power quality Power system harmonics Power electronics Harmonic analysis Nonlinear systems
Harmonic analysis studies of modern power systems commonly employ Norton and Thévenin equivalents at harmonic frequencies for the nonlinear devices. This approach neglects the so-called nonlinear interaction phenomenon. This paper addresses the difference between the results from the commonly-used model and the actual harmonic distortion measured in a low-voltage installation. A number of indices are introduced to quantify the nonlinear interaction. These indices allow a quantification of the extent to which the commonlyused model is also to predict harmonic voltages and currents in a modern low-voltage installation. The proposed model and the subsequent mathematical analysis are illustrated through measurements from different combinations of PV inverters and LED lamps using different technologies. The results show that deviation is dependent on the used technology, network impedance, and source voltage waveform. Other findings are that nonlinear interaction happens mainly in the low harmonic orders and impacts are more perceived on the harmonics phase angle. Possible explanations for these observations are discussed.
1. Introduction With the proliferation of power electronic devices in the power system, issues related to harmonic distortion have been regularly addressed during the last decades. More recently, the subject has gained additional focus due to the appearance of electrical vehicle (EV) charging and power inverters to interface solar and wind power plants. Depending on system complexity and desired accuracy, different harmonic analysis methods in time and frequency domain are described in the literature [1,2]. Time domain methods [3,4], provide by far the most accurate solution. However, limitations arise for analysis of a power system with multiple devices and multiple operational states. Harmonic analysis in the frequency domain presents fewer constraints enabling analysis with a larger number of devices and is therefore commonly used in studies with multiple harmonic sources [5–8]. In this context, the direct method [9,10] provides the simplest solution: the frequency response for a given bus is obtained from the solution of the network equation [Y ] V = I , where [Y ] is the network admittance matrix, V the nodal voltage vector to be solved, and I is the vector of current injections. Although this method is broadly used, it cannot provide adequate results for high levels of voltage distortion
⁎
[11], since the devices’ current harmonics are dependent on their terminal voltage and operating current as well [12]. To include these dependencies, but without having to use a detailed time domain model, several harmonic analysis methods in the frequency domain and methods combining the individual advantages of frequency and time domain have been developed [11,13–16]. These methods, however, suffer from practical limitations when the power system contains a considerable number of power electronic devices and buses. Accuracy will depend on the details and accuracy of the individual device models. Each type of device is different; large amounts of input data and additional measurements are needed before one even can get started. For instance, several of the accurate frequency-domain methods, such as the iterative harmonic analysis (IHA) [15], present slow convergence, narrow stability margin and, high computational effort dependent on the system complexity [1]. Similarly, the harmonic domain (HD) method suffers from its algorithm complexity [17]. Their computational advantage compared to time-domain methods becomes less with the increasing number of devices. Study cases employing these methods are limited to a few nonlinear devices that far from cover the actual complexity and extension of nonlinear devices in real power systems. All these drawbacks diminish from its adoption by the
Corresponding author. E-mail address: [email protected] (T. Busatto).
https://doi.org/10.1016/j.epsr.2019.106166 Received 3 October 2019; Received in revised form 16 November 2019; Accepted 11 December 2019 Available online 25 December 2019 0378-7796/ © 2019 Elsevier B.V. All rights reserved.
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of PVI and LEDs connected to an emulated power grid. As a first step in the method, the installation is modeled by only the current sources and impedances for the different parts of the system. Next, the values for current sources and impedances in the harmonic range are obtained from dedicated measurements. The current spectrum for the devices is obtained from the measurements using a known voltage distortion and quasi-zero impedance of the emulated grid. The device currents are combined with the measured impedances to obtain the Norton equivalent. Finally, the complex currents and voltages at the points of interest are calculated from the resulting linear model. To determine the nonlinearity deviation due to the nonlinear interaction, the currents and voltages from simulation and measurement are compared in different operating scenarios. The uncertainties from the linear model and measurements are taken into consideration as described in Appendix A. Any deviation above the uncertainty range accounts to nonlinearity deviation for the device or installation. Harmonic components phase angle for all channels is referenced to the ascending zero-crossing of the point of common coupling (PCC) voltage channel (see Fig. 2 for the location of PCC). A 5 Hz frequency resolution was adopted and harmonics calculated following the recommendations in [23].
industry. The afore-mentioned frequency and hybrid frequency-time domain methods have focused mainly on the harmonic models of individual devices, limiting the discussion on how harmonic analysis can be performed in an existing power system with a large number of nonlinear devices. Due to the constant inclusion of new power electronics to the grid, there is a growing demand for harmonic studies in a broader scope. A modern low-voltage (LV) installation can easily contain more than 100 power electronic devices connected close to each other. Modeling each of these devices in detail is in practice not possible. Additionally, LV installations are not static presenting often variations during operation. This means that stochastic methods are needed, most commonly based on the random generation of many different cases [18–20]. With so many cases and devices to be modeled, anything more complicated than the direct method will be difficult to solve. As there is, in fact, no suitable alternative when a large number of devices are involved, it is appropriate to evaluate and quantify the accuracy of the direct method. A simple but general model for the interaction between a single device and the rest of the grid is presented in [21] and recommended to be used by an international working group [22]; next to two types of linear interaction (primary and secondary emission), the term “nonlinear interaction” is introduced. This term refers to any interaction that cannot be explained from time-independent Norton/Thévenin models. The term is however not quantitatively explained in [21]. This paper proposes a method to quantify to which extent the direct method employing Norton/Thévenin equivalent harmonic sources is a suitable representation for an LV installation. The proposed method was applied to 9 different installations of photovoltaic PV inverters (PVI) and LED lamps (LED). Considering voltage source and network impedance variants, a total of 486 different combinations were studied. Measurements in the experimental test setup have been compared with results obtained by employing the direct method, referred from here on as the linear model. Difference between measurement and the linear model is an indication of “nonlinear interaction” as introduced in [21]. Considering the linear model and measuring results, any difference between the results reveals the presence of nonlinearity in the system. Throughout this paper, this difference (and thus the nonlinear interaction) is quantified through an index referred to as “nonlinearity deviation”. Separate indices are defined for individual harmonic frequencies and for covering the whole frequency range of interest. The study demonstrates to which extent the results obtained from the direct method, using Norton/Thévenin equivalent for the harmonic sources, is an accurate representation for the harmonic propagation in an installation with a large number of power electronic devices. Additionally, the most important factors that contribute to nonlinear interaction are identified in different operating scenarios, changes in the grid characteristics, and combinations of different technologies. This paper does not deal with the origins and causes of nonlinearities, instead, it is a first step in revealing to which extent nonlinearity deviations impact harmonic voltage and current distortion and to which extent the direct method gives reasonably accurate results. The methodology to quantify the nonlinear interaction is introduced in Section 2. Sections 3 and 4 describe the laboratory test setup and the used linear model, respectively. Section 5 summarizes the results, followed by the discussion in Section 6, where the results are further interpreted. Finally, Section 7 presents the main conclusions.
2.1. Quantifying the nonlinear interaction Nonlinearity deviation can be evaluated by the difference between magnitudes and phases angle of the harmonic components obtained from measurements and linear model. However, the individual evaluation of these harmonic characteristics can lead to misinterpretation of the real extension of nonlinearity deviations. To quantify the deviation consider YH and YˆH as the harmonic components in the complex plane ℂ obtained from measurements and linear model respectively, given by:
⎧YH , h = ah + bh j ⎨YˆH , h = aˆh + bˆh j ⎩
(1)
where the symbol Y is replaced, as required, by I for currents, or by U for voltages; H qualifies the variable as harmonic, and h is the harmonic order. Since ℂ is isomorphic to ℝ2 , the Euclidean distance between harmonic components, dYH , h , described in (2), is used as a measure of nonlinearity deviation:
dYH , h = |YH , h − YˆH , h| =
(bh − bˆh)2 + (ah − aˆh)2
(2)
To compare different scenarios, the authors have introduced the Total Harmonic Nonlinearity Deviation factor, ΔTHNLY , based on the subgroup concept introduced in [23], but considering as input the values obtained from (2) and uncertainty: hmax
ΔTHNLY =
∑ h=2
2
⎛⎜ DYH , h ⎞⎟ ⎝ YH ,1 ⎠
(3)
In (3), DYH , h is the effective (or certain) distance (i.e., harmonic components distance minus the uncertainty) given by:
0 if dYH , h < U (dYH , h) DYH , h = ⎧ ⎨ dY − U (dY ) if dYH , h ≥ U (dYH , h) H , h H , h ⎩
(4)
where U (dYH , h) is the expanded total distance uncertainty given by the linear model and actual measurements:
2. Methodology
U (dYH , h) = U (YH , h) + U (YˆH , h)
The methodological approach taken in this study is based on the comparison of results obtained from a linear model allowing the application of the direct method and measurements performed in a controlled but realistic scenario. A simplified LV installation, containing the essential elements for the purposes of the analysis is considered. The test setup consists of a set
(5)
3. Experimental setup The photo and the simplified schematic of the measurement setup are show in Figs. 1 and 2 , respectively. The setup consists of a 10 kW 2
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Table 2 Network impedances (phase + neutral). Z1
Z2
Z3
Z4
≈0 Ω
0.17 + j 0.10 Ω
0.40 + j 0.25 Ω
0.40 + j 0.04 Ω
between 50° and 90° commonly found in low-power LEDs where the requirements for power factor are less strict. LED B and C also differ in impedance characteristics. 3. Network impedances: the different network impedances considered in this study are listed in Table 2. The first one is a quasi-zero impedance (Z1). It is defined by the Power Amplifier internal impedance plus cable impedances. The second is about half the reference impedance (Z2), given by IEC/TR 60725 [30]. To emulate a weak grid, a reference impedance (Z3) is used. A typical Swedish network impedance value (Z4) [31] is included with a predominantly resistive characteristic. 4. Source supply voltage waveform: besides the ideal sinusoidal voltage waveform, V1 (i.e., the reference), two additional voltage waveforms, flat-top, V2, and pointed-top, V3, are considered. Both are based on LV networks measurements which specification is given in the Appendix of [29].
Fig. 1. Photo of the experimental measurement setup.
4. Linear model Based on the test setup shown in Fig. 2, Fig. 3 shows the linear model used to predict current and voltage harmonics. The PVI, LED, and grid currents, IPVI , ILED, and IGRID, respectively, are obtained from the following set of linear equations: Fig. 2. Simplified test-setup schematic.
IˆPVI ⎤ IsPVI. ZPVI ⎡ ⎤ ⎡ ZPVI 0 ZGRID ⎤ ⎡ IˆLED ⎥ ⎢ IsLED. ZLED ⎥ ⎢ 0 ZLED ZGRID ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ 1 ⎥ 0 ˆ ⎦⎢ ⎣− 1 − 1 ⎦ ⎦⎣ ⎣ IGRID ⎥
PV array emulator [24], which is connected to the PVI under test. The AC side of the PVI is connected to an adjustable line impedance emulator [25] and the source supply is emulated by a programmable 45 kVA three-phase power amplifier [26]. The LEDs are connected between the PVI and the line impedance (PCC) through a board with several lamp sockets. All the measurements were performed by a data acquisition (DAQ) system [27] equipped with high-bandwidth and high-accuracy acquisition modules and voltage/current probes [28] placed at the interested monitoring locations. Five test-setup variants with respective sub-variants, summarized in Table 1, were tested during the experiment.
VˆPCC = VS − IˆGRID. ZGRID
The magnitude and phase angles of ZPVI , ZLEDs , and ZGRID used in the linear model are shown in Fig. 4. Note the magnitude difference among the LED types and the larger phase variation for the PVIs. The complex impedances were obtained as follows: 1. LED impedance: measurements were performed in the Pehr Högström Laboratory with Luleå University of Technology in Skellefteå employing an invasive steady-state-based method using short-time current injections [32] at harmonic frequencies. The impedance of one single LED was obtained, and assuming linearity, the equivalent impedance for different numbers of lamps in parallel was obtained. Results were verified by supplementary measurements revealing errors below 5.0% and 1.0° for magnitude and phase, respectively.
Table 1 Tested variants and sub-variants. Sub-variant ID
Test points
PVIs LEDs Number of lamps Network impedance Source supply voltage waveform
I1, I2, I3 A, B, C 1, 3, 6, 15, 24, 50 Z1, Z2, Z3, Z4 V1, V2, V3
3 3 6 4 3
(7)
4.1. Estimating the impedances
1. PVIs: are the same inverters described in [29], where further detail on primary emission can be found. The set covers some of the inverter topologies used in small residential and commercial applications. I1 is a 4.6 kW transformerless inverter, while I2 and I3 are 4.6 and 10 kW inverters and use a low- and high-frequency transformer for galvanic isolation, respectively. 2. LEDs: three LED lamps commonly found in the market (i.e., year 2017), differentiating mainly by the current waveform and impedance characteristics, were selected. LED A has a power factor correction (PFC) feature presenting a more sinusoidal current waveform, while LEDs B and C have a typical marked current peak
Variant name
(6)
where ZPVI , ZLEDs , and ZS , are the internal impedances for PVI and LED, and network impedance, respectively. IsPVI and IsLEDs are the source currents from LEDs and PVI. From the calculated grid current, IGrid , the PCC voltage is obtained using the following expression:
Fig. 3. Linear model equivalent circuit. 3
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Fig. 6. Difference between IPVI harmonics components H5 and H9 obtained from linear model and measurements.
considered, dYH , h becomes DYH , h and values for DYH,5 and DYH,9 are 13.58 and 12.68 mA, respectively. The figure shows a slight change in magnitude and a substantial phase shift in both harmonics. Harmonic H9, for instance, presents a small magnitude deviation (4.19 mA), but a phase shift of about 32° is observed. Note the difference in uncertainties, which are dependent on the harmonic RMS values, see Appendix A. 5.1. Devices technology impact on PVI nonlinearity Fig. 4. Magnitude and phase impedance characteristics for the Linear Model components.
An examination on the nonlinearity deviation in dependence of the device's technology is given by the data in Table 3, which lists the ΔTHNL IPVI (%) for different device combination and number of lamps. Significant differences can be observed in at least three aspects from the results in Table 3. First, PVIs present different levels of nonlinearity deviation. This can be observed by maximum ΔTHNL IPVI (i.e., italicized values) for PVI I1, I2 and I3 which are close to 5%, 30%, and 12%, respectively. From the results, PVI I1 and PVI I2 present the smallest and largest nonlinearity deviations, respectively, due to the LED operation. Second, LED A and B cause the lowest and the highest deviation, respectively, on the inverter nonlinearity. This is observed by ΔTHNL ≈ 0 in cases where LED A is used, and ΔTHNL with high values when LED B is used. Finally, in general the inverters nonlinearity deviation tends to increase as the number of lamps increases, with some exceptions (e.g., 15lamps I2-A and 3-lamps I2-C, where ΔTHNL is greater than the ΔTHNL for 50-lamps).
2. PVI input impedance: characteristics for PVI I1, I2 and I3 were extracted from [33], in which impedance values were obtained from measurements performed at TU Dresden. 3. Network impedance: estimated by assuming an impedance R + jωL with constant R and L . The measurements were done in a controlled environment where the network impedance is known with reasonable accuracy. 4.2. Estimating the source currents The source currents were obtained by using the simplified schema shown in Fig. 5. To obtain the source currents, the Z1 (Zs ≈ 0 Ω) network impedance was used, emulating a terminal short circuit at harmonic frequencies, and the voltage waveform fixed to V1, V2, or V3. Consequently, the current from the source becomes equal to the measured one minus the current through the (known) source impedance driven by the (known) harmonic voltage distortion. For each background voltage, a particular spectrum for the source current is given, being assessed distinct operating points around which the linearization is made.
5.1.1. Frequency-dependency of PVI Nonlinearity Taking as an example the combination I2-B (i.e., the one that presents the highest ΔTHNL variation, Fig. 7 illustrates the frequency-dependency on the nonlinearity deviation for IPVI as the number of lamps changes. As shown in Fig. 7 (top graph), higher values of dIH are observed at the low order harmonics, and the values decrease as the frequency increases. Note that when no lamps are considered (i.e., middle graph, when there is no interaction between PVI and LED), dIH , is inside the uncertainty range for most of the harmonic components. Fig. 8 shows in detail how the dIH for one individual harmonic (H9) changes as a function of the number of lamps. Note that dIH tends to
5. Results An example of nonlinearity deviation caused by nonlinear interaction between PVIs and LEDs is presented in Fig. 6, where two particular harmonic components of a PVI in the complex plane ℂ are detailed. The results shown in Fig. 6 were obtained from a scenario considering PVI I1 at 50% of its rated power and 15 LED C, network impedance Z3, and sinusoidal voltage waveform (V1). The same setup will be used as a reference in the following analysis unless otherwise specified. In the figure, the solid line represents the distance in the complex plane, dYH , h , as introduced in Section 2.1. Once the uncertainty is
Table 3 ΔTHNL IPVI (%) for diverse devices combination.
Fig. 5. Simplified circuit schematic for obtaining the current sources. PVI (left) and LED (right). 4
Lamps
I1-A
I1-B
I1-C
I2-A
I2-B
I2-C
I3-A
I3-B
I3-C
None 1 3 6 15 24 50
0.0 0.0 0.0 0.0 0.0 0.2 0.0
0.0 0.0 0.1 0.1 0.0 1.0 4.9
0.0 0.0 0.1 0.0 0.0 0.5 1.8
1.0 11.3 8.0 0.8 18.0 1.9 2.1
1.0 5.8 5.5 11.9 8.9 10.8 28.9
1.0 10.4 15.1 1.7 2.3 12.3 13.4
2.1 0.2 0.2 0.1 2.0 0.1 1.7
2.1 0.8 1.1 0.8 1.6 4.4 11.3
2.1 0.1 0.1 2.0 0.4 2.5 5.1
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Fig. 9. LED dIH as function of number of lamps for the combination PVI I2 and LED B.
the LED operation individually, the last three columns show the ΔTHNL when PVI is not considered. From Table 4, certain similarities to ΔTHNL IPVI values (see Table 3) are observed. ΔTHNL ILED tends to increase as the number of lamps increases, and the LED nonlinearity, from the lowest to highest deviation, is observed when LED A, C, and B, and PVI I1, I3, and I2 are used, respectively. Further, ΔTHNL ILED is in general much higher than the ΔTHNL IPVI . Also, if we consider the ΔTHNL ILED for the LED operation alone (3 last columns in Table 4), we can conclude that the isolated LED operation has the most significant contribution towards increased values.
Fig. 7. dIH as function of number of lamps for PVI I2 and LEDs B: (top) all number of lamps, (middle) no lamps detail, and (bottom) 50 lamps detail.
5.3. Frequency-dependency of LED Nonlinearity The detailed analysis for the combinations I2-B (i.e., the one that presents the highest nonlinearity deviation) is shown as an example in Fig. 9, where the nonlinearity deviation, dIH , for different numbers of lamps is presented in the frequency domain. Because LED B has a high current harmonic content, it creates an additional voltage distortion at the PCC depending on the grid impedance. Consequently, this extra distortion will change the LED harmonic current emissions itself and of other devices connected at the PCC. Part of the changes originates from the secondary emissions and part from the nonlinearity deviation because not all the devices operating states are included in the linear model. This means that the device's nonlinearity can deviate not only because of the devices interactions but also by the device operation alone. Fig. 10 illustrates this phenomena in more detail taken as an example the highest ΔTHNL ILED (i.e., 317.1%) listed in Table 4, given by 50 lamps and the same combination as in Fig. 9. In the figure, two viewpoints are illustrated. The first shows the LED dIH , considering the presence and absence of PVI, while the second shows the magnitude of the current harmonics for the linear model and actual measurements. From Fig. 10, the first clear observation is that LED dIH is small above the 15th harmonic order. Second, dIH changes substantially when PVI I2 operation is considered, confirming that the harmonic interactions between PVI and LED create changes in the LED current harmonics emission. Third, values for dIH are not strongly correlated with the difference between RMS current harmonics obtained from the linear model and actual measurements. The THDI difference between the two sets of harmonic components is only 3.13%. This suggests that the main nonlinearity deviation is in the phase angle instead on the magnitude of
Fig. 8. Nonlinearity deviation of an individual harmonic as function of number of lamps when PVI I2 and LEDs B are considered: (left) dIH variation and (right) its polar plot representation.
increase as the number of lamps increases, mainly by the changes in the harmonic phase angle. Since the uncertainty is also dependent on the RMS value, no certainty can be given for the dIH when none or 6 lamps are considered. Note also, from the polar plot, that the linear model harmonics have a regular trend at about 120° , while the actual harmonic measurements present an irregular pattern, varying in magnitude and phase. 5.2. Devices technology impact on LED nonlinearity Table 4 shows ΔTHNL ILED (%) of different devices combination and number of lamps. To evaluate the nonlinearity deviation attributed to Table 4 ΔTHNL ILED for diverse device combinations. Lamps
I1-A
I1-B
I1-C
I2-A
I2-B
I2-C
I3-A
I3-B
I3-C
A
B
C
None 1 3 6 15 24 50
0.0 0.0 0.0 0.0 0.0 6.6 1.7
0.0 0.0 49.9 60.0 73.8 204.4 281.5
0.0 0.0 70.0 1.3 54.5 131.9 147.4
0.0 0.0 0.0 2.4 2.4 14.3 0.8
0.0 3.6 15.9 96.7 106.3 166.5 317.1
0.0 0.0 40.2 3.9 23.7 65.3 97.6
0.0 0.0 0.0 0.0 14.3 0.8 0.3
0.0 0.0 0.6 1.8 29.2 172.9 169.0
0.0 0.0 25.5 26.2 15.4 77.9 85.7
0.0 0.0 0.0 0.0 7.4 3.8 0.8
0.0 0.0 0.0 15.6 68.9 104.5 204.0
0.0 0.0 0.0 0.0 18.7 98.1 97.3
5
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Fig. 10. LED B nonlinearity deviation considering 50 lamps: (top) dIH for ILED with and without PVI I2 operation, and (bottom) ILED RMS current harmonics obtained from the linear model and measurements.
Fig. 12. (a) VPCC H5 obtained from the linear model and measurements for combination I1-C for different voltage source waveform. (b)–(d) show details for the voltage waveforms V1, V2, and V3, respectively.
nearly the same. However, as it is observed in Table 5, when the nonsinusoidal voltage waveforms V2, and V3 are considered, the source currents change, resulting in a ΔTHNL increase for most of the PVI-LED combinations and channels. For instance, ΔTHNL ILED for the combination I1-A more than doubles if we compare V1 to V2 and V3. Also, ΔTHNL IPVI , for V2 and V3 is much higher than for V1. Fig. 12 illustrates the impact that different voltage waveforms have on one individual voltage harmonic component as a function of the number of lamps. The different voltage waveforms place, by secondary emission, the H5 harmonic component at distinct locations in the polar plot (see Fig. 12a). Fig. 12b–d shows different characteristics for the harmonics described by the linear model and measurements. For instance, V1 creates a predominant phase shift, while V2 and V3 impact more the magnitude than phase (note that V2 decreases the harmonic magnitudes, compared to the linear model, as the number of lamps increases). The examples given by Table 5 and Fig. 12 implies that nonlinearity deviations are created by the nonlinear interactions when a distorted voltage waveform at PCC is present. In other words, when more than one device is considered in the system, the voltage distortion at the device's terminals is prone to be changed by the multiple devices’ current harmonics. This partially happens because of the changes caused by the secondary emission into the devices, covered by the linear model, and partially from a harmonic scenario not predicted by the individual device-source supply combinations in the linear model. Consequently, a different operating point is attained by the devices, resulting in unaccounted harmonic current emission levels over the system.
Fig. 11. Detail of an individual harmonic variation as function of number of lamps when PVI I2 and LEDs B are considered: (left) dIH variation and (right) its polar plot representation.
the harmonic components. This is confirmed by taking one harmonic order as an example. Fig. 11 shows dIH for H7 and its harmonic polar plot obtained from the linear model and actual measurements. As Fig. 11 outlines, for this LED type in particular, dIH,7 increases as the number of lamps increases. This occurs because of the difference between the phase angle of the harmonic components from measurements and linear model increases. For instance, for 24 and 50 lamps, the phase angle difference is about 15° and 60°, respectively. 5.4. Impact of voltage source distortion Looking to the impact that the voltage source distortion creates on devices nonlinearity (i.e, nonlinearity dependence on voltage), Table 5 reports the ΔTHNL for IPVI , ILED, and VPCC for 24 lamps in diverse PVILED combinations. Because the different source supply voltages are covered by the linear model for all the possible test setup combinations, the harmonics obtained from the linear model and actual measurements tend to be Table 5 ΔTHNL (%) for IPVI , ILED , and IGRID in diverse devices combination and 24 lamps. Ch.
Vs
I1-A
I1-B
I1-C
I2-A
I2-B
I2-C
I3-A
I3-B
I3-C
IPVI
V1 V2 V3 V1 V2 V3 V1 V2 V3
0.2 5.9 6.4 6.6 12.3 15.3 0.5 7.8 9.1
1.0 6.7 8.4 204.4 230.3 301.9 15.8 21.4 26.1
0.5 5.8 6.7 131.9 118.5 174.2 9.8 14.4 17.7
1.9 20.5 6.5 14.3 21.4 16.6 3.5 27.1 11.0
10.8 14.7 17.5 166.5 265.2 344.3 20.7 30.0 38.4
14.4 13.0 6.4 79.3 141.1 140.8 18.3 23.6 16.5
0.1 40.8 46.8 0.8 9.4 15.3 0.2 50.8 56.5
4.4 37.8 44.2 172.9 146.1 170.0 11.7 52.6 58.8
2.5 40.0 46.3 77.9 56.2 118.8 4.1 48.8 58.1
5.5. Impact of network impedance
ILED
IGRID
Table 6 lists the ΔTHNL for IPVI , ILED, and VPCC as a function of the network impedance. A test setup with PVI I1 combined with 15 and 50 LEDs A, B and C is considered. Table 6 shows that higher ΔTHNL is observed mainly for LEDs B and C. Channels ILED and VPCC are the most affected. Considering the impedance Z3 as reference (i.e., the one with higher magnitude), Z2 and Z4 create less nonlinearity deviation. For instance, ΔTHNL ILED for 6
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interaction. As the number of lamps increases, the nonlinearity deviation increases creating changes mainly in the phase angle of the harmonic components. This finding is in agreement with Mansoor [36] and Yong [39] on loads employing single-phase diode bridge rectifier, which showed that the harmonic cancellation due phase angle dispersion tends to increase with the number of loads in parallel. The results, however, show also increases in emission compared to the linear model. The nonlinearity deviation in PVIs is less compared to LEDs, but some of the assessed inverters are more susceptible to nonlinearity deviations than others. For instance, PVI I2 is the one that exhibited non-zero ΔTHNL IPVI for all LED types as observed in Table 3.
Table 6 ΔTHNL (%) for IPVI , ILED , and VPCC considering LEDs and PVI I1. Ch. (a) 15 lamps IPVI
ILED VPCC
50 lamps IPVI
ILED VPCC
Zgrid
I1-A
I1-B
I1-C
Z3 Z4 Z3 Z4 Z3 Z4
0.0 0.1 0.0 0.0 0.0 0.0
0.0 0.0 73.8 13.0 0.0 0.0
0.0 0.0 54.5 44.3 0.0 0.0
Z2 Z3 Z2 Z3 Z2 Z3
0.0 0.0 1.8 1.7 0.0 0.0
0.3 4.9 157.1 281.5 0.0 2.0
0.2 1.8 103.0 147.4 0.0 0.8
6.3. Voltage source impact The nonlinearity deviation varies depending on the distorted voltage waveform at PCC. The impact of the voltage source waveform is bigger for LEDs, especially for LEDs without PFC. In some cases, voltage source distortion can increase the nonlinearity deviation more than having tens of lamps under a pure sinusoidal waveform (e.g., PVI I3 reaches ΔTHNL IPVI > 40 % with V2 and V3, while the maximum value with 50 lamps and pure sinusoidal voltage is equal to 11.3%.). In [35], this impact is mainly attributed to the operation of the LEDs without PFC (i.e., fitted with diode bridge rectifiers), where lower-order voltage harmonics have a significant effect on the harmonic currents. This study corroborates [35] showing how a realistic installation can create significant nonlinearity deviation at higher-harmonic orders (i.e., 15h harmonic, see Fig. 9).
the combination I1-B drops from 73.8% to 13.0% when Z3 and Z4 are considered, respectively. The lower the network magnitude impedance, the less susceptible to changes in the PCC voltage. Consequently, less nonlinear interactions will happen between devices. 6. Discussion 6.1. Suitability of the method Results have shown the ability of the used method in identifying and quantifying nonlinear interaction as introduced in [21]. The method also allows a quantification of the error when using the direct method with Norton/Thévenin models for devices. The employment of the linear model proved to be a suitable tool due to its simplicity of use and results consistent with measurement results. In general, the Norton/Thévenin equivalent to describe a power electronic device is an acceptable representation. The linear model and measurements give similar results as verified by the harmonic polar plots. This finding supports previous research into this area which has found that Norton equivalent is accurate enough to forecast the harmonic currents caused by a distorted grid [34]. However, the devices nonlinearity deviation can change depending on the operational conditions. This is for example illustrated by the non-zero ΔTHNL IPVI values in Tables 3–6 . The device emission characteristics may be modified during operation, resulting in current emissions higher than those predicted by the linear model. If we take as reference the results from Fig. 10 we can see for instance, that the 9th and 15th harmonic orders show a deviation in the magnitude of about 5% and −75%, respectively, compared to the linear model. This shows that nonlinearity deviations can reduce the devices’ current emissions, but the opposite can also happen, potentially leading to harmonic issues. In an extreme condition, for instance, depending on the level of nonlinearity deviation, power electronic devices with closed-loop control are subjected to an unexpected shift in reference operating point which can lead to instability. The Mansoor's model [35] and other studies on emission by multiple passive diode rectifiers [36–38] all show a lower emission compared to the linear model. The method proposed in this paper allows for identifying cases in which the emission is significantly higher than according to the linear model. This could be an indication of possible high harmonic levels or even instability.
6.4. Network impedance impact The network impedance has also a significant impact on the nonlinearity deviation as presented in Table 6. The higher the network magnitude impedance, the more susceptible to changes is the PCC voltage, and consequently higher nonlinearity deviations are prone to happen. Our results are in line with previous findings in the literature [40], where it was mathematically proven that harmonic Norton models are sensitive to changes in the network impedance. Changes in the current harmonic emission were observed in [29] by evaluating the THDI under different network impedances. The study presented in this paper makes a clearer distinction between changes due to linear phenomena (primary and secondary emission) and changes due to nonlinear phenomena (nonlinear interaction). 6.5. Uncertainties not considered Although the main sources of errors have been considered in the model, some assumptions have been made:
• Uncertainty originating from components temperature-dependence
• • •
6.2. Technologies impact From the results, the devices fitted with current control loop (e.g., LED PFC, or PVI current control loop) exhibited minor deviations in nonlinearity. Although results are based on a small number of samples, simple designs like LEDs B and C seem to create more nonlinear 7
and time-dependence have been neglected. The measurements were taken in a laboratory environment with temperature control using the same procedure, equipment, and defined input variants. A minimum thermal stabilization time of 60 min was used for the LED [41] (measurements were performed starting from the highest to the lowest number of lamps). Any difference between LEDs belonging to the same type was neglected. The angular influence uncertainty of current probes was verified and assumed to be small (less than 1.0°), and therefore neglected. Finally, the impedance variation within the cycle and possible uncertainty impact due to the use of FFT algorithm were also neglected.
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emission changes mainly in the low order harmonics. The harmonic phase angle is the most affected harmonic characteristic. Additionally, changes in the network impedance and voltage source waveform play an important role in determining the nonlinear interaction. From the evidence of this study, the Norton/Thévenin equivalent to describe the power electronic device is still a good model for harmonic analysis. However, as the number of devices or the system order increase, more nonlinear interactions will happen and additional uncertainty in the results caused by the power electronics nonlinearity deviation should be considered.
6.6. Limitations and need for further research A limitation of the method is the need for high accuracy measuring equipment in order to reduce the uncertainty. As the number of different devices increases, the linear model combined uncertainty will also increase, reaching values that cannot be distinguished from the nonlinear interaction. Finally, further research is needed in identifying the origins and causes of nonlinearity deviation within the devices. Promising research goes towards extending the analysis of nonlinearity created by diode rectifiers, power electronics current control-loop and switching devices dynamics.
Conflict of interest
7. Conclusion
None declared. Acknowledgments
This study has used a linear model to verify the extent of nonlinear interaction between devices in the same LV installation. It was observed that the interaction of different power electronic devices creates nonlinearity deviation, resulting in current harmonics
The authors would like to thank the Swedish Energy Agency (Energimyndigheten) for supporting this work.
Appendix A. Evaluating and expressing the uncertainty The evaluation method and the expression of uncertainty used in this study follow the guidelines given by [42]. In total nine sources of uncertainty were considered: three impedances (ZPVI , ZLED, and ZGRID), two current sources (IsPVI and IsLED ), one voltage source (VS ) and three direct measurements (IPVI , ILED , and IPCC ). Type B standard uncertainties, u2 (x i ) , (function of amplitude and frequency) were obtained from the manufacturers manuals for the DAQ [27], current probes [43], power amplifier [26], and line impedance [25]. LED and PVI impedance uncertainties were extracted from [44], and source currents estimated by combining the measured current, voltage source, and line impedance uncertainties. Because we are comparing different results from the linear model and actual measurements, the uncertainty estimation method results in two distinct parts: the measurements uncertainties, U (Y ) , and linear model uncertainties, U (Yˆ ) . The first considers the uncertainty associated to the direct measurements, while the second considers the combined standard uncertainty of the output quantity, u (y ) described in the general form as: N
u2 (y ) =
2
∂Y ⎞
∑⎛X ⎜
i=1
⎝
i
⎟
u2 (x i )
(A.1)
⎠
where Y is the output quantity, defined by the linear model equations: IPVI , ILED, IGRID, and VPCC as function of the input quantities Xi (i.e., ZPVI , ZLED, ZGRID , IsPVI , and IsLED ), and u (x i ) is standard uncertainty for the input quantities Xi . In practice, due the simplicity of the linear equations, the partial derivatives ∂Y (often called sensitivity coefficients) imply that the combined ∂X uncertainties can be obtained by simply adding the absolute uncertainties when input quantities are added/subtracted, and by adding the relative uncertainties, when the input quantities are multiplied/divided. As a final step, the expanded uncertainty, U (Y ) , is calculated using U (Y ) = k . u (y ) assuming a normal distribution and coverage factor of k = 2 , which corresponds to an interval having a level of confidence of about 95%.
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Deviations between the commonly-used model and measurements of harmonic distortion in low-voltage installations
T
Tatiano Busattoa,*, Vineetha Ravindrana, Anders Larssona, Sarah K. Rönnberga, Math H.J. Bollena, Jan Meyerb a b
Electric Power Engineering, Luleå University of Technology, Skellefteå, Sweden Electric Power Engineering, Technische Universität Dresden, Dresden, Germany
A R T I C LE I N FO
A B S T R A C T
Keywords: Power quality Power system harmonics Power electronics Harmonic analysis Nonlinear systems
Harmonic analysis studies of modern power systems commonly employ Norton and Thévenin equivalents at harmonic frequencies for the nonlinear devices. This approach neglects the so-called nonlinear interaction phenomenon. This paper addresses the difference between the results from the commonly-used model and the actual harmonic distortion measured in a low-voltage installation. A number of indices are introduced to quantify the nonlinear interaction. These indices allow a quantification of the extent to which the commonlyused model is also to predict harmonic voltages and currents in a modern low-voltage installation. The proposed model and the subsequent mathematical analysis are illustrated through measurements from different combinations of PV inverters and LED lamps using different technologies. The results show that deviation is dependent on the used technology, network impedance, and source voltage waveform. Other findings are that nonlinear interaction happens mainly in the low harmonic orders and impacts are more perceived on the harmonics phase angle. Possible explanations for these observations are discussed.
1. Introduction With the proliferation of power electronic devices in the power system, issues related to harmonic distortion have been regularly addressed during the last decades. More recently, the subject has gained additional focus due to the appearance of electrical vehicle (EV) charging and power inverters to interface solar and wind power plants. Depending on system complexity and desired accuracy, different harmonic analysis methods in time and frequency domain are described in the literature [1,2]. Time domain methods [3,4], provide by far the most accurate solution. However, limitations arise for analysis of a power system with multiple devices and multiple operational states. Harmonic analysis in the frequency domain presents fewer constraints enabling analysis with a larger number of devices and is therefore commonly used in studies with multiple harmonic sources [5–8]. In this context, the direct method [9,10] provides the simplest solution: the frequency response for a given bus is obtained from the solution of the network equation [Y ] V = I , where [Y ] is the network admittance matrix, V the nodal voltage vector to be solved, and I is the vector of current injections. Although this method is broadly used, it cannot provide adequate results for high levels of voltage distortion
⁎
[11], since the devices’ current harmonics are dependent on their terminal voltage and operating current as well [12]. To include these dependencies, but without having to use a detailed time domain model, several harmonic analysis methods in the frequency domain and methods combining the individual advantages of frequency and time domain have been developed [11,13–16]. These methods, however, suffer from practical limitations when the power system contains a considerable number of power electronic devices and buses. Accuracy will depend on the details and accuracy of the individual device models. Each type of device is different; large amounts of input data and additional measurements are needed before one even can get started. For instance, several of the accurate frequency-domain methods, such as the iterative harmonic analysis (IHA) [15], present slow convergence, narrow stability margin and, high computational effort dependent on the system complexity [1]. Similarly, the harmonic domain (HD) method suffers from its algorithm complexity [17]. Their computational advantage compared to time-domain methods becomes less with the increasing number of devices. Study cases employing these methods are limited to a few nonlinear devices that far from cover the actual complexity and extension of nonlinear devices in real power systems. All these drawbacks diminish from its adoption by the
Corresponding author. E-mail address: [email protected] (T. Busatto).
https://doi.org/10.1016/j.epsr.2019.106166 Received 3 October 2019; Received in revised form 16 November 2019; Accepted 11 December 2019 Available online 25 December 2019 0378-7796/ © 2019 Elsevier B.V. All rights reserved.
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of PVI and LEDs connected to an emulated power grid. As a first step in the method, the installation is modeled by only the current sources and impedances for the different parts of the system. Next, the values for current sources and impedances in the harmonic range are obtained from dedicated measurements. The current spectrum for the devices is obtained from the measurements using a known voltage distortion and quasi-zero impedance of the emulated grid. The device currents are combined with the measured impedances to obtain the Norton equivalent. Finally, the complex currents and voltages at the points of interest are calculated from the resulting linear model. To determine the nonlinearity deviation due to the nonlinear interaction, the currents and voltages from simulation and measurement are compared in different operating scenarios. The uncertainties from the linear model and measurements are taken into consideration as described in Appendix A. Any deviation above the uncertainty range accounts to nonlinearity deviation for the device or installation. Harmonic components phase angle for all channels is referenced to the ascending zero-crossing of the point of common coupling (PCC) voltage channel (see Fig. 2 for the location of PCC). A 5 Hz frequency resolution was adopted and harmonics calculated following the recommendations in [23].
industry. The afore-mentioned frequency and hybrid frequency-time domain methods have focused mainly on the harmonic models of individual devices, limiting the discussion on how harmonic analysis can be performed in an existing power system with a large number of nonlinear devices. Due to the constant inclusion of new power electronics to the grid, there is a growing demand for harmonic studies in a broader scope. A modern low-voltage (LV) installation can easily contain more than 100 power electronic devices connected close to each other. Modeling each of these devices in detail is in practice not possible. Additionally, LV installations are not static presenting often variations during operation. This means that stochastic methods are needed, most commonly based on the random generation of many different cases [18–20]. With so many cases and devices to be modeled, anything more complicated than the direct method will be difficult to solve. As there is, in fact, no suitable alternative when a large number of devices are involved, it is appropriate to evaluate and quantify the accuracy of the direct method. A simple but general model for the interaction between a single device and the rest of the grid is presented in [21] and recommended to be used by an international working group [22]; next to two types of linear interaction (primary and secondary emission), the term “nonlinear interaction” is introduced. This term refers to any interaction that cannot be explained from time-independent Norton/Thévenin models. The term is however not quantitatively explained in [21]. This paper proposes a method to quantify to which extent the direct method employing Norton/Thévenin equivalent harmonic sources is a suitable representation for an LV installation. The proposed method was applied to 9 different installations of photovoltaic PV inverters (PVI) and LED lamps (LED). Considering voltage source and network impedance variants, a total of 486 different combinations were studied. Measurements in the experimental test setup have been compared with results obtained by employing the direct method, referred from here on as the linear model. Difference between measurement and the linear model is an indication of “nonlinear interaction” as introduced in [21]. Considering the linear model and measuring results, any difference between the results reveals the presence of nonlinearity in the system. Throughout this paper, this difference (and thus the nonlinear interaction) is quantified through an index referred to as “nonlinearity deviation”. Separate indices are defined for individual harmonic frequencies and for covering the whole frequency range of interest. The study demonstrates to which extent the results obtained from the direct method, using Norton/Thévenin equivalent for the harmonic sources, is an accurate representation for the harmonic propagation in an installation with a large number of power electronic devices. Additionally, the most important factors that contribute to nonlinear interaction are identified in different operating scenarios, changes in the grid characteristics, and combinations of different technologies. This paper does not deal with the origins and causes of nonlinearities, instead, it is a first step in revealing to which extent nonlinearity deviations impact harmonic voltage and current distortion and to which extent the direct method gives reasonably accurate results. The methodology to quantify the nonlinear interaction is introduced in Section 2. Sections 3 and 4 describe the laboratory test setup and the used linear model, respectively. Section 5 summarizes the results, followed by the discussion in Section 6, where the results are further interpreted. Finally, Section 7 presents the main conclusions.
2.1. Quantifying the nonlinear interaction Nonlinearity deviation can be evaluated by the difference between magnitudes and phases angle of the harmonic components obtained from measurements and linear model. However, the individual evaluation of these harmonic characteristics can lead to misinterpretation of the real extension of nonlinearity deviations. To quantify the deviation consider YH and YˆH as the harmonic components in the complex plane ℂ obtained from measurements and linear model respectively, given by:
⎧YH , h = ah + bh j ⎨YˆH , h = aˆh + bˆh j ⎩
(1)
where the symbol Y is replaced, as required, by I for currents, or by U for voltages; H qualifies the variable as harmonic, and h is the harmonic order. Since ℂ is isomorphic to ℝ2 , the Euclidean distance between harmonic components, dYH , h , described in (2), is used as a measure of nonlinearity deviation:
dYH , h = |YH , h − YˆH , h| =
(bh − bˆh)2 + (ah − aˆh)2
(2)
To compare different scenarios, the authors have introduced the Total Harmonic Nonlinearity Deviation factor, ΔTHNLY , based on the subgroup concept introduced in [23], but considering as input the values obtained from (2) and uncertainty: hmax
ΔTHNLY =
∑ h=2
2
⎛⎜ DYH , h ⎞⎟ ⎝ YH ,1 ⎠
(3)
In (3), DYH , h is the effective (or certain) distance (i.e., harmonic components distance minus the uncertainty) given by:
0 if dYH , h < U (dYH , h) DYH , h = ⎧ ⎨ dY − U (dY ) if dYH , h ≥ U (dYH , h) H , h H , h ⎩
(4)
where U (dYH , h) is the expanded total distance uncertainty given by the linear model and actual measurements:
2. Methodology
U (dYH , h) = U (YH , h) + U (YˆH , h)
The methodological approach taken in this study is based on the comparison of results obtained from a linear model allowing the application of the direct method and measurements performed in a controlled but realistic scenario. A simplified LV installation, containing the essential elements for the purposes of the analysis is considered. The test setup consists of a set
(5)
3. Experimental setup The photo and the simplified schematic of the measurement setup are show in Figs. 1 and 2 , respectively. The setup consists of a 10 kW 2
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Table 2 Network impedances (phase + neutral). Z1
Z2
Z3
Z4
≈0 Ω
0.17 + j 0.10 Ω
0.40 + j 0.25 Ω
0.40 + j 0.04 Ω
between 50° and 90° commonly found in low-power LEDs where the requirements for power factor are less strict. LED B and C also differ in impedance characteristics. 3. Network impedances: the different network impedances considered in this study are listed in Table 2. The first one is a quasi-zero impedance (Z1). It is defined by the Power Amplifier internal impedance plus cable impedances. The second is about half the reference impedance (Z2), given by IEC/TR 60725 [30]. To emulate a weak grid, a reference impedance (Z3) is used. A typical Swedish network impedance value (Z4) [31] is included with a predominantly resistive characteristic. 4. Source supply voltage waveform: besides the ideal sinusoidal voltage waveform, V1 (i.e., the reference), two additional voltage waveforms, flat-top, V2, and pointed-top, V3, are considered. Both are based on LV networks measurements which specification is given in the Appendix of [29].
Fig. 1. Photo of the experimental measurement setup.
4. Linear model Based on the test setup shown in Fig. 2, Fig. 3 shows the linear model used to predict current and voltage harmonics. The PVI, LED, and grid currents, IPVI , ILED, and IGRID, respectively, are obtained from the following set of linear equations: Fig. 2. Simplified test-setup schematic.
IˆPVI ⎤ IsPVI. ZPVI ⎡ ⎤ ⎡ ZPVI 0 ZGRID ⎤ ⎡ IˆLED ⎥ ⎢ IsLED. ZLED ⎥ ⎢ 0 ZLED ZGRID ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ 1 ⎥ 0 ˆ ⎦⎢ ⎣− 1 − 1 ⎦ ⎦⎣ ⎣ IGRID ⎥
PV array emulator [24], which is connected to the PVI under test. The AC side of the PVI is connected to an adjustable line impedance emulator [25] and the source supply is emulated by a programmable 45 kVA three-phase power amplifier [26]. The LEDs are connected between the PVI and the line impedance (PCC) through a board with several lamp sockets. All the measurements were performed by a data acquisition (DAQ) system [27] equipped with high-bandwidth and high-accuracy acquisition modules and voltage/current probes [28] placed at the interested monitoring locations. Five test-setup variants with respective sub-variants, summarized in Table 1, were tested during the experiment.
VˆPCC = VS − IˆGRID. ZGRID
The magnitude and phase angles of ZPVI , ZLEDs , and ZGRID used in the linear model are shown in Fig. 4. Note the magnitude difference among the LED types and the larger phase variation for the PVIs. The complex impedances were obtained as follows: 1. LED impedance: measurements were performed in the Pehr Högström Laboratory with Luleå University of Technology in Skellefteå employing an invasive steady-state-based method using short-time current injections [32] at harmonic frequencies. The impedance of one single LED was obtained, and assuming linearity, the equivalent impedance for different numbers of lamps in parallel was obtained. Results were verified by supplementary measurements revealing errors below 5.0% and 1.0° for magnitude and phase, respectively.
Table 1 Tested variants and sub-variants. Sub-variant ID
Test points
PVIs LEDs Number of lamps Network impedance Source supply voltage waveform
I1, I2, I3 A, B, C 1, 3, 6, 15, 24, 50 Z1, Z2, Z3, Z4 V1, V2, V3
3 3 6 4 3
(7)
4.1. Estimating the impedances
1. PVIs: are the same inverters described in [29], where further detail on primary emission can be found. The set covers some of the inverter topologies used in small residential and commercial applications. I1 is a 4.6 kW transformerless inverter, while I2 and I3 are 4.6 and 10 kW inverters and use a low- and high-frequency transformer for galvanic isolation, respectively. 2. LEDs: three LED lamps commonly found in the market (i.e., year 2017), differentiating mainly by the current waveform and impedance characteristics, were selected. LED A has a power factor correction (PFC) feature presenting a more sinusoidal current waveform, while LEDs B and C have a typical marked current peak
Variant name
(6)
where ZPVI , ZLEDs , and ZS , are the internal impedances for PVI and LED, and network impedance, respectively. IsPVI and IsLEDs are the source currents from LEDs and PVI. From the calculated grid current, IGrid , the PCC voltage is obtained using the following expression:
Fig. 3. Linear model equivalent circuit. 3
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Fig. 6. Difference between IPVI harmonics components H5 and H9 obtained from linear model and measurements.
considered, dYH , h becomes DYH , h and values for DYH,5 and DYH,9 are 13.58 and 12.68 mA, respectively. The figure shows a slight change in magnitude and a substantial phase shift in both harmonics. Harmonic H9, for instance, presents a small magnitude deviation (4.19 mA), but a phase shift of about 32° is observed. Note the difference in uncertainties, which are dependent on the harmonic RMS values, see Appendix A. 5.1. Devices technology impact on PVI nonlinearity Fig. 4. Magnitude and phase impedance characteristics for the Linear Model components.
An examination on the nonlinearity deviation in dependence of the device's technology is given by the data in Table 3, which lists the ΔTHNL IPVI (%) for different device combination and number of lamps. Significant differences can be observed in at least three aspects from the results in Table 3. First, PVIs present different levels of nonlinearity deviation. This can be observed by maximum ΔTHNL IPVI (i.e., italicized values) for PVI I1, I2 and I3 which are close to 5%, 30%, and 12%, respectively. From the results, PVI I1 and PVI I2 present the smallest and largest nonlinearity deviations, respectively, due to the LED operation. Second, LED A and B cause the lowest and the highest deviation, respectively, on the inverter nonlinearity. This is observed by ΔTHNL ≈ 0 in cases where LED A is used, and ΔTHNL with high values when LED B is used. Finally, in general the inverters nonlinearity deviation tends to increase as the number of lamps increases, with some exceptions (e.g., 15lamps I2-A and 3-lamps I2-C, where ΔTHNL is greater than the ΔTHNL for 50-lamps).
2. PVI input impedance: characteristics for PVI I1, I2 and I3 were extracted from [33], in which impedance values were obtained from measurements performed at TU Dresden. 3. Network impedance: estimated by assuming an impedance R + jωL with constant R and L . The measurements were done in a controlled environment where the network impedance is known with reasonable accuracy. 4.2. Estimating the source currents The source currents were obtained by using the simplified schema shown in Fig. 5. To obtain the source currents, the Z1 (Zs ≈ 0 Ω) network impedance was used, emulating a terminal short circuit at harmonic frequencies, and the voltage waveform fixed to V1, V2, or V3. Consequently, the current from the source becomes equal to the measured one minus the current through the (known) source impedance driven by the (known) harmonic voltage distortion. For each background voltage, a particular spectrum for the source current is given, being assessed distinct operating points around which the linearization is made.
5.1.1. Frequency-dependency of PVI Nonlinearity Taking as an example the combination I2-B (i.e., the one that presents the highest ΔTHNL variation, Fig. 7 illustrates the frequency-dependency on the nonlinearity deviation for IPVI as the number of lamps changes. As shown in Fig. 7 (top graph), higher values of dIH are observed at the low order harmonics, and the values decrease as the frequency increases. Note that when no lamps are considered (i.e., middle graph, when there is no interaction between PVI and LED), dIH , is inside the uncertainty range for most of the harmonic components. Fig. 8 shows in detail how the dIH for one individual harmonic (H9) changes as a function of the number of lamps. Note that dIH tends to
5. Results An example of nonlinearity deviation caused by nonlinear interaction between PVIs and LEDs is presented in Fig. 6, where two particular harmonic components of a PVI in the complex plane ℂ are detailed. The results shown in Fig. 6 were obtained from a scenario considering PVI I1 at 50% of its rated power and 15 LED C, network impedance Z3, and sinusoidal voltage waveform (V1). The same setup will be used as a reference in the following analysis unless otherwise specified. In the figure, the solid line represents the distance in the complex plane, dYH , h , as introduced in Section 2.1. Once the uncertainty is
Table 3 ΔTHNL IPVI (%) for diverse devices combination.
Fig. 5. Simplified circuit schematic for obtaining the current sources. PVI (left) and LED (right). 4
Lamps
I1-A
I1-B
I1-C
I2-A
I2-B
I2-C
I3-A
I3-B
I3-C
None 1 3 6 15 24 50
0.0 0.0 0.0 0.0 0.0 0.2 0.0
0.0 0.0 0.1 0.1 0.0 1.0 4.9
0.0 0.0 0.1 0.0 0.0 0.5 1.8
1.0 11.3 8.0 0.8 18.0 1.9 2.1
1.0 5.8 5.5 11.9 8.9 10.8 28.9
1.0 10.4 15.1 1.7 2.3 12.3 13.4
2.1 0.2 0.2 0.1 2.0 0.1 1.7
2.1 0.8 1.1 0.8 1.6 4.4 11.3
2.1 0.1 0.1 2.0 0.4 2.5 5.1
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Fig. 9. LED dIH as function of number of lamps for the combination PVI I2 and LED B.
the LED operation individually, the last three columns show the ΔTHNL when PVI is not considered. From Table 4, certain similarities to ΔTHNL IPVI values (see Table 3) are observed. ΔTHNL ILED tends to increase as the number of lamps increases, and the LED nonlinearity, from the lowest to highest deviation, is observed when LED A, C, and B, and PVI I1, I3, and I2 are used, respectively. Further, ΔTHNL ILED is in general much higher than the ΔTHNL IPVI . Also, if we consider the ΔTHNL ILED for the LED operation alone (3 last columns in Table 4), we can conclude that the isolated LED operation has the most significant contribution towards increased values.
Fig. 7. dIH as function of number of lamps for PVI I2 and LEDs B: (top) all number of lamps, (middle) no lamps detail, and (bottom) 50 lamps detail.
5.3. Frequency-dependency of LED Nonlinearity The detailed analysis for the combinations I2-B (i.e., the one that presents the highest nonlinearity deviation) is shown as an example in Fig. 9, where the nonlinearity deviation, dIH , for different numbers of lamps is presented in the frequency domain. Because LED B has a high current harmonic content, it creates an additional voltage distortion at the PCC depending on the grid impedance. Consequently, this extra distortion will change the LED harmonic current emissions itself and of other devices connected at the PCC. Part of the changes originates from the secondary emissions and part from the nonlinearity deviation because not all the devices operating states are included in the linear model. This means that the device's nonlinearity can deviate not only because of the devices interactions but also by the device operation alone. Fig. 10 illustrates this phenomena in more detail taken as an example the highest ΔTHNL ILED (i.e., 317.1%) listed in Table 4, given by 50 lamps and the same combination as in Fig. 9. In the figure, two viewpoints are illustrated. The first shows the LED dIH , considering the presence and absence of PVI, while the second shows the magnitude of the current harmonics for the linear model and actual measurements. From Fig. 10, the first clear observation is that LED dIH is small above the 15th harmonic order. Second, dIH changes substantially when PVI I2 operation is considered, confirming that the harmonic interactions between PVI and LED create changes in the LED current harmonics emission. Third, values for dIH are not strongly correlated with the difference between RMS current harmonics obtained from the linear model and actual measurements. The THDI difference between the two sets of harmonic components is only 3.13%. This suggests that the main nonlinearity deviation is in the phase angle instead on the magnitude of
Fig. 8. Nonlinearity deviation of an individual harmonic as function of number of lamps when PVI I2 and LEDs B are considered: (left) dIH variation and (right) its polar plot representation.
increase as the number of lamps increases, mainly by the changes in the harmonic phase angle. Since the uncertainty is also dependent on the RMS value, no certainty can be given for the dIH when none or 6 lamps are considered. Note also, from the polar plot, that the linear model harmonics have a regular trend at about 120° , while the actual harmonic measurements present an irregular pattern, varying in magnitude and phase. 5.2. Devices technology impact on LED nonlinearity Table 4 shows ΔTHNL ILED (%) of different devices combination and number of lamps. To evaluate the nonlinearity deviation attributed to Table 4 ΔTHNL ILED for diverse device combinations. Lamps
I1-A
I1-B
I1-C
I2-A
I2-B
I2-C
I3-A
I3-B
I3-C
A
B
C
None 1 3 6 15 24 50
0.0 0.0 0.0 0.0 0.0 6.6 1.7
0.0 0.0 49.9 60.0 73.8 204.4 281.5
0.0 0.0 70.0 1.3 54.5 131.9 147.4
0.0 0.0 0.0 2.4 2.4 14.3 0.8
0.0 3.6 15.9 96.7 106.3 166.5 317.1
0.0 0.0 40.2 3.9 23.7 65.3 97.6
0.0 0.0 0.0 0.0 14.3 0.8 0.3
0.0 0.0 0.6 1.8 29.2 172.9 169.0
0.0 0.0 25.5 26.2 15.4 77.9 85.7
0.0 0.0 0.0 0.0 7.4 3.8 0.8
0.0 0.0 0.0 15.6 68.9 104.5 204.0
0.0 0.0 0.0 0.0 18.7 98.1 97.3
5
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Fig. 10. LED B nonlinearity deviation considering 50 lamps: (top) dIH for ILED with and without PVI I2 operation, and (bottom) ILED RMS current harmonics obtained from the linear model and measurements.
Fig. 12. (a) VPCC H5 obtained from the linear model and measurements for combination I1-C for different voltage source waveform. (b)–(d) show details for the voltage waveforms V1, V2, and V3, respectively.
nearly the same. However, as it is observed in Table 5, when the nonsinusoidal voltage waveforms V2, and V3 are considered, the source currents change, resulting in a ΔTHNL increase for most of the PVI-LED combinations and channels. For instance, ΔTHNL ILED for the combination I1-A more than doubles if we compare V1 to V2 and V3. Also, ΔTHNL IPVI , for V2 and V3 is much higher than for V1. Fig. 12 illustrates the impact that different voltage waveforms have on one individual voltage harmonic component as a function of the number of lamps. The different voltage waveforms place, by secondary emission, the H5 harmonic component at distinct locations in the polar plot (see Fig. 12a). Fig. 12b–d shows different characteristics for the harmonics described by the linear model and measurements. For instance, V1 creates a predominant phase shift, while V2 and V3 impact more the magnitude than phase (note that V2 decreases the harmonic magnitudes, compared to the linear model, as the number of lamps increases). The examples given by Table 5 and Fig. 12 implies that nonlinearity deviations are created by the nonlinear interactions when a distorted voltage waveform at PCC is present. In other words, when more than one device is considered in the system, the voltage distortion at the device's terminals is prone to be changed by the multiple devices’ current harmonics. This partially happens because of the changes caused by the secondary emission into the devices, covered by the linear model, and partially from a harmonic scenario not predicted by the individual device-source supply combinations in the linear model. Consequently, a different operating point is attained by the devices, resulting in unaccounted harmonic current emission levels over the system.
Fig. 11. Detail of an individual harmonic variation as function of number of lamps when PVI I2 and LEDs B are considered: (left) dIH variation and (right) its polar plot representation.
the harmonic components. This is confirmed by taking one harmonic order as an example. Fig. 11 shows dIH for H7 and its harmonic polar plot obtained from the linear model and actual measurements. As Fig. 11 outlines, for this LED type in particular, dIH,7 increases as the number of lamps increases. This occurs because of the difference between the phase angle of the harmonic components from measurements and linear model increases. For instance, for 24 and 50 lamps, the phase angle difference is about 15° and 60°, respectively. 5.4. Impact of voltage source distortion Looking to the impact that the voltage source distortion creates on devices nonlinearity (i.e, nonlinearity dependence on voltage), Table 5 reports the ΔTHNL for IPVI , ILED, and VPCC for 24 lamps in diverse PVILED combinations. Because the different source supply voltages are covered by the linear model for all the possible test setup combinations, the harmonics obtained from the linear model and actual measurements tend to be Table 5 ΔTHNL (%) for IPVI , ILED , and IGRID in diverse devices combination and 24 lamps. Ch.
Vs
I1-A
I1-B
I1-C
I2-A
I2-B
I2-C
I3-A
I3-B
I3-C
IPVI
V1 V2 V3 V1 V2 V3 V1 V2 V3
0.2 5.9 6.4 6.6 12.3 15.3 0.5 7.8 9.1
1.0 6.7 8.4 204.4 230.3 301.9 15.8 21.4 26.1
0.5 5.8 6.7 131.9 118.5 174.2 9.8 14.4 17.7
1.9 20.5 6.5 14.3 21.4 16.6 3.5 27.1 11.0
10.8 14.7 17.5 166.5 265.2 344.3 20.7 30.0 38.4
14.4 13.0 6.4 79.3 141.1 140.8 18.3 23.6 16.5
0.1 40.8 46.8 0.8 9.4 15.3 0.2 50.8 56.5
4.4 37.8 44.2 172.9 146.1 170.0 11.7 52.6 58.8
2.5 40.0 46.3 77.9 56.2 118.8 4.1 48.8 58.1
5.5. Impact of network impedance
ILED
IGRID
Table 6 lists the ΔTHNL for IPVI , ILED, and VPCC as a function of the network impedance. A test setup with PVI I1 combined with 15 and 50 LEDs A, B and C is considered. Table 6 shows that higher ΔTHNL is observed mainly for LEDs B and C. Channels ILED and VPCC are the most affected. Considering the impedance Z3 as reference (i.e., the one with higher magnitude), Z2 and Z4 create less nonlinearity deviation. For instance, ΔTHNL ILED for 6
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interaction. As the number of lamps increases, the nonlinearity deviation increases creating changes mainly in the phase angle of the harmonic components. This finding is in agreement with Mansoor [36] and Yong [39] on loads employing single-phase diode bridge rectifier, which showed that the harmonic cancellation due phase angle dispersion tends to increase with the number of loads in parallel. The results, however, show also increases in emission compared to the linear model. The nonlinearity deviation in PVIs is less compared to LEDs, but some of the assessed inverters are more susceptible to nonlinearity deviations than others. For instance, PVI I2 is the one that exhibited non-zero ΔTHNL IPVI for all LED types as observed in Table 3.
Table 6 ΔTHNL (%) for IPVI , ILED , and VPCC considering LEDs and PVI I1. Ch. (a) 15 lamps IPVI
ILED VPCC
50 lamps IPVI
ILED VPCC
Zgrid
I1-A
I1-B
I1-C
Z3 Z4 Z3 Z4 Z3 Z4
0.0 0.1 0.0 0.0 0.0 0.0
0.0 0.0 73.8 13.0 0.0 0.0
0.0 0.0 54.5 44.3 0.0 0.0
Z2 Z3 Z2 Z3 Z2 Z3
0.0 0.0 1.8 1.7 0.0 0.0
0.3 4.9 157.1 281.5 0.0 2.0
0.2 1.8 103.0 147.4 0.0 0.8
6.3. Voltage source impact The nonlinearity deviation varies depending on the distorted voltage waveform at PCC. The impact of the voltage source waveform is bigger for LEDs, especially for LEDs without PFC. In some cases, voltage source distortion can increase the nonlinearity deviation more than having tens of lamps under a pure sinusoidal waveform (e.g., PVI I3 reaches ΔTHNL IPVI > 40 % with V2 and V3, while the maximum value with 50 lamps and pure sinusoidal voltage is equal to 11.3%.). In [35], this impact is mainly attributed to the operation of the LEDs without PFC (i.e., fitted with diode bridge rectifiers), where lower-order voltage harmonics have a significant effect on the harmonic currents. This study corroborates [35] showing how a realistic installation can create significant nonlinearity deviation at higher-harmonic orders (i.e., 15h harmonic, see Fig. 9).
the combination I1-B drops from 73.8% to 13.0% when Z3 and Z4 are considered, respectively. The lower the network magnitude impedance, the less susceptible to changes in the PCC voltage. Consequently, less nonlinear interactions will happen between devices. 6. Discussion 6.1. Suitability of the method Results have shown the ability of the used method in identifying and quantifying nonlinear interaction as introduced in [21]. The method also allows a quantification of the error when using the direct method with Norton/Thévenin models for devices. The employment of the linear model proved to be a suitable tool due to its simplicity of use and results consistent with measurement results. In general, the Norton/Thévenin equivalent to describe a power electronic device is an acceptable representation. The linear model and measurements give similar results as verified by the harmonic polar plots. This finding supports previous research into this area which has found that Norton equivalent is accurate enough to forecast the harmonic currents caused by a distorted grid [34]. However, the devices nonlinearity deviation can change depending on the operational conditions. This is for example illustrated by the non-zero ΔTHNL IPVI values in Tables 3–6 . The device emission characteristics may be modified during operation, resulting in current emissions higher than those predicted by the linear model. If we take as reference the results from Fig. 10 we can see for instance, that the 9th and 15th harmonic orders show a deviation in the magnitude of about 5% and −75%, respectively, compared to the linear model. This shows that nonlinearity deviations can reduce the devices’ current emissions, but the opposite can also happen, potentially leading to harmonic issues. In an extreme condition, for instance, depending on the level of nonlinearity deviation, power electronic devices with closed-loop control are subjected to an unexpected shift in reference operating point which can lead to instability. The Mansoor's model [35] and other studies on emission by multiple passive diode rectifiers [36–38] all show a lower emission compared to the linear model. The method proposed in this paper allows for identifying cases in which the emission is significantly higher than according to the linear model. This could be an indication of possible high harmonic levels or even instability.
6.4. Network impedance impact The network impedance has also a significant impact on the nonlinearity deviation as presented in Table 6. The higher the network magnitude impedance, the more susceptible to changes is the PCC voltage, and consequently higher nonlinearity deviations are prone to happen. Our results are in line with previous findings in the literature [40], where it was mathematically proven that harmonic Norton models are sensitive to changes in the network impedance. Changes in the current harmonic emission were observed in [29] by evaluating the THDI under different network impedances. The study presented in this paper makes a clearer distinction between changes due to linear phenomena (primary and secondary emission) and changes due to nonlinear phenomena (nonlinear interaction). 6.5. Uncertainties not considered Although the main sources of errors have been considered in the model, some assumptions have been made:
• Uncertainty originating from components temperature-dependence
• • •
6.2. Technologies impact From the results, the devices fitted with current control loop (e.g., LED PFC, or PVI current control loop) exhibited minor deviations in nonlinearity. Although results are based on a small number of samples, simple designs like LEDs B and C seem to create more nonlinear 7
and time-dependence have been neglected. The measurements were taken in a laboratory environment with temperature control using the same procedure, equipment, and defined input variants. A minimum thermal stabilization time of 60 min was used for the LED [41] (measurements were performed starting from the highest to the lowest number of lamps). Any difference between LEDs belonging to the same type was neglected. The angular influence uncertainty of current probes was verified and assumed to be small (less than 1.0°), and therefore neglected. Finally, the impedance variation within the cycle and possible uncertainty impact due to the use of FFT algorithm were also neglected.
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emission changes mainly in the low order harmonics. The harmonic phase angle is the most affected harmonic characteristic. Additionally, changes in the network impedance and voltage source waveform play an important role in determining the nonlinear interaction. From the evidence of this study, the Norton/Thévenin equivalent to describe the power electronic device is still a good model for harmonic analysis. However, as the number of devices or the system order increase, more nonlinear interactions will happen and additional uncertainty in the results caused by the power electronics nonlinearity deviation should be considered.
6.6. Limitations and need for further research A limitation of the method is the need for high accuracy measuring equipment in order to reduce the uncertainty. As the number of different devices increases, the linear model combined uncertainty will also increase, reaching values that cannot be distinguished from the nonlinear interaction. Finally, further research is needed in identifying the origins and causes of nonlinearity deviation within the devices. Promising research goes towards extending the analysis of nonlinearity created by diode rectifiers, power electronics current control-loop and switching devices dynamics.
Conflict of interest
7. Conclusion
None declared. Acknowledgments
This study has used a linear model to verify the extent of nonlinear interaction between devices in the same LV installation. It was observed that the interaction of different power electronic devices creates nonlinearity deviation, resulting in current harmonics
The authors would like to thank the Swedish Energy Agency (Energimyndigheten) for supporting this work.
Appendix A. Evaluating and expressing the uncertainty The evaluation method and the expression of uncertainty used in this study follow the guidelines given by [42]. In total nine sources of uncertainty were considered: three impedances (ZPVI , ZLED, and ZGRID), two current sources (IsPVI and IsLED ), one voltage source (VS ) and three direct measurements (IPVI , ILED , and IPCC ). Type B standard uncertainties, u2 (x i ) , (function of amplitude and frequency) were obtained from the manufacturers manuals for the DAQ [27], current probes [43], power amplifier [26], and line impedance [25]. LED and PVI impedance uncertainties were extracted from [44], and source currents estimated by combining the measured current, voltage source, and line impedance uncertainties. Because we are comparing different results from the linear model and actual measurements, the uncertainty estimation method results in two distinct parts: the measurements uncertainties, U (Y ) , and linear model uncertainties, U (Yˆ ) . The first considers the uncertainty associated to the direct measurements, while the second considers the combined standard uncertainty of the output quantity, u (y ) described in the general form as: N
u2 (y ) =
2
∂Y ⎞
∑⎛X ⎜
i=1
⎝
i
⎟
u2 (x i )
(A.1)
⎠
where Y is the output quantity, defined by the linear model equations: IPVI , ILED, IGRID, and VPCC as function of the input quantities Xi (i.e., ZPVI , ZLED, ZGRID , IsPVI , and IsLED ), and u (x i ) is standard uncertainty for the input quantities Xi . In practice, due the simplicity of the linear equations, the partial derivatives ∂Y (often called sensitivity coefficients) imply that the combined ∂X uncertainties can be obtained by simply adding the absolute uncertainties when input quantities are added/subtracted, and by adding the relative uncertainties, when the input quantities are multiplied/divided. As a final step, the expanded uncertainty, U (Y ) , is calculated using U (Y ) = k . u (y ) assuming a normal distribution and coverage factor of k = 2 , which corresponds to an interval having a level of confidence of about 95%.
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