Transformada de Laplace - Aplicações em Circuitos Elétricos e Análise de Sistemas Lineares

ESCRITA E REFLEXÃO 1 Professor Wallace da Silva Carvalho Disciplina Circuitos Elétricos II GEAD0277ONLINE01 Contextualização da proposta A transformada de Laplace e amplamente conhecida e utilizada principalmente nas ciências exatas e engenharias Encarada como um ritual de passagem pelos estudantes de graduação ela pode ser usada para análise de sistemas lineares invariantes no tempo tais como circuitos elétricos osciladores harmônicos dispositivos ópticos e sistema mecânicos Nessas aplicações costumase interpretála como transformações do domínio do tempo para o domínio de frequências A vantagem mais interessante desta transformação e que as integrações e derivações tornamse multiplicações e divisões Ela permite fazer a resolução de equações diferenciais em forma de equações polinomiais que são muito mais simples de resolver O desenvolvimento da transformada de Laplace devese a muitos nomes além do próprio Laplace como Cauchy por seus trabalhos em cálculos de resíduos e explorações em métodos simbólicos utilizando operadores diferenciais Importante ressaltar que um grande contributo para que a teoria pudesse se tornar um método viável para solução de problemas práticos foi dado pelo intrépido e obscuro inglês Oliver Heaviside 1850 1925 Heaviside homem simples e sem instrução formal foi uma das trágicas figuras da ciência ao mesmo tempo amado e odiado por homens de ciência do seu tempo Seu peculiar modo de trabalho que chamava de matemática experimental e seu estilo debochado causaram a ele muitos embaraços Com o inestimável auxílio de GF Fitzgerald e OJ Lodge Heaviside contribuiu para formalizar a teoria eletromagnética de Maxwell que originalmente totalizava 38 equações em apenas 4 equações fundamentais e contribuiu para que o cálculo vetorial se firmasse como ferramenta básica do eletromagnetismo em oposição à teoria dos quaternions de Hamilton Também foi quem primeiro resolveu o problema da onda viajante em uma linha de transmissão sem distorções o que tornou possível a comunicação transatlântica de telégrafos Em física foi o primeiro a teorizar a existência de uma camada condutiva na atmosfera ionosfera ou camada de Heaviside Kennely que permite que uma onda eletromagnética viaje segundo a curvatura da Terra Foi também quem sugeriu que uma carga elétrica em movimento aumenta sua massa com o aumento da velocidade uma das premissas da teoria da relatividade Chegou inclusive a prever a propriedade da supercondutividade nos materiais Proposta de Trabalho A Transformada de Laplace é fundamental para o estudo de alguns fenômenos físicos Por ser uma ferramenta muito eficiente de resolução de Equações Diferenciais Ordinárias Lineares de Segunda Ordem Em geral o método de Transformada de Laplace consiste em resolver equações diferenciais como se fossem equações algébricas Desta forma podese chegar a uma função de variável diferente da primeira que possui uma determinada e desejável propriedade que a primeira função não possuía Em seguida fazendo o caminho inverso o qual é chamado de transformada inversa podese obter o resultado esperado para a primeira função em sua variável original Diante deste processo apresenta aplicações da Transformada de Laplace em situações reais na prática da engenharia abordando exemplos com apoio de ARTIGOS CIENTÍFICOS disponíveis na literatura Deixe o link do artigo para determinada consulta e verificação Orientações Atividade exige o cumprimento das normas da ABNT em sua execução Apresente uma fundamentação teórica pertinente ao assunto Norma culta da língua portuguesa Apresente o link do artigo Envio em PDF Atividade Aplicações da transformada de La Place em situações reais e práticas da engenharia elétrica 1 Introdução Para que se entenda o conceito de transformada de La Place é preciso identificar o que significa uma transformada Basicamente neste caso a transformada serve para transformar equações diferenciais de difícil resolução em equações algébricas de fácil resolução Ainda a transformada de La Place especificamente fornece uma maneira fácil de solucionar problemas de circuitos envolvendo condições iniciais permitindo respostas completas referentes à solução de problemas SADIKU 2020 Neste contexto tal operação é definida como sendo uma transformação integral de uma função no domínio do tempo para o domínio da frequência complexa Em linhas gerais tratase de uma ferramenta capaz de tornar simples a solução de problemas complexos É aplicada em diversos problemas e contextos práticos da engenharia elétrica permitindo análises criteriosas e bem detalhadas 2 Fundamentação Teórica A transformada de La Place é definida pela equação abaixo 1 Nesta equação L é o símbolo que representa a transformada de La Place Já ft é uma determinada função descrita no domínio do tempo t Por sua vez Fs é a função ft reescrita no domínio da frequência complexa s Para aplicar a transformada é necessário solucionar uma integral no domínio do tempo Geralmente o processo de solução via La Place consiste em detectar a função no domínio temporal aplicar a transformada de La Place manipular as expressões encontradas e realizar a transformada inversa de La Place para obter a resposta no domínio do tempo mas agora de maneira muito mais simples Normalmente isto é feito com o auxílio de tabelas e formulações prontas Além disso a transformada de La Place apresenta uma série de propriedades As propriedades mais conhecidas são 1 Linearidade 2 Deslocamento no tempo 3 Fatores de escala 4 Diferenciação no tempo 5 Deslocamento da frequência 6 Integração no tempo etc Nesse contexto segundo SADIKU 2020 existem diversas aplicações para a transformada de La Place Geralmente a principal aplicação é na modelagem de elementos de circuito a fim de se obter soluções factíveis no domínio do tempo Mas também é aplicada na análise de sistemas de controle de sistemas lineares de potência análise de respostas transitórias análise de circuitos eletrônicos etc 3 Aplicações Há diversas aplicações práticas do uso da transformada de La Place Ao realizar um levantamento bibliográfico é possível se deparar com aplicações simples e complexas A seguir serão demonstrados alguns exemplos de aplicações mais específicas onde a transformada é realizada a fim de se obter respostas adequação e controle do sistema elétrico Primeiramente em IVANCHEKO M 2021 os autores exploram o assunto distorção harmônica elevada em redes elétricas e cálculos precisos para quantificar este fenômeno Nesse contexto é sugerida uma técnica de solução baseado na transformada de La Place para calcular o nível de corrente e seu efeito cumulativo na distorção harmônica total da rede É concluído que a técnica adotada pode ser generalizada para análise de outros tipos de rede por ter alta precisão e garantir uma análise adequada no que diz respeito à qualidade de energia Segundo em TOSSANI F 2019 o autor inicia narrando sobre as dificuldades intrínsecas na análise de transitórios eletromagnéticos aplicados a cabos submarinos ou subterrâneos haja vista que exigem uma série de aplicações de cálculos analíticos Por conta disso é sugerida uma metodologia baseada na transformada de La Place na chamada fórmula de Sunde para a análise precisa de cabos submarinos com múltiplos condutores Aplicando a transformada de La Place o processo de simulação se torna mais simples e preciso Por fim outro exemplo de aplicação é encontrado em ISHII T 2017 Neste artigo os autores propõe a aplicação da transformada de La Place juntamente com um método conhecido como método logarítimo de amostragem segmentada Neste caso a transformada de La Place é útil para simplificar a análise no domínio da frequência O objetivo disso é estimar a impedância interna de baterias elétricas a fim de realizar diagnósticos sobre sua vida útil 4 Conclusão Neste trabalho demonstrouse de maneira geral a relevância do uso da transformada de La Place em aplicações práticas que envolvem a engenharia elétrica Além disso mostrouse três exemplos de aplicações práticas envolvendo a aplicação desta ferramenta Estes exemplos vão desde aplicações práticas complexas na análise da qualidade de energia de uma rede e a análise transitória eletromagnéticas de cabos subterrâneos até estimações acerca de diagnósticos de baterias elétricas Com isso foi possível perceber a utilidade desta importante ferramenta conhecida como transformada de La Place assim como sua relevância nas mais diversas áreas da engenharia elétrica 5 Referencial Bibliográfico 1 SADIKU Matthew N O Circuitos elétricos São Paulo ABC 2020 2 M A Ivanchenko and D V Dvorkin Laplace Transform to Assess Harmonic Distortions at Resonance 2021 International Conference on Industrial Engineering Applications and Manufacturing ICIEAM Sochi Russia 2021 pp 15 Link httpsieeexploreieeeorgdocument9446412 3 F Tossani F Napolitano and A Borghetti Inverse Laplace Transform of Sundes Formula for the Ground Impedance of Buried Cables 2019 IEEE International Conference on Environment and Electrical Engineering and 2019 IEEE Industrial and Commercial Power Systems Europe EEEIC ICPS Europe Genova Italy 2019 pp 15 Link httpsieeexploreieeeorgdocument8783499 3 N Nagaoka and T Ishii A logarithmic segmented Laplace transform and its application to a battery diagnosis 2017 52nd International Universities Power Engineering Conference UPEC Heraklion Greece 2017 pp 15 Link httpsieeexploreieeeorgdocument8232026 Atividade Aplicações da transformada de La Place em situações reais e práticas da engenharia elétrica 1 Introdução Para que se entenda o conceito de transformada de La Place é preciso identificar o que significa uma transformada Basicamente neste caso a transformada serve para transformar equações diferenciais de difícil resolução em equações algébricas de fácil resolução Ainda a transformada de La Place especificamente fornece uma maneira fácil de solucionar problemas de circuitos envolvendo condições iniciais permitindo respostas completas referentes à solução de problemas SADIKU 2020 Neste contexto tal operação é definida como sendo uma transformação integral de uma função no domínio do tempo para o domínio da frequência complexa Em linhas gerais tratase de uma ferramenta capaz de tornar simples a solução de problemas complexos É aplicada em diversos problemas e contextos práticos da engenharia elétrica permitindo análises criteriosas e bem detalhadas 2 Fundamentação Teórica A transformada de La Place é definida pela equação abaixo L f t F s 0 inf f t e st dt 1 Nesta equação L é o símbolo que representa a transformada de La Place Já ft é uma determinada função descrita no domínio do tempo t Por sua vez Fs é a função ft reescrita no domínio da frequência complexa s Para aplicar a transformada é necessário solucionar uma integral no domínio do tempo Geralmente o processo de solução via La Place consiste em detectar a função no domínio temporal aplicar a transformada de La Place manipular as expressões encontradas e realizar a transformada inversa de La Place para obter a resposta no domínio do tempo mas agora de maneira muito mais simples Normalmente isto é feito com o auxílio de tabelas e formulações prontas Além disso a transformada de La Place apresenta uma série de propriedades As propriedades mais conhecidas são 1 Linearidade 2 Deslocamento no tempo 3 Fatores de escala 4 Diferenciação no tempo 5 Deslocamento da frequência 6 Integração no tempo etc Nesse contexto segundo SADIKU 2020 existem diversas aplicações para a transformada de La Place Geralmente a principal aplicação é na modelagem de elementos de circuito a fim de se obter soluções factíveis no domínio do tempo Mas também é aplicada na análise de sistemas de controle de sistemas lineares de potência análise de respostas transitórias análise de circuitos eletrônicos etc 3 Aplicações Há diversas aplicações práticas do uso da transformada de La Place Ao realizar um levantamento bibliográfico é possível se deparar com aplicações simples e complexas A seguir serão demonstrados alguns exemplos de aplicações mais específicas onde a transformada é realizada a fim de se obter respostas adequação e controle do sistema elétrico Primeiramente em IVANCHEKO M 2021 os autores exploram o assunto distorção harmônica elevada em redes elétricas e cálculos precisos para quantificar este fenômeno Nesse contexto é sugerida uma técnica de solução baseado na transformada de La Place para calcular o nível de corrente e seu efeito cumulativo na distorção harmônica total da rede É concluído que a técnica adotada pode ser generalizada para análise de outros tipos de rede por ter alta precisão e garantir uma análise adequada no que diz respeito à qualidade de energia Segundo em TOSSANI F 2019 o autor inicia narrando sobre as dificuldades intrínsecas na análise de transitórios eletromagnéticos aplicados a cabos submarinos ou subterrâneos haja vista que exigem uma série de aplicações de cálculos analíticos Por conta disso é sugerida uma metodologia baseada na transformada de La Place na chamada fórmula de Sunde para a análise precisa de cabos submarinos com múltiplos condutores Aplicando a transformada de La Place o processo de simulação se torna mais simples e preciso Por fim outro exemplo de aplicação é encontrado em ISHII T 2017 Neste artigo os autores propõe a aplicação da transformada de La Place juntamente com um método conhecido como método logarítimo de amostragem segmentada Neste caso a transformada de La Place é útil para simplificar a análise no domínio da frequência O objetivo disso é estimar a impedância interna de baterias elétricas a fim de realizar diagnósticos sobre sua vida útil 4 Conclusão Neste trabalho demonstrouse de maneira geral a relevância do uso da transformada de La Place em aplicações práticas que envolvem a engenharia elétrica Além disso mostrouse três exemplos de aplicações práticas envolvendo a aplicação desta ferramenta Estes exemplos vão desde aplicações práticas complexas na análise da qualidade de energia de uma rede e a análise transitória eletromagnéticas de cabos subterrâneos até estimações acerca de diagnósticos de baterias elétricas Com isso foi possível perceber a utilidade desta importante ferramenta conhecida como transformada de La Place assim como sua relevância nas mais diversas áreas da engenharia elétrica 5 Referencial Bibliográfico 1 SADIKU Matthew N O Circuitos elétricos São Paulo ABC 2020 2 M A Ivanchenko and D V Dvorkin Laplace Transform to Assess Harmonic Distortions at Resonance 2021 International Conference on Industrial Engineering Applications and Manufacturing ICIEAM Sochi Russia 2021 pp 15 Link httpsieeexploreieeeorgdocument9446412 3 F Tossani F Napolitano and A Borghetti Inverse Laplace Transform of Sundes Formula for the Ground Impedance of Buried Cables 2019 IEEE International Conference on Environment and Electrical Engineering and 2019 IEEE Industrial and Commercial Power Systems Europe EEEIC ICPS Europe Genova Italy 2019 pp 15 Link httpsieeexploreieeeorgdocument8783499 3 N Nagaoka and T Ishii A logarithmic segmented Laplace transform and its application to a battery diagnosis 2017 52nd International Universities Power Engineering Conference UPEC Heraklion Greece 2017 pp 15 Link httpsieeexploreieeeorgdocument8232026 9781538623442173100 2017 IEEE A Logarithmic Segmented Laplace Transform and Its Application to a Battery Diagnosis Tomohiro Ishii Doshisha University Naoto Nagaoka Doshisha University duq0317mail4doshishaacjp nnagaokamaildoshishaacjp AbstractA Laplace transform with a logarithmic segmented samplingmethod is proposed in this paper The method gives a long observation time with a small number of samples in comparison with a conventional discrete Laplace transform DLT with an equally spaced sampling The number of samples is decreased without the reduction in the analysis range The proposed method is applied to an estimation of a battery internal impedance and its results are compared with those obtained by the conventional method The computational time of the proposed method is reduced to 238 The maximum difference between the theoretical and calculated battery impedances is less than 2 The algorithm decreasing the number of samples is realized without reducing the sensitivity for a battery diagnosis system The proposed method realizes a diagnosis system at a low cost because the computational load for the battery diagnosis is greatly reduced Index TermsLaplace transform Lithium ion battery Transient characteristic I INTRODUCTION Lithiumion Liion battery is widely used in equipment from consumer devices to industrial systems The battery capacity is decreased and the risk of damage or fire is increased by its deterioration The battery can be safely and efficiently used with a diagnosis system The deterioration has been estimated from a frequency characteristic of the internal impedance measured by an alternativecurrent superimposition method 1 The method requires a removal of the battery from the equipment It is not allowed in a system requiring high reliability such as largescale storage systems that have to supply electricity continuously The conventional method also needs expensive measuring instruments As a simple diagnosis method alternating the steady state measurement some techniques using a discrete Laplace transform DLT have been proposed 23 However the method needs a long computational time since a long observation time is required for analyzing the battery voltage and current waveforms to obtain the impedance in a wide frequency range This paper presents a method using the logarithmic segmented Laplace transform that is useful to deal with a wide frequency band and a long observation time Since the algorithm of the proposed method is simple a low cost and compact diagnosis system can be realized for portable devices as well as largescale storage systems II PROPOSED SAMPLING METHOD AND LAPLACE TRANSFORM A Frequency characteristics of Liion battery A Liion battery ICR18650PD of 225 Ah in capacity is used as a sample in this paper Fig 1 shows a frequency characteristic of the internal impedance of the battery as a ColeCole plot The characteristic is measured by an alternativecurrent superimposition method which accurately measures the frequency characteristics 3 However the method is not practical because the battery which is the power source need to be removed from the equipment In order to obtain the impedance during an operation the difference between the terminal voltage and the internal voltage and the current waveforms of the battery have to be transformed into a frequency domain The ratio between the transformed data gives the internal impedance To obtain the impedance in a low frequency range as shown in Fig 1 the responses in a long observation time has to be covered by the numerical Fourier or Laplace transform The sampling interval becomes small for obtaining the characteristic in a high frequency region The number of samples becomes large if the conventional equally spaced Fourier or Laplace transform is employed TABLE I shows a sampling condition for obtaining the frequency characteristics shown in Fig 1 The number of samples is 20001 For realizing an embedded diagnostic system the requirements are not practical in terms of the memory requirement and the computational time Fig 1 Frequency characteristics of internal impedance ColeCole plot 01 Hz 562 Hz Low frequency TABLE I SAMPLING CONDITION OF CONVENTIONAL METHOD Sampling time μs Maximum observation time s Frequency step Hz Maximum observation frequency kHz Number of samples 500 10 01 1 20001 B Sampling method Fig 2 shows a set of the transient voltage and current waveforms at around an end of a discharging with a current of 1 C 225 A The voltage suddenly increases at the end of the discharging and gradually increases toward an internal voltage The number of samples for the equally spaced sampling becomes 20001 if the sampling condition shown in TABLE I is applied In order to reduce the number of samples a logarithmic time segmentation is applied in this paper 45 The time response is divided into two regions in this paper The necessary conditions are described below for obtaining the frequency range as shown Fig 1 Δt1 12fmax T2 1fmin 1 where Δt1 is the time step of the 1st section T2 is the maximum observation times of the 2nd section fmax is the maximum observation frequency and fmin is the frequency resolution The number of samples of each section is defined by the following equation Nk 1Nreg 2log2fmaxfmin ε ε 2 2 where Nk is the number of samples of the kth section Nreg is the number of regions and ε is the overlap factor TABLE II shows a proposed sampling condition The voltage waveform shown in Fig 2 is replotted in the condition given by TABLE II as shown in Fig 3 The number of samples is reduced from 20001 to 1024 by the proposed sampling TABLE II SAMPLING CONDITION OF PROPOSED METHOD Overlap factor ε2 Number of samples Sampling time ms Maximum observation time s 1st section 512 05 02555 2nd section 512 20 1024 Fig 3 Simulated transient voltage waveform with proposed sampling condition C Logarithmic segmented Laplace transform Fourier transform of a time function ft is defined by 3 Fω from to ft ejwt dt 3 where Fω is the frequency response of the time function ft For a numerical Fourier transformation the truncation error of the above Fourier integral at the maximum observation time Tmax is inevitable An exponential window function which is commonly used in a transient analysis field is adopted to reduce the truncation error The window function wet is given by 4 wet 0 t0 expαt t0 4 where α is called as shifting constant of the Fourier integral 6 The Fourier transform with the exponential window function is given by the following equation Fω from to ft wet expjωt dt from 0 to ft expαt expjωt dt from 0 to ft expαjω t dt 5 Fs from 0 to Tmax ft expst dt where s αjω where s is Laplace operator Fig 2 Transient voltage and current waveform When the integral region 0 t Tmax in 5 is divided into two regions the above equation can be expressed as follows Fs from T00 to T1 ft est dt from T1 to T2Tmax ft est dt from T00 to T1 ft est dt from 0 to T2 ft est dt from 0 to T1 ft est dt Fs Δt1 Σ from i0 to T1Δt11 fΔt1i esΔt1i Δt2 Σ from i0 to T2Δt21 fΔt2i esΔt2i Δt2 Σ from i0 to T1Δt21 fΔt2i esΔt2i 6 where Tk is the maximum observation times of the kth section and Δtk is the time step of the kth section III BATTERY DIAGNOSIS A Simulation In this section the proposed method is evaluated using a simulated transient voltage responses expressed by analytical functions The transient characteristic of a Liion battery can be expressed by an equivalent circuit consisting of some RC parallel circuits 78 In this paper a simplified equivalent circuit illustrated in Fig 4 89 is used for evaluating the proposed calculation method The circuit consists of an internal voltage v0 with a series resistor RB0 connected in series with an RC parallel circuit whose resistance and capacitance are RB1 and CB1 The impedance ZBs of the circuit is obtained from the voltage vZt and current iBt ZBs 𝓩vZt 𝓩iBt 7 The voltage vZt due to the constant discharging current IB is given by the following equation vZt vBt v0t RB0 IB RB1 IB 1 expt RB1 CB1 8 TABLE III shows the parameters of the equivalent circuit for this demonstration Fig 5 shows a simulated transient voltage by 8 using an equivalent circuit shown in Fig 4 Fig 4 Equivalent circuit of Liion battery Fig 5 Simulated transient voltage waveform TABLE III PARAMETERS OF EQUIVALENT CIRCUIT Series resistance RB0 mΩ Parallel resistance RB1 mΩ Parallel capacitance CB1 F 40 14 45 Fig 6 shows the theoretical impedance obtained by 9 and the calculated impedances ZBs RB0 RB1 1 sRB1 CB1 9 The maximum difference of 194 is observed at 783 Hz Fig 7 shows the effect of the deterioration on the internal impedance These impedances are obtained using a new battery and a deteriorated battery after 500 chargingdischarging cycles and measured by an alternativecurrent superimposition method The state of charge SOC at the end of the discharging is 80 and the ambient temperature is 25 C The maximum difference of the internal impedance between an unused battery and a deteriorated battery is 575 at 10 Hz Since the difference of the impedance due to the deterioration is large enough compared to the difference between the theoretical and calculated results the accuracy of the proposed method is satisfactory for the estimation of the impedance of the Liion battery Fig 6 Comparison of accuracy of internal impedance Fig 7 Deterioration dependence of internal impedance measured by alternativecurrent superimposition method B Computational time The computational efficiency of the proposed method will realize a costefficient impedanceestimation circuit of the battery In this section a computational time of the proposed method is evaluated by a comparison of that of the conventional method The calculations are carried out using Maple TABLE IV shows specifications of the computer used in measurements of the computational time It is measured by time command installed in the Maple library to measure the centralprocessingunit CPU time TABLE V shows the computational times of the conventional method and the proposed method Since the number of samples with the proposed method is further reduced in comparison with the conventional method the computational time is reduced to 238 of that of the conventional method as shown in TABLE V Any common microcomputer can be used for realizing the proposed method TABLE IV COMPUTER USED IN MEASUREMENT OF COMPUTATIONAL TIME Processer IntelR CoreTM i76700 RAM 800 GB Clock frequency 340 GHz TABLE V COMPARISON OF COMPUTATIONAL TIME Conventional Proposed Computational time s 9828 0234 The number of samples point 20001 1024 C Experimental and Estimated results In this section an application of the proposed method to a battery deterioration estimation is demonstrated Fig 8 shows the effect of the deterioration on the voltage waveform after a discharge The state of charge SOC at the end of the discharging is 80 and the ambient temperature around the battery is 25 C The voltage of the deteriorated battery is greater than that of an unused battery The battery deterioration can be diagnosed from the time response against a pulse current discharging ie without any timetofrequency transformation However the diagnosis method cannot be applied to the voltage response against arbitrary current waveforms Fig 9 shows an operational characteristic of a battery cell used in an electric bike Fig 10 shows the effect of the deterioration on the internal impedance estimated by the proposed method TABLE VI shows the sampling condition The results show that the method proposed in this paper can be applied to a deterioration diagnosis of the Liion battery Since the number of samples of the method is further reduced in comparison with the conventional method the method is suitable to practical applications Fig 8 Effect of deterioration on voltage after discharge Fig 9 Measured voltage and current waveform of electric bike Fig 10 Deterioration dependence of internal impedance given by proposed method 575 0 cycle 500 cycle deterioration current A deterioration TABLE VI SAMPLING CONDITION OF PROPOSED METHOD FOR ANALYZING OPERATIONAL CHARACTERISTIC OF BATTERY CELL USED IN ELECTRIC BIKE Overlap factor ε2 Number of samples Sampling time ms Maximum observation time ms 1st section 128 05 64 2nd section 128 8 1024 IV CONCLUSION A logarithmic segmented Laplace transform is proposed in this paper The proposed method gives a long observational time with small number of samples in comparison with the conventional discrete Laplace transform with equally spaced sampling The proposed method is applied to estimate an internal impedance of a Liion battery The difference between the theoretical and calculated impedances is less than 2 Since the accuracy of the method is satisfactory for detecting the battery deterioration a deterioration diagnosis system for the battery can be designed based on the internal impedance estimation The high computational efficiency of the proposed method will realize a costefficient diagnosis system of the battery A diagnosis circuit can be realized using a microcomputer due to the simple algorithm of the method The proposed system can be installed into equipment for a wide range of use from consumer devices to industrial systems REFERENCES 1 T Momma M Matsuda D Mukoyama and T Osaka Ac impedance analysis of lithium ion battery under temperature control Journal of Power Sources vol205 pp483486 May 2012 2 K Takano K Nozaki Y Saito A Negishi K Kato and Y Yamaguchi Simulation study of electrical dynamic characteristics of lithiumion battery J Power Sources Vol 90 pp 214 223 2000 3 T Noro N Narita T Higo and N Nagaoka A Deterioration Diagnosis Method for Lithiumion Battery Based on Phase characteristics of Internal Impedance Proc University Power Engineering Conference 50th International Staffordshire Sep 2015 4 A Ametani S Yamaoka On logarithmic Fourier transform IEE Japan Research Committee of Information Processing Report IP7717 1977 5 A Ametani and K Imanishi Development of exponential Fourier transform and its application to electrical transients Proc IEE Vol1261 pp5156 1979 6 N Nagaoka and A Ametani A Development of a Generalized Frequencydomain ProgramFTP IEE Trans Power Delivery Vol3 pp19962004 1988 7 T Hirai A Ohnishi N Nagaoka N Mori A Ametani and S Umeda Automatic EquivalentCircuit Estimation System for LithiumIon Battery University Power Engineering Conference 43rd International Padova Sep 2008 8 N Nagaoka An Estimation Method of LiIon Battery Impedance Using ZTransform Thirteenth IEEE Workshop on Control and Modeling for Power Electronics COMPEL12 Kyoto June 2012 9 N Nagaoka A numerical model of Lithiumion battery for a life estimation Proc University Power Engineering Conference 48th International Dublin Sep 2013 Paper ID 136 9781728106533193100 2019 IEEE Inverse Laplace Transform of Sundes Formula for the Ground Impedance of Buried Cables F Tossani F Napolitano ABorghetti Department of Electrical Electronic and Information Engineering University of Bologna Italy fabiotossaniuniboit fabionapolitanouniboit albertoborghettiuniboit AbstractThe timedomain calculation of electromagnetic transients in multiconductor lossy overhead lines and buried cables requires the evaluation of the transient ground resistance matrix For the case of overhead lines analytical expressions for the transient ground resistance obtained by solving the inverse Laplace transform of Sundes formula have been recently presented This paper presents the expressions obtained by the analytical inverse Laplace transform of Sundes formula for the case of buried cables The results provided by the proposed analytical expressions agree with those given by the numerical inverse transform of Sundes formula The new expressions are adopted for the calculation of the perunitlength voltage drop in a multiconductor underground line The voltage drop waveforms are compared with those given by recently proposed timedomain analytical expressions that neglect displacement currents KeywordsMulticonductor lines buried cables ground impedance transient ground resistance inverse Laplace transform I INTRODUCTION The calculation of electromagnetic transients in overhead lines and buried cables requires the detailed representation of the earth return losses This paper is focused on the assessment of the series self and mutual per unit length impedances to be included in the timedomain formulation of the telegraphers equations When a frequency domain approach is adopted the perunitlength parameters affected by the ground losses are the ground impedance g Z also called earth return impedance and the ground admittance of the line The time domain counterpart of g Z is the transient ground resistance TGR of the line denoted with g The ij th element of the TGR matrix is defined as 1 g ij g ij Z s t L s 1 where L1 denotes the inverse Laplace transform ILT and s is the complex Laplace variable J R Carson was the first to investigate the concept of ground impedance of an overhead line and to provide an expression for g Z in the frequency domain 1 whilst the corresponding expression for buried wires was developed by F Pollaczek 2 Both Carsons and Pollaczeks formulae were derived by assuming a low frequency approach neglecting the displacement currents in the propagation constant Displacement currents are taken into account by E D Sunde who proposed two expressions for the evaluation of the earthreturn impedance of wires with infinite length above and below the earth surface in 3 These expressions correspond to Carsons and Pollaczeks ones when displacement currents are neglected 4 5 All these expressions involve integrals over a semi infinite interval that are not suitable for a fastnumerical evaluation Therefore even with modern numerical integration routines a direct timedomain formulation of the TGR is preferred instead of the numerical inverse transform of the results in frequencydomain ie the numerical computation of 1 The analytical ILT of Carsons formula was derived in 6 for the case of a single conductor line and later extended to a multiconductor line in 7 8 The analytical ILT of Pollaczeks formula has been recently derived by Araneo and Celozzi in 9 The analytical ILT of Sundes formula for the case of overhead lines has been presented in 10 In this paper a closedform of the ILT of Sundes formula for underground cables is proposed and discussed The paper is organized as follows Section II illustrates the ILT of the ground impedance matrix of overhead lines Section III is devoted to the derivation of the ILT of the ground impedance matrix of buried cables Section IV compares the results of the TGR calculated by using the proposed expressions and the numerical inverse transform of the frequencydomain results given by Sundes formula Moreover the proposed expressions are adopted for the calculation of the perunitlength voltage drop in a multiconductor underground line The voltage drop waveforms are compared with the those obtained by using the analytical expressions by Araneo and Celozzi Section V concludes the paper II TRANSIENT GROUND RESISTANCE OF OVERHEAD LINES For the geometrical configuration shown in Fig 1 the expression of the elements of the ground impedance matrix reads 0 2 2 0 cos i j h h x g ij ij g s e Z s r x dx x x ò 2 The quantity g is the propagation constant which in Laplace domain is 0 g g g s s 3 In 3 Sunde proposed a logarithmic approximation for the diagonal elements of g Z and in 8 this expression was extended to the offdiagonal elements 2 As shown in 8 this approximation is accurate for a wide frequency range and for typical values of ground electrical parameters within the limits of the transmission line approximation The ILT of these logarithmic formulae can be found in 11 12 Fig 1 Geometrical configuration of an overhead multiconductor line Expression 2 can be rearranged and split in two terms 0 1 2 2 g ij g Z s Int Int s 4 where 2 2 2 1 0 Re 1 ij g h y g Int e y dy 5 2 0 2 2 2 2 2 cos i j i j ij i j i h h x ij j Int e x r x dx h h r h h r 6 and 2 2 i j ij ij h h r h j 7 The ILT of 5 is given by 2 1 1 2 0 0 1 2 0 Re 1 ij g h y g L Int L e y dy 8 To solve the ILT on the right side of 8 we can differentiate with respect to k equation 225 2 given in 13 obtaining 1 2 2 1 2 2 2 0 2 2 2 2 2 at k s s a L ak a I t k a e t k I t k t k t k e 9 where t is the Heaviside step function and t is the Dirac delta function By using the properties of Dirac and Heaviside functions 8 becomes 2 2 1 0 0 1 2 2 2 2 1 2 2 2 2 0 Re 1 2 4 2 1 2 4 at ij g ij t ij e t L Int b b a I t k ak k dk b t k 10 where 0 ij ij g g g b h a 11 The inverse transform of the second term in 4 can be readily evaluated thus the ILT of the ground impedance matrix elements is given by 1 0 2 2 0 2 2 0 2 2 2 2 1 2 2 2 2 0 0 cos Re 1 2 4 2 1 2 4 i j h h x g ij ij g at ij i i j j t ij h e t L r x dx x x e t b b a I t k ak k dk h b t k ì ü ï ï ï ï ï ï í ý ï ï ï ï ï ï î þ ìï é ïï ê ï í ê ï ê ï ë ïïî üùï æ ö ï ç ú ï ç ï ç ú è ø ïïú ýúï ïúïïúïû ïþ ò ò 2 2 2 0 2 1 1 ij a i j g t ij r e h h r é ù ê ú ë û 12 In 10 it is shown that 12 can be recast to obtain a form more suitable for numerical computation and the quadrature formulas are discussed The limit of 12 as εrg approaches zero is equal to the well know Timotin expression for the TGR 6 derived as the ILT of Carson formula Indeed by using the asymptotic expansion of the modified Bessel function for large arguments the integral in 12 reduces to an integral of the type 2 0 ij x t f x e dx ò 13 For the case of interest we obtain 0 0 1 1 lim Re 4 1 1 4 2 g ij ij g ij ij w j t t t é ù æ ö ç ê ú ç ê ú ì ü ï ï ï ï ï ï í ý ï ï ï ï ï ï î ç ç ê ú ç ë þ è ø û 14 where 2 fc z er w e z jz is the Faddeeva function of argument jz and 2 0 ij ij g h which can be evaluated by specific numerical routines III TRANSIENT GROUND RESISTANCE OF BURIED CABLES In this Section the procedure adopted to obtain 12 is extended to the case of buried cables For the underground configuration shown in Fig 2 the diagonal terms of the ground impedance matrix elements are 2 3 2 2 2 2 0 0 0 2 2 2 0 4 2 2 cos i g ii g i g i i h x i g s Z s K r K r h e r x dx x x 15 whilst the offdiagonal components are given by 2 2 0 0 0 2 2 2 0 2 2 cos ij g ij g ij g ij H x ij g s Z s K d K D e r x dx x x 16 where 0 K is the modified Bessel function of the second kind and order 0 and ri is the radius of the ith conductor With rij we will refer to the horizontal mutual distance between ith and jth conductor and 2 2 2 2 2 2 2 4 i j ij ij ij i j ij ij i j ij ij h h H d r h h D r h h r H 17 The integral 2 2 0 2 2 0 2 cos g ij ij ij g H x e J s r x dx x x 18 in 15 and 16 is known as Pollaczek integral if the displacement current are neglected in g Fig 2 Geometrical configuration of an underground multiconductor line We will proceed with the ILT of 16 the ILT of 15 can be obtained by setting ii i r r and ii i h h for the diagonal elements The ILT of the integral component of 16 can be evaluated by using the same approach adopted for the overhead lines ie by splitting the integral in two terms and by using 9 Following this procedure one gets 2 2 2 2 1 2 2 1 1 0 2 2 0 2 2 0 2 2 2 2 2 2 2 cos 1 2 Re 2 2 1 1 2 1 ij g ij t ij ij g at ij ij ij ij ij B ij H x ak a I t k dk t k J s e L L r x dx s x x y t y t e t B r B y t j H y t y k y k r y k j H y k 19 where 0 ij ij g B H 20 and 2 2 0 2 0 2 ij i ij g i j g j D y jk D H k r k 21 For the Bessel component we will make use of the integral definition of the modified Bessel function of the second kind 14eq 9624 The ILT of the two Bessel functions in 15 and 16 is 0 1 0 0 0 2 2 2 0 2 2 1 2 2 2 2 0 0 2 1 2 1 2 2 ij g g ij at ij g t d ij g ij g L K d e t d ak a I t k dk t k k d t d 22 Finally the expression of the TGR is 0 0 0 2 2 2 0 2 2 1 2 2 2 2 0 0 2 2 2 0 2 2 1 2 2 2 2 0 0 0 1 2 1 2 2 1 2 1 2 2 2 ij g ij g at g ij ij g t d ij g at ij g t D ij g ij g ij g t e t d ak a I t k dk t k k d e t D ak a I t k dk t k k D t t d t D 2 2 1 2 2 2 2 0 2 2 2 2 2 2 1 Re 2 2 1 1 2 1 ij t at ij ij ij ij ij B ij ak a I t k dk t k y t y t e B r B y t j H y t y k y k r y k j H y k 23 IV NUMERICAL TEST COMPARISONS AND DISCUSSION Let us now consider a twoconductor line with h1 h2 1 m having radius of 15 cm and mutual horizontal distance r12 1 m The time domain response obtained by the inverse transform of 18 and of the two Bessel components is shown in Fig 3 and Fig 4 respectively The ground parameters are σg 1 mSm and εrg 10 The curves obtained by the numerical inverse Fourier transform IFT of 16 with and without taking into account the displacement currents are also reported The results provided by the proposed analytical ILT are in good agreement with those calculated by using the numerical transform The numerical IFT of the various components of 16 is computationally inefficient with respect to the direct evaluation of 23 in time domain and may result in numerical inaccuracies due to the oscillating behavior of the Bessel functions of the second type as shown in Fig 4 in correspondence of the discontinuities It is worth recalling that transmission line approximation is questionable for the evaluation of the earlytime response of the line The time above which such a low frequency approach is acceptable is proportional to the ratio εgσg 15 In this respect the validity of Sundes formula has been discussed in 16 17 Fig 5 shows the comparison between the TGR calculated by using the proposed formulas and the one by Araneo and Celozzi 9 for different values of the relative ground permittivity As the value of εrg decreases the discontinuities present in Sundes approach shift towards minor time values and the two expressions lead to similar results Fig 3 Comparison between 19 and the numerical IFT of Jijjω with and without displacement currents h1 h2 1 m and r12 1 m σg 1 mSm and εrg 10 Fig 4 Bessel component of the TGR h1 h2 1 m and r12 1 m σg 1 mSm and εrg 10 Fig 5 Comparison between the TGR calculate by using 23 and the one by Araneo and Celozzi 9 h1 h2 1 m and r12 1 m σg 1 mSm 1 Note that in 9 Hij hihj It can be shown that the limit of 23 as εrg approaches zero leads to the expression by Araneo and Celozzi for the TGR Indeed the lower integration limit in both 19 and 22 tends to zero and the expressions can be recast to obtain a linear combination of integrals of the type of 13 again by using the asymptotic expansion of the modified Bessel function for large arguments and after some algebraic manipulations For the case of interest we obtain 0 0 4 2 2 4 2 2 2 1 4 e 1 rfc 2 2 lim 1 2 2 2 4 16 4 4 d D H H H g t t H D H r H H g i t t D r D r t j ij ij ij D e e e e t t e D t t t r H D t t t 24 where 2 2 2 0 0 0 2 0 2 H ij g D ij g d ij g ij g H r r ij i ij i j j H D d r H D r D 25 which is indeed the expression by Araneo and Celozzi1 From 19 we can observe that the contribution of the integral component is null before a time 0 2 ij g H where Hij is the mean depth of the ith and jth conductors This means that the mutual effect of the TGR between two conductors is not instantaneous but propagates with the speed of the electromagnetic field in the ground For the case of a single conductor line or diagonal elements of the matrix the effect of the integral component 19 is null before a time that is twice the depth of the cable divided by the propagation speed in the considered medium This effect is clearly in contrast with the assumption of transverse electromagnetic propagation Furthermore 19 is affected by a singularity that depends on the values assumed by y For the sake of simplicity let us consider the iith element and let us neglect the values of the radius of the conductor With these assumptions we can conclude that 19 is singular at 0 2 i g t h As for the case of the integral component the time response associated to each of the Bessel functions is null prior to a certain time 0 ij g d or 0 ij g D for the other Bessel component at which 22 features a singularity The per unit length voltage drop v g i x t in the ith conductor due to the TGR is 4 0 t j g i g ij g ij j j i x v x t v x t t d 26 We have calculated the timedomain values of v g12 by assuming σg 1 mSm and the propagation of a current in one conductor characterized by a waveform consisting of a linear front with rise time tr followed by a constant amplitude of 1 A In particular Fig 6 compares the results obtained by calculating the TGR with 23 for different values of the m m gij m relative ground permittivity and by using 24 Three different values of tr are considered tr 100 ns Fig 6a tr 500 ns Fig 6b and tr 1 μs Fig 6c The same geometrical layout of conductors considered so far is assumed A trapezoidal integration method has been used to evaluate the convolution The value of the integration time step is set to 05 ns in order to mitigate the effect of the discontinuities and spikes characteristic of 23 Furthermore since also 24 has a singularity at the origin we set its value to 0 at t 0 to compute the convolution Fig 6a shows that for the case of εrg 10 the voltage drop calculated by using 23 is null before a time interval of approximately 105 ns which corresponds to the time needed by the electromagnetic field to travel the distance of 1 m in the considered ground The two approaches give similar results as the value of the ground permittivity decreases a b c Fig 6 Perunitlength voltage drop due to the mutual TGR for different currents a tr100 ns b tr 500 ns c tr 1 μs V CONCLUSION The paper presents the analytical solution of the inverse Laplace transformation of Sundes formula for the case of underground multiconductor lines The timedomain analytical solution agrees with the numerical inverse transform of Sundes expression The advantage of using the proposed expression with respect to the use of Sundes one is the significantly reduced computational effort The proposed timedomain expression is affected by the intrinsic limitations of Sundes formula already analyzed in the literature on the subject REFERENCES 1 J R Carson Wave Propagation in Overhead Wires with Ground Return Bell Syst Tech J no 5 pp 539554 1926 2 F Pollaczek Uber das feld einer unendlichen langenwechsel stromdurchflossenen einfachleitung Elect Nachr Tech vol 3 no 9 pp 339360 1926 3 E D Sunde Earth conduction effects in transmission systems New York D Van Nostrand Company 1968 4 F Rachidi S L Loyka C A Nucci and M Ianoz A new expression for the ground transient resistance matrix elements of multiconductor overhead transmission lines Electr Power Syst Res vol 65 no 1 pp 4146 2003 5 N Theethayi R Thottappillil M Paolone C Nucci and F Rachidi External impedance and admittance of buried horizontal wires for transient studies using transmission line analysis IEEE Trans Dielectr Electr Insul vol 14 no 3 pp 751761 Jun 2007 6 A Timotin Longitudinal transient parameters of a unifilar line with ground return Rev Roum Se Techn Electrotechn Energ vol 12 no 4 pp 523535 1967 7 D Orzan Timedomain low frequency approximation for the off diagonal terms of the ground impedance matrix IEEE Trans Electromagn Compat vol 39 no 1 p 64 1997 8 F Rachidi C A Nucci and M Ianoz Transient Analysis of Multiconductor Lines Above a Lossy Ground IEEE Trans Power Deliv vol 14 no 1 pp 294302 1999 9 R Araneo and S Celozzi Ground Transient Resistance of Underground Cables IEEE Trans Electromagn Compat vol 58 no 3 pp 931934 June 2016 10 F Tossani F Napolitano and A Borghetti Inverse Laplace Transform of the Ground Impedance Matrix of Overhead Lines IEEE Trans Electromagn Compat vol 60 no 6 pp 20332036 2018 11 F Tossani F Napolitano F Rachidi and C A Nucci An Improved Approach for the Calculation of the Transient Ground Resistance Matrix of Multiconductor Lines IEEE Trans Power Deliv vol 31 no 3 pp 11421149 2016 12 F Tossani F Napolitano and A Borghetti New Integral Formulas for the Elements of the Transient Ground Resistance Matrix of Multiconductor Lines IEEE Trans Electromagn Compat vol 59 no 1 pp 193198 2017 13 A P Prudnikov Y A Brychkov and O I Marichev Integrals and Series vol5 Inverse Laplace Transforms New York Gordon and Breach 1992 14 M Abramowitz and I Stegun Handbook of mathematical functions 1964 15 S L Loyka On calculation of the ground transient resistance of overhead lines IEEE Trans Electromagn Compat vol 41 no 3 pp 193195 1999 16 T A Papadopoulos D A Tsiamitros and G K Papagiannis Impedances and admittances of underground cables for the homogeneous earth case IEEE Trans Power Deliv vol 25 no 2 pp 961969 2010 17 A Ametani Y Miyamoto Y Baba and N Nagaoka Wave Propagation on an Overhead Multiconductor in a HighFrequency Region IEEE Trans Electromagn Compat vol 56 no 6 pp 1638 1648 2014 0 05 1 15 2 25 3 35 4 time s 107 0 1 2 3 4 5 6 7 vgij Vm Sunde ILT AraneoCelozzi εrg10 εrg1 vgij Vm vgij Vm 2021 International Conference on Industrial Engineering Applications and Manufacturing ICIEAM 9781728145877213100 2021 IEEE Laplace Transform to Assess Harmonic Distortions at Resonance Mikhail A Ivanchenko Department of Power Grid Development JSC STC UPS Power system development Moscow Russia ivanchenkomasoupsru Dmitry V Dvorkin Department of Power Grid Development JSC STC UPS Power system development Moscow Russia dvorkindvsoupsru AbstractA highharmonic distortion is a wellknown negative influence on a grid mostly caused by a nonlinear loads operation And although this phenomenon is widelystudied the practical approaches for its quantitative assessment differ Previously the authors proposed a Laplacetransformbased approach to calculate a current level of a cumulative effect of all highharmonics also known as Total Harmonic Distortion THD on power quality in the grid The conducted experiments showed that the developed method allows for a more precise THD assessment than the standardized ones based on the Fourier transform However additional experiments resulted in a conclusion that this method may provide significantly higher accuracy than the traditional Fouriertransformbased method for THD assessment at series electrical resonance This paper provides additional theoretical analysis and case studies on a model of a single substation with a capacitor bank to verify the previously developed method and to compare its results with the Fouriertransformbased method Abstract Keywordspower quality highharmonics Laplace transform THD I INTRODUCTION Nonlinear loads operating in a grid cause highharmonic current and voltage deviations affecting the grids equipment measurement systems generation units and consumers devices For example the most dispersed variations of these negative effects are transformers and line thermal overheats relay systems misfunction operational failures caused by them etc To avoid these unfavorable cases it is strictly required to keep actual levels of highharmonic voltages and currents within a prescribed level which is called the maximum possible THD level Depending on the standards in 14 the maximum possible values may vary however the main principle for the actual distortion level to be assessed and compared with the standardized one is the same in all documents It is based on the Fourier series and results in 2 2 1 100 n l l F I THD I 1 where 𝑙𝑙 harmonic order 𝐼𝐼1 fundamental frequency current 𝐼𝐼𝑙𝑙 current on the frequency 𝑙𝑙 and 𝑛𝑛 maximum order to be considered which depends on the standard to be used Additionally THD also can be presented as a ratio between the fundamental frequency current and highharmonics current as described in 5 2 2 2 1 100 n l l R n l l I THD I 2 where all labels and indices are similar to ones in 1 Using these equations it is possible to quantitatively assess the effect of all highorder harmonics registered in the grid Fig 1 However the main advantage of them is that they allow for the fast identification of energy emitted by a harmonic set without vector representation so the magnitudes of all high harmonics are only required as a matter of convenience for the following example there are only the 5th and the 7th harmonics considered 2 5 7 2 5 5 7 7 2 sin5 2 sin7 I i t i t I I t I t ω ω ω ϕ ω ϕ 3 where 𝐼𝐼5 and 𝐼𝐼7 are RMSvalues of the 5th and the 7th harmonics respectively 𝜑𝜑5 and 𝜑𝜑7 are their initial phases andωis a rate of phase change The squared RMSvalue of the nonsinusoidal signal in 3 results in 2 2 2 0 2 2 5 5 7 7 0 2 2 5 7 5 7 5 7 5 7 5 7 5 7 5 7 1 1 2 2 sin5 sin7 sin 4 sin 2 2 2 sin 24 sin 2 12 I i t d t I t I t d t I I I I I I π π ω ω π π ω ϕ ω ϕ ω π ϕ ϕ ϕ ϕ π π π π π π ϕ ϕ ϕ ϕ π π 4 where components sin4𝜋𝜋 𝜑𝜑5 𝜑𝜑7 sin𝜑𝜑5 𝜑𝜑7 and sin24𝜋𝜋 𝜑𝜑5 𝜑𝜑7 sin𝜑𝜑5 𝜑𝜑7 are always equal to 0 204 2021 International Conference on Industrial Engineering Applications and Manufacturing ICIEAM 9781728145877213100 2021 IEEE DOI 101109ICIEAM5122620219446412 Authorized licensed use limited to Univ of Calif Santa Barbara Downloaded on June 302021 at 195658 UTC from IEEE Xplore Restrictions apply 2021 International Conference on Industrial Engineering Applications and Manufacturing ICIEAM Fig 1 High harmonics calculation 𝑇𝑇𝑇𝑇𝑇𝑇𝐹𝐹 vs 𝑇𝑇𝑇𝑇𝑇𝑇𝑅𝑅 in different 𝑛𝑛 Eq4 shows that the RMSvalue of the nonsinusoidal signal in 3 does not depend on the initial phases of its harmonic components which allows for the practical implementation of the approaches in 1 and 2 In its term the main disadvantages of these approaches in 1 and 2 is that the choice of the maximum highharmonic order 𝑛𝑛 to be considered in THD calculations may affect the methodical error 100 0 n n n THD THD THD δ 5 Previously in 13 possible values of the methodical error in 5 has been considered which couraged us to propose another operatormatrix state variable OMSV method to estimate THD without harmonic neglection However new research showed that within a resonance the OMSV method might result in higher accuracy so this paper provides additional case studies It worth mentioning that the Fourierbased methods for the THD assessment are common and the modern scientific society keep studying them 712 so the alternative approach presented in this paper can be used in further for additional verification II BASIC THESIS The calculations below were performed by the OMSV method which has been previously presented in 13 and 14 and this section briefly describes the primary thesis The statespace of the system is a space containing vectors 𝐱𝐱 for each time 𝑡𝑡 If an electrical circuit allows representation by the state space and described by differential equations they can be reduced to the Cauchy form t t x A x B J 6 where 𝐱𝐱𝑡𝑡 is a state variable matrix 𝐉𝐉𝑡𝑡 is a current sources matrix 𝐀𝐀 is a square matrix 𝑤𝑤 𝑤𝑤 𝑤𝑤 is a number of the state variables and 𝐁𝐁 is a circuit parameters matrix In the operator form 6 results in 1 0 p p t X 1 A B J X 7 The Convolution theorem can find the original form of 6 It is assumed that 1 2 F p p e F p p t A 1 A B J B J 8 and 0 0 t t t e t d e τ τ τ A A x B J x 9 where 𝐱𝐱0 is an initial condition matrix column The general solution is presented by the following sum 15 t t t x x x 10 where 𝐱𝐱𝑡𝑡 is a forced component and 𝐱𝐱𝑡𝑡 is a free component out of the consideration in a steadystate mode The free component can be found as follows 205 Authorized licensed use limited to Univ of Calif Santa Barbara Downloaded on June 302021 at 195658 UTC from IEEE Xplore Restrictions apply 2021 International Conference on Industrial Engineering Applications and Manufacturing ICIEAM xt eAtx0 eAtx0 x0 11 For periodic function Jt x0 results in 6 x0 1 eAT1 0t eAtB Jτdτ 12 The Laplace transform of Jτ matrix A replaces operator p is given by Fp τ 0 epl ft τ dt 13 Then x0 B JA 0 14 where JA 0 represents the Laplace image of the input signals Over a period this component results in xτ B JA τ 15 where JA τ is an operator image Then the solution of 15 means to find the Jτ t matrix Any matrix function can be obtained as follows fA k1 to n fλk Pk 16 where λk is a circuit eigenvalues represented by an A matrix Pk is a matrix projection of A for each λk which is given as follows detλ 1 A 0 17 Matrix projection Pk can be determined as Pk j1jk to n A λj 1 j1jk to n λk λj 18 Eventually the solution of 15 in the steadystate mode results in xτ k1 to m Pk B Jλk τ 19 The formula for calculation of the THD level takes this form THDL I2 I12 I1 100 20 where I is an RMSvalue of the nonsinusoidal current calculated according to 19 I1 is an RMSvalue of the current of the first harmonic I1 is an RMSvalue of the current of the first harmonic voltage in the original circuit If the eigenfrequency of the circuit is much higher than the fundamental frequency then THDL I2I1 100 21 Starting now the OMSV method is called Laplacetransformbased and the THD factor calculated according to the algorithm in 621 is named THDL III TEST SCHEME All the experiments below were conducted on the test scheme shown in Fig 2 for the frequency of the initial measurements It represents an external supplying system Us and XS 672 Ohm a transmission line ZTL rTL jXTL 12 j4 Ohm a substation transformer XT1 44 Ohm a capacitor bank XC 2200 Ohm to keep the voltage at a point of common coupling PCC within the needed range and a feeder supplying a nonlinear load ZL via a converter transformer XT2 308 Ohm Hereinafter it is assumed the external system provides sinusoidal voltage ust 2Us sin ωt Although no commutations are performed the structure of this scheme varies depending on the considering frequency If l 1 the fundamental frequency the test scheme is supplied by the sinusoidal voltage source representing a swing bus Fig 3 l 2 the frequency of the considering high harmonics the test scheme is supplied by a current source in Fig 4 Fig 2 Test scheme on the initial measurements frequency Fig 3 Test scheme on the fundamental frequency 2021 International Conference on Industrial Engineering Applications and Manufacturing ICIEAM Fig 4 Test scheme on the lth harmonics frequency It is of interest to analyze the effect of the capacitor bank on the THD level at PCC On the fundamental frequency the current at PCC can be found as follows IPCC1 Ês Żeq Ês jXS1 ŻTL1 jXT11 Żeq1 22 where Żeq1 jXC1 jXT21 jXL1 jXT21 jXL1 jXC1 23 In its term on the lth harmonics frequency the current at PCC results in IPCCl Jl0 jXCl jXSl ŻTLl jXT1l jXCl 24 As seen from 2224 if XC decreases the capacitance increases and the capacitor bank generates more reactive power the current at PCC increases on the fundamental frequency and takes lower values on the high harmonics frequencies In other words the injection of the capacitor bank primarily to keep the power factor within a prescribed range allows for stabilizing the current curve and minimizing its THD factor The capacitor bank may suffer a malfunction from the high harmonic currents so an optimal solution allowing to stabilize the current curve and prevent possible capacitor banks malfunctions is required IV SIMULATIONS In this section the mathematical experiments were conducted in order to test the theoretical model The nonlinear load generates a nonharmonic current Jt l0 to Illt and its form is shown in Fig 5 For the THDF calculations it is assumed that n 50 In this experiment the capacitor banks capacitance varies from 120 Ohm to 4120 Ohm In Fig 6 it is shown that even without resonance there is a significant difference in THD calculation based on the Fourier and Laplace transform The relative error Fig 7 fluctuates from 15 20 and it is caused by the neglection of the high harmonics of order 51 and above in THDF calculations Fig 5 Time function Jt Wherein the Fourierbased THD calculation does not allow to register the resonance on the frequencies higher than 2500 Hz which gives a critical error XC 2620 Ohm and XC 4250 Ohm As a result at resonance the relative error takes values in range 100 450 which is unacceptable Fig 6 THDF vs THDL within the change of XC Fig 7 Methodical error dispersion within the change of XC Therefore it is proved that the traditional standardized Fourierbased approach for the THD assessment may be inapplicable in the real grid at resonance due to various uncertainties in grid parameters V CONCLUSION This article provides an additional theoretical and experimental analysis for a Laplacetransformbased THD assessment method The theoretical consideration allowed the conclusion that the THD calculation based on the Fourier 2021 International Conference on Industrial Engineering Applications and Manufacturing ICIEAM transform inevitably causes to neglect the high harmonics which is higher than the standardized one Unfortunately this neglection leads to a methodical error which can be significant up to 20 according to the experimental data even in a steadystate regime Moreover in the test scheme series electrical resonance regimes were simulated The experimental results showed that the Fouriertransformbased approach might give a critically high methodical error up to 450 due to the aforementioned neglection of all high harmonics of order 51 and above Therefore the authors recommend performing the Laplace transform in THD assessment Especially in cases of a capacitor bank placement at a real substation to avoid its malfunctions in normal operation due to possible unacceptable high order current levels which can not be registered by the Fourierbased assessment ACKNOWLEDGMENT This work was performed at JSC STC UPS Power system development by its research staff and the authors thank the company for its financial support We welcome anyone who may be interested in further projects to be done in cooperation to contact us by email REFERENCES 1 IEEE standard 11591995 Recommended Practice for Monitoring Electric Power Quality 2 IEEE 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Technologies pp 16 2012 DOI 101109APCET20126302048 12 Nicolae and D Popa Realtime implementation of some fourier transform based techniques for fundamental harmonic detection using dSPACE 2016 18th European Conf on Power Electronics and Applications pp 17 2016 DOI 101109EPE20167695696 13 M A Ivanchenko and D V Dvorkin An Alternative Approach for THD Assessment 2020 Int Youth Conf on Radio Electronics Electrical and Power Engineering pp 16 2020 DOI 101109REEPE4919820209059193 14 M A Kalugina Development of Methods for Assessing the Integral Characteristics of Nonsinusoidality in Power Supply Systems for the Selection of Resources to Improve the Quality of Electricity PhD Dissertation 1989 15 David Vernon Widder The Laplace Transform Princeton Mathematical Series PU Press Princeton 1941 208 Authorized licensed use limited to Univ of Calif Santa Barbara Downloaded on June 302021 at 195658 UTC from IEEE Xplore Restrictions apply