Revan Sombor indices: Analytical and statistical study

V. R. Kulli; J. A. Méndez-Bermúdez; José M. Rodríguez; José M. Sigarreta; V. R. Kulli; J. A. Méndez-Bermúdez; José M. Rodríguez; José M. Sigarreta

doi:10.3934/mbe.2023082

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 2: 1801-1819. doi: 10.3934/mbe.2023082

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Research article Special Issues

Revan Sombor indices: Analytical and statistical study

1.
Department of Mathematics, Gulbarga University, Gulbarga 585106, India
2.
Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico
3.
Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Madrid, Spain
4.
Facultad de Matemáticas, Universidad Autónoma de Guerrero, Carlos E. Adame No.54 Col. Garita, 39650 Acalpulco Gro., Mexico

Academic Editor: Yang Kuang

Received: 27 August 2022 Revised: 11 October 2022 Accepted: 19 October 2022 Published: 07 November 2022

In this paper, we perform analytical and statistical studies of Revan indices on graphs $G$ : $R(G) = \sum_{uv \in E(G)} F(r_u, r_v)$ , where $uv$ denotes the edge of $G$ connecting the vertices $u$ and $v$ , $r_u$ is the Revan degree of the vertex $u$ , and $F$ is a function of the Revan vertex degrees. Here, $r_u = \Delta + \delta - d_u$ with $\Delta$ and $\delta$ the maximum and minimum degrees among the vertices of $G$ and $d_u$ is the degree of the vertex $u$ . We concentrate on Revan indices of the Sombor family, i.e., the Revan Sombor index and the first and second Revan $(a, b)$ - $KA$ indices. First, we present new relations to provide bounds on Revan Sombor indices which also relate them with other Revan indices (such as the Revan versions of the first and second Zagreb indices) and with standard degree-based indices (such as the Sombor index, the first and second $(a, b)$ - $KA$ indices, the first Zagreb index and the Harmonic index). Then, we extend some relations to index average values, so they can be effectively used for the statistical study of ensembles of random graphs.

Keywords:

Citation: V. R. Kulli, J. A. Méndez-Bermúdez, José M. Rodríguez, José M. Sigarreta. Revan Sombor indices: Analytical and statistical study[J]. Mathematical Biosciences and Engineering, 2023, 20(2): 1801-1819. doi: 10.3934/mbe.2023082

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Abstract

1. Introduction

Nonlinear loads are generally source of harmonics. Harmonics can be defined as periodic state distortions of voltage and/or current waveforms. Harmonics are often used for defining distorted sine waves. Distortion is associated with currents and voltages of different amplitudes and frequencies ^[1]. Conventional analysis assumes power system as linear system. Today, nonlinear loads inject great amount of harmonic currents in system. Regarding system impedance, voltage harmonics can occur. This can cause potential problems in power system. As stated in ^[2,3], harmonics have harmful effects in: conductors, electrical machines, different types of electrical devices etc. They can cause series or parallel resonances, considering corresponding grid properties ^[1,2]. Harmonics can cause increased technical losses in grid (decreasing grid efficiency). Problem solution of harmonics is in filtering of them (active or passive filtering ^[1]). Mathematically, harmonic can be stated as a sinusoidal component of a periodic wave having a frequency that is an integral multiple of the fundamental frequency ^[1,2].

Voltage and current can be presented by Fourier series (when steady-state harmonics are present):

$u\left( t \right) = \sum \nolimits_{h = 1}^\infty {u_h}\left( t \right) = \sum \nolimits_{h = 1}^\infty \sqrt 2 {U_h}{\rm{sin}}\left( {h{\omega _0}t + {\theta _h}} \right)$

(1)

$i\left( t \right) = \sum \nolimits_{h = 1}^\infty {i_h}\left( t \right) = \sum \nolimits_{h = 1}^\infty \sqrt 2 {I_h}{\rm{sin}}\left( {h{\omega _0}t + {\delta _h}} \right)$

(2)

where V_h and I_h are Root Mean Square—RMS values for h-th order of harmonic voltage and current, respectively.

Applying orthogonal relation ^[1,2], the RMS values for (1) and (2) are:

${U_{RMS}} = \sqrt {\frac{1}{T}\mathop \smallint _0^T {u^2}\left( t \right)dt} = \sqrt {\sum \nolimits_{h = 1}^\infty U_h^2}$

(3)

${I_{RMS}} = \sqrt {\frac{1}{T}\mathop \smallint _0^T {i^2}\left( t \right)dt} = \sqrt {\sum \nolimits_{h = 1}^\infty I_h^2}$

(4)

Unlike load, power system is robust enough to withstand considerable amount of harmonic currents without causing problems. It is due to smaller impedance of system compared to load impedance. Power system is not significant source of harmonic. But, it can become contributor of problems by way of resonance (when multiple distortions exist) ^[1]. LED bulbs represent a common single phase low power nonlinear electronically based loads. There are various approaches for nonlinear loads modeling in harmonics analysis. The most widely used approach to nonlinear loads modeling is current injection method ^[4]. In addition, the circuit modeling of electronically based loads, represented in ^[5] overcome some problems related to current injection method. However, parameter estimation algorithm should be applied for determination nonlinear model parameters of these kind of loads ^[6]. For simplicity, the mentioned current injection method is preferred in this paper. Primary, all nonlinear loads are presented as harmonic current injectors. Then, harmonic voltage at each system bus in power system can be obtained by solving following impedance matrix or nodal admittance equations (for all orders of harmonics under consideration):

${{\rm{U}}_{\rm{h}}} = {{\rm{Z}}_{\rm{h}}} \cdot {{\rm{I}}_{\rm{h}}}$

(5)

${{\rm{I}}_{\rm{h}}} = {{\rm{Y}}_{\rm{h}}} \cdot {{\rm{U}}_{\rm{h}}}$

(6)

where V_h is the vector consisting of h-th harmonic voltage at each bus that must be determined. Z_h is system harmonic impedance matrix. Y_h is system harmonic admittance matrix and I_h is vector of measured or estimated harmonic current representing harmonic-generating loads at connected buses. Z_h can be obtained by using Z-bus building algorithm for each harmonic of interest or from the inverse of Y_h. This approach is usually used in conjunction with fundamental frequency load flow analysis. Providing the grid harmonic impedance or admittance and harmonic currents injected by nonlinear loads (for consideration harmonics), the individual and total harmonic voltage distortions at each bus can be computed. More complicated analysis is necessary for unbalanced system.

There are multiple indices used to describe the effects of harmonics on power system components. Those are described in ^[1]. In case of this paper, Total Harmonic Distortion (THD)—Distortion Factor will be presented for voltage (THD_U) and current (THD_I). Those are the most used harmonic index. Where:

$TH{D_U} = \frac{{\sqrt {\mathop \sum \nolimits_{h = 2}^\infty U_h^2} }}{{{U_1}}}$

(7)

$TH{D_I} = \frac{{\sqrt {\mathop \sum \nolimits_{h = 2}^\infty I_h^2} }}{{{I_1}}}$

(8)

Those indices are defined as ratio of the RMS value of harmonic components to the RMS of the fundamental component. It is usually expressed in percent values. For perfect sine wave, fundamental frequency, THD values are zero. The connection of nonlinear load in the grid will change the total distorted harmonic voltage at the point of common coupling (PCC). Generally, THD_U is a function of load currents I, THD_I and system impedance Z at concerned PCC:

$\Delta TH{D_U} = f\left( {I, TH{D_I}, Z} \right)$

(9)

The component of current I supplying nonlinear load produces across supply grid equivalent impedance Z voltage drop ΔU (Eq 10).

$\Delta U = Z \cdot I$

(10)

On Figure 1 degradation of grid voltage caused by nonlinear load is shown ^[7].

Figure 1. Voltage distortion at the PCC—degradation of grid voltage caused by nonlinear load.

Individual harmonics
THD	3	5	7	9	11	13	15	17
8.0%	5.0%	6.0%	5.0%	1.5%	3.5%	3.0%	0.5%	2.0%

I_SC/I_L	< 11	11 ≤ h ≤ 17
< 20	4.0	2.0
20–50	7.0	3.5
50–100	10.0	4.5
100–1000	12.0	5.5
> 1000	15.0	7.0

ID	LED lamp type	Voltage input (V)	I_max. (mA)	Power (W)
1	Panel (circle)	85–265	280	8–12
2	Panel (square)	220–240	300	9–12
3	Downlight	85–245	300	5–9
4	Downlight	85–264	300	12–18
5	Downlight	85–264	300	4
6	Floodlight	100–265	900	30

ID	THD_U (%)	THD_I (%)
1	< 5	85–265
2	< 5	220–240
3	< 5	85–245
4	< 5	85–264
5	< 5	85–264
6	< 5	100–265

Harmonics	1	2	3	4	5	6	7	8
Amplitude [V]	317.97	0.9631	1.0554	0.3630	2.6777	0.2181	1.7832	0.2398
Phase angle [°]	-1.0	179.9	149.2	188.4	227.2	151.5	32.0	168.5
Harmonics	9	10	11	12	13	14	15
Amplitude [V]	1.6621	0.1075	1.3569	0.1213	0.8587	0.1059	0.7248
Phase angle [°]	201.6	187.7	4.4	165.9	137.7	147.6	252.9

Harmonics	1	2	3	4	5	6	7	8
Amplitude [V]	0.1229	0.0082	0.1060	0.0076	0.0791	0.0097	0.0548	0.0082
Harmonics	9	10	11	12	13	14	15
Amplitude [V]	0.0457	0.0020	0.0447	0.0039	0.0424	0.0066	0.0390

Harmonics	1	2	3	4	5	6	7	8
Amplitude [V]	0.1289	0.0010	0.1089	0.0014	0.0787	0.0013	0.0542	0.0010
Harmonics	9	10	11	12	13	14	15
Amplitude [V]	0.0511	0.0011	0.0565	0.0017	0.0528	0.0021	0.0383

Harmonics	1	2	3	4	5	6	7	8
Amplitude [%]	100	6.67	86.25	6.18	64.36	7.89	44.59	6.67
Harmonics	9	10	11	12	13	14	15
Amplitude [%]	37.18	1.62	36.37	3.17	34.49	5.37	31.73

Harmonics	1/p	1/n	3/p	3/n	5/p	5/n	7/p	7/n
Amplitude [%]	100	100	8.81	300	16.15	50	19.50	103
Harmonics	9/p	9/n	11/p	11/n	13/p	13/n	15/p	15/n
Amplitude [%]	3.70	58.5	6.16	20	1.70	3.60	0.65	5

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Individual harmonics
2	3	5	7	9	9–39
2%	30%·cosϕ	10%	7%	5%	3%

Harmonics	1	2	3	4	5	6	7	8
Amplitude [%]	0.1289	0.78	84.48	1.09	61.06	1.01	42.05	0.78
Harmonics	9	10	11	12	13	14	15
Amplitude [%]	39.64	0.85	43.83	1.32	40.96	1.63	29.72

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Individual harmonics
2	3	5	7	9	9–39
2%	30%·cosϕ	10%	7%	5%	3%

Mathematical Biosciences and Engineering

Revan Sombor indices: Analytical and statistical study

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Abstract

1. Introduction

2. Power quality Standards review: harmonics

3. Harmonic emission of LED lighting systems—literature review

4. Case study—single LED bulb

5. Case study—luminaries (street lighting systems)

5.1. Great LED lighting system-analysis

5.2. Comparison of LED and sodium small lighting system‑analysis

5.2.1. Comparison of great and small LED lighting system

6. Conclusions

Conflict of interest

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