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Multivariable polynomial fitting of controlled single-phase nonlinear load of input current total harmonic distortion

  • Roman Sikora , Przemysław Markiewicz EMAIL logo and Wiesława Pabjańczyk
Published/Copyright: April 18, 2018
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Open Physics
From the journal Open Physics Volume 16 Issue 1

Abstract

The power systems usually include a number of nonlinear receivers. Nonlinear receivers are the source of disturbances generated to the power system in the form of higher harmonics. The level of these disturbances describes the total harmonic distortion coefficient THD. Its value depends on many factors. One of them are the deformation and change in RMS value of supply voltage. A modern LED luminaire is a nonlinear receiver as well. The paper presents the results of the analysis of the influence of change in RMS value of supply voltage and the level of dimming of the tested luminaire on the value of the current THD. The analysis was made using a mathematical model based on multivariable polynomial fitting.

Keywords: curve fitting; total harmonics distortion; single-phase nonlinear loads; LED luminaire
PACS: 88.05.-b; 88.05.Tg

1 Introduction

Nonlinear receivers are the source of many negative phenomena occurring in the power system. They receive the distorted currents from the power system which, when flowing by the power systems elements, result in voltage drops caused by higher harmonics of current. Consequently, this causes distortion of the supply voltage. The distorted current causes increased heating of wires and other power systems elements, as well as insulation aging, increased power losses, increased probability of resonance, and many more. In three-phase systems, another problem concern the occurrence of current in neutral conductor.

Nowadays, more devices are equipped with SMPS power supplies. Such devices are receivers with nonlinear current-voltage characteristics e.g. LED luminaires that are equipped with SMPS power supplies. Modern luminaire solutions allow to control of the luminous flux. This is accomplished by controlling the current or output voltage of the power supply. Usually, most of power supplies are equipped with PFC systems which are used to improve the power factor. They are generally designed for increasing the rated power of the power supply. In the case of power supplies, the influence of the level of control on the generated disturbances is observed. With the decrease of the dimming level of control gear, THD coefficient of the current increases and the power factor decreases. In addition, the THDI is influenced by a number of factors: the power supply design and algorithm, the level of the supply voltage distortion and its RMS value, the control level.

The voltage and current THD coefficients are described by (1) and (2).

THDU=UHU1=UU12−1(1)
THDI=IHI1=II12−1(2)

According to standard [23] for steady-state and non-sinusoidal currents and voltages, the RMS values of voltage U and current I have two components:

U2=U02+UH2(3)
I2=I02+IH2(4)

A component with a lower index 0 denotes a constant component. For other components, the index is the actual harmonic or voltage number.

UH2=U02+∑H≠1UH2=U2−U12(5)
IH2=I02+∑H≠1IH2=I2−I12(6)

A new problem with the generation of harmonics is the fact that increasingly responsible for this phenomenon are the receivers with low power although present in large quantities. In particular, it concerns adjustable light sources and luminaires dimmed light sources and luminaires.

Many analytical methods are used to evaluate this effect. One method uses a simplified schematic diagram of a nonlinear load described by time differential equations based on the knowledge of basic electrical laws [1]. Another method include replacing the nonlinear receiver with a nonlinear resistance [4]. In the literature [12, 17, 19], a method of creating a mathematical model of a nonlinear equipment based on measurements is presented. It is based on accepting the input and output quantities which characterize the given nonlinear equipment in the studied phenomena. The method described was used to assess the impact of computer power supplies on the quality of electricity in the mains [12, 14, 17, 18, 20].

As a tested object the LED luminaire with controlled power and light flux was selected. For the analysis, the methods described in [6, 7, 8] using the measurement results of a real object were used. Using these methods a mathematical model was developed. Input values of the model are the RMS value of the supply voltage and the control level. The output value of the model is the value of the THDI. Mathematical model is a n-th degree polynomial of two input variables. The model was developed using the Curve Fitting Toolbox available in MATLAB™.

2 Laboratory results

LED luminaire equipped with power supply with analogue control input made of standard 1 - 10V has been subjected to the laboratory tests. This luminaire has rated power equal to 32W. The RMS value of the supply voltage was varied in the range 230V ± 10% (from 207 to 253 V). The limits of voltage variations aredescribed in EN 50160 [21] for acceptable voltage changes in low voltage networks. The second input values was the input control level of the luminaire, determined by the value of the control voltage from 1V to 10V, with a step equal to 1 volt. The output value was the value of the THDI coefficient. Measurements were made using the FLUKE 1760 which is anelectrical power quality analyzer. The laboratory power supply as a source of DC control voltage was used.

Figure 1 shows the relation of the power of the lighting fitting as a function of the control voltage UC and of the supply voltage US. When analyzing its waveform, it can be seen that in the range of UC voltage from 1V to 8V the dependency is linear. In the range of 8V to 10V it can be assumed that the luminaire is working at full power. Table 1 summarizes the results of the measurement of the power of the luminaire P as a function of the control voltage UC. For the tested luminaire, 1V of control voltage ofthe luminaire is achievedat 19.32% of the rated power. This corresponds to the lower limit of regulation, at which the luminaire achieves a minimum power of 6.28W.

Figure 1 Relation of the power of the lighting fixture P as a function of the supply voltage US and control voltage UC
Figure 1

Relation of the power of the lighting fixture P as a function of the supply voltage US and control voltage UC

Table 1

Relation between the luminaire power P and control voltage Uc

UC (V)P(W)P(%)
16.2819.32
29.0527.85
312.8439.54
416.6251.18
520.6863.65
624.4875.36
728.3687.29
832.48100.00
932.48100.00
1032.48100.00

In Figure 2 the relationship between the THDI of the US supply voltage and the UC control voltage is presented. Figure 2 shows two different areas of variation. For control voltages greater than approx. 5V, the THDI value of the supply current varies in a small range. In the control voltage range of 1V to 5V, the THDI coefficient shows a strong upward trend.

Figure 2 Relation of THDI coefficient as a function of supply voltage US and control voltage UC
Figure 2

Relation of THDI coefficient as a function of supply voltage US and control voltage UC

The presented dependence is characteristic for the tested luminaire. Other types of luminaires with control systems may had different dependencies whilethe general nature of the dependency changes are similar.

For the illuminated luminaire, the characteristics of the THDI coefficient are described by two polynomials according to (7):

THDI=THDI−15UC,US∀UC∈1;5∧∀US∈207;253THDI−510UC,US∀UC∈5;10∧∀US∈207;253(7)

Polynomials THDI–15 and THDI–510 are presented as the sum of the product of the supply voltage US, the control voltage UC and the coefficients aij and bij(8)(9), respectively:

THDI−15UC,US=∑i=0m∑j=0maijUCiUSj(8)
THDI−510UC,US=∑i=0m∑j=0mbijUCiUSj(9)

The coefficients aij and bij can be written in matrix form as A and B (4):

A=a00a01a02a03a04a05a10a11a12a13a140a20a21a22a2300a30a31a32000a40a410000000000m×mB=b00b01b02b03b04b05b10b11b12b13b140b20b21b22b2300b30b31b32000b40b410000b5000000m×m(10)

For the study, the luminaire coefficients aij and bij take the following values.

A=5.909⋅105−1.306⋅104115.40−0.50980.001127−9.97⋅10−71792−63.240.6836−0.36714.222⋅10−601235−20.340.09958−0.00015270083.91−0.36710.0001371000−6.3220.025640000000000B=−1.639⋅10536.36−32.170.142−0.00031272.749⋅10−7−136120.32−0.12160.0003184−3.284⋅10−6056.74−0.32980.001436−7.035⋅10−700−4.7820.001031−4.391⋅10−50000.33670.00061420000−0.0130400000

The polynomials THDI–15 and THDI–510 from the dependence (1) were obtained using the two-variable polynomial fitting method to the measurement results using the Curve Fitting Toolbox [7]. The proposed tool for data vectors US, UC and THDI have found the vector of polynomial coefficients of the assumed degrees. Polynomial coefficients were selected to obtain a minimum value of the mean square deviation between the given function THDI = f(US, UC), and approximating polynomial THDI–15(US, UC) for US ∈ 〈1;5) and THDI–510(US, UC) for US ∈ 〈5;10〉.

For the obtained approximating polynomials (Figures 3 and 4) the mean square errors were respectively 0.95 and 0.99.

Figure 3 Graph presenting the approximating polynomial THDI–15(US,UC)
Figure 3

Graph presenting the approximating polynomial THDI–15(US,UC)

Figure 4 Graph presenting the approximating polynomial THDI–510(US,UC)
Figure 4

Graph presenting the approximating polynomial THDI–510(US,UC)

Verification of the correctness of the model has been made in order to determine the error values at the calculation points. The calculation results are shown in Figure 5 for the THDI–15 polynomial approximation and Figure 6 for the THDI–510 polynomial.

Figure 5 Graph presenting the polynomial approximation error of THDI–15(US,UC)
Figure 5

Graph presenting the polynomial approximation error of THDI–15(US,UC)

Figure 6 Graph presenting polynomial approximation error of THDI–510(US,UC)
Figure 6

Graph presenting polynomial approximation error of THDI–510(US,UC)

Due to the high nonlinearity according to THDI = f(US, UC) in range of the control voltage from 1V to 5V the error values are within the range of 0.10% to 26.73%.

The maximum error value occurred for UC = 3V and US = 245V. An error graph is shown in Figure 5. Within the control voltage range of 5 to 10V, the maximum error value is 2.52%, as illustrated in Figure 6.

3 Computational example designation of THDI coefficient

The proposed method of estimating the value of THDI of the power supply was used to calculate this factor for the tested luminaire and the assumed daily schedule of its operation. The luminaire works at full power from 16.00 to 23.00 and from 5.00 to 7.00. At night from 23.00 to 5.00 in the morning it is operated at 50% of the rated power. In order to implement the adopted work schedule of luminaire, it is necessary to know the power P depending on the input value UC. This relationship is shown in Figure 7 and it is practically linear. For analytical purposes according to UC = f(P) it may therefore take advantage of the approximation as a linear function.

Figure 7 Graph presenting the approximating polynomial UC(P)
Figure 7

Graph presenting the approximating polynomial UC(P)

The obtained approximation function UC = f(P) is represented by the expression (11) and obtained for the mean square error is equal to 0.9984.

UCP=10forP=1pu8.541⋅P−0.4562for0.1932≤P<1pu(11)

For the further calculations it is assumed that in order to achieve the full power of the luminaire UC = 10V and for the remaining control levels, the UC value is calculated from the relationship (11). The dependence (11) allows to determine the value of the UC control voltage to obtain the assumed power of the luminaire in any schedule of its operation.

The second input value (supply voltage US) was derived from measurements made in an existing lighting installation.

Figure 8 shows the time dependent waveforms of RMS supply voltage from measurements and the THDI values obtained from the calculations. These results correspond to a full daylight cycle with a scheduled power reduction. For the full power of the luminaire, the predicted THDI value 14%. The reduction of the power of the luminaire causes it to increase to a range of 15.5 to 17%. This corresponds to a 19% increase in THDI.

Figure 8 Results of calculations using the model
Figure 8

Results of calculations using the model

4 Conclusions

The problem of the quality of electricity is one of the most important aspects related to the operation of power devices. According to EN 61000-3-2 [21], class C receivers cannot generate harmonic currents exceeding their limit values. One of the indicators of the level of disturbances generated to the power supply in the form of higher harmonic current is the total harmonic distortion factor THDI. On the basis of the measurements it was proved that for the LED luminaire with power regulation its value depends on the level of control and the value of the supply voltage. From an operational point of view, it is important to predict THDI, values under given power conditions at a given level of control.

The applied mathematical model of the nonlinear receiver based on the results of the measurements allows for accurate estimation of the THDI level of the given receiver. The knowledge of the THDI characteristics as a function of changes in the input values allows to determine permissible changes of these values to maintain the current deformation at a given level. For the tested luminaire, this method determined the acceptable range of power regulation and luminous flux. The analysis can be carried out for the assumed range of voltage supply variations, ranging from 230V ± 10% (from 207V to 253V and full range of power P control).

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Received: 2017-11-02
Accepted: 2017-11-12
Published Online: 2018-04-18

© 2018 R. Sikora et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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https://doi.org/10.1515/phys-2018-0021
Keywords for this article
curve fitting; total harmonics distortion; single-phase nonlinear loads; LED luminaire
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  7. Three phase heat and mass transfer model for unsaturated soil freezing process: Part 2 - model validation
  8. Mathematical model for thermal and entropy analysis of thermal solar collectors by using Maxwell nanofluids with slip conditions, thermal radiation and variable thermal conductivity
  9. Constructing analytic solutions on the Tricomi equation
  10. Feynman diagrams and rooted maps
  11. New type of chaos synchronization in discrete-time systems: the F-M synchronization
  12. Unsteady flow of fractional Oldroyd-B fluids through rotating annulus
  13. A note on the uniqueness of 2D elastostatic problems formulated by different types of potential functions
  14. On the conservation laws and solutions of a (2+1) dimensional KdV-mKdV equation of mathematical physics
  15. Computational methods and traveling wave solutions for the fourth-order nonlinear Ablowitz-Kaup-Newell-Segur water wave dynamical equation via two methods and its applications
  16. Siewert solutions of transcendental equations, generalized Lambert functions and physical applications
  17. Numerical solution of mixed convection flow of an MHD Jeffery fluid over an exponentially stretching sheet in the presence of thermal radiation and chemical reaction
  18. A new three-dimensional chaotic flow with one stable equilibrium: dynamical properties and complexity analysis
  19. Dynamics of a dry-rebounding drop: observations, simulations, and modeling
  20. Modeling the initial mechanical response and yielding behavior of gelled crude oil
  21. Lie symmetry analysis and conservation laws for the time fractional simplified modified Kawahara equation
  22. Solitary wave solutions of two KdV-type equations
  23. Applying industrial tomography to control and optimization flow systems
  24. Reconstructing time series into a complex network to assess the evolution dynamics of the correlations among energy prices
  25. An optimal solution for software testing case generation based on particle swarm optimization
  26. Optimal system, nonlinear self-adjointness and conservation laws for generalized shallow water wave equation
  27. Alternative methods for solving nonlinear two-point boundary value problems
  28. Global model simulation of OH production in pulsed-DC atmospheric pressure helium-air plasma jets
  29. Experimental investigation on optical vortex tweezers for microbubble trapping
  30. Joint measurements of optical parameters by irradiance scintillation and angle-of-arrival fluctuations
  31. M-polynomials and topological indices of hex-derived networks
  32. Generalized convergence analysis of the fractional order systems
  33. Porous flow characteristics of solution-gas drive in tight oil reservoirs
  34. Complementary wave solutions for the long-short wave resonance model via the extended trial equation method and the generalized Kudryashov method
  35. A Note on Koide’s Doubly Special Parametrization of Quark Masses
  36. On right-angled spherical Artin monoid of type Dn
  37. Gas flow regimes judgement in nanoporous media by digital core analysis
  38. 4 + n-dimensional water and waves on four and eleven-dimensional manifolds
  39. Stabilization and Analytic Approximate Solutions of an Optimal Control Problem
  40. On the equations of electrodynamics in a flat or curved spacetime and a possible interaction energy
  41. New prediction method for transient productivity of fractured five-spot patterns in low permeability reservoirs at high water cut stages
  42. The collinear equilibrium points in the restricted three body problem with triaxial primaries
  43. Detection of the damage threshold of fused silica components and morphologies of repaired damage sites based on the beam deflection method
  44. On the bivariate spectral quasi-linearization method for solving the two-dimensional Bratu problem
  45. Ion acoustic quasi-soliton in an electron-positron-ion plasma with superthermal electrons and positrons
  46. Analysis of projectile motion in view of conformable derivative
  47. Computing multiple ABC index and multiple GA index of some grid graphs
  48. Terahertz pulse imaging: A novel denoising method by combing the ant colony algorithm with the compressive sensing
  49. Characteristics of microscopic pore-throat structure of tight oil reservoirs in Sichuan Basin measured by rate-controlled mercury injection
  50. An activity window model for social interaction structure on Twitter
  51. Transient thermal regime trough the constitutive matrix applied to asynchronous electrical machine using the cell method
  52. On the zagreb polynomials of benzenoid systems
  53. Integrability analysis of the partial differential equation describing the classical bond-pricing model of mathematical finance
  54. The Greek parameters of a continuous arithmetic Asian option pricing model via Laplace Adomian decomposition method
  55. Quantifying the global solar radiation received in Pietermaritzburg, KwaZulu-Natal to motivate the consumption of solar technologies
  56. Sturm-Liouville difference equations having Bessel and hydrogen atom potential type
  57. Study on the response characteristics of oil wells after deep profile control in low permeability fractured reservoirs
  58. Depiction and analysis of a modified theta shaped double negative metamaterial for satellite application
  59. An attempt to geometrize electromagnetism
  60. Structure of traveling wave solutions for some nonlinear models via modified mathematical method
  61. Thermo-convective instability in a rotating ferromagnetic fluid layer with temperature modulation
  62. Construction of new solitary wave solutions of generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony and simplified modified form of Camassa-Holm equations
  63. Effect of magnetic field and heat source on Upper-convected-maxwell fluid in a porous channel
  64. Physical cues of biomaterials guide stem cell fate of differentiation: The effect of elasticity of cell culture biomaterials
  65. Shooting method analysis in wire coating withdrawing from a bath of Oldroyd 8-constant fluid with temperature dependent viscosity
  66. Rank correlation between centrality metrics in complex networks: an empirical study
  67. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering
  68. Modeling of electric and heat processes in spot resistance welding of cross-wire steel bars
  69. Dynamic characteristics of triaxial active control magnetic bearing with asymmetric structure
  70. Design optimization of an axial-field eddy-current magnetic coupling based on magneto-thermal analytical model
  71. Thermal constitutive matrix applied to asynchronous electrical machine using the cell method
  72. Temperature distribution around thin electroconductive layers created on composite textile substrates
  73. Model of the multipolar engine with decreased cogging torque by asymmetrical distribution of the magnets
  74. Analysis of spatial thermal field in a magnetic bearing
  75. Use of the mathematical model of the ignition system to analyze the spark discharge, including the destruction of spark plug electrodes
  76. Assessment of short/long term electric field strength measurements for a pilot district
  77. Simulation study and experimental results for detection and classification of the transient capacitor inrush current using discrete wavelet transform and artificial intelligence
  78. Magnetic transmission gear finite element simulation with iron pole hysteresis
  79. Pulsed excitation terahertz tomography – multiparametric approach
  80. Low and high frequency model of three phase transformer by frequency response analysis measurement
  81. Multivariable polynomial fitting of controlled single-phase nonlinear load of input current total harmonic distortion
  82. Optimal design of a for middle-low-speed maglev trains
  83. Eddy current modeling in linear and nonlinear multifilamentary composite materials
  84. The visual attention saliency map for movie retrospection
  85. AC/DC current ratio in a current superimposition variable flux reluctance machine
  86. Influence of material uncertainties on the RLC parameters of wound inductors modeled using the finite element method
  87. Cogging force reduction in linear tubular flux switching permanent-magnet machines
  88. Modeling hysteresis curves of La(FeCoSi)13 compound near the transition point with the GRUCAD model
  89. Electro-magneto-hydrodynamic lubrication
  90. 3-D Electromagnetic field analysis of wireless power transfer system using K computer
  91. Simplified simulation technique of rotating, induction heated, calender rolls for study of temperature field control
  92. Design, fabrication and testing of electroadhesive interdigital electrodes
  93. A method to reduce partial discharges in motor windings fed by PWM inverter
  94. Reluctance network lumped mechanical & thermal models for the modeling and predesign of concentrated flux synchronous machine
  95. Special Issue Applications of Nonlinear Dynamics
  96. Study on dynamic characteristics of silo-stock-foundation interaction system under seismic load
  97. Microblog topic evolution computing based on LDA algorithm
  98. Modeling the creep damage effect on the creep crack growth behavior of rotor steel
  99. Neighborhood condition for all fractional (g, f, n′, m)-critical deleted graphs
  100. Chinese open information extraction based on DBMCSS in the field of national information resources
  101. 10.1515/phys-2018-0079
  102. CPW-fed circularly-polarized antenna array with high front-to-back ratio and low-profile
  103. Intelligent Monitoring Network Construction based on the utilization of the Internet of things (IoT) in the Metallurgical Coking Process
  104. Temperature detection technology of power equipment based on Fiber Bragg Grating
  105. Research on a rotational speed control strategy of the mandrel in a rotary steering system
  106. Dynamic load balancing algorithm for large data flow in distributed complex networks
  107. Super-structured photonic crystal fiber Bragg grating biosensor image model based on sparse matrix
  108. Fractal-based techniques for physiological time series: An updated approach
  109. Analysis of the Imaging Characteristics of the KB and KBA X-ray Microscopes at Non-coaxial Grazing Incidence
  110. Application of modified culture Kalman filter in bearing fault diagnosis
  111. Exact solutions and conservation laws for the modified equal width-Burgers equation
  112. On topological properties of block shift and hierarchical hypercube networks
  113. Elastic properties and plane acoustic velocity of cubic Sr2CaMoO6 and Sr2CaWO6 from first-principles calculations
  114. A note on the transmission feasibility problem in networks
  115. Ontology learning algorithm using weak functions
  116. Diagnosis of the power frequency vacuum arc shape based on 2D-PIV
  117. Parametric simulation analysis and reliability of escalator truss
  118. A new algorithm for real economy benefit evaluation based on big data analysis
  119. Synergy analysis of agricultural economic cycle fluctuation based on ant colony algorithm
  120. Multi-level encryption algorithm for user-related information across social networks
  121. Multi-target tracking algorithm in intelligent transportation based on wireless sensor network
  122. Fast recognition method of moving video images based on BP neural networks
  123. Compressed sensing image restoration algorithm based on improved SURF operator
  124. Design of load optimal control algorithm for smart grid based on demand response in different scenarios
  125. Face recognition method based on GA-BP neural network algorithm
  126. Optimal path selection algorithm for mobile beacons in sensor network under non-dense distribution
  127. Localization and recognition algorithm for fuzzy anomaly data in big data networks
  128. Urban road traffic flow control under incidental congestion as a function of accident duration
  129. Optimization design of reconfiguration algorithm for high voltage power distribution network based on ant colony algorithm
  130. Feasibility simulation of aseismic structure design for long-span bridges
  131. Construction of renewable energy supply chain model based on LCA
  132. The tribological properties study of carbon fabric/ epoxy composites reinforced by nano-TiO2 and MWNTs
  133. A text-Image feature mapping algorithm based on transfer learning
  134. Fast recognition algorithm for static traffic sign information
  135. Topical Issue: Clean Energy: Materials, Processes and Energy Generation
  136. An investigation of the melting process of RT-35 filled circular thermal energy storage system
  137. Numerical analysis on the dynamic response of a plate-and-frame membrane humidifier for PEMFC vehicles under various operating conditions
  138. Energy converting layers for thin-film flexible photovoltaic structures
  139. Effect of convection heat transfer on thermal energy storage unit
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