{"title":"An Affine Linear Solution of the Nonlinear Inverse Power Flow Problem in Resistive Networks","authors":"Martin Wachs, Miriam Primbs","doi":"10.1002/jnm.70026","DOIUrl":null,"url":null,"abstract":"<p>In the analysis of linear electrical networks, an inverse problem can be inferring all edge impedances only from known external voltage sources and measured resulting edge currents. Given all external edge voltages <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>u</mi>\n <mi>ext</mi>\n </msub>\n </mrow>\n <annotation>$$ {\\boldsymbol{u}}_{\\mathrm{ext}} $$</annotation>\n </semantics></math> and all resulting edge currents <span></span><math>\n <semantics>\n <mrow>\n <mi>i</mi>\n </mrow>\n <annotation>$$ \\boldsymbol{i} $$</annotation>\n </semantics></math>, we present a new calculation method for the edge resistances <span></span><math>\n <semantics>\n <mrow>\n <mi>R</mi>\n </mrow>\n <annotation>$$ \\boldsymbol{R} $$</annotation>\n </semantics></math>, with the assumption that the reactance is everywhere zero (e.g., a resistive network). Our considerations are based on affine subspaces and their intersection. We show, that in case of having a sequence of <span></span><math>\n <semantics>\n <mrow>\n <mi>l</mi>\n <mo>≥</mo>\n <mn>3</mn>\n </mrow>\n <annotation>$$ l\\ge 3 $$</annotation>\n </semantics></math> measurements <span></span><math>\n <semantics>\n <mrow>\n <mfenced>\n <msub>\n <mi>u</mi>\n <msub>\n <mi>ext</mi>\n <mn>1</mn>\n </msub>\n </msub>\n <msub>\n <mi>i</mi>\n <mn>1</mn>\n </msub>\n </mfenced>\n <mo>,</mo>\n <mo>…</mo>\n <mo>,</mo>\n <mfenced>\n <msub>\n <mi>u</mi>\n <msub>\n <mi>ext</mi>\n <mi>l</mi>\n </msub>\n </msub>\n <msub>\n <mi>i</mi>\n <mi>l</mi>\n </msub>\n </mfenced>\n </mrow>\n <annotation>$$ \\left({\\boldsymbol{u}}_{{\\operatorname{ext}}_1},{\\boldsymbol{i}}_1\\right),\\dots, \\left({\\boldsymbol{u}}_{{\\operatorname{ext}}_l},{\\boldsymbol{i}}_l\\right) $$</annotation>\n </semantics></math>, we can calculate <span></span><math>\n <semantics>\n <mrow>\n <mi>R</mi>\n </mrow>\n <annotation>$$ \\boldsymbol{R} $$</annotation>\n </semantics></math> uniquely in every such network. For a sufficiently large but still small cuboid grid, we can reduce the number of needed measurements to 2.</p>","PeriodicalId":50300,"journal":{"name":"International Journal of Numerical Modelling-Electronic Networks Devices and Fields","volume":"38 2","pages":""},"PeriodicalIF":1.6000,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jnm.70026","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal of Numerical Modelling-Electronic Networks Devices and Fields","FirstCategoryId":"5","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jnm.70026","RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"ENGINEERING, ELECTRICAL & ELECTRONIC","Score":null,"Total":0}
引用次数: 0
Abstract
In the analysis of linear electrical networks, an inverse problem can be inferring all edge impedances only from known external voltage sources and measured resulting edge currents. Given all external edge voltages and all resulting edge currents , we present a new calculation method for the edge resistances , with the assumption that the reactance is everywhere zero (e.g., a resistive network). Our considerations are based on affine subspaces and their intersection. We show, that in case of having a sequence of measurements , we can calculate uniquely in every such network. For a sufficiently large but still small cuboid grid, we can reduce the number of needed measurements to 2.
期刊介绍:
Prediction through modelling forms the basis of engineering design. The computational power at the fingertips of the professional engineer is increasing enormously and techniques for computer simulation are changing rapidly. Engineers need models which relate to their design area and which are adaptable to new design concepts. They also need efficient and friendly ways of presenting, viewing and transmitting the data associated with their models.
The International Journal of Numerical Modelling: Electronic Networks, Devices and Fields provides a communication vehicle for numerical modelling methods and data preparation methods associated with electrical and electronic circuits and fields. It concentrates on numerical modelling rather than abstract numerical mathematics.
Contributions on numerical modelling will cover the entire subject of electrical and electronic engineering. They will range from electrical distribution networks to integrated circuits on VLSI design, and from static electric and magnetic fields through microwaves to optical design. They will also include the use of electrical networks as a modelling medium.