Battery_State_Estimation.rar_battery_estimation

<html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta charset="utf-8"> <meta name="generator" content="pdf2htmlEX"> <meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"> <link rel="stylesheet" href="https://static.pudn.com/base/css/base.min.css"> <link rel="stylesheet" href="https://static.pudn.com/base/css/fancy.min.css"> <link rel="stylesheet" href="https://static.pudn.com/prod/directory_preview_static/6250a1e374bc5c01056d2748/raw.css"> <script src="https://static.pudn.com/base/js/compatibility.min.js"></script> <script src="https://static.pudn.com/base/js/pdf2htmlEX.min.js"></script> <script> try{ pdf2htmlEX.defaultViewer = new pdf2htmlEX.Viewer({}); }catch(e){} </script> <title></title> </head> <body> <div id="sidebar" style="display: none"> <div id="outline"> </div> </div> <div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6250a1e374bc5c01056d2748/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">IEEE<span class="_ _0"> </span>TRANSACTIONS<span class="_ _0"> </span>ON<span class="_ _0"> </span>CONTR<span class="_ _1"></span>OL<span class="_ _0"> </span>SYS<span class="_ _1"></span>TEMS<span class="_ _0"> </span>T<span class="_ _2"></span>ECHNOLOGY<span class="_ _3"></span>,<span class="_ _0"> </span>V<span class="_ _1"></span>OL.<span class="_ _0"> </span>25,<span class="_ _0"> </span>NO.<span class="_ _0"> </span>2,<span class="_ _0"> </span>MARCH<span class="_ _0"> </span>2017<span class="_ _4"> </span>453</div><div class="t m0 x2 h3 y2 ff1 fs1 fc0 sc0 ls1 ws0">Battery<span class="_ _5"> </span>State<span class="_ _5"> </span>Estimation<span class="_ _5"> </span>for<span class="_ _6"> </span>a<span class="_ _5"> </span>Single<span class="_ _6"> </span>Particle</div><div class="t m0 x3 h3 y3 ff1 fs1 fc0 sc0 ls2 ws0">Model<span class="_ _6"> </span>W<span class="_ _3"></span>ith<span class="_ _6"> </span>Electrolyte<span class="_ _5"> </span>Dynamics</div><div class="t m0 x4 h4 y4 ff1 fs2 fc0 sc0 ls3 ws0">Scott<span class="_ _7"> </span>J.<span class="_ _8"> </span>Moura,<span class="_ _7"> </span><span class="ff2 ls4">Member<span class="_ _9"></span>,<span class="_ _7"> </span>IEEE<span class="_ _a"></span><span class="ff1 ls5">,<span class="_ _7"> </span>Federico<span class="_ _7"> </span>Bribiesca<span class="_ _7"> </span>Ar<span class="_ _1"></span>gomedo,<span class="_ _7"> </span>Reinhardt<span class="_ _7"> </span>Klein,</span></span></div><div class="t m0 x5 h4 y5 ff1 fs2 fc0 sc0 ls6 ws0">Anahita<span class="_ _8"> </span>Mirta<span class="_ _2"></span>bata<span class="_ _2"></span>bae<span class="_ _2"></span>i,<span class="_ _8"> </span>and<span class="_ _8"> </span>Miros<span class="_ _2"></span>lav<span class="_ _8"> </span>Krstic,<span class="_ _8"> </span><span class="ff2 ls4">F<span class="_ _3"></span>ellow<span class="_ _3"></span>,<span class="_ _7"> </span>IEEE</span></div><div class="t m0 x6 h5 y6 ff3 fs3 fc0 sc0 ls7 ws0">Abstract<span class="_ _2"></span><span class="ff4 ls8">—<span class="_ _b"></span>This<span class="_ _7"> </span>paper<span class="_ _7"> </span>studies<span class="_ _7"> </span>a<span class="_ _c"> </span>state<span class="_ _c"> </span>estimation<span class="_ _7"> </span>scheme<span class="_ _7"> </span>for<span class="_ _c"> </span>a</span></div><div class="t m0 x1 h5 y7 ff4 fs3 fc0 sc0 ls9 ws0">reduced<span class="_ _d"> </span>electrochemical<span class="_ _d"> </span>b<span class="_ _2"></span>attery<span class="_ _d"> </span>model,<span class="_ _d"> </span>u<span class="_ _2"></span>sing<span class="_ _d"> </span>voltage<span class="_ _d"> </span>and<span class="_ _d"> </span>cu<span class="_ _2"></span>r<span class="_ _1"></span>-</div><div class="t m0 x1 h5 y8 ff4 fs3 fc0 sc0 lsa ws0">rent<span class="_ _c"> </span>measu<span class="_ _2"></span>rements.<span class="_ _c"> </span>Real<span class="_ _2"></span>-time<span class="_ _7"> </span>electrochemical<span class="_ _7"> </span>state<span class="_ _7"> </span>information</div><div class="t m0 x1 h5 y9 ff4 fs3 fc0 sc0 lsb ws0">enables<span class="_"> </span>high-ﬁdelity<span class="_"> </span>monitoring<span class="_"> </span>and<span class="_ _0"> </span>high-perf<span class="_ _1"></span>ormanc<span class="_ _1"></span>e<span class="_ _e"> </span>opera<span class="_ _1"></span>tion</div><div class="t m0 x1 h5 ya ff4 fs3 fc0 sc0 lsc ws0">in<span class="_ _c"> </span>adv<span class="_ _1"></span>anced<span class="_ _c"> </span>battery<span class="_ _c"> </span>mana<span class="_ _1"></span>gement<span class="_ _c"> </span>syst<span class="_ _1"></span>ems,<span class="_ _c"> </span>fo<span class="_ _1"></span>r<span class="_ _c"> </span>applications<span class="_ _c"> </span>s<span class="_ _1"></span>uch</div><div class="t m0 x1 h5 yb ff4 fs3 fc0 sc0 lsd ws0">as<span class="_ _f"> </span>consumer<span class="_ _f"> </span>electronics,<span class="_ _f"> </span>el<span class="_ _2"></span>ectriﬁed<span class="_ _f"> </span>vehicles,<span class="_ _f"> </span>and<span class="_ _f"> </span>grid<span class="_ _f"> </span>energy</div><div class="t m0 x1 h5 yc ff4 fs3 fc0 sc0 lse ws0">stora<span class="_ _1"></span>ge.<span class="_ _10"> </span>This<span class="_ _10"> </span>paper<span class="_ _11"> </span>deri<span class="_ _1"></span>ves<span class="_ _11"> </span>a<span class="_ _10"> </span>single<span class="_ _10"> </span>partic<span class="_ _1"></span>le<span class="_ _10"> </span>model<span class="_ _11"> </span>(SPM)</div><div class="t m0 x1 h5 yd ff4 fs3 fc0 sc0 lsf ws0">with<span class="_ _12"> </span>electrolyte<span class="_ _12"> </span>that<span class="_ _12"> </span>achieves<span class="_ _12"> </span>higher<span class="_ _12"> </span>predictive<span class="_ _12"> </span>accuracy<span class="_ _12"> </span>than</div><div class="t m0 x1 h5 ye ff4 fs3 fc0 sc0 ls10 ws0">the<span class="_ _13"> </span>SPM.<span class="_ _13"> </span>Next,<span class="_ _13"> </span>we<span class="_ _13"> </span>p<span class="_ _2"></span>ropose<span class="_ _12"> </span>an<span class="_ _13"> </span>esti<span class="_ _2"></span>mation<span class="_ _13"> </span>scheme<span class="_ _13"> </span>and<span class="_ _13"> </span>p<span class="_ _2"></span>rov<span class="_ _1"></span>e</div><div class="t m0 x1 h5 yf ff4 fs3 fc0 sc0 ls11 ws0">estimatio<span class="_ _1"></span>n<span class="_ _e"> </span>error<span class="_ _e"> </span>system<span class="_ _e"> </span>stability<span class="_ _3"></span>,<span class="_ _c"> </span>a<span class="_ _1"></span>ssuming<span class="_ _e"> </span>that<span class="_ _e"> </span>the<span class="_ _e"> </span>total<span class="_ _e"> </span>amount</div><div class="t m0 x1 h5 y10 ff4 fs3 fc0 sc0 ls12 ws0">of<span class="_ _12"> </span>lithium<span class="_ _12"> </span>in<span class="_ _12"> </span>the<span class="_ _13"> </span>cell<span class="_ _12"> </span>is<span class="_ _12"> </span>known.<span class="_ _12"> </span>The<span class="_ _12"> </span>state<span class="_ _12"> </span>estimatio<span class="_ _1"></span>n<span class="_ _13"> </span>scheme</div><div class="t m0 x1 h5 y11 ff4 fs3 fc0 sc0 ls13 ws0">exploits<span class="_ _14"> </span>the<span class="_"> </span>dynam<span class="_ _1"></span>ical<span class="_"> </span>pro<span class="_ _1"></span>perties,<span class="_"> </span>such<span class="_ _14"> </span>as<span class="_"> </span>mar<span class="_ _1"></span>ginal<span class="_"> </span>stability<span class="_ _3"></span>,<span class="_"> </span>loc<span class="_ _1"></span>al</div><div class="t m0 x1 h5 y12 ff4 fs3 fc0 sc0 ls14 ws0">in<span class="_ _1"></span>ver<span class="_ _1"></span>tibility<span class="_ _3"></span>,<span class="_ _13"> </span>and<span class="_ _12"> </span>conserv<span class="_ _1"></span>ation<span class="_ _12"> </span>of<span class="_ _13"> </span>lithium.<span class="_ _12"> </span>Simulations<span class="_ _12"> </span>demo<span class="_ _1"></span>n-</div><div class="t m0 x1 h5 y13 ff4 fs3 fc0 sc0 ls15 ws0">strate<span class="_ _c"> </span>the<span class="_ _c"> </span>algo<span class="_ _1"></span>rithm’<span class="_ _1"></span>s<span class="_ _c"> </span>perfo<span class="_ _1"></span>rmance<span class="_ _c"> </span>and<span class="_ _c"> </span>limitatio<span class="_ _1"></span>ns.</div><div class="t m0 x6 h5 y14 ff3 fs3 fc0 sc0 ls16 ws0">Index<span class="_ _13"> </span>T<span class="_ _3"></span>erms<span class="ff4 lsa">—<span class="_ _b"></span>Batteries,<span class="_ _12"> </span>b<span class="_ _2"></span>attery<span class="_ _12"> </span>managemen<span class="_ _2"></span>t<span class="_ _13"> </span>systems,<span class="_ _13"> </span>elec-</span></div><div class="t m0 x1 h5 y15 ff4 fs3 fc0 sc0 ls17 ws0">trochemical<span class="_ _5"> </span>devices,<span class="_ _6"> </span>n<span class="_ _2"></span>onlin<span class="_ _2"></span>ear<span class="_ _6"> </span>dyn<span class="_ _2"></span>amical<span class="_ _5"> </span>systems,<span class="_ _5"> </span>partial</div><div class="t m0 x1 h5 y16 ff4 fs3 fc0 sc0 ls18 ws0">differe<span class="_ _1"></span>ntial<span class="_ _c"> </span>equations,<span class="_ _c"> </span>stability<span class="_ _15"> </span>analysis,<span class="_ _15"> </span>state<span class="_ _c"> </span>estimation.</div><div class="t m0 x7 h6 y17 ff1 fs4 fc0 sc0 ls19 ws0">I.<span class="_ _12"> </span>I<span class="fs5 ls1a">NT<span class="_ _2"></span>R<span class="_ _1"></span>ODUCTION</span></div><div class="t m0 x1 h7 y18 ff4 fs6 fc0 sc0 ls1b ws0">T</div><div class="t m0 x8 h6 y19 ff1 fs4 fc0 sc0 ls1c ws0">HIS<span class="_ _d"> </span>P<span class="_ _3"></span>APER<span class="_ _d"> </span>stud<span class="_ _2"></span>ies<span class="_ _d"> </span>a<span class="_ _12"> </span>state<span class="_ _d"> </span>estimation<span class="_ _12"> </span>algorithm<span class="_ _d"> </span>b<span class="_ _2"></span>ased</div><div class="t m0 x8 h6 y1a ff1 fs4 fc0 sc0 ls1d ws0">upon<span class="_ _8"> </span>the<span class="_ _8"> </span>single<span class="_ _d"> </span>particl<span class="_ _1"></span>e<span class="_ _d"> </span>model<span class="_ _8"> </span>with<span class="_ _8"> </span>electrolyte<span class="_ _8"> </span>(SPMe)</div><div class="t m0 x1 h6 y1b ff1 fs4 fc0 sc0 ls1e ws0">dynamics—an<span class="_ _8"> </span>electrochemical<span class="_ _7"> </span>battery<span class="_ _8"> </span>model.<span class="_ _d"> </span>The<span class="_ _8"> </span>algorithm</div><div class="t m0 x1 h6 y1c ff1 fs4 fc0 sc0 ls1f ws0">features<span class="_ _16"> </span>proper<span class="_ _2"></span>ties<span class="_ _10"> </span>such<span class="_ _16"> </span>as<span class="_ _16"> </span>stability<span class="_ _16"> </span>and<span class="_ _16"> </span>con<span class="_ _2"></span>servation<span class="_ _10"> </span>o<span class="_ _2"></span>f</div><div class="t m0 x1 h6 y1d ff1 fs4 fc0 sc0 ls20 ws0">lithium.</div><div class="t m0 x1 h8 y1e ff2 fs4 fc0 sc0 ls21 ws0">A.<span class="_ _c"> </span>Backgr<span class="_ _3"></span>ound<span class="_ _c"> </span>and<span class="_ _c"> </span>Motivati<span class="_ _1"></span>on</div><div class="t m0 x6 h6 y1f ff1 fs4 fc0 sc0 ls22 ws0">Batteries<span class="_"> </span>are<span class="_ _15"> </span>ubiquitous.<span class="_"> </span>They<span class="_"> </span>po<span class="_ _1"></span>wer<span class="_"> </span>a<span class="_ _15"> </span>spectrum<span class="_"> </span>of<span class="_"> </span>d<span class="_ _2"></span>evices<span class="_ _1"></span>,</div><div class="t m0 x1 h6 y20 ff1 fs4 fc0 sc0 ls1e ws0">including<span class="_"> </span>consumer<span class="_"> </span>electronics,<span class="_"> </span>e<span class="ls23">lectriﬁed<span class="_ _15"> </span>vehicles,<span class="_"> </span>and<span class="_ _15"> </span>smart</span></div><div class="t m0 x1 h6 y21 ff1 fs4 fc0 sc0 ls24 ws0">grid<span class="_ _8"> </span>systems.<span class="_ _8"> </span>Control<span class="_ _8"> </span>system<span class="_ _d"> </span>technologies<span class="_ _7"> </span>that<span class="_ _d"> </span>enhance<span class="_ _8"> </span>bat-</div><div class="t m0 x1 h6 y22 ff1 fs4 fc0 sc0 ls25 ws0">tery<span class="_ _d"> </span>performance<span class="_ _7"> </span>are<span class="_ _8"> </span>of<span class="_ _d"> </span>extreme<span class="_ _8"> </span>interest.<span class="_ _8"> </span>In<span class="_ _d"> </span>particular,<span class="_ _7"> </span>elec-</div><div class="t m0 x1 h6 y23 ff1 fs4 fc0 sc0 ls26 ws0">trochem<span class="_ _2"></span>ical<span class="_ _f"> </span>model-<span class="_ _2"></span>based<span class="_ _f"> </span>contro<span class="_ _2"></span>l<span class="_ _f"> </span>systems<span class="_ _f"> </span>p<span class="_ _2"></span>rovide<span class="_ _f"> </span>v<span class="_ _2"></span>isibility</div><div class="t m0 x1 h6 y24 ff1 fs4 fc0 sc0 ls27 ws0">into<span class="_ _17"> </span>operating<span class="_ _f"> </span>regimes<span class="_ _f"> </span>that<span class="_ _17"> </span>induce<span class="_ _17"> </span>degradation.<span class="_ _13"> </span>This<span class="_ _11"> </span>visi<span class="_ _1"></span>-</div><div class="t m0 x1 h6 y25 ff1 fs4 fc0 sc0 ls1f ws0">bility<span class="_ _13"> </span>enables<span class="_ _13"> </span>a<span class="_ _13"> </span>larger<span class="_ _12"> </span>o<span class="_ _2"></span>perational<span class="_ _12"> </span>e<span class="_ _2"></span>n<span class="_ _1"></span>velope<span class="_ _12"> </span>to<span class="_ _13"> </span>incr<span class="_ _2"></span>ease<span class="_ _13"> </span>the</div><div class="t m0 x9 h9 y26 ff1 fs5 fc0 sc0 ls28 ws0">Manuscript<span class="_ _d"> </span>received<span class="_ _d"> </span>July<span class="_ _d"> </span>28,<span class="_ _d"> </span>2015;<span class="_ _d"> </span>re<span class="_ _1"></span>vised<span class="_ _d"> </span>February<span class="_ _d"> </span>7,<span class="_ _d"> </span>2016;<span class="_ _d"> </span>accepted</div><div class="t m0 x1 h9 y27 ff1 fs5 fc0 sc0 ls29 ws0">May<span class="_ _8"> </span>7,<span class="_ _8"> </span>2016.<span class="_ _d"> </span>Date<span class="_ _8"> </span>of<span class="_ _d"> </span>publication<span class="_ _d"> </span>June<span class="_ _8"> </span>10,<span class="_ _d"> </span>2016;<span class="_ _8"> </span>date<span class="_ _d"> </span>of<span class="_ _8"> </span>current<span class="_ _d"> </span>version</div><div class="t m0 x1 h9 y28 ff1 fs5 fc0 sc0 ls29 ws0">February<span class="_ _0"> </span>8,<span class="_ _0"> </span>2017.<span class="_ _0"> </span>Manuscript<span class="_ _0"> </span>recei<span class="_ _1"></span>ved<span class="_ _0"> </span>in<span class="_ _0"> </span>ﬁnal<span class="_ _0"> </span>form<span class="_ _0"> </span>May<span class="_ _0"> </span>17,<span class="_"> </span>2016.<span class="_ _0"> </span>T<span class="_ _2"></span>his<span class="_ _0"> </span>work</div><div class="t m0 x1 h9 y29 ff1 fs5 fc0 sc0 ls2a ws0">was<span class="_ _0"> </span>supported<span class="_ _0"> </span>in<span class="_ _0"> </span>part<span class="_ _0"> </span>by<span class="_ _0"> </span>the<span class="_ _0"> </span>National<span class="_ _e"> </span>Science<span class="_ _e"> </span>Foundation<span class="_ _0"> </span>within<span class="_ _e"> </span>the<span class="_ _0"> </span>Division</div><div class="t m0 x1 h9 y2a ff1 fs5 fc0 sc0 ls2b ws0">of<span class="_ _e"> </span>Electrical,<span class="_ _c"> </span>Communications<span class="_ _c"> </span>an<span class="ls29">d<span class="_ _e"> </span>Cyber<span class="_ _15"> </span>Systems<span class="_ _15"> </span>under<span class="_ _e"> </span>Grant<span class="_ _15"> </span>1408107<span class="_ _e"> </span>and</span></div><div class="t m0 x1 h9 y2b ff1 fs5 fc0 sc0 ls2c ws0">in<span class="_ _c"> </span>part<span class="_ _8"> </span>by<span class="_ _c"> </span>the<span class="_ _7"> </span>U.<span class="_ _2"></span>S.<span class="_ _c"> </span>D<span class="_ _2"></span>epartment<span class="_ _8"> </span>of<span class="_ _c"> </span>E<span class="_ _2"></span>nergy<span class="_ _c"> </span>through<span class="_ _7"> </span>the<span class="_ _8"> </span>Advanced<span class="_ _7"> </span>Research</div><div class="t m0 x1 h9 y2c ff1 fs5 fc0 sc0 ls2d ws0">Projects<span class="_ _13"> </span>Agency-Energy<span class="_ _13"> </span>under<span class="_ _f"> </span>G<span class="ls1">rant<span class="_ _12"> </span>DE<span class="_ _2"></span>-AR0000278.<span class="_ _13"> </span>Recommended<span class="_ _f"> </span>by</span></div><div class="t m0 x1 h9 y2d ff1 fs5 fc0 sc0 ls2c ws0">Associate<span class="_ _15"> </span>Editor<span class="_ _15"> </span>S<span class="_ _2"></span>.<span class="_ _e"> </span>V<span class="_ _3"></span>arigonda.</div><div class="t m0 x9 h9 y2e ff1 fs5 fc0 sc0 ls2e ws0">S.<span class="_"> </span>J<span class="_ _2"></span>.<span class="_"> </span>Moura<span class="_ _0"> </span>is<span class="_"> </span>with<span class="_ _0"> </span>the<span class="_ _0"> </span>Department<span class="_ _0"> </span>of<span class="_ _0"> </span>Ci<span class="_ _1"></span>vil<span class="_ _0"> </span>and<span class="_"> </span>Environmental<span class="_ _0"> </span>Engineering,</div><div class="t m0 x1 h9 y2f ff1 fs5 fc0 sc0 ls2f ws0">Uni<span class="_ _1"></span>versity<span class="_ _12"> </span>of<span class="_ _12"> </span>California<span class="_ _13"> </span>at<span class="_ _12"> </span>Berkeley<span class="_ _3"></span>,<span class="_ _12"> </span>Berkeley<span class="_ _3"></span>,<span class="_ _f"> </span>CA<span class="_ _12"> </span>94720<span class="_ _d"> </span>USA<span class="_ _12"> </span>(e-mail:</div><div class="t m0 x1 h9 y30 ff1 fs5 fc0 sc0 ls30 ws0">smoura@berkeley<span class="_ _3"></span>.edu).</div><div class="t m0 x9 h9 y31 ff1 fs5 fc0 sc0 ls31 ws0">F<span class="_ _3"></span>.<span class="_ _c"> </span>Bribies<span class="_ _2"></span>ca<span class="_ _7"> </span>Argomedo<span class="_ _7"> </span>is<span class="_ _7"> </span>with<span class="_ _7"> </span>the<span class="_ _7"> </span>Laboratoir<span class="_ _2"></span>e<span class="_ _7"> </span>Am<span class="_ _2"></span>père,<span class="_ _7"> </span>Institut<span class="_ _7"> </span>N<span class="_ _2"></span>ational</div><div class="t m0 x1 h9 y32 ff1 fs5 fc0 sc0 ls32 ws0">des<span class="_ _c"> </span>Sciences<span class="_ _7"> </span>Appliquées<span class="_ _7"> </span>de<span class="_ _c"> </span>L<span class="_ _1"></span>yon,<span class="_ _15"> </span>U<span class="_ _2"></span>niv<span class="_ _1"></span>ersité<span class="_ _7"> </span>de<span class="_ _c"> </span>L<span class="_ _1"></span>yon,<span class="_ _c"> </span>V<span class="_ _1"></span>illeurbanne<span class="_ _7"> </span>69621,</div><div class="t m0 x1 h9 y33 ff1 fs5 fc0 sc0 ls33 ws0">France<span class="_ _e"> </span>(e-mail:<span class="_ _15"> </span>federico.bribiesc<span class="_ _1"></span>a@insa-lyon.fr).</div><div class="t m0 x9 h9 y34 ff1 fs5 fc0 sc0 ls34 ws0">R.<span class="_ _f"> </span>Klein<span class="_ _17"> </span>and<span class="_ _f"> </span>A.<span class="_ _f"> </span>Mirtabatabaei<span class="_ _11"> </span>are<span class="_ _17"> </span>with<span class="_ _f"> </span>the<span class="_ _17"> </span>Research<span class="_ _11"> </span>and<span class="_ _f"> </span>T<span class="_ _3"></span>echnol-</div><div class="t m0 x1 h9 y35 ff1 fs5 fc0 sc0 ls28 ws0">ogy<span class="_ _17"> </span>Center<span class="_ _1"></span>,<span class="_ _11"> </span>Robert<span class="_ _17"> </span>Bosch<span class="_ _17"> </span>LL<span class="_ _2"></span>C,<span class="_ _f"> </span>Palo<span class="_ _17"> </span>A<span class="_ _2"></span>lto,<span class="_ _17"> </span>CA<span class="_ _17"> </span>94304<span class="_ _17"> </span>USA<span class="_ _17"> </span>(e-m<span class="_ _2"></span>ail:</div><div class="t m0 x1 h9 y36 ff1 fs5 fc0 sc0 ls28 ws0">reinhardt.klein@us.bos<span class="_ _2"></span>ch.com;<span class="_ _7"> </span>ana<span class="ls35">hita.mirtabatabaei<span class="_ _2"></span>@us.bos<span class="_ _2"></span>ch.com<span class="_ _2"></span>).</span></div><div class="t m0 x9 h9 y37 ff1 fs5 fc0 sc0 ls36 ws0">M.<span class="_"> </span>Krstic<span class="_ _0"> </span>is<span class="_"> </span>with<span class="_"> </span>the<span class="_ _0"> </span>Department<span class="_"> </span>of<span class="_"> </span>M<span class="_ _2"></span>ech<span class="ls35">anical<span class="_ _0"> </span>and<span class="_"> </span>Aer<span class="_ _2"></span>ospace<span class="_ _0"> </span>Engineering,</span></div><div class="t m0 x1 h9 y38 ff1 fs5 fc0 sc0 ls37 ws0">Uni<span class="_ _1"></span>versity<span class="_ _0"> </span>of<span class="_ _0"> </span>California<span class="_ _0"> </span>at<span class="_ _0"> </span>San<span class="_"> </span>Diego,<span class="_ _0"> </span>La<span class="_"> </span>J<span class="_ _2"></span>olla,<span class="_ _0"> </span>CA<span class="_"> </span>92093-0411<span class="_ _0"> </span>USA<span class="_ _0"> </span>(e-mail:</div><div class="t m0 x1 h9 y39 ff1 fs5 fc0 sc0 ls2f ws0">krstic@ucsd.edu).</div><div class="t m0 x9 h9 y3a ff1 fs5 fc0 sc0 ls2f ws0">Color<span class="_ _7"> </span>versions<span class="_ _8"> </span>of<span class="_ _8"> </span>one<span class="_ _7"> </span>or<span class="_ _8"> </span>more<span class="_ _7"> </span>of<span class="_ _8"> </span>the<span class="_ _8"> </span>ﬁgures<span class="_ _7"> </span>in<span class="_ _8"> </span>this<span class="_ _8"> </span>paper<span class="_ _8"> </span>are<span class="_ _7"> </span>available</div><div class="t m0 x1 h9 y3b ff1 fs5 fc0 sc0 ls35 ws0">online<span class="_ _15"> </span>at<span class="_ _e"> </span>http://ieeexplore.<span class="_ _2"></span>ieee.org.</div><div class="t m0 x9 h9 y3c ff1 fs5 fc0 sc0 ls38 ws0">Digital<span class="_ _15"> </span>Object<span class="_ _15"> </span>Identiﬁer<span class="_ _15"> </span>10.<span class="_ _2"></span>1109/TCST<span class="_ _1"></span>.2016.2571663</div><div class="t m0 xa h6 y6 ff1 fs4 fc0 sc0 ls39 ws0">performance<span class="_ _15"> </span>with<span class="_ _7"> </span>respect<span class="_ _7"> </span>to<span class="_ _7"> </span>en<span class="_ _2"></span><span class="ls3a">ergy<span class="_ _c"> </span>capacity<span class="_ _3"></span>,<span class="_ _7"> </span>power<span class="_ _c"> </span>capacity<span class="_ _1"></span>,</span></div><div class="t m0 xa h6 y3d ff1 fs4 fc0 sc0 ls25 ws0">and<span class="_ _8"> </span>fast<span class="_ _d"> </span>charge<span class="_ _8"> </span>rates<span class="_ _8"> </span>[1].<span class="_ _d"> </span>Electrochemical<span class="_ _8"> </span>model-based<span class="_ _8"> </span>state</div><div class="t m0 xa h6 y3e ff1 fs4 fc0 sc0 ls1f ws0">estimation<span class="_ _17"> </span>is<span class="_ _11"> </span>p<span class="_ _2"></span>articularly<span class="_ _17"> </span>cha<span class="ls25">llenging<span class="_ _17"> </span>for<span class="_ _11"> </span>se<span class="_ _1"></span>veral<span class="_ _17"> </span>technical</span></div><div class="t m0 xa h6 y3f ff1 fs4 fc0 sc0 ls3b ws0">reasons.<span class="_"> </span>First,<span class="_ _e"> </span>the<span class="_ _15"> </span>dynamics<span class="_"> </span>are<span class="_ _15"> </span>governed<span class="_"> </span>by<span class="_"> </span>a<span class="_ _15"> </span>system<span class="_ _15"> </span>of<span class="_ _e"> </span>non-</div><div class="t m0 xa h6 y40 ff1 fs4 fc0 sc0 ls27 ws0">linear<span class="_ _c"> </span>partial<span class="_ _15"> </span>different<span class="_ _1"></span>ial<span class="_ _15"> </span>algebraic<span class="_ _c"> </span>equations<span class="_ _15"> </span>[2],<span class="_ _c"> </span>[3].<span class="_ _c"> </span>Second,</div><div class="t m0 xa h6 y41 ff1 fs4 fc0 sc0 ls1c ws0">local<span class="_ _15"> </span>state<span class="_ _7"> </span>observability<span class="_ _15"> </span>does<span class="_ _c"> </span>not<span class="_ _c"> </span>hold<span class="_ _c"> </span>globally<span class="_ _c"> </span>[4].<span class="_ _c"> </span>Third,<span class="_ _15"> </span>th<span class="_ _2"></span>e</div><div class="t m0 xa h6 y42 ff1 fs4 fc0 sc0 ls22 ws0">model<span class="_ _15"> </span>parameters<span class="_ _15"> </span>vary<span class="_ _15"> </span>widely<span class="_ _15"> </span>w<span class="ls3a">ith<span class="_ _c"> </span>electrode<span class="_ _c"> </span>chemistry<span class="_ _3"></span>,<span class="_ _c"> </span>elec-</span></div><div class="t m0 xa h6 y43 ff1 fs4 fc0 sc0 ls3c ws0">trolyte,<span class="_ _e"> </span>packag<span class="_ _2"></span>ing,<span class="_"> </span>an<span class="_ _2"></span>d<span class="_ _e"> </span>time.<span class="_ _15"> </span>Finally<span class="_ _1"></span>,<span class="_"> </span>the<span class="_ _15"> </span>cells<span class="_ _15"> </span>in<span class="_"> </span>b<span class="_ _2"></span>attery<span class="_ _e"> </span>p<span class="_ _2"></span>acks</div><div class="t m0 xa h6 y44 ff1 fs4 fc0 sc0 ls1d ws0">are<span class="_ _7"> </span>generally<span class="_ _c"> </span>heterogeneous<span class="_ _c"> </span>w<span class="ls3d">ith<span class="_ _8"> </span>respect<span class="_ _7"> </span>to<span class="_ _8"> </span>their<span class="_ _7"> </span>parameters,</span></div><div class="t m0 xa h6 y45 ff1 fs4 fc0 sc0 ls3e ws0">temperature,<span class="_ _12"> </span>and<span class="_ _12"> </span>state<span class="_ _13"> </span>of<span class="_ _13"> </span>char<span class="ls1c">ge<span class="_ _12"> </span>(SOC).<span class="_ _13"> </span>Th<span class="_ _2"></span>is<span class="_ _12"> </span>m<span class="_ _2"></span>otiv<span class="_ _1"></span>ates<span class="_ _12"> </span>an</span></div><div class="t m0 xa h6 y46 ff1 fs4 fc0 sc0 ls27 ws0">intimate<span class="_ _d"> </span>understanding<span class="_ _d"> </span>of<span class="_ _12"> </span>the<span class="_ _12"> </span>mathemat<span class="_ _1"></span>ical<span class="_ _d"> </span>model<span class="_ _12"> </span>structure</div><div class="t m0 xa h6 y47 ff1 fs4 fc0 sc0 ls3f ws0">for<span class="_ _15"> </span>an<span class="_ _c"> </span>observer<span class="_ _e"> </span>design.<span class="_ _c"> </span>In<span class="_ _15"> </span>this<span class="_ _c"> </span>paper<span class="_ _1"></span>,<span class="_ _15"> </span>we<span class="_ _c"> </span>address<span class="_ _15"> </span>the<span class="_ _15"> </span>ﬁrst<span class="_ _c"> </span>two</div><div class="t m0 xa h6 y48 ff1 fs4 fc0 sc0 ls40 ws0">technical<span class="_ _c"> </span>challenges.</div><div class="t m0 xa h8 y49 ff2 fs4 fc0 sc0 ls41 ws0">B.<span class="_ _c"> </span>Relevant<span class="_ _7"> </span>Literatur<span class="_ _1"></span>e</div><div class="t m0 xb h6 y4a ff1 fs4 fc0 sc0 ls42 ws0">Over<span class="_ _7"> </span>the<span class="_ _8"> </span>past<span class="_ _7"> </span>decad<span class="_ _2"></span>e,<span class="_ _7"> </span>the<span class="_ _7"> </span>en<span class="_ _2"></span>gineering<span class="_ _7"> </span>literature<span class="_ _7"> </span>o<span class="_ _2"></span>n<span class="_ _7"> </span>battery</div><div class="t m0 xa h6 y4b ff1 fs4 fc0 sc0 ls43 ws0">estimation<span class="_ _f"> </span>has<span class="_ _f"> </span>grown<span class="_ _13"> </span>c<span class="_ _2"></span>onsiderably<span class="_ _f"> </span>rich<span class="_ _f"> </span>with<span class="_ _f"> </span>various<span class="_ _13"> </span>alg<span class="_ _2"></span>o-</div><div class="t m0 xa h6 y4c ff1 fs4 fc0 sc0 ls3b ws0">rithms,<span class="_ _13"> </span>models,<span class="_ _13"> </span>and<span class="_ _f"> </span>applications.<span class="_ _13"> </span>One<span class="_ _f"> </span>may<span class="_ _f"> </span>categorize<span class="_ _13"> </span>this</div><div class="t m0 xa h6 y4d ff1 fs4 fc0 sc0 ls44 ws0">literature<span class="_ _c"> </span>by<span class="_ _7"> </span>the<span class="_ _7"> </span>battery<span class="_ _7"> </span>models<span class="_ _7"> </span>that<span class="_ _c"> </span>each<span class="_ _7"> </span>alg<span class="_ _2"></span>orithm<span class="_ _7"> </span>employs.</div><div class="t m0 xa h6 y4e ff1 fs4 fc0 sc0 ls45 ws0">Note<span class="_"> </span>that<span class="_ _15"> </span>these<span class="_"> </span>studies<span class="_"> </span>inherently<span class="_"> </span>assume<span class="_"> </span>the<span class="_ _15"> </span>pack<span class="_"> </span>or<span class="_ _15"> </span>modules</div><div class="t m0 xa h6 y4f ff1 fs4 fc0 sc0 ls46 ws0">can<span class="_"> </span>be<span class="_ _e"> </span>conceptualized<span class="_"> </span>as<span class="_"> </span>an<span class="_ _e"> </span>aggregate<span class="_"> </span>single<span class="_"> </span>cell.<span class="_"> </span>This<span class="_ _e"> </span>is<span class="_ _e"> </span>valid</div><div class="t m0 xa h6 y50 ff1 fs4 fc0 sc0 ls1e ws0">when<span class="_ _12"> </span>cell<span class="_ _d"> </span>balancing,<span class="_ _12"> </span>binni<span class="_ _1"></span>ng,<span class="_ _d"> </span>and<span class="_ _12"> </span>temperature<span class="_ _12"> </span>management</div><div class="t m0 xa h6 y51 ff1 fs4 fc0 sc0 ls3f ws0">maintain<span class="_ _15"> </span>homogeneity<span class="_ _15"> </span>across<span class="_ _c"> </span>the<span class="_ _c"> </span>pack<span class="_ _7"> </span>or<span class="_ _c"> </span>module.</div><div class="t m0 xb h6 y52 ff1 fs4 fc0 sc0 ls43 ws0">The<span class="_"> </span>ﬁrst<span class="_"> </span>category<span class="_"> </span>utilizes<span class="_ _14"> </span>eq<span class="_ _2"></span>uiv<span class="_ _1"></span>alent<span class="_"> </span>circuit<span class="_"> </span>models<span class="_"> </span>(ECMs).</div><div class="t m0 xa h6 y53 ff1 fs4 fc0 sc0 ls47 ws0">These<span class="_ _15"> </span>models<span class="_ _c"> </span>use<span class="_ _7"> </span>circuit<span class="_ _15"> </span>elements<span class="_ _c"> </span>to<span class="_ _7"> </span>mimic<span class="_ _c"> </span>the<span class="_ _c"> </span>input–output</div><div class="t m0 xa h6 y54 ff1 fs4 fc0 sc0 ls24 ws0">beha<span class="_ _1"></span>vior<span class="_ _c"> </span>of<span class="_ _7"> </span>batteries<span class="_ _c"> </span>[5].<span class="_ _7"> </span>The<span class="_ _7"> </span>seminal<span class="_ _c"> </span>paper<span class="_ _7"> </span>by<span class="_ _7"> </span>Plett<span class="_ _7"> </span>[6]<span class="_ _c"> </span>was</div><div class="t m0 xa h6 y55 ff1 fs4 fc0 sc0 ls48 ws0">one<span class="_ _7"> </span>of<span class="_ _8"> </span>the<span class="_ _8"> </span>ﬁrst<span class="_ _7"> </span>to<span class="_ _8"> </span>apply<span class="_ _7"> </span>extended<span class="_ _7"> </span>Kalman<span class="_ _7"> </span>ﬁltering<span class="_ _7"> </span>(EKF)<span class="_ _7"> </span>to</div><div class="t m0 xa h6 y56 ff1 fs4 fc0 sc0 ls1d ws0">ECMs<span class="_ _c"> </span>for<span class="_ _7"> </span>simultaneous<span class="_ _15"> </span>state<span class="_ _7"> </span>and<span class="_ _c"> </span>parameter<span class="_ _c"> </span>estimation.<span class="_ _c"> </span>Over</div><div class="t m0 xa h6 y57 ff1 fs4 fc0 sc0 ls49 ws0">the<span class="_"> </span>past<span class="_ _e"> </span>decade,<span class="_ _e"> </span>a<span class="_ _e"> </span>variety<span class="_"> </span>of<span class="_ _e"> </span>ECM-based<span class="_"> </span>algorithms<span class="_"> </span>hav<span class="_ _1"></span>e<span class="_"> </span>been</div><div class="t m0 xa h6 y58 ff1 fs4 fc0 sc0 ls3 ws0">de<span class="_ _1"></span>veloped,<span class="_ _7"> </span>including<span class="_ _8"> </span>linear<span class="_ _d"> </span>parameter<span class="_ _8"> </span>varying<span class="_ _8"> </span>observers<span class="_ _7"> </span>[7],</div><div class="t m0 xa h6 y59 ff1 fs4 fc0 sc0 ls4a ws0">sliding-mode<span class="_"> </span>observers<span class="_ _15"> </span>[8],<span class="_ _15"> </span>polynomial<span class="_ _15"> </span>chaos<span class="_ _15"> </span>[9],<span class="_ _c"> </span>neural<span class="_ _c"> </span>net-</div><div class="t m0 xa h6 y5a ff1 fs4 fc0 sc0 ls3b ws0">works<span class="_ _13"> </span>[10],<span class="_ _f"> </span>unscented<span class="_ _13"> </span>Kalman<span class="_ _f"> </span>ﬁlters<span class="_ _f"> </span>(KFs<span class="_ _1"></span>)<span class="_ _f"> </span>[11],<span class="_ _f"> </span>adapti<span class="_ _1"></span>ve</div><div class="t m0 xa h6 y5b ff1 fs4 fc0 sc0 ls4b ws0">KFs<span class="_ _c"> </span>[12],<span class="_ _15"> </span>and<span class="_ _7"> </span>particle<span class="_ _15"> </span>ﬁlters<span class="_ _c"> </span>[13].</div><div class="t m0 xb h6 y5c ff1 fs4 fc0 sc0 ls4c ws0">The<span class="_ _17"> </span>second<span class="_ _f"> </span>categor<span class="_ _2"></span>y<span class="_ _f"> </span>of<span class="_ _17"> </span>the<span class="_ _17"> </span>literatu<span class="_ _2"></span>re<span class="_ _f"> </span>con<span class="_ _2"></span>siders<span class="_ _f"> </span>electro-</div><div class="t m0 xa h6 y5d ff1 fs4 fc0 sc0 ls1d ws0">chemical<span class="_ _8"> </span>models,<span class="_ _8"> </span>which<span class="_ _8"> </span>account<span class="_ _8"> </span>for<span class="_ _d"> </span>the<span class="_ _8"> </span>diffusi<span class="_ _1"></span>on,<span class="_ _7"> </span>intercala-</div><div class="t m0 xa h6 y5e ff1 fs4 fc0 sc0 ls45 ws0">tion,<span class="_"> </span>and<span class="_ _c"> </span>electrochemical<span class="_"> </span>kinetics.<span class="_"> </span>Although<span class="_ _e"> </span>these<span class="_ _15"> </span>models<span class="_ _e"> </span>can</div><div class="t m0 xa h6 y5f ff1 fs4 fc0 sc0 ls4d ws0">accurately<span class="_ _7"> </span>predict<span class="_ _8"> </span>internal<span class="_ _7"> </span>state<span class="_ _d"> </span><span class="ls3a">variables<span class="_ _1"></span>,<span class="_ _8"> </span>their<span class="_ _8"> </span>mathematical</span></div><div class="t m0 xa h6 y60 ff1 fs4 fc0 sc0 ls3b ws0">structure<span class="_ _7"> </span>renders<span class="_ _d"> </span>observer<span class="_ _7"> </span>design<span class="_ _d"> </span>challenging.<span class="_ _7"> </span>Consequently<span class="_ _3"></span>,</div><div class="t m0 xa h6 y61 ff1 fs4 fc0 sc0 ls4e ws0">most<span class="_ _17"> </span>approaches<span class="_ _f"> </span>develop<span class="_ _f"> </span>estim<span class="ls4f">ators<span class="_ _11"> </span>for<span class="_ _17"> </span>the<span class="_ _17"> </span>reduced-order</span></div><div class="t m0 xa h6 y62 ff1 fs4 fc0 sc0 ls24 ws0">models.<span class="_ _e"> </span>The<span class="_ _15"> </span>model<span class="_ _15"> </span>reduction<span class="_ _e"> </span>and<span class="_ _15"> </span>the<span class="_ _15"> </span>observer<span class="_ _e"> </span>design<span class="_ _15"> </span>process</div><div class="t m0 xa h6 y63 ff1 fs4 fc0 sc0 ls1c ws0">are<span class="_ _7"> </span>intimately<span class="_ _7"> </span>intertwined,<span class="_ _7"> </span>as<span class="_ _7"> </span>simpler<span class="_ _7"> </span>mod<span class="_ _2"></span>els<span class="_ _c"> </span>ease<span class="_ _7"> </span>estim<span class="_ _2"></span>ation</div><div class="t m0 xa h6 y64 ff1 fs4 fc0 sc0 ls1c ws0">design<span class="_ _8"> </span>at<span class="_ _7"> </span>the<span class="_ _8"> </span>expense<span class="_ _7"> </span>o<span class="_ _2"></span>f<span class="_ _7"> </span>ﬁdelity<span class="_ _1"></span>.<span class="_ _7"> </span>Ideally<span class="_ _1"></span>,<span class="_ _7"> </span>one<span class="_ _8"> </span>seeks<span class="_ _8"> </span>to<span class="_ _7"> </span>d<span class="_ _2"></span>eriv<span class="_ _1"></span>e</div><div class="t m0 xa h6 y65 ff1 fs4 fc0 sc0 ls50 ws0">a<span class="_ _13"> </span>provably<span class="_ _12"> </span>stable<span class="_ _13"> </span>estimato<span class="_ _2"></span>r<span class="_ _13"> </span>for<span class="_ _13"> </span>the<span class="_ _f"> </span>highest<span class="_ _12"> </span>ﬁd<span class="_ _2"></span>elity<span class="_ _13"> </span>electro-</div><div class="t m0 xa h6 y66 ff1 fs4 fc0 sc0 ls1d ws0">chemical<span class="_ _12"> </span>battery<span class="_ _12"> </span>model<span class="_ _12"> </span>possible.<span class="_ _12"> </span>The<span class="_ _12"> </span>ﬁrst<span class="_ _13"> </span>wa<span class="_ _1"></span>ve<span class="_ _12"> </span>of<span class="_ _13"> </span>studies</div><div class="t m0 xa h6 y67 ff1 fs4 fc0 sc0 ls51 ws0">utilize<span class="_ _d"> </span>th<span class="_ _2"></span>e<span class="_ _d"> </span>SPM<span class="_ _12"> </span>for<span class="_ _12"> </span>an<span class="_ _d"> </span>estimator<span class="_ _d"> </span>d<span class="_ _2"></span>esign<span class="_ _12"> </span>[4],<span class="_ _d"> </span>[14<span class="_ _2"></span>]–[1<span class="_ _2"></span>7].<span class="_ _d"> </span>The</div><div class="t m0 xa h6 y68 ff1 fs4 fc0 sc0 ls52 ws0">SPM<span class="_ _f"> </span>idealizes<span class="_ _13"> </span>each<span class="_ _f"> </span>electrode<span class="_ _13"> </span><span class="ls53">as<span class="_ _f"> </span>a<span class="_ _f"> </span>single<span class="_ _13"> </span>spherical<span class="_ _13"> </span>porous</span></div><div class="t m0 xa h6 y69 ff1 fs4 fc0 sc0 ls24 ws0">particle<span class="_ _e"> </span>by<span class="_ _c"> </span>neglect<span class="_ _1"></span>ing<span class="_ _15"> </span>the<span class="_ _15"> </span>electrolyte<span class="_ _e"> </span>dynamics.<span class="_ _15"> </span>This<span class="_ _15"> </span>model<span class="_ _15"> </span>is</div><div class="t m0 xa h6 y6a ff1 fs4 fc0 sc0 ls54 ws0">suitable<span class="_"> </span>for<span class="_"> </span>low<span class="_"> </span><span class="ff2 ls1b">C<span class="_ _18"></span></span><span class="ls55">-rates,<span class="_"> </span>but<span class="_"> </span>loses<span class="_"> </span>ﬁdelity<span class="_"> </span>at<span class="_"> </span><span class="ff2 ls1b">C<span class="_ _18"></span></span><span class="ls27">-rates<span class="_"> </span>abov<span class="_ _1"></span>e<span class="_"> </span><span class="ff2 ls1b">C<span class="_ _a"></span></span><span class="ls20">/2</span></span></span></div><div class="t m0 xc h9 y6b ff1 fs5 fc0 sc0 ls56 ws0">1063-6536<span class="_ _15"> </span>©<span class="_ _e"> </span>2016<span class="_ _15"> </span>I<span class="ls57">EEE.<span class="_ _0"> </span>Personal<span class="_ _15"> </span>u<span class="ls58">se<span class="_ _e"> </span>is<span class="_ _15"> </span>perm<span class="ls2b">itted,<span class="_ _15"> </span>but<span class="_ _e"> </span>republication/redis<span class="_ _2"></span>tri<span class="ls38">bution<span class="_ _7"> </span>requires<span class="_ _15"> </span>IE<span class="_ _2"></span>EE<span class="_ _e"> </span>permis<span class="_ _2"></span>sion.</span></span></span></span></div><div class="t m0 xd h9 y6c ff1 fs5 fc0 sc0 ls28 ws0">See<span class="_ _e"> </span>http://w<span class="_ _2"></span>ww<span class="_ _1"></span>.ieee.org/publications_s<span class="_ _2"></span>tandards<span class="_ _2"></span>/publica<span class="_ _2"></span>tions/<span class="_ _2"></span>rights/i<span class="_ _2"></span>ndex.html<span class="_ _7"> </span>for<span class="_ _15"> </span>more<span class="_ _15"> </span>information.</div><div class="t m0 xe ha y6d ff5 fs7 fc0 sc0 ls1b ws0">Authorized licensed use limited to: Harbin 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<div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6250a1e374bc5c01056d2748/bg2.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls59 ws0">454<span class="_ _4"> </span>IEEE<span class="_ _0"> </span>TRANSACTIONS<span class="_ _14"> </span>ON<span class="_ _0"> </span>CONT<span class="_ _2"></span>ROL<span class="_ _14"> </span>SYSTEMS<span class="_ _0"> </span>TECHNOL<span class="_ _2"></span>OGY<span class="_ _3"></span>,<span class="_ _14"> </span>VOL.<span class="_ _0"> </span>25,<span class="_ _0"> </span>NO.<span class="_ _0"> </span>2,<span class="_ _0"> </span>MARCH<span class="_ _0"> </span>2017</div><div class="t m0 x1 h6 y6e ff1 fs4 fc0 sc0 ls5a ws0">(see<span class="_ _11"> </span>Section<span class="_ _10"> </span>II-C).<span class="_ _11"> </span>Recently<span class="_ _3"></span>,<span class="_ _11"> </span>there<span class="_ _10"> </span>has<span class="_ _11"> </span>been<span class="_ _10"> </span>progress<span class="_ _11"> </span>on</div><div class="t m0 x1 h6 y6f ff1 fs4 fc0 sc0 ls3 ws0">extendi<span class="_ _1"></span>ng<span class="_ _e"> </span>the<span class="_ _c"> </span>SPM<span class="_ _15"> </span>to<span class="_ _15"> </span>include<span class="_ _e"> </span>electrolyte<span class="_ _e"> </span>dynamics<span class="_ _e"> </span>[18]–[22].</div><div class="t m0 x1 h6 y70 ff1 fs4 fc0 sc0 ls4f ws0">In<span class="_ _8"> </span>sev<span class="_ _1"></span>eral<span class="_ _8"> </span>cases,<span class="_ _8"> </span>state<span class="_ _8"> </span>observers<span class="_ _7"> </span>have<span class="_ _7"> </span>been<span class="_ _d"> </span>designed<span class="_ _7"> </span>via<span class="_ _8"> </span>lin-</div><div class="t m0 x1 h6 y71 ff1 fs4 fc0 sc0 ls24 ws0">earization<span class="_ _15"> </span>and<span class="_ _c"> </span>Luenberger<span class="_ _e"> </span>observers<span class="_ _15"> </span>[23]<span class="_ _15"> </span>or<span class="_ _c"> </span>EKFs<span class="_ _c"> </span>[24].<span class="_ _15"> </span>State</div><div class="t m0 x1 h6 y72 ff1 fs4 fc0 sc0 ls40 ws0">estimation<span class="_ _7"> </span>designs<span class="_ _8"> </span>ha<span class="_ _1"></span>ve<span class="_ _7"> </span>also<span class="_ _8"> </span>em<span class="ls5b">erged<span class="_ _7"> </span>for<span class="_ _7"> </span>other<span class="_ _8"> </span>electrochem-</span></div><div class="t m0 x1 h6 y73 ff1 fs4 fc0 sc0 ls4f ws0">ical<span class="_ _d"> </span>models<span class="_ _8"> </span>that<span class="_ _d"> </span>incorporate<span class="_ _8"> </span>electrolyte<span class="_ _8"> </span>dynamics.<span class="_ _8"> </span>Examples</div><div class="t m0 x1 h6 y74 ff1 fs4 fc0 sc0 ls3f ws0">include<span class="_ _f"> </span>spectral<span class="_ _17"> </span>methods<span class="_ _17"> </span>with<span class="_ _17"> </span>output<span class="_ _17"> </span>error<span class="_ _f"> </span>injection<span class="_ _17"> </span>[25],</div><div class="t m0 x1 h6 y75 ff1 fs4 fc0 sc0 ls3f ws0">residue<span class="_ _11"> </span>grouping<span class="_ _11"> </span>with<span class="_ _16"> </span>K<span class="_ _1"></span>alman<span class="_ _10"> </span>ﬁltering<span class="_ _10"> </span>[26],<span class="_ _10"> </span>semisepara-</div><div class="t m0 x1 h6 y76 ff1 fs4 fc0 sc0 ls43 ws0">ble<span class="_ _17"> </span>str<span class="_ _2"></span>uctures<span class="_ _11"> </span>with<span class="_ _17"> </span>an<span class="_ _11"> </span>EKF<span class="_ _11"> </span>[27],<span class="_ _17"> </span>d<span class="_ _2"></span>iscrete-time<span class="_ _11"> </span>realization</div><div class="t m0 x1 h6 y77 ff1 fs4 fc0 sc0 ls1d ws0">algorithms<span class="_ _c"> </span>with<span class="_ _7"> </span>an<span class="_ _8"> </span>EKF<span class="_ _7"> </span>[28],<span class="_ _7"> </span>and<span class="_ _7"> </span>composite<span class="_ _c"> </span>electrodes<span class="_ _7"> </span>with</div><div class="t m0 x1 h6 y78 ff1 fs4 fc0 sc0 ls1b ws0">nonlinear<span class="_ _7"> </span>ﬁlters<span class="_ _c"> </span>[2<span class="_ _2"></span>9].</div><div class="t m0 x6 h6 y79 ff1 fs4 fc0 sc0 ls3e ws0">In<span class="_ _11"> </span>all<span class="_ _10"> </span>the<span class="_ _11"> </span>aforementioned<span class="_ _17"> </span>estimation<span class="_ _11"> </span>studies<span class="_ _11"> </span>for<span class="_ _11"> </span>SPMe</div><div class="t m0 x1 h6 y7a ff1 fs4 fc0 sc0 ls3b ws0">dynamics,<span class="_ _12"> </span>a<span class="_ _17"> </span>rigorous<span class="_ _12"> </span>analysis<span class="_ _f"> </span>of<span class="_ _13"> </span>observer<span class="_ _13"> </span>estimation<span class="_ _13"> </span>error</div><div class="t m0 x1 h6 y7b ff1 fs4 fc0 sc0 ls5c ws0">stability<span class="_ _d"> </span>an<span class="_ _2"></span>d<span class="_ _d"> </span>conservation<span class="_ _d"> </span>of<span class="_ _d"> </span>lithiu<span class="_ _2"></span>m<span class="_ _d"> </span>is<span class="_ _12"> </span>lacking.<span class="_ _d"> </span>In<span class="_ _d"> </span>additio<span class="_ _2"></span>n,</div><div class="t m0 x1 h6 y7c ff1 fs4 fc0 sc0 ls1d ws0">all<span class="_ _12"> </span>these<span class="_ _d"> </span>methods<span class="_ _d"> </span>are<span class="_ _12"> </span>reliant<span class="_ _12"> </span>on<span class="_ _d"> </span>a<span class="_ _12"> </span>particular<span class="_ _12"> </span>numeri<span class="_ _1"></span>cal<span class="_ _12"> </span>dis<span class="_ _1"></span>-</div><div class="t m0 x1 h6 y7d ff1 fs4 fc0 sc0 ls41 ws0">cretization<span class="_ _8"> </span>scheme.<span class="_ _8"> </span>That<span class="_ _8"> </span>is,<span class="_ _8"> </span>th<span class="_ _2"></span>ey<span class="_ _7"> </span>d<span class="_ _2"></span>iscretize<span class="_ _8"> </span>the<span class="_ _8"> </span>par<span class="_ _2"></span>tial<span class="_ _8"> </span>differ<span class="_ _1"></span>-</div><div class="t m0 x1 h6 y7e ff1 fs4 fc0 sc0 ls3 ws0">ential<span class="_"> </span>equations<span class="_"> </span>(PDEs)<span class="_ _15"> </span>immediately<span class="_"> </span>and,<span class="_"> </span>then,<span class="_ _e"> </span>apply<span class="_ _15"> </span>analysis</div><div class="t m0 x1 h6 y7f ff1 fs4 fc0 sc0 ls48 ws0">and<span class="_ _f"> </span>estimation<span class="_ _f"> </span>synthesis<span class="_ _f"> </span>in<span class="_ _17"> </span>the<span class="_ _f"> </span>ﬁnite-dimensional<span class="_ _13"> </span>domain.</div><div class="t m0 x1 h6 y80 ff1 fs4 fc0 sc0 ls3f ws0">This<span class="_ _c"> </span>paper<span class="_ _7"> </span>performs<span class="_ _15"> </span>the<span class="_ _7"> </span>analysis<span class="_ _c"> </span>and<span class="_ _c"> </span>estimation<span class="_ _c"> </span>synthesis<span class="_ _c"> </span>on</div><div class="t m0 x1 h6 y81 ff1 fs4 fc0 sc0 ls3f ws0">the<span class="_ _8"> </span>PDEs<span class="_ _d"> </span>before<span class="_ _8"> </span>discretization.<span class="_ _7"> </span>The<span class="_ _d"> </span>advant<span class="_ _1"></span>ages<span class="_ _8"> </span>are<span class="_ _d"> </span>tw<span class="_ _1"></span>ofold:</div><div class="t m0 x1 h6 y82 ff1 fs4 fc0 sc0 ls4b ws0">1)<span class="_ _12"> </span>one<span class="_ _12"> </span>does<span class="_ _12"> </span>not<span class="_ _12"> </span>ha<span class="_ _1"></span>ve<span class="_ _12"> </span>to<span class="_ _12"> </span>re<span class="_ _1"></span>visit<span class="_ _12"> </span>the<span class="_ _12"> </span>esti<span class="_ _1"></span>mator<span class="_ _12"> </span>design<span class="_ _12"> </span>if<span class="_ _12"> </span>the</div><div class="t m0 x1 h6 y83 ff1 fs4 fc0 sc0 ls3f ws0">discreti<span class="_ _1"></span>zation<span class="_ _12"> </span>method<span class="_ _d"> </span>is<span class="_ _12"> </span>altered<span class="_ _d"> </span>and<span class="_ _12"> </span>2)<span class="_ _12"> </span>the<span class="_ _12"> </span>physical<span class="_ _d"> </span>signiﬁ-</div><div class="t m0 x1 h6 y84 ff1 fs4 fc0 sc0 ls48 ws0">cance<span class="_ _15"> </span>of<span class="_ _15"> </span>the<span class="_ _15"> </span>equations<span class="_ _e"> </span>(and<span class="_ _c"> </span>of<span class="_ _15"> </span>the<span class="_ _15"> </span>phenomena<span class="_ _15"> </span>they<span class="_"> </span>represent)</div><div class="t m0 x1 h6 y85 ff1 fs4 fc0 sc0 ls1c ws0">is<span class="_ _f"> </span>retained<span class="_ _f"> </span>regardless<span class="_ _f"> </span>of<span class="_ _f"> </span>the<span class="_ _f"> </span>discretization<span class="_ _f"> </span>method<span class="_ _f"> </span>used<span class="_ _f"> </span>at</div><div class="t m0 x1 h6 y86 ff1 fs4 fc0 sc0 ls4a ws0">the<span class="_ _d"> </span>implementation<span class="_ _8"> </span>stage,<span class="_ _d"> </span>thus<span class="_ _d"> </span>yielding<span class="_ _d"> </span>valuabl<span class="_ _1"></span>e<span class="_ _d"> </span>insights<span class="_ _d"> </span>as</div><div class="t m0 x1 h6 y87 ff1 fs4 fc0 sc0 ls4a ws0">a<span class="_ _d"> </span>by-product<span class="_ _d"> </span>of<span class="_ _12"> </span>t<span class="_ _1"></span>he<span class="_ _12"> </span>desi<span class="_ _1"></span>gn<span class="_ _d"> </span>process,<span class="_ _d"> </span>which<span class="_ _d"> </span>are<span class="_ _12"> </span>los<span class="_ _1"></span>t<span class="_ _12"> </span>in<span class="_ _d"> </span>other</div><div class="t m0 x1 h6 y88 ff1 fs4 fc0 sc0 ls1d ws0">approaches.<span class="_"> </span>Unfortunately<span class="_ _3"></span>,<span class="_"> </span>it<span class="_ _e"> </span>becomes<span class="_ _e"> </span>increasingly<span class="_"> </span>difﬁcult<span class="_"> </span>to</div><div class="t m0 x1 h6 y89 ff1 fs4 fc0 sc0 ls55 ws0">prove<span class="_"> </span>estimation<span class="_"> </span>err<span class="_ _2"></span>or<span class="_"> </span>stability<span class="_ _e"> </span>as<span class="_ _e"> </span>mo<span class="_ _2"></span>del<span class="_"> </span>comp<span class="_ _2"></span>lexity<span class="_"> </span>increases,</div><div class="t m0 x1 h6 y8a ff1 fs4 fc0 sc0 ls41 ws0">as<span class="_ _c"> </span>hig<span class="_ _2"></span>hlighted<span class="_ _c"> </span>in<span class="_ _7"> </span>the<span class="_ _7"> </span>referenced<span class="_ _7"> </span>literature.<span class="_ _c"> </span>The<span class="_ _7"> </span>cor<span class="_ _2"></span>e<span class="_ _c"> </span>difﬁculty</div><div class="t m0 x1 h6 y8b ff1 fs4 fc0 sc0 ls43 ws0">is<span class="_ _c"> </span>lack<span class="_ _c"> </span>of<span class="_ _c"> </span>com<span class="_ _2"></span>plete<span class="_ _15"> </span>ob<span class="_ _2"></span>servability<span class="_ _15"> </span>from<span class="_ _c"> </span>voltage<span class="_ _15"> </span>measurem<span class="_ _2"></span>ents.</div><div class="t m0 x1 h8 y8c ff2 fs4 fc0 sc0 ls5d ws0">C.<span class="_ _c"> </span>Main<span class="_ _c"> </span>Contrib<span class="_ _1"></span>utions</div><div class="t m0 x6 h6 y8d ff1 fs4 fc0 sc0 ls5b ws0">In<span class="_ _8"> </span>this<span class="_ _8"> </span>paper,<span class="_ _7"> </span>we<span class="_ _d"> </span>adv<span class="_ _1"></span>ance<span class="_ _8"> </span>the<span class="_ _8"> </span>aforementioned<span class="_ _7"> </span>research<span class="_ _8"> </span>by</div><div class="t m0 x1 h6 y8e ff1 fs4 fc0 sc0 ls24 ws0">designing<span class="_ _7"> </span>a<span class="_ _8"> </span>PDE-based<span class="_ _7"> </span>observer<span class="_ _7"> </span>for<span class="_ _7"> </span>an<span class="_ _8"> </span>SPMe.<span class="_ _8"> </span>The<span class="_ _8"> </span>observer</div><div class="t m0 x1 h6 y8f ff1 fs4 fc0 sc0 ls24 ws0">design<span class="_ _8"> </span>is<span class="_ _d"> </span>unique,<span class="_ _8"> </span>since<span class="_ _d"> </span>it<span class="_ _d"> </span>exploi<span class="_ _1"></span>ts<span class="_ _8"> </span>fundamental<span class="_ _8"> </span>electrochem-</div><div class="t m0 x1 h6 y90 ff1 fs4 fc0 sc0 ls3 ws0">istry<span class="_ _7"> </span>dynamic<span class="_ _7"> </span>properties,<span class="_ _c"> </span>yielding<span class="_ _7"> </span>a<span class="_ _7"> </span>deeper<span class="_ _8"> </span>insight<span class="_ _c"> </span>in<span class="_ _8"> </span>battery</div><div class="t m0 x1 h6 y91 ff1 fs4 fc0 sc0 ls5e ws0">state<span class="_ _f"> </span>estimation.<span class="_ _13"> </span>W<span class="_ _3"></span>e<span class="_ _f"> </span>additiona<span class="_ _2"></span>lly<span class="_ _12"> </span>p<span class="_ _2"></span>rove<span class="_ _12"> </span>the<span class="_ _f"> </span>stability<span class="_ _f"> </span>of<span class="_ _13"> </span>the</div><div class="t m0 x1 h6 y92 ff1 fs4 fc0 sc0 ls55 ws0">estimation<span class="_ _d"> </span>error<span class="_ _d"> </span>system<span class="_ _d"> </span>an<span class="_ _2"></span>d<span class="_ _d"> </span>conservation<span class="_ _8"> </span>of<span class="_ _d"> </span>lithiu<span class="_ _2"></span>m.<span class="_ _d"> </span>Conse-</div><div class="t m0 x1 h6 y93 ff1 fs4 fc0 sc0 ls27 ws0">quently<span class="_ _3"></span>,<span class="_ _12"> </span>the<span class="_ _12"> </span>article’<span class="_ _3"></span>s<span class="_ _12"> </span>main<span class="_ _12"> </span>contrib<span class="_ _1"></span>utions<span class="_ _12"> </span>are<span class="_ _12"> </span>summarized<span class="_ _d"> </span>as</div><div class="t m0 x1 h6 y94 ff1 fs4 fc0 sc0 ls51 ws0">follows.</div><div class="t m0 x6 h6 y95 ff1 fs4 fc0 sc0 ls40 ws0">1)<span class="_ _11"> </span>Deri<span class="_ _1"></span>vation<span class="_ _16"> </span>o<span class="_ _2"></span>f<span class="_ _19"> </span>an<span class="_ _19"> </span>SPMe<span class="_ _19"> </span>oriented<span class="_ _19"> </span>tow<span class="_ _1"></span>ard<span class="_ _19"> </span>the<span class="_ _19"> </span>state</div><div class="t m0 xf h6 y96 ff1 fs4 fc0 sc0 ls55 ws0">estimation<span class="_ _f"> </span>design.<span class="_ _f"> </span>Similar<span class="_ _f"> </span>m<span class="_ _2"></span>odels<span class="_ _f"> </span>have<span class="_ _13"> </span>been<span class="_ _f"> </span>derived</div><div class="t m0 xf h6 y97 ff1 fs4 fc0 sc0 ls3f ws0">in<span class="_ _12"> </span>[18]–[22].<span class="_ _12"> </span>W<span class="_ _9"></span>e<span class="_ _f"> </span>additional<span class="_ _1"></span>ly<span class="_ _12"> </span>analyze<span class="_ _12"> </span>dynamical<span class="_ _12"> </span>sys-</div><div class="t m0 xf h6 y98 ff1 fs4 fc0 sc0 ls1d ws0">tem<span class="_ _d"> </span>properties<span class="_ _8"> </span>that<span class="_ _d"> </span>enable<span class="_ _8"> </span>a<span class="_ _d"> </span>prov<span class="_ _1"></span>ably<span class="_ _8"> </span>conv<span class="_ _1"></span>ergent<span class="_ _7"> </span>state</div><div class="t m0 xf h6 y99 ff1 fs4 fc0 sc0 ls5f ws0">estimator<span class="_ _3"></span>.</div><div class="t m0 x6 h6 y9a ff1 fs4 fc0 sc0 ls60 ws0">2)<span class="_ _11"> </span>A<span class="_ _7"> </span>PDE-based<span class="_ _c"> </span>state<span class="_ _7"> </span>estimation<span class="_ _7"> </span>design<span class="_ _7"> </span>that<span class="_ _c"> </span>explo<span class="_ _2"></span>its<span class="_ _15"> </span>m<span class="_ _2"></span>ar-</div><div class="t m0 xf h6 y9b ff1 fs4 fc0 sc0 ls61 ws0">ginal<span class="_ _15"> </span>stability<span class="_ _1"></span>,<span class="_ _15"> </span>conservation<span class="_ _e"> </span>o<span class="_ _2"></span>f<span class="_ _15"> </span>lithium,<span class="_ _c"> </span>and<span class="_ _15"> </span>outp<span class="_ _2"></span>ut<span class="_ _15"> </span>func-</div><div class="t m0 xf h6 y9c ff1 fs4 fc0 sc0 ls50 ws0">tion<span class="_ _e"> </span>invertibility<span class="_"> </span>to<span class="_ _e"> </span>ren<span class="_ _2"></span>der<span class="_"> </span>co<span class="_ _2"></span>n<span class="_ _1"></span>vergent<span class="_"> </span>estimates.<span class="_ _e"> </span>W<span class="_ _3"></span>e<span class="_"> </span>a<span class="_ _2"></span>lso</div><div class="t m0 xf h6 y9d ff1 fs4 fc0 sc0 ls60 ws0">include<span class="_ _12"> </span>a<span class="_ _13"> </span>rigoro<span class="_ _2"></span>us<span class="_ _12"> </span>stability<span class="_ _12"> </span>an<span class="_ _2"></span>alysis<span class="_ _12"> </span>of<span class="_ _12"> </span>th<span class="_ _2"></span>e<span class="_ _12"> </span>estimation</div><div class="t m0 xf h6 y9e ff1 fs4 fc0 sc0 ls47 ws0">error<span class="_ _15"> </span>systems.</div><div class="t m0 x1 h8 y9f ff2 fs4 fc0 sc0 ls5c ws0">D.<span class="_ _7"> </span>Outline</div><div class="t m0 x6 h6 ya0 ff1 fs4 fc0 sc0 ls62 ws0">The<span class="_ _16"> </span>remainder<span class="_ _10"> </span>of<span class="_ _16"> </span>this<span class="_ _16"> </span>paper<span class="_ _1a"> </span>is<span class="_ _1a"> </span>organized<span class="_ _1a"> </span>as<span class="_ _1a"> </span>follows.</div><div class="t m0 x1 h6 ya1 ff1 fs4 fc0 sc0 ls62 ws0">Section<span class="_ _13"> </span>II<span class="_ _13"> </span>deriv<span class="_ _1"></span>es<span class="_ _13"> </span>the<span class="_ _13"> </span>SPMe<span class="_ _13"> </span>and<span class="_ _13"> </span>analyzes<span class="_ _12"> </span>relev<span class="_ _1"></span>ant<span class="_ _12"> </span>conser-</div><div class="t m0 x1 h6 ya2 ff1 fs4 fc0 sc0 ls63 ws0">vation<span class="_ _7"> </span>properties,<span class="_ _8"> </span>in<span class="_ _1"></span>vertability<span class="_ _3"></span>,<span class="_ _7"> </span>and<span class="_ _8"> </span>accuracy<span class="_ _7"> </span>relative<span class="_ _c"> </span>to<span class="_ _8"> </span>o<span class="_ _2"></span>ther</div><div class="t m0 x1 h6 ya3 ff1 fs4 fc0 sc0 ls5f ws0">electrochemical<span class="_ _14"> </span>models.<span class="_"> </span>S<span class="_ _1"></span>ection<span class="_"> </span><span class="ls20">III<span class="_"> </span>deri<span class="_ _1"></span>ves<span class="_ _14"> </span>th<span class="_ _2"></span>e<span class="_"> </span>state<span class="_"> </span>estimation</span></div><div class="t m0 x1 h6 ya4 ff1 fs4 fc0 sc0 ls64 ws0">scheme<span class="_ _8"> </span>for<span class="_ _7"> </span>each<span class="_ _8"> </span>constitutive<span class="_ _c"> </span>subsystem.<span class="_ _8"> </span>Section<span class="_ _7"> </span>IV<span class="_ _8"> </span>analyzes</div><div class="t m0 xa h9 ya5 ff1 fs5 fc0 sc0 ls65 ws0">Fig.<span class="_"> </span>1.<span class="_ _17"> </span>Schematic<span class="_ _0"> </span>of<span class="_"> </span>the<span class="_"> </span>DFN<span class="_"> </span>model<span class="_"> </span>[2]<span class="ls37">.<span class="_"> </span>The<span class="_"> </span>model<span class="_"> </span>considers<span class="_ _0"> </span>two<span class="_"> </span>phases:<span class="_"> </span>the</span></div><div class="t m0 xa h9 ya6 ff1 fs5 fc0 sc0 ls66 ws0">solid<span class="_ _0"> </span>and<span class="_ _0"> </span>the<span class="_ _0"> </span>electr<span class="_ _2"></span>olyte.<span class="_ _e"> </span>I<span class="_ _2"></span>n<span class="_ _0"> </span>the<span class="_ _0"> </span>solid,<span class="_ _0"> </span>states<span class="_ _e"> </span>evolv<span class="_ _1"></span>e<span class="_ _0"> </span>in<span class="_ _0"> </span>the<span class="_ _15"> </span><span class="ff2 ls1b">x<span class="_ _e"> </span></span><span class="ls34">and<span class="_ _0"> </span><span class="ff2 ls1b">r<span class="_ _15"> </span></span><span class="ls29">dim<span class="_ _2"></span>ensions.</span></span></div><div class="t m0 xa h9 ya7 ff1 fs5 fc0 sc0 ls67 ws0">In<span class="_ _15"> </span>the<span class="_ _15"> </span>electrolyte,<span class="_ _7"> </span>states<span class="_ _c"> </span>evolv<span class="_ _1"></span>e<span class="_ _c"> </span>in<span class="_ _15"> </span>the<span class="_ _c"> </span><span class="ff2 ls1b">x<span class="_ _7"> </span></span><span class="ls2e">dim<span class="_ _2"></span>ension<span class="_ _15"> </span>only.<span class="_ _e"> </span>T<span class="_ _2"></span>he<span class="_ _15"> </span>cell<span class="_ _15"> </span>is<span class="_ _15"> </span>divided</span></div><div class="t m0 xa h9 ya8 ff1 fs5 fc0 sc0 ls2c ws0">into<span class="_ _e"> </span>three<span class="_ _15"> </span>regions:<span class="_ _c"> </span>anode,<span class="_ _15"> </span>separator<span class="_ _1"></span>,<span class="_ _15"> </span>and<span class="_ _15"> </span>cathode.</div><div class="t m0 xa h6 ya9 ff1 fs4 fc0 sc0 ls68 ws0">the<span class="_ _8"> </span>stability<span class="_ _d"> </span>of<span class="_ _8"> </span>the<span class="_ _8"> </span>estimatio<span class="ls69">n<span class="_ _8"> </span>error<span class="_ _8"> </span>dynamics.<span class="_ _8"> </span>The<span class="_ _8"> </span>estimator</span></div><div class="t m0 xa h6 yaa ff1 fs4 fc0 sc0 ls61 ws0">is<span class="_ _f"> </span>demo<span class="_ _2"></span>nstrated<span class="_ _f"> </span>via<span class="_ _f"> </span>simulations<span class="_ _f"> </span>in<span class="_ _17"> </span>Section<span class="_ _f"> </span>V.<span class="_ _f"> </span>Finally<span class="_ _1"></span>,<span class="_ _f"> </span>the</div><div class="t m0 xa h6 yab ff1 fs4 fc0 sc0 ls27 ws0">conclusion<span class="_ _15"> </span>is<span class="_ _c"> </span>drawn<span class="_ _c"> </span>in<span class="_ _c"> </span>Section<span class="_ _c"> </span>VI.</div><div class="t m0 x10 h6 yac ff1 fs4 fc0 sc0 ls19 ws0">II.<span class="_ _12"> </span>S</div><div class="t m0 x11 h6 yad ff1 fs5 fc0 sc0 ls6a ws0">INGLE<span class="_"> </span><span class="fs4 ls1b">P</span><span class="ls1a">AR<span class="_ _3"></span>TICLE<span class="_"> </span><span class="fs4 ls1b">M<span class="_ _a"></span></span>ODEL<span class="_"> </span><span class="fs4 ls1b">W<span class="_ _a"></span></span><span class="ls6b">ITH<span class="_"> </span><span class="fs4 ls1b">E<span class="_ _a"></span></span><span class="ls6c">LECTR<span class="_ _1"></span>OL<span class="_ _3"></span>YTE</span></span></span></div><div class="t m0 xa h8 yae ff2 fs4 fc0 sc0 ls1d ws0">A.<span class="_ _c"> </span>Doyle–Fuller<span class="_ _1"></span>–Newm<span class="_ _1"></span>an<span class="_ _c"> </span>Model</div><div class="t m0 xb h6 yaf ff1 fs4 fc0 sc0 ls50 ws0">In<span class="_ _11"> </span>this<span class="_ _11"> </span>section,<span class="_ _17"> </span>we<span class="_ _11"> </span>de<span class="_ _2"></span>scribe<span class="_ _11"> </span>the<span class="_ _17"> </span>assu<span class="_ _2"></span>mption<span class="_ _2"></span>s<span class="_ _17"> </span>an<span class="_ _2"></span>d<span class="_ _17"> </span>step<span class="_ _2"></span>s</div><div class="t m0 xa h6 yb0 ff1 fs4 fc0 sc0 ls55 ws0">followed<span class="_ _13"> </span>to<span class="_ _17"> </span>derive<span class="_ _f"> </span>the<span class="_ _f"> </span>SPMe<span class="_ _f"> </span>m<span class="_ _2"></span>odel.<span class="_ _f"> </span>First,<span class="_ _17"> </span>we<span class="_ _17"> </span>summar<span class="_ _2"></span>ize</div><div class="t m0 xa h6 yb1 ff1 fs4 fc0 sc0 ls1f ws0">the<span class="_ _7"> </span>Doyle–Fuller–Newman<span class="_ _c"> </span>(DFN)<span class="_ _8"> </span>model<span class="_ _7"> </span>in<span class="_ _8"> </span>Fig.<span class="_ _8"> </span>1<span class="_ _8"> </span>to<span class="_ _7"> </span>predict</div><div class="t m0 xa h6 yb2 ff1 fs4 fc0 sc0 ls5e ws0">the<span class="_ _c"> </span>evolution<span class="_ _15"> </span>of<span class="_ _7"> </span>lithium<span class="_ _7"> </span>concentr<span class="_ _2"></span>ation<span class="_ _c"> </span>in<span class="_ _c"> </span>the<span class="_ _7"> </span>solid<span class="_ _7"> </span><span class="ff2 ls1b">c</span></div><div class="t m0 x12 hb yb3 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x12 hc yb4 ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 x13 h6 yb5 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _b"> </span><span class="ff2">r</span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _18"></span></span>)<span class="ff1">,</span></div><div class="t m0 xa h6 yb6 ff1 fs4 fc0 sc0 ls5c ws0">lithium<span class="_ _7"> </span>concentratio<span class="_ _2"></span>n<span class="_ _c"> </span>in<span class="_ _7"> </span>the<span class="_ _7"> </span>electrolyte<span class="_ _7"> </span><span class="ff2 ls1b">c</span></div><div class="t m0 x14 hc yb7 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x15 h6 yb8 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="ff1 ls64">,<span class="_ _8"> </span>solid<span class="_ _c"> </span>electric</span></div><div class="t m0 xa h6 yb9 ff1 fs4 fc0 sc0 ls6d ws0">potential<span class="_"> </span><span class="ff7 ls1b">φ</span></div><div class="t m0 x16 hb yba ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x16 hc ybb ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 x17 h6 ybc ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="ff1 ls6e">,<span class="_ _c"> </span>electrolyte<span class="_ _15"> </span>electric<span class="_ _c"> </span>potential<span class="_ _15"> </span></span>φ</div><div class="t m0 x18 hc ybd ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x19 h6 ybc ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="ff1 ls5d">,<span class="_ _c"> </span>ionic</span></div><div class="t m0 xa h6 ybe ff1 fs4 fc0 sc0 ls24 ws0">current<span class="_"> </span><span class="ff2 ls1b">i</span></div><div class="t m0 x1a hb ybf ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x1a hc yc0 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x1b h6 yc1 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="ff1 ls6f">,<span class="_ _15"> </span>and<span class="_"> </span>molar<span class="_ _e"> </span>ion<span class="_ _e"> </span>ﬂuxes<span class="_ _d"> </span></span><span class="ff2">j</span></div><div class="t m0 x1c hb ybf ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x1d hc yc0 ff2 fs8 fc0 sc0 ls1b ws0">n</div><div class="t m0 x1e h6 yc1 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="ff1 ls70">.<span class="_ _15"> </span>The<span class="_ _e"> </span>governing</span></div><div class="t m0 xa h6 yc2 ff1 fs4 fc0 sc0 ls3f ws0">equations<span class="_ _15"> </span>are</div><div class="t m0 x1f h8 yc3 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">c</span></div><div class="t m0 x20 hb yc4 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x20 hc yc5 ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 x21 h8 yc6 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">t</span></div><div class="t m0 x22 h8 yc7 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _b"> </span><span class="ff2">r</span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _18"></span></span>)<span class="_"> </span><span class="ff6">=</span></div><div class="t m0 x23 h6 yc8 ff1 fs4 fc0 sc0 ls1b ws0">1</div><div class="t m0 x24 h8 yc6 ff2 fs4 fc0 sc0 ls1b ws0">r</div><div class="t m0 x25 hd yc9 ff1 fs8 fc0 sc0 ls1b ws0">2</div><div class="t m0 x26 he yca ff7 fs4 fc0 sc0 ls1b ws0">∂</div><div class="t m0 x27 h8 yc6 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _a"></span><span class="ff2">r</span></div><div class="t m0 x28 hf ycb ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x29 h8 yc7 ff2 fs4 fc0 sc0 ls1b ws0">D</div><div class="t m0 x2a hb ycc ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x2a hc ycd ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 x2b h8 yce ff2 fs4 fc0 sc0 ls1b ws0">r</div><div class="t m0 x2c hd ycc ff1 fs8 fc0 sc0 ls1b ws0">2</div><div class="t m0 x2d h8 yca ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">c</span></div><div class="t m0 x2e hb yc4 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x2e hc yc5 ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 x2f h8 yc6 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _a"></span><span class="ff2">r</span></div><div class="t m0 x30 h8 yc7 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _b"> </span><span class="ff2">r</span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _18"></span></span>)</div><div class="t m0 x15 hf ycb ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x31 h6 yc7 ff1 fs4 fc0 sc0 ls6d ws0">(1)</div><div class="t m0 x1f he ycf ff7 fs4 fc0 sc0 ls1b ws0">ε</div><div class="t m0 x32 hc yd0 ff2 fs8 fc0 sc0 ls1b ws0">j</div><div class="t m0 x21 hc yd1 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x20 h8 yd2 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">c</span></div><div class="t m0 x33 hc yd3 ff2 fs8 fc0 sc0 ls1b ws0">j</div><div class="t m0 x34 hc yd4 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x35 h8 yd5 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">t</span></div><div class="t m0 x36 h8 yd6 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_"> </span><span class="ff6">=</span></div><div class="t m0 x37 he yd7 ff7 fs4 fc0 sc0 ls1b ws0">∂</div><div class="t m0 x24 h8 yd5 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">x</span></div><div class="t m0 x26 hf yd8 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x38 h8 yd6 ff2 fs4 fc0 sc0 ls1b ws0">D</div><div class="t m0 x39 hd yd9 ff1 fs8 fc0 sc0 ls71 ws0">eff</div><div class="t m0 x39 hc yda ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x3a hf ydb ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x3b h8 ydc ff2 fs4 fc0 sc0 ls1b ws0">c</div><div class="t m0 x3c hc yd0 ff2 fs8 fc0 sc0 ls1b ws0">j</div><div class="t m0 x3d hc yd1 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x3e hf ydb ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x3f h8 ydd ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">c</span></div><div class="t m0 x40 hc yd3 ff2 fs8 fc0 sc0 ls1b ws0">j</div><div class="t m0 x2e hc yd4 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x2f h8 yd5 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">x</span></div><div class="t m0 x41 h8 yd6 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _a"></span><span class="ff6">+</span></div><div class="t m0 x42 h6 yd7 ff1 fs4 fc0 sc0 ls1b ws0">1<span class="_ _a"></span><span class="ff6">−<span class="_ _a"></span><span class="ff2">t</span></span></div><div class="t m0 x43 hd yde ff1 fs8 fc0 sc0 ls1b ws0">0</div><div class="t m0 x44 hc ydf ff2 fs8 fc0 sc0 ls1b ws0">c</div><div class="t m0 x45 h8 yd5 ff2 fs4 fc0 sc0 ls1b ws0">F</div><div class="t m0 x46 h8 yd6 ff2 fs4 fc0 sc0 ls1b ws0">i</div><div class="t m0 x18 hc yd0 ff2 fs8 fc0 sc0 ls1b ws0">j</div><div class="t m0 x47 hc yd1 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x19 h8 ye0 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)</div><div class="t m0 x48 hf ye1 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x31 h6 ye0 ff1 fs4 fc0 sc0 ls6d ws0">(2)</div><div class="t m0 xa h6 ye2 ff1 fs4 fc0 sc0 ls6d ws0">for<span class="_ _13"> </span><span class="ff2 ls1b">j<span class="_ _8"> </span><span class="ff6 ls72">∈{<span class="_ _1c"></span>−<span class="_ _1c"></span><span class="ff7 ls1b">,<span class="_ _1b"> </span><span class="ff1 ls2f">sep</span>,<span class="_ _1b"> </span><span class="ff6 ls49">+}<span class="_"> </span><span class="ff1 ls45">and</span></span></span></span></span></div><div class="t m0 x1f he ye3 ff7 fs4 fc0 sc0 ls1b ws0">σ</div><div class="t m0 x49 hd ye4 ff1 fs8 fc0 sc0 ls71 ws0">eff<span class="ff7 ls1b">,<span class="ff6">±</span></span></div><div class="t m0 x36 h10 ye5 ff6 fs4 fc0 sc0 ls1b ws0">·</div><div class="t m0 x11 he ye6 ff7 fs4 fc0 sc0 ls73 ws0">∂φ</div><div class="t m0 x4a hb ye7 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x4a hc ye8 ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 x1b h8 ye9 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">x</span></div><div class="t m0 x4b h8 yea ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _7"> </span><span class="ff6">=<span class="_"> </span><span class="ff2">i</span></span></div><div class="t m0 x39 hb ye4 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x4c hc yeb ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x4d h6 ye5 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _0"> </span><span class="ff6">−<span class="_ _15"> </span><span class="ff2">I<span class="_ _b"> </span></span></span>(<span class="ff2">t<span class="_ _18"></span></span>)<span class="_ _1d"> </span><span class="ff1 ls6d">(3)</span></div><div class="t m0 x4e he yec ff7 fs4 fc0 sc0 ls1b ws0">κ</div><div class="t m0 x4f hd yed ff1 fs8 fc0 sc0 ls71 ws0">eff</div><div class="t m0 x20 h8 yee ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">c</span></div><div class="t m0 x50 hc yef ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x51 h10 yee ff7 fs4 fc0 sc0 ls1b ws0">)<span class="_ _0"> </span><span class="ff6">·</span></div><div class="t m0 x52 he yf0 ff7 fs4 fc0 sc0 ls73 ws0">∂φ</div><div class="t m0 x53 hc yf1 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x54 h8 yf2 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">x</span></div><div class="t m0 x4b h8 yf3 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _7"> </span><span class="ff6 ls74">=−<span class="_ _1e"></span><span class="ff2 ls1b">i</span></span></div><div class="t m0 x55 hb yed ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x55 hc yf4 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x56 h8 yee ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _0"> </span><span class="ff6">+<span class="_ _0"> </span></span>κ</div><div class="t m0 x57 hd yed ff1 fs8 fc0 sc0 ls75 ws0">eff</div><div class="t m0 x1c h8 yee ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">c</span></div><div class="t m0 x14 hc yef ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x15 h10 yee ff7 fs4 fc0 sc0 ls1b ws0">)<span class="_ _14"> </span><span class="ff6">·</span></div><div class="t m0 x58 h6 yf0 ff1 fs4 fc0 sc0 ls5d ws0">2R<span class="_ _3"></span>T</div><div class="t m0 x44 h8 yf2 ff2 fs4 fc0 sc0 ls1b ws0">F</div><div class="t m0 x59 hf yf5 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x5a h6 yf6 ff1 fs4 fc0 sc0 ls1b ws0">1<span class="_ _0"> </span><span class="ff6">−<span class="_ _14"> </span><span class="ff2">t</span></span></div><div class="t m0 x5b hd yed ff1 fs8 fc0 sc0 ls1b ws0">0</div><div class="t m0 x5c hc yf4 ff2 fs8 fc0 sc0 ls1b ws0">c</div><div class="t m0 x5d hf yf7 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x28 h10 yf8 ff6 fs4 fc0 sc0 ls1b ws0">×</div><div class="t m0 x4d hf yf9 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x5e h6 yf8 ff1 fs4 fc0 sc0 ls1b ws0">1<span class="_ _0"> </span><span class="ff6">+</span></div><div class="t m0 x5f h6 yfa ff2 fs4 fc0 sc0 ls1b ws0">d<span class="_"> </span><span class="ff1 ls20">ln<span class="_ _8"> </span></span>f</div><div class="t m0 x60 hc yfb ff2 fs8 fc0 sc0 ls1b ws0">c<span class="ff7">/</span>a</div><div class="t m0 x61 h6 yfc ff2 fs4 fc0 sc0 ls1b ws0">d<span class="_"> </span><span class="ff1 ls20">ln<span class="_ _1b"> </span></span>c</div><div class="t m0 x62 hc yfd ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x63 h8 yfe ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)</div><div class="t m0 x64 hf yff ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x65 h6 y100 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _0"> </span><span class="ff1 ls20">ln<span class="_ _14"> </span></span><span class="ff2">c</span></div><div class="t m0 x66 hc y101 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x67 h8 yfc ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">x</span></div><div class="t m0 x68 h8 y102 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)</div><div class="t m0 x31 h6 y103 ff1 fs4 fc0 sc0 ls6d ws0">(4)</div><div class="t m0 x52 h8 y104 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _a"></span><span class="ff2">i</span></div><div class="t m0 x4a hb y105 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x4a hc y106 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x54 h8 y107 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">x</span></div><div class="t m0 x4b h8 y108 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _7"> </span><span class="ff6">=<span class="_"> </span><span class="ff2">a</span></span></div><div class="t m0 x69 hb y109 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x3a h8 y10a ff2 fs4 fc0 sc0 ls76 ws0">Fj</div><div class="t m0 x6a hb y109 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x3c hc y10b ff2 fs8 fc0 sc0 ls1b ws0">n</div><div class="t m0 x2d h6 y10a ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _1f"> </span><span class="ff1 ls6d">(5)</span></div><div class="t m0 x6b h8 y10c ff2 fs4 fc0 sc0 ls1b ws0">j</div><div class="t m0 x17 hb y10d ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x4a hc y10e ff2 fs8 fc0 sc0 ls1b ws0">n</div><div class="t m0 x4b h8 y10f ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _7"> </span><span class="ff6">=</span></div><div class="t m0 x4c h6 y110 ff1 fs4 fc0 sc0 ls1b ws0">1</div><div class="t m0 x6c h8 y111 ff2 fs4 fc0 sc0 ls1b ws0">F</div><div class="t m0 x4d h8 y112 ff2 fs4 fc0 sc0 ls1b ws0">i</div><div class="t m0 x2a hb y113 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x3a hd y114 ff1 fs8 fc0 sc0 ls1b ws0">0</div><div class="t m0 x2b h8 y10f ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)</div><div class="t m0 x6d hf y115 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x6e h8 y10f ff2 fs4 fc0 sc0 ls1b ws0">e</div><div class="t m0 x6f h11 y116 ff7 fs9 fc0 sc0 ls1b ws0">α</div><div class="t m0 x70 h12 y117 ff2 fs9 fc0 sc0 ls1b ws0">a</div><div class="t m0 x62 h12 y116 ff2 fs9 fc0 sc0 ls1b ws0">F</div><div class="t m0 x71 h13 y118 ff1 fs9 fc0 sc0 ls77 ws0">RT</div><div class="t m0 x1d h14 y10d ff7 fs8 fc0 sc0 ls1b ws0">η</div><div class="t m0 x63 h15 y119 ff6 fs9 fc0 sc0 ls1b ws0">±</div><div class="t m0 x14 hc y10d ff7 fs8 fc0 sc0 ls1b ws0">(<span class="_ _2"></span><span class="ff2">x<span class="_ _a"></span></span>,<span class="ff2">t<span class="_ _18"></span></span>)</div><div class="t m0 x72 h8 y10f ff6 fs4 fc0 sc0 ls1b ws0">−<span class="_ _14"> </span><span class="ff2">e</span></div><div class="t m0 x18 hb y10d ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 x73 h11 y116 ff7 fs9 fc0 sc0 ls1b ws0">α</div><div class="t m0 x74 h12 y117 ff2 fs9 fc0 sc0 ls1b ws0">c</div><div class="t m0 x75 h12 y116 ff2 fs9 fc0 sc0 ls1b ws0">F</div><div class="t m0 x76 h13 y118 ff1 fs9 fc0 sc0 ls77 ws0">RT</div><div class="t m0 x68 h14 y10d ff7 fs8 fc0 sc0 ls1b ws0">η</div><div class="t m0 x77 h15 y119 ff6 fs9 fc0 sc0 ls1b ws0">±</div><div class="t m0 x78 hc y10d ff7 fs8 fc0 sc0 ls1b ws0">(<span class="_ _2"></span><span class="ff2">x<span class="_ _a"></span></span>,<span class="ff2">t<span class="_ _18"></span></span>)</div><div class="t m0 x79 hf y11a ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x31 h6 y11b ff1 fs4 fc0 sc0 ls6d ws0">(6)</div><div class="t m0 x6b h8 y11c ff2 fs4 fc0 sc0 ls1b ws0">i</div><div class="t m0 x17 hb y11d ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x4a hd y11e ff1 fs8 fc0 sc0 ls1b ws0">0</div><div class="t m0 x4b h8 y11f ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _7"> </span><span class="ff6">=<span class="_"> </span><span class="ff2">k</span></span></div><div class="t m0 x69 hb y120 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x55 hf y121 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x7a h8 y122 ff2 fs4 fc0 sc0 ls1b ws0">c</div><div class="t m0 x2b hb y120 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x2b hd y123 ff1 fs8 fc0 sc0 ls78 ws0">ss</div><div class="t m0 x3e h8 y11f ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)</div><div class="t m0 x30 hf y124 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x6f h14 y125 ff7 fs8 fc0 sc0 ls1b ws0">α</div><div class="t m0 x70 h12 y126 ff2 fs9 fc0 sc0 ls1b ws0">c</div><div class="t m0 x28 h10 y127 ff6 fs4 fc0 sc0 ls1b ws0">×</div><div class="t m0 x7b hf y128 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x55 h8 y127 ff2 fs4 fc0 sc0 ls1b ws0">c</div><div class="t m0 x5e hc y129 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x2b h8 y127 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)</div><div class="t m0 x6d hf y128 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x6e h8 y127 ff2 fs4 fc0 sc0 ls1b ws0">c</div><div class="t m0 x7c hb y12a ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x7c hd y12b ff2 fs8 fc0 sc0 ls1b ws0">s<span class="_ _2"></span><span class="ff7">,<span class="ff1 ls79">max</span></span></div><div class="t m0 x7d h8 y127 ff6 fs4 fc0 sc0 ls1b ws0">−<span class="_ _0"> </span><span class="ff2">c</span></div><div class="t m0 x45 hb y12a ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x45 hd y12b ff1 fs8 fc0 sc0 ls78 ws0">ss</div><div class="t m0 x72 h8 y127 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)</div><div class="t m0 x73 hf y128 ff8 fs4 fc0 sc0 ls7a ws0"></div><div class="t m0 x66 h14 y12c ff7 fs8 fc0 sc0 ls1b ws0">α</div><div class="t m0 x68 h12 y12d ff2 fs9 fc0 sc0 ls1b ws0">a</div><div class="t m0 x31 h6 y127 ff1 fs4 fc0 sc0 ls6d ws0">(7)</div><div class="t m0 x7e he y12e ff7 fs4 fc0 sc0 ls1b ws0">η</div><div class="t m0 x17 hb y12f ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x4b h8 y130 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _7"> </span><span class="ff6">=<span class="_"> </span></span>φ</div><div class="t m0 x29 hb y12f ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x69 hc y131 ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 x3a h8 y130 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _0"> </span><span class="ff6">−<span class="_ _0"> </span></span>φ</div><div class="t m0 x6f hc y132 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x70 h8 y130 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _0"> </span><span class="ff6">−<span class="_ _14"> </span><span class="ff2">U</span></span></div><div class="t m0 x7f hb y12f ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x47 hf y133 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x59 h8 y134 ff2 fs4 fc0 sc0 ls1b ws0">c</div><div class="t m0 x5a hb y12f ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x5a hd y131 ff1 fs8 fc0 sc0 ls78 ws0">ss</div><div class="t m0 x12 h8 y130 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)</div><div class="t m0 x80 hf y135 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x28 h8 y136 ff6 fs4 fc0 sc0 ls1b ws0">−<span class="_ _0"> </span><span class="ff2 ls7b">FR</span></div><div class="t m0 x6a hb y137 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x2c hc y138 ff2 fs8 fc0 sc0 ls1b ws0">f</div><div class="t m0 x81 h8 y139 ff2 fs4 fc0 sc0 ls1b ws0">j</div><div class="t m0 x82 hb y13a ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x83 hc y13b ff2 fs8 fc0 sc0 ls1b ws0">n</div><div class="t m0 x40 h6 y139 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _20"> </span><span class="ff1 ls6d">(8)</span></div><div class="t m0 x84 h8 y13c ff2 fs4 fc0 sc0 ls1b ws0">c</div><div class="t m0 x17 hb y13d ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x17 hd y13e ff1 fs8 fc0 sc0 ls78 ws0">ss</div><div class="t m0 x4b h8 y13f ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _7"> </span><span class="ff6">=<span class="_"> </span><span class="ff2">c</span></span></div><div class="t m0 x39 hb y13d ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x39 hc y13e ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 x55 hf y140 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x7a h8 y141 ff2 fs4 fc0 sc0 ls1b ws0">x<span class="_ _18"></span><span class="ff7">,<span class="_ _0"> </span></span>R</div><div class="t m0 x81 hb y13d ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x81 hc y13e ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 x61 h8 y13f ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t</span></div><div class="t m0 x41 hf y142 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x31 h6 y13f ff1 fs4 fc0 sc0 ls6d ws0">(9)</div><div class="t m0 xe ha y6d ff5 fs7 fc0 sc0 ls1b ws0">Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on January 19,2022 at 04:38:28 UTC from IEEE Xplore. Restrictions apply. </div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>

<div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6250a1e374bc5c01056d2748/bg3.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls7c ws0">MOURA<span class="_ _0"> </span><span class="ff2 ls7d">et<span class="_ _0"> </span>al.</span><span class="ls7e">:<span class="_ _15"> </span>BA<span class="_ _3"></span>TTER<span class="_ _3"></span>Y<span class="_ _14"> </span>ST<span class="_ _3"></span>A<span class="_ _3"></span>TE<span class="_ _0"> </span>ESTI<span class="_ _1"></span>MA<span class="_ _3"></span>TION<span class="_ _14"> </span>FOR<span class="_ _0"> </span>A<span class="_ _0"> </span>SINGLE<span class="_ _14"> </span>P<span class="_ _3"></span>AR<span class="_ _1"></span>TICLE<span class="_ _0"> </span>MODEL<span class="_"> </span>WITH<span class="_ _0"> </span>ELECTRO<span class="_ _1"></span>L<span class="_ _3"></span>YTE<span class="_"> </span>DY<span class="_ _1"></span>NA<span class="_ _1"></span>MICS<span class="_ _21"> </span>455</span></div><div class="t m0 x1 h6 y6e ff1 fs4 fc0 sc0 ls46 ws0">where<span class="_ _d"> </span><span class="ff2 ls1b">D</span></div><div class="t m0 x85 hd y143 ff1 fs8 fc0 sc0 ls71 ws0">eff</div><div class="t m0 x85 hc y144 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x86 h8 y6e ff6 fs4 fc0 sc0 ls1b ws0">=<span class="_ _8"> </span><span class="ff2">D</span></div><div class="t m0 x87 hc y145 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x88 h8 y6e ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">c</span></div><div class="t m0 x89 hc y145 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x8a h10 y6e ff7 fs4 fc0 sc0 ls1b ws0">)<span class="_ _0"> </span><span class="ff6">·<span class="_ _0"> </span></span><span class="ls63">(ε</span></div><div class="t m0 x8b hc y146 ff2 fs8 fc0 sc0 ls1b ws0">j</div><div class="t m0 x8c hc y147 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x8d he y6e ff7 fs4 fc0 sc0 ls1b ws0">)</div><div class="t m0 x8e hd y143 ff1 fs8 fc0 sc0 ls7f ws0">brug</div><div class="t m0 x5 h6 y6e ff1 fs4 fc0 sc0 ls1b ws0">,<span class="_ _7"> </span><span class="ff7">σ</span></div><div class="t m0 x8f hd y143 ff1 fs8 fc0 sc0 ls71 ws0">eff</div><div class="t m0 x90 h10 y6e ff6 fs4 fc0 sc0 ls1b ws0">=<span class="_"> </span><span class="ff7">σ<span class="_ _7"> </span></span>·<span class="_ _0"> </span><span class="ff7 ls63">(ε</span></div><div class="t m0 x91 hc y146 ff2 fs8 fc0 sc0 ls1b ws0">j</div><div class="t m0 x92 hc y147 ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 x93 h10 y6e ff6 fs4 fc0 sc0 ls1b ws0">+<span class="_ _0"> </span><span class="ff7">ε</span></div><div class="t m0 x94 hc y146 ff2 fs8 fc0 sc0 ls1b ws0">j</div><div class="t m0 x94 hc y148 ff2 fs8 fc0 sc0 ls1b ws0">f</div><div class="t m0 x95 he y6e ff7 fs4 fc0 sc0 ls1b ws0">)</div><div class="t m0 x96 hd y143 ff1 fs8 fc0 sc0 ls7f ws0">brug</div><div class="t m0 x97 h6 y6e ff1 fs4 fc0 sc0 ls80 ws0">,a<span class="_ _22"></span>n<span class="_ _22"></span>d</div><div class="t m0 x1 he y149 ff7 fs4 fc0 sc0 ls1b ws0">κ</div><div class="t m0 x98 hd y2 ff1 fs8 fc0 sc0 ls71 ws0">eff</div><div class="t m0 x99 h8 y14a ff6 fs4 fc0 sc0 ls1b ws0">=<span class="_"> </span><span class="ff7 ls81">κ(<span class="_ _3"></span><span class="ff2 ls1b">c</span></span></div><div class="t m0 x9a hc y14b ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x9b h10 y14a ff7 fs4 fc0 sc0 ls1b ws0">)<span class="_ _0"> </span><span class="ff6">·<span class="_ _0"> </span></span><span class="ls63">(ε</span></div><div class="t m0 x87 hc y14c ff2 fs8 fc0 sc0 ls1b ws0">j</div><div class="t m0 x9c hc y14d ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x9d he y14a ff7 fs4 fc0 sc0 ls1b ws0">)</div><div class="t m0 x9e hd y2 ff1 fs8 fc0 sc0 ls7f ws0">brug</div><div class="t m0 x9f h6 y14a ff1 fs4 fc0 sc0 ls5f ws0">are<span class="_ _7"> </span>the<span class="_ _7"> </span>effecti<span class="_ _1"></span>ve<span class="_ _c"> </span>electrolyte<span class="_ _8"> </span>diffus<span class="_ _1"></span>ivity<span class="_ _3"></span>,</div><div class="t m0 x1 h6 y14e ff1 fs4 fc0 sc0 ls82 ws0">effect<span class="_ _1"></span>iv<span class="_ _1"></span>e<span class="_ _7"> </span>solid<span class="_ _8"> </span>conducti<span class="_ _1"></span>vity<span class="_ _3"></span>,<span class="_ _7"> </span>and<span class="_ _7"> </span>effecti<span class="_ _1"></span>ve<span class="_ _7"> </span>electrolyte<span class="_ _7"> </span>conduc-</div><div class="t m0 x1 h6 y14f ff1 fs4 fc0 sc0 ls4b ws0">ti<span class="_ _1"></span>vity<span class="_ _c"> </span>gi<span class="_ _1"></span>ven<span class="_ _c"> </span>by<span class="_ _c"> </span>the<span class="_ _c"> </span>Bruggeman<span class="_ _c"> </span>relationshi<span class="_ _1"></span>p.</div><div class="t m0 x6 h6 y150 ff1 fs4 fc0 sc0 ls1e ws0">The<span class="_ _15"> </span>boundary<span class="_ _15"> </span>conditions<span class="_ _15"> </span>for<span class="_ _c"> </span>solid-phase<span class="_ _15"> </span>dif<span class="_ _1"></span>fusion<span class="_ _15"> </span>PDE<span class="_ _c"> </span>(1)</div><div class="t m0 x1 h6 y151 ff1 fs4 fc0 sc0 ls83 ws0">are</div><div class="t m0 xa0 h8 y152 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">c</span></div><div class="t m0 xa1 hb y153 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 xa1 hc y154 ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 x88 h8 y155 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _a"></span><span class="ff2">r</span></div><div class="t m0 xa2 h6 y156 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff1">0</span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _7"> </span><span class="ff6">=<span class="_"> </span><span class="ff1 ls5d">0<span class="_ _23"> </span>(10)</span></span></div><div class="t m0 xa3 h8 y157 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">c</span></div><div class="t m0 xa4 hb y158 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 xa0 hc y159 ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 xa5 h8 y15a ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _a"></span><span class="ff2">r</span></div><div class="t m0 xc hf y15b ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 xa6 h8 y15c ff2 fs4 fc0 sc0 ls1b ws0">x<span class="_ _18"></span><span class="ff7">,<span class="_ _0"> </span></span>R</div><div class="t m0 xa7 hb y15d ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 xa7 hc y15e ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 x8b h8 y15f ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t</span></div><div class="t m0 xa8 hf y160 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 xa9 h10 y15f ff6 fs4 fc0 sc0 ls74 ws0">=−</div><div class="t m0 xaa h6 y161 ff1 fs4 fc0 sc0 ls1b ws0">1</div><div class="t m0 xab h8 y162 ff2 fs4 fc0 sc0 ls1b ws0">D</div><div class="t m0 xac hb y163 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 xac hc y164 ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 xad h8 y15f ff2 fs4 fc0 sc0 ls1b ws0">j</div><div class="t m0 xae hb y15d ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 xaf hc y15e ff2 fs8 fc0 sc0 ls1b ws0">n</div><div class="t m0 xb0 h6 y15f ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span><span class="ls63">).<span class="_ _24"> </span><span class="ff1 ls84">(11)</span></span></div><div class="t m0 x1 h6 y165 ff1 fs4 fc0 sc0 ls24 ws0">The<span class="_ _12"> </span>boundary<span class="_ _d"> </span>conditions<span class="_ _12"> </span>for<span class="_ _12"> </span>the<span class="_ _12"> </span>el<span class="ls83">ectrolyte-phase<span class="_ _d"> </span>diffusion</span></div><div class="t m0 x1 h6 y166 ff1 fs4 fc0 sc0 ls4e ws0">PDE<span class="_ _c"> </span>(2)<span class="_ _7"> </span>are<span class="_ _c"> </span>given<span class="_ _15"> </span>by</div><div class="t m0 xa6 h8 y167 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">c</span></div><div class="t m0 xb1 hb y168 ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 xb1 hc y169 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x8a h8 y16a ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">x</span></div><div class="t m0 xa7 h6 y16b ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff1">0</span></div><div class="t m0 x8d hb y16c ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 xb2 h8 y16d ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _7"> </span><span class="ff6">=</span></div><div class="t m0 xab h8 y16e ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">c</span></div><div class="t m0 xb3 hb y168 ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 xb3 hc y169 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xb4 h8 y16a ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">x</span></div><div class="t m0 xb5 h6 y16b ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff1">0</span></div><div class="t m0 xb0 hb y16c ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 xb6 h6 y16d ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_"> </span><span class="ff6">=<span class="_ _0"> </span><span class="ff1 ls5d">0<span class="_ _25"> </span>(12)</span></span></div><div class="t m0 xe h8 y16f ff2 fs4 fc0 sc0 ls1b ws0">D</div><div class="t m0 xf hd y170 ff6 fs8 fc0 sc0 ls1b ws0">−<span class="ff7">,<span class="ff1 ls71">ef<span class="_ _1"></span>f</span></span></div><div class="t m0 xf hc y171 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xb7 h8 y172 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">c</span></div><div class="t m0 xb8 hc y173 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xb9 h8 y172 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="_ _a"></span><span class="ff2">L</span></div><div class="t m0 xba hb y170 ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 x9d he y172 ff7 fs4 fc0 sc0 ls63 ws0">))</div><div class="t m0 x89 h8 y174 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">c</span></div><div class="t m0 xbb hb y175 ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 x3 hc y176 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xbc h8 y177 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">x</span></div><div class="t m0 xbd h8 y178 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="_ _a"></span><span class="ff2">L</span></div><div class="t m0 x8d hb y170 ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 xb2 h8 y172 ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _7"> </span><span class="ff6">=<span class="_ _d"> </span><span class="ff2">D</span></span></div><div class="t m0 xac hd y179 ff1 fs8 fc0 sc0 ls78 ws0">sep<span class="ff7 ls1b">,</span><span class="ls71">ef<span class="_ _1"></span>f</span></div><div class="t m0 xac hc y17a ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xbe h8 y172 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">c</span></div><div class="t m0 xbf hc y173 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xc0 h6 y172 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff1">0</span></div><div class="t m0 x93 hd y170 ff1 fs8 fc0 sc0 ls78 ws0">sep</div><div class="t m0 xc1 he y172 ff7 fs4 fc0 sc0 ls63 ws0">))</div><div class="t m0 xab h10 y17b ff6 fs4 fc0 sc0 ls1b ws0">×</div><div class="t m0 xc2 h8 y17c ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">c</span></div><div class="t m0 xb5 hd y17d ff1 fs8 fc0 sc0 ls78 ws0">sep</div><div class="t m0 xb5 hc y17e ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xc3 h8 y17f ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">x</span></div><div class="t m0 xbe h6 y180 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff1">0</span></div><div class="t m0 xc4 hd y181 ff1 fs8 fc0 sc0 ls78 ws0">sep</div><div class="t m0 x93 h6 y182 ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _26"> </span><span class="ff1 ls84">(13)</span></div><div class="t m0 xc5 h8 y183 ff2 fs4 fc0 sc0 ls1b ws0">D</div><div class="t m0 xc6 hd y184 ff1 fs8 fc0 sc0 ls78 ws0">sep<span class="ff7 ls1b">,</span><span class="ls71">ef<span class="_ _1"></span>f</span></div><div class="t m0 xc6 hc y185 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xc7 h8 y186 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">c</span></div><div class="t m0 xc8 hc y187 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x9a h8 y186 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="_ _a"></span><span class="ff2">L</span></div><div class="t m0 xb9 hd y188 ff1 fs8 fc0 sc0 ls78 ws0">sep</div><div class="t m0 xba he y186 ff7 fs4 fc0 sc0 ls63 ws0">))</div><div class="t m0 xc9 h8 y189 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">c</span></div><div class="t m0 xbc hd y18a ff1 fs8 fc0 sc0 ls78 ws0">sep</div><div class="t m0 xbc hc y18b ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xca h8 y18c ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">x</span></div><div class="t m0 xd h8 y18d ff7 fs4 fc0 sc0 ls1b ws0">(<span class="_ _a"></span><span class="ff2">L</span></div><div class="t m0 xcb hd y188 ff1 fs8 fc0 sc0 ls78 ws0">sep</div><div class="t m0 xcc h8 y186 ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _14"> </span><span class="ff6">=<span class="_ _0"> </span><span class="ff2">D</span></span></div><div class="t m0 xcd hd y188 ff6 fs8 fc0 sc0 ls1b ws0">+<span class="ff7">,<span class="ff1 ls71">ef<span class="_ _1"></span>f</span></span></div><div class="t m0 xcd hc y18e ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xce h8 y186 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">c</span></div><div class="t m0 xcf hc y187 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xd0 h8 y186 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="_ _a"></span><span class="ff2">L</span></div><div class="t m0 xd1 hb y188 ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 xd2 he y186 ff7 fs4 fc0 sc0 ls63 ws0">))</div><div class="t m0 xc1 h8 y189 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">c</span></div><div class="t m0 xd3 hb y18f ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 xd3 hc y190 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xd4 h8 y18c ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">x</span></div><div class="t m0 xd5 h8 y18d ff7 fs4 fc0 sc0 ls1b ws0">(<span class="_ _a"></span><span class="ff2">L</span></div><div class="t m0 xd6 hb y188 ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 xd7 h8 y186 ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)</div><div class="t m0 xd8 h6 y191 ff1 fs4 fc0 sc0 ls84 ws0">(14)</div><div class="t m0 x3 h8 y192 ff2 fs4 fc0 sc0 ls1b ws0">c</div><div class="t m0 x9f hc y193 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xbd h8 y194 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="_ _a"></span><span class="ff2">L</span></div><div class="t m0 x8d hb y195 ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 xb2 h8 y194 ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _7"> </span><span class="ff6">=<span class="_ _7"> </span><span class="ff2">c</span></span></div><div class="t m0 xb4 hc y193 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xac h6 y194 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff1">0</span></div><div class="t m0 xad hd y195 ff1 fs8 fc0 sc0 ls78 ws0">sep</div><div class="t m0 xb0 h6 y194 ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _27"> </span><span class="ff1 ls84">(15)</span></div><div class="t m0 xd9 h8 y196 ff2 fs4 fc0 sc0 ls1b ws0">c</div><div class="t m0 x3 hc y197 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x9f h8 y198 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="_ _a"></span><span class="ff2">L</span></div><div class="t m0 x8c hd y199 ff1 fs8 fc0 sc0 ls78 ws0">sep</div><div class="t m0 xb2 h8 y198 ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _7"> </span><span class="ff6">=<span class="_ _7"> </span><span class="ff2">c</span></span></div><div class="t m0 xb4 hc y197 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xac h8 y198 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="_ _a"></span><span class="ff2">L</span></div><div class="t m0 xb5 hb y199 ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 xda h6 y198 ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span><span class="ls63">).<span class="_ _28"> </span><span class="ff1 ls84">(16)</span></span></div><div class="t m0 x1 h6 y19a ff1 fs4 fc0 sc0 ls24 ws0">The<span class="_ _12"> </span>boundary<span class="_ _12"> </span>conditions<span class="_ _12"> </span>for<span class="_ _12"> </span>the<span class="_ _12"> </span>el<span class="ls25">ectrolyte-phase<span class="_ _12"> </span>potential</span></div><div class="t m0 x1 h6 y19b ff1 fs4 fc0 sc0 ls3 ws0">ordinary<span class="_ _15"> </span>different<span class="_ _1"></span>ial<span class="_ _15"> </span>equation<span class="_ _c"> </span>(ODE)<span class="_ _c"> </span>(4)<span class="_ _c"> </span>are<span class="_ _c"> </span>given<span class="_ _15"> </span>by</div><div class="t m0 xdb he y19c ff7 fs4 fc0 sc0 ls1b ws0">φ</div><div class="t m0 xdc hc y19d ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xb1 h6 y19e ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff1">0</span></div><div class="t m0 xdd hb y19f ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 xde h6 y19e ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _7"> </span><span class="ff6">=<span class="_"> </span><span class="ff1 ls5d">0<span class="_ _29"> </span>(17)</span></span></div><div class="t m0 x89 he y1a0 ff7 fs4 fc0 sc0 ls1b ws0">φ</div><div class="t m0 xa2 hc y1a1 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xbb h8 y1a2 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="_ _a"></span><span class="ff2">L</span></div><div class="t m0 xdd hb y1a3 ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 xde h8 y1a2 ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _7"> </span><span class="ff6">=<span class="_"> </span></span>φ</div><div class="t m0 x8f hc y1a1 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xdf h6 y1a2 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff1">0</span></div><div class="t m0 xb3 hd y1a3 ff1 fs8 fc0 sc0 ls78 ws0">sep</div><div class="t m0 xae h6 y1a2 ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _2a"> </span><span class="ff1 ls84">(18)</span></div><div class="t m0 xc he y1a4 ff7 fs4 fc0 sc0 ls1b ws0">φ</div><div class="t m0 xa6 hc y1a5 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xa2 h8 y1a6 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="_ _a"></span><span class="ff2">L</span></div><div class="t m0 xe0 hd y1a7 ff1 fs8 fc0 sc0 ls78 ws0">sep</div><div class="t m0 xde h8 y1a6 ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _7"> </span><span class="ff6">=<span class="_"> </span></span>φ</div><div class="t m0 x8f hc y1a5 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xdf h8 y1a6 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="_ _a"></span><span class="ff2">L</span></div><div class="t m0 xe1 hb y1a7 ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 xb5 h6 y1a6 ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span><span class="ls63">).<span class="_ _2b"> </span><span class="ff1 ls84">(19)</span></span></div><div class="t m0 x1 h6 y1a8 ff1 fs4 fc0 sc0 ls3 ws0">The<span class="_ _d"> </span>boundary<span class="_ _d"> </span>conditions<span class="_ _d"> </span>for<span class="_ _d"> </span>the<span class="_ _d"> </span><span class="ls85">ionic<span class="_ _d"> </span>current<span class="_ _12"> </span>ODE<span class="_ _d"> </span>(5)<span class="_ _d"> </span>are</span></div><div class="t m0 x1 h6 y1a9 ff1 fs4 fc0 sc0 ls5d ws0">gi<span class="_ _1"></span>ven<span class="_ _15"> </span>by</div><div class="t m0 xa0 h8 y1aa ff2 fs4 fc0 sc0 ls1b ws0">i</div><div class="t m0 xc9 hb y1ab ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 x88 hc y1ac ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xa1 h6 y1ad ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff1">0</span></div><div class="t m0 xdc hb y1ab ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 xe2 h8 y1ad ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_"> </span><span class="ff6">=<span class="_ _0"> </span><span class="ff2">i</span></span></div><div class="t m0 xe3 hb y1ab ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 xe3 hc y1ac ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xe4 h6 y1ad ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff1">0</span></div><div class="t m0 x8f hb y1ab ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 xb4 h6 y1ad ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_"> </span><span class="ff6">=<span class="_ _0"> </span><span class="ff1 ls5d">0<span class="_ _2c"> </span>(20)</span></span></div><div class="t m0 x1 h6 y1ae ff1 fs4 fc0 sc0 ls47 ws0">and<span class="_ _c"> </span>also<span class="_ _c"> </span>note<span class="_ _c"> </span>that<span class="_ _c"> </span><span class="ff2 ls1b">i</span></div><div class="t m0 x89 hc y1af ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x8a h6 y1b0 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_"> </span><span class="ff6">=<span class="_"> </span><span class="ff2">I<span class="_ _b"> </span></span></span>(<span class="ff2">t<span class="_ _18"></span></span>)<span class="_ _7"> </span><span class="ff1 ls6d">for<span class="_ _7"> </span></span><span class="ff2">x<span class="_ _8"> </span><span class="ff6 ls86">∈[<span class="_ _1c"></span><span class="ff1 ls1b">0</span></span></span></div><div class="t m0 xbf hd y1b1 ff1 fs8 fc0 sc0 ls78 ws0">sep</div><div class="t m0 xe5 h8 y1b0 ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _0"> </span><span class="ff2">L</span></div><div class="t m0 xe6 hd y1b1 ff1 fs8 fc0 sc0 ls78 ws0">sep</div><div class="t m0 x95 h6 y1b0 ff6 fs4 fc0 sc0 ls1b ws0">]<span class="ff1">.</span></div><div class="t m0 x6 h6 y1b2 ff1 fs4 fc0 sc0 ls87 ws0">The<span class="_ _e"> </span>model<span class="_ _15"> </span>input<span class="_"> </span>is<span class="_ _15"> </span>the<span class="_ _15"> </span>applied<span class="_"> </span>current<span class="_ _e"> </span>density<span class="_ _7"> </span><span class="ff2 ls1b">I<span class="_ _b"> </span><span class="ff7">(</span>t<span class="_ _18"></span><span class="ff7">)<span class="_"> </span></span></span><span class="ls3e">[A/m</span></div><div class="t m0 xe7 hd y1b3 ff1 fs8 fc0 sc0 ls1b ws0">2</div><div class="t m0 xe8 h6 y1b4 ff1 fs4 fc0 sc0 ls3f ws0">],</div><div class="t m0 x1 h6 y1b5 ff1 fs4 fc0 sc0 ls3b ws0">and<span class="_ _13"> </span>the<span class="_ _f"> </span>output<span class="_ _13"> </span>is<span class="_ _f"> </span>the<span class="_ _f"> </span>v<span class="_ _1"></span>oltage<span class="_ _13"> </span>measured<span class="_ _13"> </span>across<span class="_ _13"> </span>the<span class="_ _f"> </span>current</div><div class="t m0 x1 h6 y1b6 ff1 fs4 fc0 sc0 ls4d ws0">collectors</div><div class="t m0 xa3 h8 y1b7 ff2 fs4 fc0 sc0 ls1b ws0">V<span class="_ _1b"> </span><span class="ff7">(</span>t<span class="_ _a"></span><span class="ff7">)<span class="_"> </span><span class="ff6">=<span class="_ _0"> </span></span>φ</span></div><div class="t m0 xbd hb y1b8 ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 xe0 hc y1b9 ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 x8c h6 y1ba ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff1">0</span></div><div class="t m0 x8e hb y1b8 ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 xe9 h8 y1ba ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _0"> </span><span class="ff6">−<span class="_ _0"> </span></span>φ</div><div class="t m0 xac hb y1b8 ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 xac hc y1b9 ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 xc3 h6 y1ba ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff1">0</span></div><div class="t m0 xea hb y1b8 ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 xeb h6 y1ba ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span><span class="ls63">).<span class="_ _2d"> </span><span class="ff1 ls84">(21)</span></span></div><div class="t m0 x1 h6 y1bb ff1 fs4 fc0 sc0 ls85 ws0">A<span class="_ _d"> </span>complete<span class="_ _8"> </span>exposition<span class="_ _8"> </span>on<span class="_ _d"> </span>the<span class="_ _8"> </span>m<span class="ls69">odel<span class="_ _d"> </span>equations<span class="_ _8"> </span>and<span class="_ _d"> </span>notation</span></div><div class="t m0 x1 h6 y1bc ff1 fs4 fc0 sc0 ls48 ws0">can<span class="_ _c"> </span>be<span class="_ _c"> </span>found<span class="_ _15"> </span>in<span class="_ _c"> </span>[2]<span class="_ _c"> </span>and<span class="_ _c"> </span>[3].<span class="_ _c"> </span>Symbols<span class="_ _15"> </span>are<span class="_ _c"> </span>deﬁned<span class="_ _c"> </span>in<span class="_ _c"> </span>T<span class="_ _3"></span>able<span class="_ _15"> </span>III</div><div class="t m0 x1 h6 y1bd ff1 fs4 fc0 sc0 ls1e ws0">of<span class="_ _11"> </span>the<span class="_ _17"> </span>Appendix.<span class="_ _17"> </span>Note<span class="_ _11"> </span>the<span class="_ _11"> </span>mathematical<span class="_ _17"> </span>structure,<span class="_ _17"> </span>which</div><div class="t m0 x1 h6 y1be ff1 fs4 fc0 sc0 ls25 ws0">contains<span class="_ _13"> </span>linear<span class="_ _f"> </span>PDEs<span class="_ _f"> </span>(1),<span class="_ _13"> </span>quasi-linear<span class="_ _13"> </span>PDEs<span class="_ _17"> </span>(2),<span class="_ _13"> </span>ODEs<span class="_ _f"> </span>in</div><div class="t m0 x1 h6 y1bf ff1 fs4 fc0 sc0 ls24 ws0">space<span class="_"> </span>(3)–(5),<span class="_ _14"> </span>and<span class="_"> </span>nonlinear<span class="_"> </span>al<span class="_ _1"></span>gebraic<span class="_"> </span>const<span class="_ _1"></span>raints<span class="_"> </span>(6)–(8).<span class="_ _14"> </span>This</div><div class="t m0 x1 h6 y1c0 ff1 fs4 fc0 sc0 ls24 ws0">presents<span class="_ _8"> </span>a<span class="_ _d"> </span>formidable<span class="_ _7"> </span>task<span class="_ _d"> </span>for<span class="_ _d"> </span>model-based<span class="_ _7"> </span>state<span class="_ _d"> </span>est<span class="_ _1"></span>imation.</div><div class="t m0 x1 h6 y1c1 ff1 fs4 fc0 sc0 ls1e ws0">Consequentl<span class="_ _1"></span>y<span class="_ _3"></span>,<span class="_ _d"> </span>we<span class="_ _12"> </span>s<span class="_ _1"></span>eek<span class="_ _12"> </span>an<span class="_ _d"> </span>appropriately<span class="_ _8"> </span>reduced<span class="_ _d"> </span>model<span class="_ _d"> </span>that</div><div class="t m0 x1 h6 y1c2 ff1 fs4 fc0 sc0 ls60 ws0">maintains<span class="_ _8"> </span>pr<span class="_ _2"></span>ediction<span class="_ _8"> </span>ﬁdelity—at<span class="_ _8"> </span>hig<span class="_ _2"></span>h<span class="_ _8"> </span><span class="ff2 ls1b">C<span class="_ _a"></span></span><span class="ls64">-<span class="_ _2"></span>rates<span class="_ _8"> </span>in<span class="_ _8"> </span>par<span class="_ _2"></span>ticular—</span></div><div class="t m0 x1 h6 y1c3 ff1 fs4 fc0 sc0 ls3b ws0">yet<span class="_ _c"> </span>enables<span class="_ _c"> </span>a<span class="_ _c"> </span>provably<span class="_ _e"> </span>conv<span class="_ _1"></span>ergent<span class="_ _e"> </span>state<span class="_ _c"> </span>observer<span class="_ _3"></span>.</div><div class="t m0 x1 h8 y1c4 ff2 fs4 fc0 sc0 ls1e ws0">B.<span class="_ _c"> </span>SPMe<span class="_ _7"> </span>Model<span class="_ _c"> </span>Derivation</div><div class="t m0 x6 h6 y1c5 ff1 fs4 fc0 sc0 ls48 ws0">The<span class="_ _c"> </span>SPMe<span class="_ _c"> </span>is<span class="_ _7"> </span>deri<span class="_ _1"></span>ved<span class="_ _15"> </span>under<span class="_ _c"> </span>the<span class="_ _c"> </span>following<span class="_ _15"> </span>assumptions<span class="_ _1"></span>.</div><div class="t m0 xec h6 y1c6 ff2 fs4 fc0 sc0 ls2f ws0">[A1]:<span class="_ _11"> </span><span class="ff1 ls53">The<span class="_ _7"> </span>solid-phase<span class="_ _7"> </span>Li<span class="_ _7"> </span>concen<span class="ls23">tration<span class="_ _c"> </span>in<span class="_ _8"> </span>each<span class="_ _7"> </span>electrode<span class="_ _7"> </span>is</span></span></div><div class="t m0 xed h6 y1c7 ff1 fs4 fc0 sc0 ls4b ws0">constant<span class="_ _12"> </span>in<span class="_ _12"> </span>spatia<span class="_ _1"></span>l<span class="_ _12"> </span>coordinate<span class="_ _12"> </span><span class="ff2 ls1b">x<span class="_ _18"></span></span><span class="ls55">,<span class="_ _12"> </span>u<span class="_ _2"></span>niform<span class="_ _2"></span>ly<span class="_ _12"> </span>in<span class="_ _12"> </span>time.</span></div><div class="t m0 xed h6 y1c8 ff1 fs4 fc0 sc0 ls63 ws0">Mathematically<span class="_ _1"></span>,<span class="_ _7"> </span><span class="ff2 ls1b">c</span></div><div class="t m0 x8b hb y1c9 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x8b hc y1ca ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 x8e h6 y1cb ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _b"> </span><span class="ff2">r</span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _18"></span></span>)<span class="_ _8"> </span><span class="ff1 ls45">and<span class="_ _17"> </span></span><span class="ff2">j</span></div><div class="t m0 xee hb y1c9 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 xb0 hc y1ca ff2 fs8 fc0 sc0 ls1b ws0">n</div><div class="t m0 xef h6 y1cb ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _d"> </span><span class="ff1 ls6d">are<span class="_ _8"> </span>constant</span></div><div class="t m0 xed h6 y1cc ff1 fs4 fc0 sc0 ls20 ws0">in<span class="_ _7"> </span>the<span class="_ _8"> </span><span class="ff2 ls1b">x<span class="_ _18"></span></span><span class="ls1c">-directio<span class="_ _2"></span>n.</span></div><div class="t m0 xa h9 y1cd ff1 fs5 fc0 sc0 ls28 ws0">Fig.<span class="_ _c"> </span>2.<span class="_ _2e"> </span>Block<span class="_ _c"> </span>diagram<span class="_ _7"> </span>of<span class="_ _c"> </span>SPMe.<span class="_ _c"> </span>Note<span class="_ _c"> </span>that<span class="_ _7"> </span>the<span class="_ _7"> </span><span class="ff2 ls1b">c</span></div><div class="t m0 xf0 h16 y1ce ff6 fsa fc0 sc0 ls1b ws0">+</div><div class="t m0 xf0 h17 y1cf ff2 fsa fc0 sc0 ls1b ws0">s</div><div class="t m0 xf1 h18 y1cd ff7 fs5 fc0 sc0 ls1b ws0">,<span class="_ _b"> </span><span class="ff2">c</span></div><div class="t m0 xf2 h16 y1ce ff6 fsa fc0 sc0 ls1b ws0">−</div><div class="t m0 xf2 h17 y1cf ff2 fsa fc0 sc0 ls1b ws0">s</div><div class="t m0 xf3 h18 y1cd ff7 fs5 fc0 sc0 ls1b ws0">,<span class="_ _b"> </span><span class="ff2">c</span></div><div class="t m0 x46 h17 y1d0 ff2 fsa fc0 sc0 ls1b ws0">e</div><div class="t m0 x59 h9 y1cd ff1 fs5 fc0 sc0 ls2d ws0">subsystems<span class="_ _7"> </span>are</div><div class="t m0 xa h9 y1d1 ff1 fs5 fc0 sc0 ls2d ws0">independent<span class="_ _15"> </span>of<span class="_ _15"> </span>one<span class="_ _e"> </span>another<span class="_ _1"></span>.</div><div class="t m0 xa h9 y1d2 ff1 fs5 fc0 sc0 ls88 ws0">Fig.<span class="_ _e"> </span>3.<span class="_ _5"> </span>Simpliﬁed<span class="_ _15"> </span>form<span class="_ _15"> </span>of<span class="_ _e"> </span><span class="ff2 ls1b">i</span></div><div class="t m0 x4c h17 y1d3 ff2 fsa fc0 sc0 ls1b ws0">e</div><div class="t m0 x29 h9 y1d2 ff7 fs5 fc0 sc0 ls1b ws0">(<span class="_ _2"></span><span class="ff2">x<span class="_ _a"></span></span>,<span class="_ _b"> </span><span class="ff2">t<span class="_ _18"></span></span>)<span class="_"> </span><span class="ff1 ls89">in<span class="_ _15"> </span>the<span class="_ _e"> </span>SP<span class="_ _2"></span>Me<span class="_ _e"> </span>m<span class="_ _2"></span>odel.</span></div><div class="t m0 xf4 h6 y1d4 ff2 fs4 fc0 sc0 ls2f ws0">[A2]:<span class="_ _11"> </span><span class="ff1 ls3b">The<span class="_ _13"> </span>exchange<span class="_ _12"> </span>current<span class="_ _13"> </span>density<span class="_ _12"> </span>term<span class="_ _12"> </span></span><span class="ls1b">i</span></div><div class="t m0 x43 hb y1d5 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x44 hd y1d6 ff1 fs8 fc0 sc0 ls1b ws0">0</div><div class="t m0 xf5 h6 y1d4 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _f"> </span><span class="ff1 ls53">can<span class="_ _13"> </span>be</span></div><div class="t m0 x36 h6 y1d7 ff1 fs4 fc0 sc0 ls3e ws0">approximated<span class="_ _d"> </span>by<span class="_ _12"> </span>its<span class="_ _12"> </span>averaged<span class="_ _d"> </span>value</div><div class="t m0 xf3 h10 y1d8 ff6 fs4 fc0 sc0 ls1b ws0">¯</div><div class="t m0 xf3 h8 y1d7 ff2 fs4 fc0 sc0 ls1b ws0">i</div><div class="t m0 x7f hb y1d9 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x43 hd y1da ff1 fs8 fc0 sc0 ls1b ws0">0</div><div class="t m0 xf6 h6 yae ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">t<span class="_ _18"></span></span>)<span class="ff1 ls8a">,w<span class="_ _2f"></span>h<span class="_ _2f"></span>i<span class="_ _2f"></span>c<span class="_ _30"></span>hi<span class="_ _2f"></span>s</span></div><div class="t m0 x36 h6 y1db ff1 fs4 fc0 sc0 ls4b ws0">independent<span class="_ _15"> </span>of<span class="_ _7"> </span><span class="ff2 ls1b">x<span class="_ _18"></span><span class="ff1">.</span></span></div><div class="t m0 xf4 h6 y1dc ff2 fs4 fc0 sc0 ls2f ws0">[A3]:<span class="_ _11"> </span><span class="ff1 ls1f">The<span class="_ _17"> </span>total<span class="_ _17"> </span>moles<span class="_ _f"> </span>o<span class="_ _2"></span>f<span class="_ _f"> </span>lithiu<span class="_ _2"></span>m<span class="_ _f"> </span>in<span class="_ _17"> </span>the<span class="_ _17"> </span>electrolyte<span class="_ _f"> </span></span><span class="ls1b">n</span></div><div class="t m0 xf7 hd y1dd ff1 fs8 fc0 sc0 ls7f ws0">Li<span class="ff7 ls1b">,<span class="ff2">e</span></span></div><div class="t m0 x36 h6 y1de ff1 fs4 fc0 sc0 ls47 ws0">and<span class="_ _7"> </span>in<span class="_ _7"> </span>the<span class="_ _7"> </span>solid<span class="_ _7"> </span>phase<span class="_ _7"> </span><span class="ff2 ls1b">n</span></div><div class="t m0 xf8 hd y1df ff1 fs8 fc0 sc0 ls7f ws0">Li<span class="ff7 ls1b">,<span class="ff2">s</span></span></div><div class="t m0 x60 h6 y1de ff1 fs4 fc0 sc0 ls4b ws0">are<span class="_ _7"> </span>both<span class="_ _7"> </span>conserved.<span class="_ _15"> </span>This</div><div class="t m0 x36 h6 y1e0 ff1 fs4 fc0 sc0 ls20 ws0">assumptio<span class="_ _2"></span>n,<span class="_ _15"> </span>to<span class="_ _2"></span>gether<span class="_ _c"> </span>with<span class="_ _7"> </span>Assumptio<span class="_ _2"></span>n<span class="_ _c"> </span>[A1],<span class="_ _c"> </span>makes<span class="_ _c"> </span>it</div><div class="t m0 x36 h6 y1e1 ff1 fs4 fc0 sc0 ls4a ws0">possibl<span class="_ _1"></span>e<span class="_ _15"> </span>to<span class="_"> </span>write<span class="_ _e"> </span>the<span class="_ _15"> </span>ﬂuxes<span class="_ _d"> </span><span class="ff2 ls1b">j</span></div><div class="t m0 x70 hb y1e2 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x71 hc y1e3 ff2 fs8 fc0 sc0 ls1b ws0">n</div><div class="t m0 xf9 h6 y1e4 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_"> </span><span class="ff1 ls3b">as<span class="_"> </span>proportional<span class="_"> </span>to</span></div><div class="t m0 x36 h6 y1e5 ff1 fs4 fc0 sc0 ls24 ws0">current<span class="_ _8"> </span><span class="ff2 ls1b">I<span class="_ _1b"> </span><span class="ff7">(</span>t<span class="_ _a"></span><span class="ff7">)<span class="ff1">.</span></span></span></div><div class="t m0 xf4 h6 y1e6 ff2 fs4 fc0 sc0 ls2f ws0">[A4]:<span class="_ _11"> </span><span class="ff1 ls84">The<span class="_ _8"> </span>constants<span class="_ _7"> </span><span class="ff7 ls1b">α</span></span></div><div class="t m0 x69 hc y1e7 ff2 fs8 fc0 sc0 ls1b ws0">a</div><div class="t m0 x3a h10 y1e8 ff6 fs4 fc0 sc0 ls1b ws0">=<span class="_ _7"> </span><span class="ff7">α</span></div><div class="t m0 x2d hc y1e7 ff2 fs8 fc0 sc0 ls1b ws0">c</div><div class="t m0 x61 h6 y1e8 ff1 fs4 fc0 sc0 ls49 ws0">(hereafter<span class="_ _7"> </span>denoted<span class="_ _8"> </span>simply<span class="_ _8"> </span>by</div><div class="t m0 x36 h6 y1e9 ff7 fs4 fc0 sc0 ls1b ws0">α<span class="_ _2"></span><span class="ff1">).<span class="_ _c"> </span>This<span class="_ _c"> </span>assump<span class="_ _2"></span>tion<span class="_ _15"> </span>is<span class="_ _c"> </span>almost<span class="_ _c"> </span>always<span class="_ _15"> </span>true<span class="_ _c"> </span>in<span class="_ _15"> </span>p<span class="_ _2"></span>ractice.</span></div><div class="t m0 xb h6 y1ea ff1 fs4 fc0 sc0 ls50 ws0">These<span class="_ _f"> </span>assump<span class="_ _2"></span>tions<span class="_ _f"> </span>ultimately<span class="_ _17"> </span>render<span class="_ _f"> </span>a<span class="_ _17"> </span>model<span class="_ _f"> </span>consisting</div><div class="t m0 xa h6 y1eb ff1 fs4 fc0 sc0 ls22 ws0">of:<span class="_ _f"> </span>1)<span class="_ _f"> </span>two<span class="_ _f"> </span>linear<span class="_ _17"> </span>spherical<span class="_ _f"> </span>dif<span class="_ _1"></span>fusion<span class="_ _f"> </span>PDEs<span class="_ _f"> </span>modeling<span class="_ _f"> </span>each</div><div class="t m0 xa h6 y1ec ff1 fs4 fc0 sc0 ls53 ws0">electrode’<span class="_ _3"></span>s<span class="_ _f"> </span>solid<span class="_ _f"> </span>concentration<span class="_ _f"> </span>dynamics;<span class="_ _f"> </span>2)<span class="_ _f"> </span>a<span class="_ _17"> </span>quasi-linear</div><div class="t m0 xa h6 y1ed ff1 fs4 fc0 sc0 ls27 ws0">dif<span class="_ _1"></span>fusion<span class="_ _c"> </span>equation<span class="_ _7"> </span>(across<span class="_ _7"> </span>three<span class="_ _7"> </span>domains)<span class="_ _7"> </span>modeling<span class="_ _7"> </span>the<span class="_ _7"> </span>elec-</div><div class="t m0 xa h6 y1ee ff1 fs4 fc0 sc0 ls47 ws0">trolyte<span class="_ _f"> </span>concentration<span class="_ _13"> </span>dynamics;<span class="_ _f"> </span>and<span class="_ _17"> </span>3)<span class="_ _17"> </span>a<span class="_ _17"> </span>nonlinear<span class="_ _f"> </span>output</div><div class="t m0 xa h6 y1ef ff1 fs4 fc0 sc0 ls47 ws0">function<span class="_ _17"> </span>mapping<span class="_ _17"> </span>boundary<span class="_ _17"> </span>values<span class="_ _f"> </span>of<span class="_ _11"> </span>solid<span class="_ _17"> </span>concentration,</div><div class="t m0 xa h6 y1f0 ff1 fs4 fc0 sc0 ls3e ws0">electrolyte<span class="_ _15"> </span>concentration,<span class="_ _c"> </span>and<span class="_ _c"> </span>c<span class="ls48">urrent<span class="_ _c"> </span>to<span class="_ _7"> </span>vol<span class="_ _1"></span>tage<span class="_ _c"> </span>(see<span class="_ _c"> </span>Fig.<span class="_ _c"> </span>2).</span></div><div class="t m0 xb h6 y1f1 ff1 fs4 fc0 sc0 ls53 ws0">W<span class="_ _3"></span>e<span class="_"> </span>now<span class="_ _e"> </span>introduce<span class="_ _e"> </span>the<span class="_ _15"> </span>resulting<span class="_ _e"> </span>SPMe<span class="_ _15"> </span>equations.<span class="_ _e"> </span>Whenev<span class="_ _1"></span>er</div><div class="t m0 xa h6 y1f2 ff1 fs4 fc0 sc0 ls3b ws0">Assumpti<span class="_ _1"></span>ons<span class="_ _8"> </span>[A1]–[A3]<span class="_ _7"> </span>remove<span class="_ _8"> </span>the<span class="_ _8"> </span>spatial<span class="_ _8"> </span>dependence<span class="_ _8"> </span>of<span class="_ _d"> </span>a</div><div class="t m0 xa h6 y1f3 ff1 fs4 fc0 sc0 ls1b ws0">variable,<span class="_ _8"> </span>an<span class="_ _d"> </span>overline<span class="_ _8"> </span>is<span class="_ _d"> </span>added<span class="_ _d"> </span>to<span class="_ _d"> </span>the<span class="_ _d"> </span>variable<span class="_ _8"> </span>name<span class="_ _d"> </span>to<span class="_ _d"> </span>av<span class="_ _1"></span>oid</div><div class="t m0 xa h6 y1f4 ff1 fs4 fc0 sc0 ls8b ws0">confusion.<span class="_ _13"> </span>The<span class="_ _f"> </span>ﬁrst<span class="_ _f"> </span>step<span class="_ _f"> </span>is<span class="_ _17"> </span>to<span class="_ _f"> </span>combine<span class="_ _f"> </span>Ass<span class="_ _1"></span>umption<span class="_ _f"> </span>[A1],</div><div class="t m0 xa h6 y1f5 ff1 fs4 fc0 sc0 ls69 ws0">ODE<span class="_ _8"> </span>(5),<span class="_ _8"> </span>and<span class="_ _8"> </span>its<span class="_ _d"> </span>boundary<span class="_ _7"> </span>conditions<span class="_ _8"> </span>(20)<span class="_ _8"> </span>to<span class="_ _8"> </span>express<span class="_ _7"> </span>molar</div><div class="t m0 xa h6 y1f6 ff1 fs4 fc0 sc0 ls47 ws0">ion<span class="_ _c"> </span>ﬂux<span class="_ _c"> </span>as<span class="_ _c"> </span>proportional<span class="_ _15"> </span>to<span class="_ _7"> </span>current</div><div class="t m0 x1b h10 y1f7 ff6 fs4 fc0 sc0 ls1b ws0">¯</div><div class="t m0 x52 h8 y1f8 ff2 fs4 fc0 sc0 ls1b ws0">j</div><div class="t m0 x16 hb y1f9 ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 xfa hc y1fa ff2 fs8 fc0 sc0 ls1b ws0">n</div><div class="t m0 x17 h8 y1fb ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">t<span class="_ _18"></span></span>)<span class="_ _e"> </span><span class="ff6 ls8c">=−</span></div><div class="t m0 x28 h8 y1fc ff2 fs4 fc0 sc0 ls1b ws0">I<span class="_ _b"> </span><span class="ff7">(</span>t<span class="_ _18"></span><span class="ff7">)</span></div><div class="t m0 xfb h8 y1fd ff2 fs4 fc0 sc0 ls8d ws0">Fa</div><div class="t m0 x39 hb y1fe ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 x55 h8 y1fd ff2 fs4 fc0 sc0 ls1b ws0">L</div><div class="t m0 x3b hb y1fe ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 x2c he y1fb ff7 fs4 fc0 sc0 ls1b ws0">,</div><div class="t m0 xf8 h10 y1ff ff6 fs4 fc0 sc0 ls1b ws0">¯</div><div class="t m0 xf8 h8 y1fb ff2 fs4 fc0 sc0 ls1b ws0">j</div><div class="t m0 x40 hb y1f9 ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 x6d hc y1fa ff2 fs8 fc0 sc0 ls1b ws0">n</div><div class="t m0 xfc h8 y1fb ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">t<span class="_ _18"></span></span>)<span class="_ _e"> </span><span class="ff6">=</span></div><div class="t m0 xfd h8 y1fc ff2 fs4 fc0 sc0 ls1b ws0">I<span class="_ _b"> </span><span class="ff7">(</span>t<span class="_ _18"></span><span class="ff7">)</span></div><div class="t m0 xfe h8 y1fd ff2 fs4 fc0 sc0 ls8d ws0">Fa</div><div class="t m0 xf2 hb y1fe ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 x44 h8 y1fd ff2 fs4 fc0 sc0 ls1b ws0">L</div><div class="t m0 x46 hb y1fe ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 x59 h6 y1fb ff7 fs4 fc0 sc0 ls1b ws0">.<span class="_ _31"> </span><span class="ff1 ls84">(22)</span></div><div class="t m0 xa h6 y200 ff1 fs4 fc0 sc0 ls6d ws0">Note<span class="_"> </span>the<span class="_"> </span>ionic<span class="_"> </span>current<span class="_ _0"> </span><span class="ff2 ls1b">i</span></div><div class="t m0 x28 hc y113 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x39 h6 y201 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _e"> </span><span class="ff1 ls69">has<span class="_ _e"> </span>the<span class="_ _e"> </span>trapezoidal<span class="_"> </span>shape<span class="_"> </span>sho<span class="_ _1"></span>wn</span></div><div class="t m0 xa h6 y202 ff1 fs4 fc0 sc0 ls5d ws0">in<span class="_ _f"> </span>Fig.<span class="_ _f"> </span>3.<span class="_ _17"> </span>Apply</div><div class="t m0 x27 h8 y203 ff2 fs4 fc0 sc0 ls1b ws0">j</div><div class="t m0 xff hb y204 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 xff hc y205 ff2 fs8 fc0 sc0 ls1b ws0">n</div><div class="t m0 x6c h6 y203 ff1 fs4 fc0 sc0 ls45 ws0">in<span class="_ _f"> </span>(22)<span class="_ _17"> </span>to<span class="_ _17"> </span>boundary<span class="_ _13"> </span>conditions<span class="_ _f"> </span>(10)</div><div class="t m0 xa h6 y206 ff1 fs4 fc0 sc0 ls87 ws0">and<span class="_ _d"> </span>(11)<span class="_ _d"> </span>and<span class="_ _d"> </span>Assumpt<span class="_ _1"></span>ion<span class="_ _d"> </span>[A1]<span class="_ _8"> </span>to<span class="_ _12"> </span>deri<span class="_ _1"></span>ve<span class="_ _8"> </span>the<span class="_ _d"> </span>solid<span class="_ _d"> </span>dif<span class="_ _1"></span>fusion</div><div class="t m0 xa h6 y207 ff1 fs4 fc0 sc0 ls47 ws0">equations</div><div class="t m0 x100 he y208 ff7 fs4 fc0 sc0 ls1b ws0">∂</div><div class="t m0 x101 h8 y209 ff2 fs4 fc0 sc0 ls1b ws0">c</div><div class="t m0 x50 hb y20a ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x50 hc y20b ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 x35 h8 y20c ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">t</span></div><div class="t m0 x36 h8 y20d ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">r<span class="_ _1"></span><span class="ff7">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _18"></span></span>)<span class="_"> </span><span class="ff6">=</span></span></span></div><div class="t m0 x102 h6 y20e ff1 fs4 fc0 sc0 ls1b ws0">1</div><div class="t m0 x103 h8 y20c ff2 fs4 fc0 sc0 ls1b ws0">r</div><div class="t m0 x37 hd y20f ff1 fs8 fc0 sc0 ls1b ws0">2</div><div class="t m0 x104 he y209 ff7 fs4 fc0 sc0 ls1b ws0">∂</div><div class="t m0 x105 h8 y20c ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _a"></span><span class="ff2">r</span></div><div class="t m0 x106 hf y210 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x39 h8 y20d ff2 fs4 fc0 sc0 ls1b ws0">D</div><div class="t m0 x107 hb y133 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x55 hc y211 ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 x108 h8 y212 ff2 fs4 fc0 sc0 ls1b ws0">r</div><div class="t m0 x3c hd y133 ff1 fs8 fc0 sc0 ls1b ws0">2</div><div class="t m0 x109 h8 y209 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">c</span></div><div class="t m0 x61 hb y20a ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x61 hc y20b ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 x81 h8 y20c ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _a"></span><span class="ff2">r</span></div><div class="t m0 x6e h8 y20d ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">r<span class="_ _1"></span><span class="ff7">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _18"></span></span>)</span></span></div><div class="t m0 x1c hf y210 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x80 h6 y20d ff1 fs4 fc0 sc0 ls84 ws0">(23)</div><div class="t m0 x10a he y213 ff7 fs4 fc0 sc0 ls1b ws0">∂</div><div class="t m0 x35 h8 y214 ff2 fs4 fc0 sc0 ls1b ws0">c</div><div class="t m0 x22 hb y215 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x22 hc y216 ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 x20 h8 y217 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _a"></span><span class="ff2">r</span></div><div class="t m0 x10b h6 y218 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff1">0</span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _18"></span></span>)<span class="_"> </span><span class="ff6">=<span class="_"> </span><span class="ff1">0</span></span>,</div><div class="t m0 xfb he y219 ff7 fs4 fc0 sc0 ls1b ws0">∂</div><div class="t m0 x10c h8 y214 ff2 fs4 fc0 sc0 ls1b ws0">c</div><div class="t m0 x6c hb y215 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x6c hc y216 ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 x10d h8 y217 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _a"></span><span class="ff2">r</span></div><div class="t m0 x4d hf y21a ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x2a h8 y218 ff2 fs4 fc0 sc0 ls1b ws0">R</div><div class="t m0 x3d hb y21b ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x2b hc y21c ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 x10e h8 y21d ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t</span></div><div class="t m0 x82 hf y21e ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x40 h10 y21d ff6 fs4 fc0 sc0 ls8e ws0">=±</div><div class="t m0 xfd h6 y21f ff1 fs4 fc0 sc0 ls1b ws0">1</div><div class="t m0 x10f h8 y220 ff2 fs4 fc0 sc0 ls1b ws0">D</div><div class="t m0 x110 hb y221 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x110 hc y222 ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 x15 h8 y220 ff2 fs4 fc0 sc0 ls8d ws0">Fa</div><div class="t m0 x58 hb y223 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x111 h8 y220 ff2 fs4 fc0 sc0 ls1b ws0">L</div><div class="t m0 xf6 hb y223 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x112 h6 y21d ff2 fs4 fc0 sc0 ls1b ws0">I<span class="_ _b"> </span><span class="ff7">(</span>t<span class="_ _18"></span><span class="ff7 ls63">).<span class="_ _32"> </span><span class="ff1 ls84">(24)</span></span></div><div class="t m0 xe ha y6d ff5 fs7 fc0 sc0 ls1b ws0">Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on January 19,2022 at 04:38:28 UTC from IEEE Xplore. Restrictions apply. </div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>

<div id="pf4" class="pf w0 h0" data-page-no="4"><div class="pc pc4 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6250a1e374bc5c01056d2748/bg4.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls59 ws0">456<span class="_ _4"> </span>IEEE<span class="_ _0"> </span>TRANSACTIONS<span class="_ _14"> </span>ON<span class="_ _0"> </span>CONT<span class="_ _2"></span>ROL<span class="_ _14"> </span>SYSTEMS<span class="_ _0"> </span>TECHNOL<span class="_ _2"></span>OGY<span class="_ _3"></span>,<span class="_ _14"> </span>VOL.<span class="_ _0"> </span>25,<span class="_ _0"> </span>NO.<span class="_ _0"> </span>2,<span class="_ _0"> </span>MARCH<span class="_ _0"> </span>2017</div><div class="t m0 x1 h6 y6e ff1 fs4 fc0 sc0 ls48 ws0">Next,<span class="_ _15"> </span>deriv<span class="_ _1"></span>e<span class="_ _c"> </span>the<span class="_ _c"> </span>electrolyte<span class="_ _15"> </span>dif<span class="_ _1"></span>fusion<span class="_ _15"> </span>equations<span class="_ _15"> </span>by<span class="_ _c"> </span>combining</div><div class="t m0 x1 h6 y6f ff1 fs4 fc0 sc0 ls21 ws0">PDE<span class="_ _c"> </span>(2)<span class="_ _c"> </span>with<span class="_ _c"> </span>(5),<span class="_ _c"> </span>(22),<span class="_ _c"> </span>and<span class="_ _c"> </span>Assumption<span class="_ _15"> </span>[A1]</div><div class="t m0 xc6 h8 y224 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">c</span></div><div class="t m0 x113 hb y225 ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 x99 hc y226 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x114 h8 y227 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">t</span></div><div class="t m0 x115 h8 y228 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _14"> </span><span class="ff6">=</span></div><div class="t m0 xa5 he y229 ff7 fs4 fc0 sc0 ls1b ws0">∂</div><div class="t m0 x116 h8 y227 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">x</span></div><div class="t m0 x117 hf y22a ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 xa6 h8 y22b ff2 fs4 fc0 sc0 ls1b ws0">D</div><div class="t m0 x3 hd y225 ff1 fs8 fc0 sc0 ls71 ws0">eff</div><div class="t m0 x3 hc y226 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xa7 h8 y22c ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">c</span></div><div class="t m0 xde hb y225 ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 xde hc y226 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xa8 he y22c ff7 fs4 fc0 sc0 ls1b ws0">)</div><div class="t m0 x9f he y22d ff7 fs4 fc0 sc0 ls1b ws0">ε</div><div class="t m0 xbd hb y22e ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 xbd hc y22f ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xe9 h8 y22c ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">c</span></div><div class="t m0 xe4 hb y225 ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 x118 hc y226 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x119 h8 y227 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">x</span></div><div class="t m0 x11a h8 y228 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)</div><div class="t m0 xad hf y230 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 xea h10 y228 ff6 fs4 fc0 sc0 ls1b ws0">+</div><div class="t m0 xcf hf y231 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 xd0 h6 y232 ff1 fs4 fc0 sc0 ls1b ws0">1<span class="_ _0"> </span><span class="ff6">−<span class="_ _14"> </span><span class="ff2">t</span></span></div><div class="t m0 x11b hd y233 ff1 fs8 fc0 sc0 ls1b ws0">0</div><div class="t m0 x11c hc y234 ff2 fs8 fc0 sc0 ls1b ws0">c</div><div class="t m0 xc1 hf y235 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 xcf he y22d ff7 fs4 fc0 sc0 ls1b ws0">ε</div><div class="t m0 xbf hb y22e ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 x11d hc y22f ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x91 h8 y22d ff2 fs4 fc0 sc0 ls76 ws0">FL</div><div class="t m0 x11e hb y236 ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 x11f h6 y237 ff2 fs4 fc0 sc0 ls1b ws0">I<span class="_ _b"> </span><span class="ff7">(</span>t<span class="_ _18"></span><span class="ff7">)<span class="_ _2e"> </span><span class="ff1 ls84">(25)</span></span></div><div class="t m0 xec h8 y238 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">c</span></div><div class="t m0 x120 hd y239 ff1 fs8 fc0 sc0 ls78 ws0">sep</div><div class="t m0 x120 hc y23a ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x6 h8 y23b ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">t</span></div><div class="t m0 x115 h8 y23c ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _14"> </span><span class="ff6">=</span></div><div class="t m0 xa5 he y23d ff7 fs4 fc0 sc0 ls1b ws0">∂</div><div class="t m0 x116 h8 y23b ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">x</span></div><div class="t m0 x117 hf y23e ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 xa6 h8 y23f ff2 fs4 fc0 sc0 ls1b ws0">D</div><div class="t m0 x3 hd y240 ff1 fs8 fc0 sc0 ls71 ws0">eff</div><div class="t m0 x3 hc y241 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xa7 h8 y242 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">c</span></div><div class="t m0 xde hd y239 ff1 fs8 fc0 sc0 ls78 ws0">sep</div><div class="t m0 xde hc y23a ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x121 he y242 ff7 fs4 fc0 sc0 ls1b ws0">)</div><div class="t m0 x9f he y243 ff7 fs4 fc0 sc0 ls1b ws0">ε</div><div class="t m0 xbd hd y244 ff1 fs8 fc0 sc0 ls78 ws0">sep</div><div class="t m0 xbd hc y245 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xe3 h8 y242 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">c</span></div><div class="t m0 x122 hd y239 ff1 fs8 fc0 sc0 ls78 ws0">sep</div><div class="t m0 x122 hc y23a ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x123 h8 y23b ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">x</span></div><div class="t m0 xb4 h8 y23c ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)</div><div class="t m0 xda hf y246 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 xd8 h6 y23c ff1 fs4 fc0 sc0 ls84 ws0">(26)</div><div class="t m0 xc6 h8 y247 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">c</span></div><div class="t m0 x113 hb y248 ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 x99 hc y249 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x114 h8 y24a ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">t</span></div><div class="t m0 x115 h8 y24b ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _14"> </span><span class="ff6">=</span></div><div class="t m0 xa5 he y24c ff7 fs4 fc0 sc0 ls1b ws0">∂</div><div class="t m0 x116 h8 y24a ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">x</span></div><div class="t m0 x117 hf y24d ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 xa6 h8 y24c ff2 fs4 fc0 sc0 ls1b ws0">D</div><div class="t m0 x3 hd y248 ff1 fs8 fc0 sc0 ls71 ws0">eff</div><div class="t m0 x3 hc y249 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xa7 h8 y24e ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">c</span></div><div class="t m0 xde hb y248 ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 xde hc y249 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xa8 he y24e ff7 fs4 fc0 sc0 ls1b ws0">)</div><div class="t m0 x9f he y24f ff7 fs4 fc0 sc0 ls1b ws0">ε</div><div class="t m0 xbd hb y250 ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 xbd hc y251 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xe9 h8 y24e ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">c</span></div><div class="t m0 x118 hb y248 ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 x118 hc y249 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x119 h8 y24a ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">x</span></div><div class="t m0 x11a h8 y24b ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)</div><div class="t m0 xad hf y24d ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 xea h10 y24b ff6 fs4 fc0 sc0 ls1b ws0">−</div><div class="t m0 xcf hf y252 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 xd0 h6 y253 ff1 fs4 fc0 sc0 ls1b ws0">1<span class="_ _0"> </span><span class="ff6">−<span class="_ _14"> </span><span class="ff2">t</span></span></div><div class="t m0 x11b hd y254 ff1 fs8 fc0 sc0 ls1b ws0">0</div><div class="t m0 x11c hc y255 ff2 fs8 fc0 sc0 ls1b ws0">c</div><div class="t m0 xc1 hf y256 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 xcf he y24f ff7 fs4 fc0 sc0 ls1b ws0">ε</div><div class="t m0 x11d hb y250 ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 x11d hc y251 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x91 h8 y24f ff2 fs4 fc0 sc0 ls76 ws0">FL</div><div class="t m0 x11e hb y257 ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 x11f h6 y258 ff2 fs4 fc0 sc0 ls1b ws0">I<span class="_ _b"> </span><span class="ff7">(</span>t<span class="_ _18"></span><span class="ff7">)<span class="_ _2e"> </span><span class="ff1 ls84">(27)</span></span></div><div class="t m0 x1 h6 y259 ff1 fs4 fc0 sc0 ls82 ws0">with<span class="_ _c"> </span>the<span class="_ _7"> </span>same<span class="_ _c"> </span>boundary<span class="_ _15"> </span>conditions<span class="_ _c"> </span>as<span class="_ _7"> </span>(12)–(16).</div><div class="t m0 x6 h6 y25a ff1 fs4 fc0 sc0 ls48 ws0">Next,<span class="_ _c"> </span>we<span class="_ _7"> </span>derive<span class="_ _c"> </span>the<span class="_ _7"> </span>nonlinear<span class="_ _c"> </span>output<span class="_ _7"> </span>function<span class="_ _c"> </span>for<span class="_ _7"> </span>terminal</div><div class="t m0 x1 h6 y25b ff1 fs4 fc0 sc0 ls8b ws0">vol<span class="_ _1"></span>tage.<span class="_ _15"> </span>From<span class="_ _15"> </span>(21),<span class="_ _15"> </span>we<span class="_ _15"> </span>note<span class="_ _15"> </span>that<span class="_ _15"> </span>the<span class="_ _15"> </span>voltage<span class="_ _15"> </span><span class="ff2 ls1b">V<span class="_ _1b"> </span><span class="ff7">(</span>t<span class="_ _a"></span><span class="ff7">)<span class="_"> </span></span></span><span class="ls87">depends<span class="_ _15"> </span>on</span></div><div class="t m0 x1 h6 y25c ff1 fs4 fc0 sc0 ls3c ws0">the<span class="_"> </span>solid<span class="_"> </span>potential<span class="_"> </span>at<span class="_"> </span>the<span class="_"> </span>current<span class="_"> </span>collectors<span class="_ _0"> </span><span class="ff7 ls1b">φ</span></div><div class="t m0 xef hb y25d ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 xef hc y25e ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 x124 h6 y25f ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="ff1 ls4f">.<span class="_"> </span>Therefore,</span></div><div class="t m0 x1 h6 y260 ff1 fs4 fc0 sc0 ls8f ws0">we<span class="_ _c"> </span>so<span class="_ _2"></span>lve<span class="_ _c"> </span>(8)<span class="_ _7"> </span>in<span class="_ _7"> </span>terms<span class="_ _7"> </span>of<span class="_ _c"> </span><span class="ff7 ls1b">φ</span></div><div class="t m0 x125 hc y261 ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 xcc h6 y262 ff1 fs4 fc0 sc0 ls63 ws0">and<span class="_ _c"> </span>spatially<span class="_ _7"> </span>averaged<span class="_ _15"> </span>q<span class="_ _2"></span>uantities</div><div class="t m0 x98 he y263 ff7 fs4 fc0 sc0 ls1b ws0">φ</div><div class="t m0 x114 hb y264 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x114 hc y265 ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 x99 h8 y266 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _15"> </span><span class="ff6 ls90">=¯<span class="_ _33"></span><span class="ff7 ls1b">η</span></span></div><div class="t m0 xa3 hb y264 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 xba h8 y266 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">t<span class="_ _18"></span></span>)<span class="_ _14"> </span><span class="ff6">+</span></div><div class="t m0 xbb h10 y267 ff6 fs4 fc0 sc0 ls1b ws0">¯</div><div class="t m0 x126 he y266 ff7 fs4 fc0 sc0 ls1b ws0">φ</div><div class="t m0 xe2 hb y264 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 xe2 hc y265 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x127 h8 y266 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _0"> </span><span class="ff6">+<span class="_ _14"> </span><span class="ff2">U</span></span></div><div class="t m0 xdf hb y264 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 xac hf y268 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 xe1 h8 y269 ff6 fs4 fc0 sc0 ls1b ws0">¯<span class="_ _30"></span><span class="ff2">c</span></div><div class="t m0 x128 hb y264 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x128 hd y265 ff1 fs8 fc0 sc0 ls78 ws0">ss</div><div class="t m0 xea h8 y266 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">t<span class="_ _18"></span></span>)</div><div class="t m0 xb6 hf y26a ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x129 h8 y266 ff6 fs4 fc0 sc0 ls1b ws0">+<span class="_ _e"> </span><span class="ff2 ls7b">FR</span></div><div class="t m0 xd4 hb y26b ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x94 hc y26c ff2 fs8 fc0 sc0 ls1b ws0">f</div><div class="t m0 x12a h10 y26d ff6 fs4 fc0 sc0 ls1b ws0">¯</div><div class="t m0 x12a h8 y26e ff2 fs4 fc0 sc0 ls1b ws0">j</div><div class="t m0 x12b hb y264 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x12c hc y265 ff2 fs8 fc0 sc0 ls1b ws0">n</div><div class="t m0 x12d h8 y266 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">t<span class="_ _18"></span></span><span class="ls63">).</span></div><div class="t m0 xd8 h6 y26f ff1 fs4 fc0 sc0 ls84 ws0">(28)</div><div class="t m0 x1 h6 y270 ff1 fs4 fc0 sc0 ls5a ws0">Next,<span class="_ _34"> </span>we<span class="_ _34"> </span>derive<span class="_ _34"> </span>each<span class="_ _34"> </span>term<span class="_ _2e"> </span>on<span class="_ _34"> </span>the<span class="_ _2e"> </span>right-hand<span class="_ _34"> </span>side</div><div class="t m0 x1 h6 y271 ff1 fs4 fc0 sc0 ls8b ws0">of<span class="_ _6"> </span>(28).<span class="_ _35"> </span>Overpot<span class="_ _1"></span>ential<span class="_ _34"> </span><span class="ff6 ls1b">¯<span class="_ _2f"></span><span class="ff7">η</span></span></div><div class="t m0 x8e hb y272 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 xe9 h6 y273 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">t<span class="_ _18"></span></span>)<span class="_ _6"> </span><span class="ff1 ls6f">is<span class="_ _36"> </span>found<span class="_ _35"> </span>by<span class="_ _6"> </span>solving<span class="_ _35"> </span>the</span></div><div class="t m0 x1 h6 y274 ff1 fs4 fc0 sc0 ls26 ws0">Butler–V<span class="_ _9"></span>olmer<span class="_ _1a"> </span>equation<span class="_ _16"> </span>(6)<span class="_ _16"> </span>in<span class="_ _16"> </span>ter<span class="_ _2"></span>ms<span class="_ _1a"> </span>of<span class="_ _5"> </span><span class="ff6 ls1b">¯<span class="_ _2f"></span><span class="ff7">η</span></span></div><div class="t m0 x12e hb y275 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x12f h6 y276 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">t<span class="_ _18"></span></span>)<span class="ff1 ls5d">,<span class="_ _16"> </span>applying</span></div><div class="t m0 x1 h6 y277 ff1 fs4 fc0 sc0 ls60 ws0">Assumption<span class="_ _2"></span>s<span class="_ _c"> </span>[A1]<span class="_ _2"></span>,<span class="_ _c"> </span>[A2],<span class="_ _7"> </span>and<span class="_ _7"> </span>[A4],<span class="_ _c"> </span>and<span class="_ _7"> </span>substitutin<span class="_ _2"></span>g<span class="_ _c"> </span>(2<span class="_ _2"></span>2)</div><div class="t m0 x9b h10 y278 ff6 fs4 fc0 sc0 ls1b ws0">¯<span class="_ _2f"></span><span class="ff7">η</span></div><div class="t m0 xb8 hb y279 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x130 h8 y27a ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">t<span class="_ _18"></span></span>)<span class="_ _e"> </span><span class="ff6">=</span></div><div class="t m0 xbc h6 y27b ff1 fs4 fc0 sc0 ls91 ws0">RT</div><div class="t m0 xdb h8 y27c ff7 fs4 fc0 sc0 ls1b ws0">α<span class="_ _18"></span><span class="ff2">F</span></div><div class="t m0 xa7 h6 y27d ff1 fs4 fc0 sc0 ls92 ws0">sinh</div><div class="t m0 xcc hd y279 ff6 fs8 fc0 sc0 ls1b ws0">−<span class="ff1">1</span></div><div class="t m0 x123 hf y27e ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 xac h8 y27f ff6 fs4 fc0 sc0 ls1b ws0">∓<span class="_ _a"></span><span class="ff2">I<span class="_ _1b"> </span><span class="ff7">(</span>t<span class="_ _18"></span><span class="ff7">)</span></span></div><div class="t m0 x131 h6 y280 ff1 fs4 fc0 sc0 ls1b ws0">2<span class="ff2">a</span></div><div class="t m0 xaa hb y281 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 xb3 h8 y280 ff2 fs4 fc0 sc0 ls1b ws0">L</div><div class="t m0 xad hb y281 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 xce h10 y282 ff6 fs4 fc0 sc0 ls1b ws0">¯</div><div class="t m0 xce h8 y283 ff2 fs4 fc0 sc0 ls1b ws0">i</div><div class="t m0 xb0 hb y1d3 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 xb0 hd y284 ff1 fs8 fc0 sc0 ls1b ws0">0</div><div class="t m0 xef h8 y280 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">t<span class="_ _18"></span></span>)</div><div class="t m0 xd1 hf y285 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x132 h6 y286 ff7 fs4 fc0 sc0 ls1b ws0">.<span class="_ _37"> </span><span class="ff1 ls84">(29)</span></div><div class="t m0 x1 h6 y287 ff1 fs4 fc0 sc0 ls41 ws0">The<span class="_ _11"> </span>electrolyte<span class="_ _17"> </span>p<span class="_ _2"></span>otential</div><div class="t m0 x8e h10 y288 ff6 fs4 fc0 sc0 ls1b ws0">¯</div><div class="t m0 x133 he y287 ff7 fs4 fc0 sc0 ls1b ws0">φ</div><div class="t m0 xcc hb y289 ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 xcc hc y28a ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xe3 h6 y28b ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _11"> </span><span class="ff1 ls8b">is<span class="_ _11"> </span>found<span class="_ _17"> </span>by<span class="_ _11"> </span>inte<span class="_ _1"></span>grating</span></div><div class="t m0 x1 h6 y28c ff1 fs4 fc0 sc0 ls55 ws0">ODE<span class="_ _c"> </span>(<span class="_ _2"></span>4)<span class="_ _c"> </span>with<span class="_ _7"> </span>resp<span class="_ _2"></span>ect<span class="_ _c"> </span>to<span class="_ _8"> </span><span class="ff2 ls1b">x<span class="_ _12"> </span></span><span class="ls62">across<span class="_ _15"> </span>the<span class="_ _c"> </span>entire<span class="_ _7"> </span>cell<span class="_ _c"> </span>width</span></div><div class="t m0 xc5 hf y28d ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x134 hd y28e ff1 fs8 fc0 sc0 ls1b ws0">0</div><div class="t m0 x120 h15 y18 ff6 fs9 fc0 sc0 ls1b ws0">+</div><div class="t m0 x98 hd y28f ff1 fs8 fc0 sc0 ls1b ws0">0</div><div class="t m0 x6 h15 y290 ff6 fs9 fc0 sc0 ls1b ws0">−</div><div class="t m0 x135 he y291 ff7 fs4 fc0 sc0 ls73 ws0">∂φ</div><div class="t m0 x2 hc y292 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x115 h8 y293 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">x</span></div><div class="t m0 xc8 h8 y294 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="ff2 ls93">dx<span class="_"> </span></span><span class="ff6">=</span></div><div class="t m0 x7 hf y295 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 xbd hd y28e ff1 fs8 fc0 sc0 ls1b ws0">0</div><div class="t m0 xdd h15 y18 ff6 fs9 fc0 sc0 ls1b ws0">+</div><div class="t m0 x9f hd y28f ff1 fs8 fc0 sc0 ls1b ws0">0</div><div class="t m0 xe0 h15 y290 ff6 fs9 fc0 sc0 ls1b ws0">−</div><div class="t m0 x125 h8 y291 ff2 fs4 fc0 sc0 ls1b ws0">i</div><div class="t m0 xa8 hb y28e ff6 fs8 fc0 sc0 ls1b ws0">±</div><div class="t m0 x136 hc y296 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x137 h8 y291 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)</div><div class="t m0 xb2 h8 y293 ff7 fs4 fc0 sc0 ls81 ws0">κ(<span class="_ _3"></span><span class="ff2 ls1b">c</span></div><div class="t m0 x123 hc y297 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x138 he y293 ff7 fs4 fc0 sc0 ls1b ws0">)</div><div class="t m0 xab h8 y294 ff2 fs4 fc0 sc0 ls93 ws0">dx<span class="_ _14"> </span><span class="ff6 ls1b">+</span></div><div class="t m0 xce hf y295 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 xb6 hd y28e ff1 fs8 fc0 sc0 ls1b ws0">0</div><div class="t m0 x11d h15 y18 ff6 fs9 fc0 sc0 ls1b ws0">+</div><div class="t m0 xeb hd y28f ff1 fs8 fc0 sc0 ls1b ws0">0</div><div class="t m0 xb6 h15 y290 ff6 fs9 fc0 sc0 ls1b ws0">−</div><div class="t m0 xd1 h6 y291 ff1 fs4 fc0 sc0 ls5d ws0">2R<span class="_ _3"></span>T</div><div class="t m0 x139 h8 y293 ff2 fs4 fc0 sc0 ls1b ws0">F</div><div class="t m0 xd4 hf y298 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x13a h6 y294 ff1 fs4 fc0 sc0 ls1b ws0">1<span class="_ _0"> </span><span class="ff6">−<span class="_ _14"> </span><span class="ff2">t</span></span></div><div class="t m0 xd6 hd y299 ff1 fs8 fc0 sc0 ls1b ws0">0</div><div class="t m0 x97 hc y29a ff2 fs8 fc0 sc0 ls1b ws0">c</div><div class="t m0 x13b hf y29b ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x7 h10 y29c ff6 fs4 fc0 sc0 ls1b ws0">×</div><div class="t m0 xe0 hf y29d ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x8c h6 y29c ff1 fs4 fc0 sc0 ls1b ws0">1<span class="_ _0"> </span><span class="ff6">+</span></div><div class="t m0 x119 h6 y29e ff2 fs4 fc0 sc0 ls1b ws0">d<span class="_"> </span><span class="ff1 ls20">ln<span class="_ _8"> </span></span>f</div><div class="t m0 xdf hc y19e ff2 fs8 fc0 sc0 ls1b ws0">c<span class="ff7">/</span>a</div><div class="t m0 x5 h6 y29f ff2 fs4 fc0 sc0 ls1b ws0">d<span class="_"> </span><span class="ff1 ls20">ln<span class="_ _1b"> </span></span>c</div><div class="t m0 xcd hc y2a0 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x13c h8 y2a1 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)</div><div class="t m0 xd0 hf y2a2 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 xd1 h6 y2a3 ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _0"> </span><span class="ff1 ls20">ln<span class="_ _1b"> </span></span><span class="ff2">c</span></div><div class="t m0 x11f hc y2a4 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xd2 h8 y29f ff7 fs4 fc0 sc0 ls1b ws0">∂<span class="_ _18"></span><span class="ff2">x</span></div><div class="t m0 x13d h8 y2a5 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="ff2 ls93">dx</span>.</div><div class="t m0 xd8 h6 y2a6 ff1 fs4 fc0 sc0 ls84 ws0">(30)</div><div class="t m0 x1 h6 y2a7 ff1 fs4 fc0 sc0 ls50 ws0">In<span class="_ _7"> </span>or<span class="_ _2"></span>der<span class="_ _7"> </span>to<span class="_ _8"> </span>analytically<span class="_ _8"> </span>integrate<span class="_ _7"> </span>(30<span class="_ _2"></span>),<span class="_ _7"> </span>we<span class="_ _8"> </span>further<span class="_ _7"> </span>assume<span class="_ _8"> </span>the</div><div class="t m0 x1 h6 y2a8 ff1 fs4 fc0 sc0 ls50 ws0">following.</div><div class="t m0 xec h6 y2a9 ff2 fs4 fc0 sc0 ls2f ws0">[A5]:<span class="_ _11"> </span><span class="ff1 ls45">The<span class="_ _d"> </span>term<span class="_ _8"> </span></span><span class="ls1b">k</span></div><div class="t m0 xa6 hc y2aa ff2 fs8 fc0 sc0 ls1b ws0">f</div><div class="t m0 xd9 h8 y2ab ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)</div><div class="t m0 x136 he y2ac ff7 fs4 fc0 sc0 ls1b ws0">.</div><div class="t m0 x133 h6 y2ab ff6 fs4 fc0 sc0 ls1b ws0">=<span class="_ _8"> </span><span class="ff7">(<span class="ff1">1<span class="_ _e"> </span></span></span>+<span class="_ _e"> </span><span class="ff7">(<span class="ff2">d<span class="_"> </span><span class="ff1 ls20">ln<span class="_ _8"> </span></span>f</span></span></div><div class="t m0 xb0 hc y2aa ff2 fs8 fc0 sc0 ls1b ws0">c<span class="ff7">/</span>a</div><div class="t m0 xc4 h6 y2ab ff7 fs4 fc0 sc0 ls1b ws0">/<span class="ff2">d<span class="_"> </span><span class="ff1 ls20">ln<span class="_ _1b"> </span></span>c</span></div><div class="t m0 x13e hc y2aa ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x13f h6 y2ab ff7 fs4 fc0 sc0 ls63 ws0">)(<span class="ff2 ls1b">x<span class="_ _b"></span><span class="ff7">,<span class="_ _1b"> </span></span>t<span class="_ _a"></span></span>))<span class="_ _d"> </span><span class="ff1 ls20">is</span></div><div class="t m0 xed h6 y2ad ff1 fs4 fc0 sc0 ls4b ws0">approximately<span class="_ _15"> </span>consta<span class="_ _1"></span>nt<span class="_ _c"> </span>in<span class="_ _7"> </span><span class="ff2 ls1b">x<span class="_ _b"></span></span><span class="ls25">,<span class="_ _c"> </span>i.e.,</span></div><div class="t m0 xbe h10 y2ae ff6 fs4 fc0 sc0 ls1b ws0">¯</div><div class="t m0 xeb h8 y2ad ff2 fs4 fc0 sc0 ls1b ws0">k</div><div class="t m0 x140 hc y2af ff2 fs8 fc0 sc0 ls1b ws0">f</div><div class="t m0 xc4 h8 y2b0 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">t<span class="_ _18"></span></span>)<span class="_ _e"> </span><span class="ff6">≈<span class="_ _15"> </span><span class="ff2">k</span></span></div><div class="t m0 x95 hc y2af ff2 fs8 fc0 sc0 ls1b ws0">f</div><div class="t m0 x96 h6 y2b0 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">x<span class="_ _b"></span></span>,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="ff1">.</span></div><div class="t m0 xec h6 y2b1 ff2 fs4 fc0 sc0 ls2f ws0">[A6]:<span class="_ _11"> </span><span class="ff1 ls45">The<span class="_ _d"> </span>term<span class="_ _8"> </span><span class="ff7 ls81">κ(<span class="_ _3"></span><span class="ff2 ls1b">c</span></span></span></div><div class="t m0 xdc hc y2b2 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xb1 h6 y2b3 ff7 fs4 fc0 sc0 ls1b ws0">)<span class="_ _d"> </span><span class="ff1 ls4b">is<span class="_ _8"> </span>approximately<span class="_ _c"> </span>constant<span class="_ _8"> </span>in<span class="_ _d"> </span></span><span class="ff2">c</span></div><div class="t m0 x141 hc y2b2 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x97 h6 y2b3 ff1 fs4 fc0 sc0 ls94 ws0">,i<span class="_ _38"></span>.<span class="_ _38"></span>e<span class="_ _38"></span>.<span class="_ _38"></span>,</div><div class="t m0 xed h8 y2b4 ff7 fs4 fc0 sc0 ls1b ws0">κ<span class="_ _7"> </span><span class="ff6">≈<span class="_ _e"> </span></span><span class="ls81">κ(<span class="_ _3"></span><span class="ff2 ls1b">c</span></span></div><div class="t m0 xba hc y2b5 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xa4 h6 y2b4 ff7 fs4 fc0 sc0 ls1b ws0">)<span class="ff1">.</span></div><div class="t m0 x1 h6 y2b6 ff1 fs4 fc0 sc0 ls47 ws0">This<span class="_ _c"> </span>yields<span class="_ _15"> </span>the<span class="_ _7"> </span>expressi<span class="_ _1"></span>on</div><div class="t m0 x142 he y2b7 ff7 fs4 fc0 sc0 ls1b ws0">φ</div><div class="t m0 xc6 hb y2b8 ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 xc6 hc y2b9 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x143 h6 y2b7 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff1">0</span></div><div class="t m0 xf hb y2ba ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 xc7 h8 y2b7 ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _0"> </span><span class="ff6">−<span class="_ _0"> </span></span>φ</div><div class="t m0 xa5 hb y2b8 ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 xa5 hc y2b9 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x87 h6 y2b7 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff1">0</span></div><div class="t m0 x4 hb y2ba ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 x8a h8 y2b7 ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _15"> </span><span class="ff6">=</span></div><div class="t m0 x133 h8 y2bb ff2 fs4 fc0 sc0 ls1b ws0">L</div><div class="t m0 x121 hb y2bc ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 x144 h6 y2bd ff6 fs4 fc0 sc0 ls1b ws0">+<span class="_ _0"> </span><span class="ff1">2<span class="_ _a"></span><span class="ff2">L</span></span></div><div class="t m0 xcd hd y2bc ff1 fs8 fc0 sc0 ls78 ws0">sep</div><div class="t m0 xb5 h8 y2bd ff6 fs4 fc0 sc0 ls1b ws0">+<span class="_ _e"> </span><span class="ff2">L</span></div><div class="t m0 x140 hb y2bc ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 xab h6 y2be ff1 fs4 fc0 sc0 ls1b ws0">2<span class="ff7">κ</span></div><div class="t m0 x145 h8 y2bf ff2 fs4 fc0 sc0 ls1b ws0">I<span class="_ _b"> </span><span class="ff7">(</span>t<span class="_ _18"></span><span class="ff7">)</span></div><div class="t m0 x146 h10 y2c0 ff6 fs4 fc0 sc0 ls1b ws0">+</div><div class="t m0 x147 h6 y2c1 ff1 fs4 fc0 sc0 ls5d ws0">2R<span class="_ _3"></span>T</div><div class="t m0 x9a h8 y2c2 ff2 fs4 fc0 sc0 ls1b ws0">F</div><div class="t m0 x148 hf y2c3 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x116 h6 y2c4 ff1 fs4 fc0 sc0 ls1b ws0">1<span class="_ _0"> </span><span class="ff6">−<span class="_ _14"> </span><span class="ff2">t</span></span></div><div class="t m0 xa6 hd y2c5 ff1 fs8 fc0 sc0 ls1b ws0">0</div><div class="t m0 x89 hc y2c6 ff2 fs8 fc0 sc0 ls1b ws0">c</div><div class="t m0 xd9 hf y2c7 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x3 h8 y2c8 ff2 fs4 fc0 sc0 ls1b ws0">k</div><div class="t m0 xd hc y2c9 ff2 fs8 fc0 sc0 ls1b ws0">f</div><div class="t m0 xdd h6 y2ca ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">t<span class="_ _18"></span></span>)<span class="ff6">[<span class="ff1 ls20">ln<span class="_ _1b"> </span></span><span class="ff2">c</span></span></div><div class="t m0 x123 hc y2c9 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x138 h6 y2ca ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff1">0</span></div><div class="t m0 x149 hb y2c5 ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 x14a h6 y2ca ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _0"> </span><span class="ff6">−<span class="_ _0"> </span><span class="ff1 ls20">ln<span class="_ _1b"> </span></span><span class="ff2">c</span></span></div><div class="t m0 x145 hc y2c9 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x12e h6 y2ca ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff1">0</span></div><div class="t m0 x11b hb y2c5 ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 x14b h6 y2ca ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="ff6">]<span class="_ _2"></span></span>.<span class="_ _39"> </span><span class="ff1 ls84">(31)</span></div><div class="t m0 x1 h6 y2cb ff1 fs4 fc0 sc0 ls6d ws0">No<span class="_ _1"></span>w<span class="_ _3"></span>,<span class="_ _13"> </span>we<span class="_ _13"> </span>combine<span class="_ _12"> </span>(22)<span class="_ _13"> </span>and<span class="_ _13"> </span>(28)–(31)<span class="_ _12"> </span>to<span class="_ _13"> </span>compute<span class="_ _13"> </span><span class="ff2 ls1b">V<span class="_ _1b"> </span><span class="ff7">(</span>t<span class="_ _18"></span><span class="ff7">)<span class="_ _f"> </span><span class="ff6">=</span></span></span></div><div class="t m0 x1 he y2cc ff7 fs4 fc0 sc0 ls1b ws0">φ</div><div class="t m0 x98 hb y2cd ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 x98 hc y2ce ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 x114 h6 y2cf ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff1">0</span></div><div class="t m0 x14c hb y2cd ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 x14d h8 y2cf ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _0"> </span><span class="ff6">−<span class="_ _0"> </span></span>φ</div><div class="t m0 xa3 hb y2cd ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 x14e hc y2ce ff2 fs8 fc0 sc0 ls1b ws0">s</div><div class="t m0 xba h6 y2cf ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff1">0</span></div><div class="t m0 x117 hb y2cd ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 x89 h8 y2cf ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)</div><div class="t m0 xc5 h8 y2d0 ff2 fs4 fc0 sc0 ls1b ws0">V<span class="_ _1b"> </span><span class="ff7">(</span>t<span class="_ _a"></span><span class="ff7">)<span class="_ _7"> </span><span class="ff6">=</span></span></div><div class="t m0 x85 h6 y2d1 ff1 fs4 fc0 sc0 ls91 ws0">RT</div><div class="t m0 x2 h8 y2d2 ff7 fs4 fc0 sc0 ls1b ws0">α<span class="_ _18"></span><span class="ff2">F</span></div><div class="t m0 x14f h6 y2d3 ff1 fs4 fc0 sc0 ls92 ws0">sinh</div><div class="t m0 xa0 hd y2d4 ff6 fs8 fc0 sc0 ls1b ws0">−<span class="ff1">1</span></div><div class="t m0 xa6 hf y2d5 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x150 h8 y2d6 ff6 fs4 fc0 sc0 ls1b ws0">−<span class="_ _a"></span><span class="ff2">I<span class="_ _1b"> </span><span class="ff7">(</span>t<span class="_ _18"></span><span class="ff7">)</span></span></div><div class="t m0 x126 h6 y2d7 ff1 fs4 fc0 sc0 ls1b ws0">2<span class="ff2">a</span></div><div class="t m0 x151 hb y2d8 ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 xde h8 y2d7 ff2 fs4 fc0 sc0 ls1b ws0">L</div><div class="t m0 x136 hb y2d8 ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 x137 h10 y2d9 ff6 fs4 fc0 sc0 ls1b ws0">¯</div><div class="t m0 xe9 h8 y2da ff2 fs4 fc0 sc0 ls1b ws0">i</div><div class="t m0 xe3 hb y2db ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 x119 hd y2dc ff1 fs8 fc0 sc0 ls1b ws0">0</div><div class="t m0 x118 h8 y2d7 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">t<span class="_ _18"></span></span>)</div><div class="t m0 x149 hf y2dd ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x2 h10 y2de ff6 fs4 fc0 sc0 ls1b ws0">−</div><div class="t m0 x152 h6 y2df ff1 fs4 fc0 sc0 ls91 ws0">RT</div><div class="t m0 x153 h8 y2e0 ff7 fs4 fc0 sc0 ls1b ws0">α<span class="_ _18"></span><span class="ff2">F</span></div><div class="t m0 xa5 h6 y2e1 ff1 fs4 fc0 sc0 ls92 ws0">sinh</div><div class="t m0 xca hd y2e2 ff6 fs8 fc0 sc0 ls1b ws0">−<span class="ff1">1</span></div><div class="t m0 xbb hf y2e3 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 xcc h8 y2e4 ff2 fs4 fc0 sc0 ls1b ws0">I<span class="_ _b"> </span><span class="ff7">(</span>t<span class="_ _18"></span><span class="ff7">)</span></div><div class="t m0 xa7 h6 y2e5 ff1 fs4 fc0 sc0 ls1b ws0">2<span class="ff2">a</span></div><div class="t m0 x125 hb y2e6 ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 xcc h8 y2e5 ff2 fs4 fc0 sc0 ls1b ws0">L</div><div class="t m0 x119 hb y2e6 ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 x123 h10 y2e7 ff6 fs4 fc0 sc0 ls1b ws0">¯</div><div class="t m0 x123 h8 y2e8 ff2 fs4 fc0 sc0 ls1b ws0">i</div><div class="t m0 x138 hb y2e9 ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 x154 hd y2ea ff1 fs8 fc0 sc0 ls1b ws0">0</div><div class="t m0 x8f h8 y2e5 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">t<span class="_ _18"></span></span>)</div><div class="t m0 xc2 hf y2eb ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x2 h8 y2ec ff6 fs4 fc0 sc0 ls1b ws0">+<span class="_ _1b"> </span><span class="ff2">U</span></div><div class="t m0 x14f hb y2ed ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 x14e hf y2ee ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x155 h8 y2ef ff6 fs4 fc0 sc0 ls1b ws0">¯<span class="_ _30"></span><span class="ff2">c</span></div><div class="t m0 x87 hb y2ed ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 x156 hd y2f0 ff1 fs8 fc0 sc0 ls78 ws0">ss</div><div class="t m0 xc9 h8 y2f1 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">t<span class="_ _18"></span></span>)</div><div class="t m0 xd9 hf y2f2 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x157 h8 y2f1 ff6 fs4 fc0 sc0 ls1b ws0">−<span class="_ _14"> </span><span class="ff2">U</span></div><div class="t m0 x125 hb y2ed ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 x158 hf y2ee ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 xa9 h8 y2ef ff6 fs4 fc0 sc0 ls1b ws0">¯<span class="_ _30"></span><span class="ff2">c</span></div><div class="t m0 xe3 hb y2ed ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 xe3 hd y2f0 ff1 fs8 fc0 sc0 ls78 ws0">ss</div><div class="t m0 xe4 h8 y2f1 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">t<span class="_ _18"></span></span>)</div><div class="t m0 x149 hf y2f2 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 xaa h10 y2f1 ff6 fs4 fc0 sc0 ls1b ws0">−</div><div class="t m0 xe1 hf y2f3 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 xda h8 y2f4 ff2 fs4 fc0 sc0 ls1b ws0">R</div><div class="t m0 xcf hb y2f5 ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 xb6 hc y2f6 ff2 fs8 fc0 sc0 ls1b ws0">f</div><div class="t m0 x159 h8 y2f7 ff2 fs4 fc0 sc0 ls1b ws0">a</div><div class="t m0 xce hb y2f8 ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 xcf h8 y2f7 ff2 fs4 fc0 sc0 ls1b ws0">L</div><div class="t m0 x11d hb y2f8 ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 x12e h10 y2f1 ff6 fs4 fc0 sc0 ls1b ws0">+</div><div class="t m0 x13e h8 y2f9 ff2 fs4 fc0 sc0 ls1b ws0">R</div><div class="t m0 xd3 hb y2f5 ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 x13d hc y2f6 ff2 fs8 fc0 sc0 ls1b ws0">f</div><div class="t m0 x11e h8 y2f7 ff2 fs4 fc0 sc0 ls1b ws0">a</div><div class="t m0 xd4 hb y2f8 ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 x95 h8 y2f7 ff2 fs4 fc0 sc0 ls1b ws0">L</div><div class="t m0 x12b hb y2f8 ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 x12d hf y2fa ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 xd8 h8 y2fb ff2 fs4 fc0 sc0 ls1b ws0">I<span class="_ _b"> </span><span class="ff7">(</span>t<span class="_ _18"></span><span class="ff7">)</span></div><div class="t m0 x2 h10 y2fc ff6 fs4 fc0 sc0 ls1b ws0">+</div><div class="t m0 x153 h8 y2fd ff2 fs4 fc0 sc0 ls1b ws0">L</div><div class="t m0 x14f hb y2fe ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 x116 h6 y2ff ff6 fs4 fc0 sc0 ls1b ws0">+<span class="_ _0"> </span><span class="ff1">2<span class="_ _a"></span><span class="ff2">L</span></span></div><div class="t m0 xdb hd y2fe ff1 fs8 fc0 sc0 ls78 ws0">sep</div><div class="t m0 xd h8 y2ff ff6 fs4 fc0 sc0 ls1b ws0">+<span class="_ _e"> </span><span class="ff2">L</span></div><div class="t m0 x8e hb y2fe ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 xc h6 y300 ff1 fs4 fc0 sc0 ls1b ws0">2<span class="ff7">κ</span></div><div class="t m0 x15a h8 y301 ff2 fs4 fc0 sc0 ls1b ws0">I<span class="_ _b"> </span><span class="ff7">(</span>t<span class="_ _18"></span><span class="ff7">)</span></div><div class="t m0 x2 h8 y302 ff6 fs4 fc0 sc0 ls1b ws0">+<span class="_ _1b"> </span><span class="ff2">k</span></div><div class="t m0 x86 hd y303 ff1 fs8 fc0 sc0 ls7f ws0">conc</div><div class="t m0 x155 h6 y304 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">t<span class="_ _18"></span></span>)<span class="ff6">[<span class="ff1 ls20">ln<span class="_ _1b"> </span></span><span class="ff2">c</span></span></div><div class="t m0 xb1 hc y303 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xd h6 y304 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff1">0</span></div><div class="t m0 x8b hb y305 ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 x133 h6 y304 ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="_ _0"> </span><span class="ff6">−<span class="_ _0"> </span><span class="ff1 ls20">ln<span class="_ _1b"> </span></span><span class="ff2">c</span></span></div><div class="t m0 xac hc y303 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xe1 h6 y304 ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff1">0</span></div><div class="t m0 xaf hb y305 ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 xb0 h6 y304 ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span>)<span class="ff6">]<span class="_ _3a"> </span><span class="ff1 ls84">(32)</span></span></div><div class="t m0 xc5 h8 y306 ff2 fs4 fc0 sc0 ls1b ws0">V<span class="_ _1b"> </span><span class="ff7">(</span>t<span class="_ _a"></span><span class="ff7">)<span class="_ _7"> </span><span class="ff6">=<span class="_"> </span></span></span>h</div><div class="t m0 xc8 hf y307 ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 x153 h8 y306 ff6 fs4 fc0 sc0 ls1b ws0">¯<span class="_ _30"></span><span class="ff2">c</span></div><div class="t m0 x86 hb y308 ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 x86 hd y309 ff1 fs8 fc0 sc0 ls78 ws0">ss</div><div class="t m0 x15b h8 y30a ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _e"> </span><span class="ff6">¯<span class="_ _30"></span><span class="ff2">c</span></span></div><div class="t m0 xba hb y308 ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 xba hd y309 ff1 fs8 fc0 sc0 ls78 ws0">ss</div><div class="t m0 x9d h8 y30a ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">c</span></div><div class="t m0 xa6 hb y308 ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 x89 hc y309 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 x7 h6 y30a ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff1">0</span></div><div class="t m0 xe0 hb y308 ff6 fs8 fc0 sc0 ls1b ws0">+</div><div class="t m0 x150 h8 y30a ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span><span class="ls63">),<span class="_ _1b"> </span></span><span class="ff2">c</span></div><div class="t m0 xe3 hb y308 ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 xe3 hc y309 ff2 fs8 fc0 sc0 ls1b ws0">e</div><div class="t m0 xe4 h6 y30a ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff1">0</span></div><div class="t m0 x8f hb y308 ff6 fs8 fc0 sc0 ls1b ws0">−</div><div class="t m0 xb4 h8 y30a ff7 fs4 fc0 sc0 ls1b ws0">,<span class="_ _1b"> </span><span class="ff2">t<span class="_ _a"></span></span><span class="ls63">),<span class="_ _0"> </span></span><span class="ff2">I<span class="_ _1b"> </span></span>(<span class="ff2">t<span class="_ _a"></span></span>)</div><div class="t m0 x11d hf y30b ff8 fs4 fc0 sc0 ls1b ws0"></div><div class="t m0 xd8 h6 y30a ff1 fs4 fc0 sc0 ls84 ws0">(33)</div><div class="t m0 x1 h6 y30c ff1 fs4 fc0 sc0 ls46 ws0">where<span class="_ _c"> </span><span class="ff2 ls1b">k</span></div><div class="t m0 xed hd y30d ff1 fs8 fc0 sc0 ls7f ws0">conc</div><div class="t m0 xb8 h6 y13f ff6 fs4 fc0 sc0 ls1b ws0">=<span class="_ _e"> </span><span class="ff7">(<span class="ff1 ls5d">2R<span class="_ _3"></span>T<span class="ff7 ls1b">/<span class="_ _18"></span><span class="ff2">F<span class="_ _a"></span></span><span class="ls63">)(</span><span class="ff1">1<span class="_"> </span><span class="ff6">−<span class="_ _14"> </span><span class="ff2">t</span></span></span></span></span></span></div><div class="t m0 x119 hd y30e ff1 fs8 fc0 sc0 ls1b ws0">0</div><div class="t m0 xa9 hc y30f ff2 fs8 fc0 sc0 ls1b ws0">c</div><div class="t m0 x5 h8 y13f ff7 fs4 fc0 sc0 ls1b ws0">)<span class="ff2">k</span></div><div class="t m0 x11a hc y30d ff2 fs8 fc0 sc0 ls1b ws0">f</div><div class="t m0 x149 h6 y13f ff7 fs4 fc0 sc0 ls1b ws0">(<span class="ff2">t<span class="_ _18"></span></span>)<span class="ff1">.</span></div><div class="t m0 xb h6 y310 ff1 fs4 fc0 sc0 ls5c ws0">This<span class="_ _8"> </span>summar<span class="_ _2"></span>izes<span class="_ _8"> </span>the<span class="_ _8"> </span>SPMe.<span class="_ _8"> </span>No<span class="_ _2"></span>te<span class="_ _8"> </span>the<span class="_ _d"> </span>signiﬁcantly<span class="_ _8"> </span>simpli-</div><div class="t m0 xa h6 y311 ff1 fs4 fc0 sc0 ls48 ws0">ﬁed<span class="_ _12"> </span>structure.<span class="_ _12"> </span>The<span class="_ _12"> </span>dynamical<span class="_ _12"> </span>equati<span class="_ _1"></span>ons<span class="_ _12"> </span>(23)<span class="_ _12"> </span>and<span class="_ _13"> </span>(25)–(27)</div><div class="t m0 xa h6 y312 ff1 fs4 fc0 sc0 ls53 ws0">are<span class="_ _7"> </span>linear<span class="_ _8"> </span>and<span class="_ _7"> </span>quasi-linear<span class="_ _7"> </span>PDEs,<span class="_ _7"> </span>r<span class="_ _2"></span>especti<span class="_ _1"></span>vely<span class="_ _3"></span>.<span class="_ _c"> </span>The<span class="_ _8"> </span>boundary</div><div class="t m0 xa h6 y313 ff1 fs4 fc0 sc0 ls47 ws0">conditions<span class="_ _13"> </span>are<span class="_ _17"> </span>all<span class="_ _17"> </span>linear<span class="_ _3"></span>.<span class="_ _17"> </span>Finally<span class="_ _3"></span>,<span class="_ _f"> </span>the<span class="_ _17"> </span>output<span class="_ _f"> </span>function<span class="_ _f"> </span>(32)</div><div class="t m0 xa h6 y314 ff1 fs4 fc0 sc0 ls53 ws0">is<span class="_ _d"> </span>nonlinear<span class="_ _d"> </span>with<span class="_ _12"> </span>respect<span class="_ _d"> </span>to<span class="_ _d"> </span>states<span class="_ _12"> </span>and<span class="_ _d"> </span>inputs.<span class="_ _d"> </span>As<span class="_ _12"> </span>sho<span class="_ _1"></span>wn<span class="_ _d"> </span>in</div><div class="t m0 xa h6 y315 ff1 fs4 fc0 sc0 ls1e ws0">Section<span class="_ _d"> </span>III,<span class="_ _12"> </span>the<span class="_ _d"> </span>SPMe<span class="_ _12"> </span>is<span class="_ _d"> </span>amenable<span class="_ _d"> </span>to<span class="_ _12"> </span>state<span class="_ _d"> </span>observer<span class="_ _d"> </span>design</div><div class="t m0 xa h6 y316 ff1 fs4 fc0 sc0 ls39 ws0">with<span class="_ _d"> </span>provable<span class="_ _8"> </span>con<span class="_ _1"></span>vergence.<span class="_ _7"> </span>Mo<span class="_ _2"></span><span class="ls5f">reover<span class="_ _3"></span>,<span class="_ _8"> </span>it<span class="_ _12"> </span>maintains<span class="_ _d"> </span>accuracy</span></div><div class="t m0 xa h6 y317 ff1 fs4 fc0 sc0 ls3b ws0">at<span class="_ _c"> </span>high<span class="_ _c"> </span><span class="ff2 ls1b">C<span class="_ _18"></span></span><span class="ls4f">-rates,<span class="_ _c"> </span>as<span class="_ _c"> </span>discussed<span class="_ _15"> </span>in<span class="_ _7"> </span>Section<span class="_ _c"> </span>II-C.</span></div><div class="t m0 xb h6 y318 ff2 fs4 fc0 sc0 ls6d ws0">Remark<span class="_ _15"> </span>1<span class="_ _13"> </span>(SPMe<span class="_ _15"> </span>V<span class="_ _9"></span>ers<span class="_ _1"></span>us<span class="_ _15"> </span>SPM<span class="_ _15"> </span>Compar<span class="_ _1"></span>ison):<span class="_"> </span><span class="ff1 ls26">Note<span class="_ _15"> </span>the<span class="_ _15"> </span>volt-</span></div><div class="t m0 xa h6 y319 ff1 fs4 fc0 sc0 ls55 ws0">age<span class="_ _d"> </span>expression<span class="_ _d"> </span>(3<span class="_ _2"></span>2)<span class="_ _d"> </span>is<span class="_ _d"> </span>id<span class="_ _2"></span>entical<span class="_ _d"> </span>to<span class="_ _d"> </span>th<span class="_ _2"></span>e<span class="_ _d"> </span>SPM<span class="_ _12"> </span>v<span class="_ _1"></span>oltage<span class="_ _d"> </span>expres-</div><div class="t m0 xa h6 y31a ff1 fs4 fc0 sc0 ls3f ws0">sion<span class="_ _f"> </span>[15],<span class="_ _f"> </span>but<span class="_ _f"> </span>the<span class="_ _17"> </span>SPMe<span class="_ _17"> </span>adds<span class="_ _f"> </span>the<span class="_ _17"> </span>last<span class="_ _f"> </span>two<span class="_ _17"> </span>terms.<span class="_ _f"> </span>These,</div><div class="t m0 xa h6 y31b ff1 fs4 fc0 sc0 ls25 ws0">respecti<span class="_ _1"></span>vely<span class="_ _3"></span>,<span class="_ _12"> </span>represent<span class="_ _13"> </span>ohmic<span class="_ _13"> </span>potential<span class="_ _12"> </span>d<span class="_ _2"></span>rop<span class="_ _12"> </span>due<span class="_ _13"> </span>electrolyte</div><div class="t m0 xa h6 y31c ff1 fs4 fc0 sc0 ls1d ws0">conducti<span class="_ _1"></span>vity<span class="_ _15"> </span>and<span class="_ _c"> </span>the<span class="_ _c"> </span>electrolyte<span class="_ _c"> </span>concentration<span class="_ _15"> </span>overpotenti<span class="_ _1"></span>al.</div><div class="t m0 xb h6 y31d ff2 fs4 fc0 sc0 ls8f ws0">Remark<span class="_ _c"> </span>2<span class="_ _13"> </span>(<span class="_ _2"></span>Compariso<span class="_ _2"></span>n<span class="_ _c"> </span>W<span class="_ _1"></span>ith<span class="_ _7"> </span>Existing<span class="_ _7"> </span>SPMe<span class="_ _7"> </span>Mod<span class="_ _2"></span>els):<span class="_ _7"> </span><span class="ff1 ls48">The</span></div><div class="t m0 xa h6 y31e ff1 fs4 fc0 sc0 ls3b ws0">SPMe<span class="_ _c"> </span>model<span class="_ _7"> </span>dev<span class="_ _1"></span>elopment<span class="_ _15"> </span>was<span class="_ _7"> </span>motiv<span class="_ _1"></span>ated<span class="_ _15"> </span>by<span class="_ _8"> </span>the<span class="_ _c"> </span>difﬁcult<span class="_ _1"></span>y<span class="_ _7"> </span>of</div><div class="t m0 xa h6 y31f ff1 fs4 fc0 sc0 ls6d ws0">extendi<span class="_ _1"></span>ng<span class="_ _e"> </span>the<span class="_ _15"> </span>results<span class="_ _15"> </span>in<span class="_ _15"> </span>[15]<span class="_ _15"> </span>using<span class="_ _e"> </span>the<span class="_ _15"> </span>models<span class="_ _15"> </span>a<span class="_ _1"></span>vail<span class="_ _1"></span>able<span class="_ _15"> </span>in<span class="_ _15"> </span>the</div><div class="t m0 xa h6 y320 ff1 fs4 fc0 sc0 ls5c ws0">literature.<span class="_ _8"> </span>For<span class="_ _7"> </span>instance,<span class="_ _8"> </span>the<span class="_ _8"> </span>model<span class="_ _8"> </span>in<span class="_ _7"> </span>[<span class="_ _2"></span>19]<span class="_ _7"> </span>d<span class="_ _2"></span>oes<span class="_ _8"> </span>not<span class="_ _7"> </span>allow<span class="_ _7"> </span>fo<span class="_ _2"></span>r</div><div class="t m0 xa h6 y321 ff1 fs4 fc0 sc0 ls47 ws0">the<span class="_ _15"> </span>output-in<span class="_ _1"></span>version<span class="_"> </span>step<span class="_ _15"> </span>and<span class="_ _c"> </span>we<span class="_ _c"> </span>require<span class="_ _15"> </span>to<span class="_ _c"> </span>design<span class="_ _15"> </span>a<span class="_ _c"> </span>provably</div><div class="t m0 xa h6 y322 ff1 fs4 fc0 sc0 ls60 ws0">conv<span class="_ _1"></span>ergent<span class="_ _e"> </span>o<span class="_ _2"></span>bserver,<span class="_"> </span>d<span class="_ _2"></span>ue<span class="_ _15"> </span>to<span class="_ _c"> </span>th<span class="_ _2"></span>e<span class="_ _c"> </span>spatial<span class="_ _c"> </span>distribution<span class="_ _15"> </span>of<span class="_ _7"> </span>the<span class="_ _15"> </span>su<span class="_ _2"></span>r-</div><div class="t m0 xa h6 y323 ff1 fs4 fc0 sc0 ls1c ws0">face<span class="_ _c"> </span>concentr<span class="_ _2"></span>ation<span class="_ _15"> </span>in<span class="_ _7"> </span>the<span class="_ _c"> </span>solid<span class="_ _7"> </span>given<span class="_ _15"> </span>by<span class="_ _c"> </span>th<span class="_ _2"></span>e<span class="_ _c"> </span>interaction<span class="_ _c"> </span>o<span class="_ _2"></span>f<span class="_ _c"> </span>the</div><div class="t m0 xa h6 y324 ff1 fs4 fc0 sc0 ls87 ws0">approximate<span class="_ _8"> </span>dif<span class="_ _1"></span>fusion<span class="_ _8"> </span>representation<span class="_ _8"> </span>and<span class="_ _d"> </span>the<span class="_ _d"> </span>polynomial<span class="_ _8"> </span>(in</div><div class="t m0 xa h6 y325 ff1 fs4 fc0 sc0 ls5a ws0">space)<span class="_ _e"> </span>approximation<span class="_ _e"> </span>of<span class="_ _15"> </span>the<span class="_ _15"> </span>electro<span class="ls62">lyte<span class="_ _15"> </span>concentra<span class="ls6e">tion<span class="_"> </span>p<span class="_ _2"></span>roﬁles.</span></span></div><div class="t m0 xb h6 y326 ff1 fs4 fc0 sc0 ls3 ws0">The<span class="_ _12"> </span>model<span class="_ _d"> </span>in<span class="_ _12"> </span>[21]<span class="_ _d"> </span>linearizes<span class="_ _12"> </span>t<span class="_ _1"></span>he<span class="_ _12"> </span>transportat<span class="_ _1"></span>ion<span class="_ _d"> </span>equations</div><div class="t m0 xa h6 y327 ff1 fs4 fc0 sc0 ls24 ws0">ﬁrst.<span class="_ _7"> </span>Then,<span class="_ _8"> </span>it<span class="_ _8"> </span>applies<span class="_ _7"> </span>a<span class="_ _8"> </span>Laplace<span class="_ _8"> </span>transformation<span class="_ _c"> </span>and<span class="_ _8"> </span>performs</div><div class="t m0 xa h6 y328 ff1 fs4 fc0 sc0 ls22 ws0">a<span class="_ _12"> </span>polynomial<span class="_ _d"> </span>approximation<span class="_ _d"> </span>across<span class="_ _12"> </span>space.<span class="_ _12"> </span>This<span class="_ _12"> </span>results<span class="_ _12"> </span>in<span class="_ _12"> </span>a</div><div class="t m0 xa h6 y329 ff1 fs4 fc0 sc0 ls27 ws0">linear<span class="_ _15"> </span>output<span class="_ _15"> </span>function,<span class="_"> </span>but<span class="_ _15"> </span>produces<span class="_ _e"> </span>a<span class="_ _c"> </span>nonphysical<span class="_ _e"> </span>state-space</div><div class="t m0 xa h6 y32a ff1 fs4 fc0 sc0 ls27 ws0">representati<span class="_ _1"></span>on.<span class="_ _c"> </span>The<span class="_ _7"> </span>model<span class="_ _7"> </span>we<span class="_ _7"> </span>propose,<span class="_ _c"> </span>in<span class="_ _7"> </span>contrast,<span class="_ _c"> </span>maintains</div><div class="t m0 xa h6 y32b ff1 fs4 fc0 sc0 ls5a ws0">the<span class="_ _0"> </span>physical<span class="_ _1b"> </span>interpretation<span class="_ _14"> </span>o<span class="_ _2"></span>f<span class="_"> </span>the<span class="_ _14"> </span>state<span class="_"> </span>s<span class="_ _1"></span>pace—a<span class="_ _0"> </span>useful<span class="_ _14"> </span>property</div><div class="t m0 xa h6 y32c ff1 fs4 fc0 sc0 ls24 ws0">for<span class="_ _17"> </span>state<span class="_ _11"> </span>estimati<span class="_ _1"></span>on.<span class="_ _11"> </span>Furthermore,<span class="_ _f"> </span>our<span class="_ _11"> </span>time-domain<span class="_ _f"> </span>model</div><div class="t m0 xa h6 y32d ff1 fs4 fc0 sc0 ls5a ws0">can<span class="_ _d"> </span>accommodate<span class="_ _8"> </span>some<span class="_ _8"> </span>time-vary<span class="ls3a">ing<span class="_ _8"> </span>coefﬁcients<span class="_ _8"> </span>in<span class="_ _d"> </span>a<span class="_ _d"> </span>much</span></div><div class="t m0 xa h6 y32e ff1 fs4 fc0 sc0 ls1c ws0">simpler<span class="_ _c"> </span>way<span class="_ _7"> </span>than<span class="_ _c"> </span>an<span class="_ _7"> </span>ap<span class="_ _2"></span>proxim<span class="_ _2"></span>ate<span class="_ _15"> </span>tran<span class="_ _2"></span>sfer<span class="_ _c"> </span>fu<span class="_ _2"></span>nction<span class="_ _c"> </span>mod<span class="_ _2"></span>el.</div><div class="t m0 xb h6 y32f ff1 fs4 fc0 sc0 ls6d ws0">The<span class="_ _0"> </span>models<span class="_ _1b"> </span>in<span class="_"> </span>[18],<span class="_ _14"> </span>[20],<span class="_"> </span>a<span class="_ _1"></span>nd<span class="_"> </span>[22]<span class="_ _14"> </span>are<span class="_"> </span>t<span class="_ _1"></span>he<span class="_"> </span>mos<span class="_ _1"></span>t<span class="_"> </span>si<span class="_ _1"></span>milar<span class="_ _14"> </span>to<span class="_"> </span>the</div><div class="t m0 xa h6 y330 ff1 fs4 fc0 sc0 ls48 ws0">SPMe<span class="_ _7"> </span>deriv<span class="_ _1"></span>ed<span class="_ _7"> </span>here,<span class="_ _8"> </span>as<span class="_ _8"> </span>they<span class="_ _7"> </span>apply<span class="_ _7"> </span>Assumption<span class="_ _c"> </span>[A1]<span class="_ _8"> </span>in<span class="_ _8"> </span>addi-</div><div class="t m0 xa h6 y331 ff1 fs4 fc0 sc0 ls48 ws0">tion<span class="_ _c"> </span>to<span class="_ _8"> </span>various<span class="_ _c"> </span>numerical<span class="_ _c"> </span>approximations.<span class="_ _15"> </span>In<span class="_ _8"> </span>[18],<span class="_ _7"> </span>bulk<span class="_ _c"> </span>solid</div><div class="t m0 xa h6 y332 ff1 fs4 fc0 sc0 ls3b ws0">concentration<span class="_ _8"> </span>is<span class="_ _12"> </span>used<span class="_ _d"> </span>in<span class="_ _d"> </span>the<span class="_ _12"> </span>v<span class="_ _1"></span>oltage<span class="_ _d"> </span>output<span class="_ _d"> </span>function<span class="_ _d"> </span>inst<span class="_ _1"></span>ead</div><div class="t m0 xa h6 y333 ff1 fs4 fc0 sc0 ls27 ws0">of<span class="_ _c"> </span>the<span class="_ _7"> </span>surface<span class="_ _c"> </span>concentration<span class="_ _c"> </span>we<span class="_ _7"> </span>use<span class="_ _7"> </span>here<span class="_ _c"> </span>(see<span class="_ _7"> </span>[18,<span class="_ _7"> </span>eq.<span class="_ _7"> </span>(26)]).</div><div class="t m0 xa h6 y334 ff1 fs4 fc0 sc0 ls3f ws0">In<span class="_ _12"> </span>the<span class="_ _13"> </span>case<span class="_ _12"> </span>of<span class="_ _13"> </span>[20],<span class="_ _12"> </span>volume<span class="_ _12"> </span>a<span class="_ _1"></span>veraging<span class="_ _d"> </span>is<span class="_ _13"> </span>performed<span class="_ _12"> </span>in<span class="_ _12"> </span>the</div><div class="t m0 xa h6 y335 ff1 fs4 fc0 sc0 ls1b ws0">electrolyte<span class="_ _d"> </span>phase,<span class="_ _d"> </span>wh<span class="_ _2"></span>ich<span class="_ _d"> </span>partially<span class="_ _d"> </span>obscu<span class="_ _2"></span>res<span class="_ _d"> </span>electrolyte<span class="_ _d"> </span>polar-</div><div class="t m0 xa h6 y336 ff1 fs4 fc0 sc0 ls63 ws0">ization.<span class="_"> </span>Han<span class="_"> </span><span class="ff2 ls5f">et<span class="_ _0"> </span>al.<span class="_"> </span></span><span class="ls3f">[22]<span class="_ _1b"> </span>used<span class="_"> </span>an<span class="_"> </span>approximation<span class="_ _1b"> </span>of<span class="_"> </span>the<span class="_ _0"> </span>s<span class="_ _1"></span>olid<span class="_"> </span>stat<span class="_ _1"></span>e</span></div><div class="t m0 xa h6 y337 ff1 fs4 fc0 sc0 ls47 ws0">dif<span class="_ _1"></span>fusion<span class="_"> </span>equat<span class="_ _1"></span>ion<span class="_"> </span>inst<span class="_ _1"></span>ead<span class="_"> </span>of<span class="_"> </span>retaining<span class="_"> </span>the<span class="_"> </span>PDE<span class="_"> </span>vers<span class="_ _1"></span>ion<span class="_"> </span>we<span class="_"> </span>use</div><div class="t m0 xa h6 y338 ff1 fs4 fc0 sc0 ls3f ws0">in<span class="_ _e"> </span>(23)<span class="_ _e"> </span>and<span class="_ _15"> </span>(24)<span class="_"> </span>(see<span class="_ _e"> </span>[22,<span class="_ _e"> </span>Sec.<span class="_ _15"> </span>2]).<span class="_"> </span>Since<span class="_ _e"> </span>our<span class="_ _15"> </span>main<span class="_"> </span>objecti<span class="_ _1"></span>ve<span class="_"> </span>is</div><div class="t m0 xa h6 y339 ff1 fs4 fc0 sc0 ls1d ws0">prov<span class="_ _1"></span>ably<span class="_ _7"> </span>con<span class="_ _1"></span>vergent<span class="_ _c"> </span>state<span class="_ _8"> </span>observers,<span class="_ _7"> </span>we<span class="_ _8"> </span>additionally<span class="_ _7"> </span>analyze</div><div class="t m0 xa h6 y33a ff1 fs4 fc0 sc0 ls4f ws0">the<span class="_ _0"> </span>SP<span class="_ _1"></span>Me’<span class="_ _3"></span>s<span class="_"> </span>dynamical<span class="_ _14"> </span>properties<span class="_ _0"> </span>in<span class="_ _14"> </span>Section<span class="_"> </span>II-D.<span class="_ _14"> </span>Furthermore,</div><div class="t m0 xa h6 y33b ff1 fs4 fc0 sc0 ls4b ws0">the<span class="_"> </span>structure<span class="_"> </span>of<span class="_ _e"> </span>our<span class="_ _15"> </span>proposed<span class="_"> </span>SP<span class="_ _1"></span>Me<span class="_ _15"> </span>woul<span class="_ _1"></span>d<span class="_ _e"> </span>allow<span class="_"> </span>for<span class="_ _e"> </span>relati<span class="_ _1"></span>vely</div><div class="t m0 xa h6 y33c ff1 fs4 fc0 sc0 ls4f ws0">simple<span class="_"> </span>e<span class="_ _1"></span>xtensions<span class="_"> </span>to<span class="_"> </span>nonhomogeneous<span class="_ _1b"> </span>(in<span class="_"> </span>space)<span class="_"> </span>transport<span class="_ _0"> </span>and</div><div class="t m0 xa h6 y33d ff1 fs4 fc0 sc0 ls3f ws0">conducti<span class="_ _1"></span>vity<span class="_ _d"> </span>terms,<span class="_ _d"> </span>albeit<span class="_ _12"> </span>rende<span class="ls3b">ring<span class="_ _d"> </span>the<span class="_ _12"> </span>integrati<span class="_ _1"></span>on<span class="_ _d"> </span>required</span></div><div class="t m0 xa h6 y33e ff1 fs4 fc0 sc0 ls3b ws0">to<span class="_ _11"> </span>obtain<span class="_ _11"> </span>the<span class="_ _11"> </span>output<span class="_ _11"> </span>equation<span class="_ _11"> </span>in<span class="_ _11"> </span>(32)<span class="_ _11"> </span>harder<span class="_ _1"></span>.<span class="_ _11"> </span>Namely<span class="_ _3"></span>,<span class="_ _11"> </span>it</div><div class="t m0 xa h6 y33f ff1 fs4 fc0 sc0 ls1b ws0">would<span class="_ _15"> </span>require<span class="_ _15"> </span>numerically<span class="_ _15"> </span>integrating<span class="_ _15"> </span>the<span class="_ _e"> </span>electr<span class="_ _2"></span>olyte<span class="_ _15"> </span>potential</div><div class="t m0 xa h6 y340 ff1 fs4 fc0 sc0 ls47 ws0">gradient<span class="_ _7"> </span>instead<span class="_ _8"> </span>of<span class="_ _d"> </span>obtaining<span class="_ _7"> </span>an<span class="_ _d"> </span>explic<span class="_ _1"></span>it<span class="_ _8"> </span>form<span class="_ _8"> </span>for<span class="_ _d"> </span>the<span class="_ _8"> </span>ohmic</div><div class="t m0 xa h6 y341 ff1 fs4 fc0 sc0 ls21 ws0">potential<span class="_ _15"> </span>drop.</div><div class="t m0 xa h8 y342 ff2 fs4 fc0 sc0 ls84 ws0">C.<span class="_ _c"> </span>Model<span class="_ _c"> </span>Comparison</div><div class="t m0 xb h6 y343 ff1 fs4 fc0 sc0 ls3c ws0">In<span class="_ _e"> </span>this<span class="_ _15"> </span>section,<span class="_ _15"> </span>we<span class="_ _e"> </span>compar<span class="_ _2"></span>e<span class="_"> </span>the<span class="_ _15"> </span>voltage<span class="_"> </span>p<span class="_ _2"></span>redictions<span class="_ _e"> </span>between</div><div class="t m0 xa h6 y344 ff1 fs4 fc0 sc0 ls8b ws0">the<span class="_ _13"> </span>SPMe,<span class="_ _13"> </span>SPM,<span class="_ _f"> </span>and<span class="_ _13"> </span>DFN<span class="_ _13"> </span>models.<span class="_ _13"> </span>Note<span class="_ _f"> </span>the<span class="_ _13"> </span>SPM<span class="_ _13"> </span>output</div><div class="t m0 xa h6 y345 ff1 fs4 fc0 sc0 ls87 ws0">vol<span class="_ _1"></span>tage<span class="_"> </span>equation<span class="_"> </span>is<span class="_ _15"> </span>equi<span class="_ _1"></span>val<span class="_ _1"></span>ent<span class="_"> </span>to<span class="_ _15"> </span>(32),<span class="_"> </span>but<span class="_"> </span>without<span class="_"> </span>the<span class="_ _15"> </span>last<span class="_"> </span>two</div><div class="t m0 xa h6 y346 ff1 fs4 fc0 sc0 ls5b ws0">terms.<span class="_"> </span>The<span class="_"> </span>model<span class="_ _e"> </span>parameters<span class="_"> </span>use<span class="ls83">d<span class="_ _e"> </span>in<span class="_ _e"> </span>this<span class="_ _e"> </span>paper<span class="_"> </span>originate<span class="_ _e"> </span>from</span></div><div class="t m0 xa h6 y347 ff1 fs4 fc0 sc0 ls1f ws0">the<span class="_ _13"> </span>publicly<span class="_ _13"> </span>av<span class="_ _1"></span>ailable<span class="_ _13"> </span>DU<span class="_ _1"></span>ALFOIL<span class="_ _13"> </span>simulation<span class="_ _13"> </span>pack<span class="_ _2"></span>age<span class="_ _12"> </span>[<span class="_ _2"></span>30]</div><div class="t m0 xe ha y6d ff5 fs7 fc0 sc0 ls1b ws0">Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on January 19,2022 at 04:38:28 UTC from IEEE Xplore. Restrictions apply. </div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>