超级电容器数学模型

2017年第5期 信息通信2017 (总第 173 期）INFORMATION&COMMUNICATIONS(Sum.No173)超级电容器动态等效模型研究夏峰利^沈谅平 '周方媛^王浩1(1.湖北大学物理与电子科学学院，湖北武汉430062;2.西安交通大学电力设备电气绝缘国家重点实验室，陕西西安710049)摘要：以超级电容器数学模型为研究对象，考虑超级电容器的充放电特性，对其等效模型进行研究，提出一种基于超级电容器充放电特性的动态等效电路模型。

在该模型中，超级电容器的等效串并联内阻随电容器两端电压和电流动态变化。

通过对比超级电容器进行充放电实验数据，结果表明该模型预测的充放电电压与实验值吻合。

关键词:超级电容器;数学模型;动态等效模型；等效电路中图分类号:TM53 文献标识码:A文章编号:1673-1131( 2017 )05-0070-03Research on the Dynamic Equivalent Model of Super CapacitorXia Fengli1,Shen Liangping u,Zhou Fangyuan1,Wang Hao1(Faculty of physics a n d electro n ic scien ce,H u b ei U niversity,W u h an,H ubei,430062) Abstract:B ased o n ch a rgin g a n d d isch argin g ch a ra cteristic,th e p ap er stu d ie d th e su p erca p a cito r eq u ivalen t m od el a n d p u t fo r­w ard a d ynam ic eq u ivalen t m odel.In th e new m odel,th e eq u ivalen t se rie s resista n ce a n d p arallel resista n ce vary w ith voltagea n d cu rr e n t of su p erca p a cito r.T h rou gh ch a rge a n d d isch a rge exp erim en t,it in d ica tes th a t th e ch a rge a n d d isch a rge voltage p re­d ica ted by new m od el a gree w ell w ith exp erim en tal values.Key word:su p erca p a cito r;m a th em a tical m odel;d ynam ic eq u ivalen t m odel;eq u ivalen t circu ii概述由于超级电容器功率密度高、充放电块、循环寿命长、低 温性能好、清洁环保等众多优点，近年来其在储能，电极材料, 电能质量改善等方面研究一直是前沿热点，使得超级电容器 在交通、工业、消费电子等方面的应用越来越广泛。

2019年9期创新前沿科技创新与应用Technology Innovation and Application基于最小二乘系法的电化学超级电容器建模与参数辨识方法刘宇航（西安工程大学电子信息学院，陕西西安710600）引言由于石油、煤矿等一次能源属于不可再生能源，随着经济的发展其储量也在日益减少，因此各国都采取了一系列的政策措施大力发展新能源产业改变现有能源结构[1]。

为保证政策改变后供电的安全性和可靠性需要先进的电能储存技术作为支撑[2]。

超级电容器是二十世纪七八十年代发展起来的一种介于常规电容器与化学电池二者之间的新型储能器件，相比传统的蓄电池与电容而言在许多方面更具优势。

它具备传统电容那样的优良的脉冲充放电性能，也具备化学电池大容量储备电荷的能力，而且相对于蓄电池更具有充放电时间短、循环寿命长等优势，并且解决了传统电容能量密度低的缺点。

超级电容器的实际应用系统所涉及到的模型分析多采用经典等效电路模型，但是，经典等效电路模型对于长时间充放电和静置的情况下，模型仿真精确度不高，经常使仿真结果和实际实验结果有很大的误差。

而且按照线性时不变常系数模型进行实际系统选型和分析计算的结果往往与真实需求相差较大。

在客观场景中使用的电力系统变得越来越繁复，让此装置的应用受到更多限制，更多的使用者期许储能系统的实用效果有更好的表现。

为此，在应用的环节之中，要尽快对此装置的动态特点和荷电状况等情况有清晰的把握。

为了达成此目的，就需要创建出对应的客观模型[4]。

1系统辨识原理创建对应的模型重点包含机理解析以及系统辨识两个方式[5]。

前者是依据对目标事物的了解，解读其中的因果关联，明确体现其内原理的规律，要对系统的运作状况有很深地了解且不具备通用性，为此，不适合用在繁复且需要运算参数超标的系统之中。

后者是以对辨别系统开展输入、输出监控得到这两个方面的信息的基本条件下，从一组安排好的模型类之内，明确出和被辨别系统对等的模型。

超级电容器性能原理及应用本文摘自: 电池论坛() 详细出处请参考：/thread-209320-1-1.html超级电容器是在19世纪60、70年代率先在美国出现，并于80年代实现市场化的一种新型的储能器件，具有超级储电能力。

它兼具普通电容器的大电流快速充放电特性与电池的储能特性，填补了普通电容器与电池之间比能量与比功率的空白。

超级电容器被称为是能量储存领域的一次革命，并将会在某些领域取代传统蓄电池。

超级电容器性能超级电容器的能量密度是传统电容器的几百倍，功率密度高出电池两个数量级，很好地弥补了电池比功率低、大电流充放电性能差和传统电容器能量密度小的缺点。

图1：超级电容器性能优势图超级电容器与铅酸、镍氢和锂电池相比，在自放电、能量密度和能量成本方面显现不足，但在效率、快充特性、温度范围、安全性、功率成本、功率密度、寿命方面，超级电容器有着其他电池不可超越的优势。

超级电容器是一种无污染的新型储能装置，寿命超长(1-50万次)、安全可靠、储能巨大，是一种理想的储能装置，具体特性如下：1、高循环寿命，循环寿命可达50万次以上，合计10年，远超电池理论上的最大循环2000-5000次;2、快速充电特性，由于不存在电能转化化学能的化学反应，充电10秒-10分钟，可达到其额定容量95%以上;3、高功率密度特性，具有优越的动力特性，可达300W/kg~5000W/kg，相当于电池的5-10倍;能较好地满足车辆在启动、加速、爬坡时对瞬间大功率的要求;4、大电流放电能力超强，过程损失小;大电流是电池的几十倍;5、超低温特性好，温度范围宽-40℃~+70℃;而一般电池是-20℃~+60℃;6、无污染，安全可靠，超级电容器是绿色能源(活性炭)，不污染环境，是理想的绿色环保电源;7、全寿命免维护：超级电容器采用全密封结构，没有水分等液体挥发，在使用过程中全寿命不需要维护;8、相符成本地，超级电容器价格比铅酸电池高1倍，但寿命比电池高10倍。

华北电力大学硕士学位论文原创性声明本人郑重声明：此处所提交的硕士学位论文《超级电容器分数阶建模及其控制方法研究》，是本人在导师指导下，在华北电力大学攻读硕士学位期间独立进行研究工作所取得的成果。

据本人所知，论文中除已注明部分外不包含他人已发表或撰写过的研究成果。

对本文的研究工作做出重要贡献的个人和集体，均已在文中以明确方式注明。

本声明的法律结果将完全由本人承担。

作者签名：日期：年月日华北电力大学硕士学位论文使用授权书《超级电容器分数阶建模及其控制方法研究》系本人在华北电力大学攻读硕士学位期间在导师指导下完成的硕士学位论文。

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本学位论文属于（请在以上相应方框内打“√”）：保密□，在年解密后适用本授权书不保密□作者签名：日期：年月日导师签名：日期：年月日摘要摘要超级电容器是一种具有发展前景的储能元件，建立其精确的模型对于进一步研究超级电容器的实际应用具有重要的意义。

本文对不同规格的超级电容器的阻抗在不同充电程度下的频率特性进行了测量，对基于超级电容器阻抗的分数阶特性进行了分析。

提出了利用分数阶矢量匹配法进行阻抗拟合，进而建立超级电容器分数阶模型的方法。

综合矢量匹配结果以及超级电容器的经典等效电路和多孔电极理论，并考虑了超级电容器内部的法拉第过程，提出一种新的超级电容器的电路模型。

通过对不同电压下的频域特性的拟合得到了参数与电压的关系，建立了超级电容器分数阶非线性模型。

通过对比模型充放电仿真与实验结果，证明了该模型的准确性。

对于超级电容器分数阶模型的阶跃响应，分别在频域利用了分数阶达尔文粒子群算法进行了基于主导极点法的分数阶PID控制器设计，在时域利用了遗传算法与NCD 优化相结合的方法进行了基于优化算法的分数阶PID控制器设计，控制结果仿真说明了分数阶控制器在超级电容器上的应用是成功的，且在对分数阶系统的控制中分数阶控制器显著优于整数阶控制器。

超级电容器建模及其在能源互联网中的应用JYoung_Dream2016/4/23目录1引言 (1)2超级电容器原理 (1)3超级电容器建模研究 (3)3.1超级电容器双电层模型 (3)3.1.1 Helmholtz双电层模型 (3)3.1.2 Gouy和Chapman双电层模型 (4)3.1.3 Stern和Grahame双电层模型 (4)3.2多孔电极传输线模型 (5)3.3等效电路模型 (5)3.3.1经典等效电路模型 (6)3.3.2梯形电路模型 (6)3.3.3多分支RC参数模型 (6)3.4超级电容器频域模型 (7)3.5 超级电容器智能模型 (8)4超级电容器储能特点 (8)5超级电容器在能源互联网中的功用 (9)5.1电网电能质量调节 (9)5.2分布式新能源发电 (10)5.3 微电网与分布式储能 (10)6参考文献 (11)超级电容器建模及其在能源互联网中的应用摘要：本文介绍了超级电容器的储能原理，综述了现有的各种超级电容器模型，分析了各种模型的特点和适用范围。

通过总结分析超级电容器的储能原理及相关模型，重点阐述了其在发展能源互联网中的作用。

1引言近几年，以可再生能源、分布式发电、储能、电动汽车等为代表的新能源技术和以物联网、大数据、云计算、移动互联网等为代表的互联网技术发展迅猛，以“新能源+互联网”为代表的第三次工业革命正在世界范围内发生，成为各国新的战略竞争焦点。

能源互联网应运而生，它是能源和互联网深度融合的结果，是第三次工业革命的核心之一。

能源互联网是以电力系统为核心，以智能电网为基础，采用先进的信息和通信技术及电力电子技术等，最大限度地接入分布式可再生能源，以及促进电力网络、交通网络、天然气网络和信息网络的融合与协调巧制，实现能源的清洁、高效利用，实现能量流、物流和信息流的优化与协调运行。

能源互联网的概念中，电能储能技术占有核心位置。

新能源技术中，风力发电与太阳能发电对环境影响小，但是发电具有间歇性和随机性，发电功率受天气影响大。

电动汽车复合电源系统超级电容器建模与仿真研究电动汽车的发展，离不开电池作为核心的复合电源系统。

然而，在实际应用中，电池存在着充电时间长、寿命短、安全隐患等诸多问题。

为了解决这些问题，超级电容器作为一种新型的储能技术受到了越来越多的关注。

本文将从电动汽车复合电源系统入手，介绍超级电容器建模与仿真研究的相关内容。

一、电动汽车复合电源系统介绍电动汽车复合电源系统由燃油发动机、电动机和电池组成。

电池是其中最重要的组成部分，因为它直接影响到电动汽车的续航里程和性能。

目前，锂离子电池、铅酸电池等成熟的储能技术被广泛应用于电动汽车中。

然而，这些电池存在着充电时间长、寿命短、能量密度低等问题，影响车辆的性能和可靠性。

二、超级电容器的优势超级电容器作为一种新型的储能技术，有很多优势。

首先，超级电容器充电时间短，可以在短时间内快速充满能量，提高了电动汽车的使用便利性。

其次，超级电容器寿命长，可以循环充放电数万次以上，大大延长了电动汽车的使用寿命。

此外，超级电容器能量密度高，可以储存更多的能量，提高了电动汽车的续航里程。

三、超级电容器建模与仿真超级电容器的建模与仿真是研究超级电容器的有效手段。

常见的建模方法有等效电路模型、特征参数模型、动态模型等。

等效电路模型是最常用的建模方法，将超级电容器模拟成一个等效电路，可以方便地分析电容器的电性能。

特征参数模型则是将超级电容器的特征参数，如电容、内阻等参数进行建模。

动态模型则是通过物理公式和传递函数等方法，建立超级电容器的时域和频域模型。

超级电容器仿真主要通过MATLAB/Simulink等软件实现，可以实时模拟超级电容器的充放电过程，并进行各种参数的分析和优化。

通过仿真和调试，可以有效提高系统的性能和可靠性。

四、结论在电动汽车复合电源系统中，超级电容器作为一种新型的储能技术，具有充电时间短、寿命长、能量密度高等优势，是改善电动汽车性能和可靠性的重要手段。

超级电容器建模与仿真是研究超级电容器的有效手段，可以分析电容器的电性能，提高系统的性能和可靠性。

超级电容模型的递推增广最小二乘辨识李月;张晓冬【摘要】In order to analyze the electrical characteristics of supercapacitor ,the model of supercapacitor should be built at first.In this paper, a new model of supercapacitor based on system identification was presented. This paper also introduces the principle of system identification and the basic principle of how to apply recursive extended least squares method to parameter identification. Based on the MATLAB,the program of recursive extended least squares is compiled. According to this program, the transfer function of the supercapacitor is estimated. Finally, the model was simulated and verified. The experimental and simulated results prove that the model presented can depict the characteristics of supercapacitor exactly.%为了更好的描述超级电容的电气特性，需要建立超级电容模型。

提出系统辨识方法进行建模，阐述了系统辨识的原理和递推增广最小二乘法算法，在此基础上利用MATLAB编写递推增广最小二乘法程序估计出超级电容的传递函数，通过仿真对辨识结果进行验证。

Modeling of SupercapacitorGanesh Madabattula1, Sanjeev K. Gupta*21,2Indian Institute of Science, Bangalore*Correspondingauthor:Dept.ofChemicalEngg.IISc,Bangalore,********************.ernet.inAbstract:Recovering and storing energy of a moving vehicle as it slows down and using it to accelerate the vehicle later can significantly increase fuel efficiencies of automobiles. Low cost batteries available today perform quite poorly in applications that require high (dis)charge currents. Use of adequately designed supercapacitors (SC) in electrical circuits in principle allows present day high energy density batteries to be used effectively, as the momentary burden of high (dis)charge currents is taken up by the supercapacitors.The high capacitance of supercapacitors is achieved by increasing the surface area offered by the electrodes by choosing appropriate porous material such as activated carbon and by reducing the distance over which electric field is experienced to Debye length by using mobile ions in an electrolyte. Supercapacitors are typically modeled as a complex RC circuit. The parameters of such a model do not easily relate to the physical processes such as movement of ions in micro and meso voids in response to applied electric field and building up of charge in double layer. The present work uses a more fundamental transport process based approach, already available in the literature, to model a SC. In this work, a 1D transport model is developed for a SC with porous activated carbon coated electrodes inserted in an aqueous electrolyte solution. The model considers diffusive and convective movement of ions in a straight narrow channel in response to concentration gradient and local electric field. The governing equations are solved using COMSOL. The model explains variation of anodic and cathodic potentials during (dis)charging, recovery of potential drop during relaxation phase after high rate of discharge, limiting current densities, and effect of electrolyte concentration and diffusivities of ions on dynamics of (dis)charging process. The approximations used to obtain 1D model were dropped and simulations were carried with full 2D domain in COMSOL. The simulation results show that 1D model for a SC is quite adequate. Keywords: Supercapacitor, double layer capacitance, energy storage, 1D model.1. IntroductionSupercapacitors (SC) are high power density energy storage devices and are used where batteries alone cannot provide energy needs at high rates. SC’s help us in recovering energy from decelerating automobiles, tides and wind etc. and supply energy when there is a momentary demand at high rates. Emergency power backup for computational facilities and aeroplanes and peak power load at rocket launching and automobiles startup are some of the examples. The most widespread use of SC is in hybrid electric automobiles where SC’s share peak energy demand and thereby help to extend the battery life. SC’s also recover energy while applying brakes, thereby increasing the fuel efficiency of vehicles. In hybrid system, SC gets charged first in a few seconds. It later charges the battery after the energy source is withdrawn.A typical SC contains two electrodes made up of high surface area activated porous carbon particles coated on a highly conductive current collector. The electrodes are separated by an ion permeable polymer membrane to avoid short circuit. The use of scaling up of SC in commercial hybrid vehicles require 5 Wh/kg energy density, 1 kW/kg power density, a RC time constant <1s and low cost(<US$2-3/Wh) (Burke, 2000). To meet these requirements, several supercapacitors are connected in series and parallel to form a network. These bundles of SC’s have high resistance, time lag, unequal energy distribution and less efficiency. In the present work, we model to an electrochemical supercapacitors with no Faradaic reactions on electrode to under movement of ions in supercapacitors better.2. Model equations for supercapacitor Present modeling approach is based on the transport processes used earlier by (Johnson andNewman (1971), Newman and Thomas-Alyea (2004) and Srinivasan and Weidner (1999) to understand ion pore resistance in a laboratory scale SC. The model uses local concentration dependent ionic conductivity. The electrolyte used is 3M H 2SO 4 solution.Figure 1 shows the schematic of a laboratory scaleSC modelled in the present work. The two sided electrodes have activated porous carbon coated on current collector and inserted in a bulk electrolyte pool. The ion permeable membrane separator is not present, as it is unnecessary for the present setup. The resistance offered by the current collectors is neglected in the model due to their high conductivity.Figures 2 and 3 show the simplified SC setup for 1D model. The ends in 1D model are connected by assuming continuity of fluxes and values of variables to make the system closed. The capacitor system can thus be separated into two independent capacitors in parallel with different internal resistance for movement of ions on account of different separation distances between inner and outer faces of positive and negative electrodes shown as L s1 and L s2 in the figures.Figure 1: Schematic representation of supercapacitor setupThe total capacitance of the setup isandFigure 2: Approximation oftwoparallelsupercapacitorsFigure 3: 1D model of supercapacitor2.1 One dimensional concentration dependent model2.1.1. Porous Electrode PhaseThe governing equations scaled with electrode thickness, L (150e −6m) are discussed next. The electronic current through electrode (A/m 2), given by Ohm’s law is:where σ is electrode conductivity (S/m).The charge conservation equation in electrodephase is (Johnson and Newman, 1971)where r.h.s represents charge conserved in double layer, with capacitance C d . The latter is taken to be constant (F/m 2). Here, S d is interfacial surface area (m 2/m 3).The current distribution in electrolyte phase in all the subdomains is given bywhere z i is valency and N i is the flux of i thspeciesThe Nernst-Plank equation (without convection) for flux of i isThe ionic current through electrolyte (A/cm 2) (Newman and Thomas-Alyea, 2004) is given bywhere first term on the r.h.s represents the flux due to electric field and the second term represents the flux due to the concentration gradient. κpn is electrolyte conductivity in porous medium, ε is porosity of electrode, z + and z − are charge numbers of cation and anion and D + and D − are diffusivities of cation and anion respectively.The charge conservation equation in electrolytephase is (Johnson and Newman, 1971)The overall charge balance with the assumption of electroneutraulity is (Johnson and Newman, 1971)Mass balance equation for i thspecies is:where R i is the source term (double layer contribution) and N i is the flux, defined as:At the negative electrode (subdomains s 5 and s 6)At the positive electrode (subdomains s 2 and s 3)where t 0 + and t 0 − are transference numbers of positive and negative ions, which accounts for fraction of current carried by positive andnegative ions.The second term on the r.h.s. of equation for C i accounts for charge accumulation near electrode surface due to the formation of double layer while charging and discharging.The salt concentration is given by2.1.2 Bulk electrolyte phaseThe concentration variation of ions in subdomains s 1, s 4 and s 7 is given bywhere κsn is conductivity of the electrolyte in bulk. As the charge accumulation in the bulk iszero,2.1.3 Initial conditionsThe initial values for variables C + and C − are:C + = C s0ν+ C − = C s0ν−2.1.3.1 Discharge cycleAt positive electrode, ϕ1=1. At negative electrode, ϕ1=0. Everywhere in the supercapacitor, ϕ2 = 0.5.2.1.3.2 Charge cycleAt positive electrode, ϕ1=0. At negative electrode, ϕ1=0. Everywhere in the supercapacitor, ϕ 2 = 0.2.2 Boundary conditionsThe model consists of seven subdomains involving 25 independent equations for two potentials ϕ1 and ϕ 2 and two concentrations C + and C −. The system of equations requires 50 independent boundary relations to completelyspecify the model. These relations are discussed next.Boundaries B 3 and B 6 where current collectors are located act as barrier to the movement of ions, hence i 2 and N i become zero at these locations.At boundaries B 9, B 11, B 12 and B 14, porous electrode face is in contact with the bulk electrolyte. It is assumed that the surface area offered at these surfaces is negligible compared to the interior surface, hence no current flows in the matrix phase at these locations(i 1 = 0).At boundaries B 2, B 5, B 4 and B 7 where electrolyte in porous electrode face meets bulk electrolyte, continuity conditions require both fluxes (N i and i 2) and absolute values (ϕ2, C + ,and C −) to be equal on both sides. These boundaries are continuous in ionic current flow. At boundary B 10, where cell current I cell is drawn (from the positive electrode during discharge)At boundary B 13, where cell current is pumped (into the negative electrode during discharge)Equations 23 and 25 represent jump conditions for flux at the respective boundaries, physically they represent a SC discharging at current I cell . Electrons equivalent to current I cell are withdrawn on one electrode and pumped into the other. The direction of movement of electrons is reversed during the charging process.At boundaries B 1 and B 8, symmetric boundary condition is applied: ϕ2 and C i at B 8 is equal to ϕ2 and C i at B 1, and the ionic flux leaving one boundary enters through the other. One more equation is required to completely specify theproblem, and it is given byat B 14. Equation 27 serves as a reference point for all the potentials in the supercapacitor.2.3 Two dimensional modelThe geometry for 2D model is similar to the one shown in Figure 1. The 2D model equations were developed based on 1D model. The initial and the boundary conditions for the simulations are the same as those for the 1D problem. The periodicity b.c. used in 1-d model is not required as the entire geometry is taken into account. Cell current boundary conditions are applied to entire current collector plate (A/m 2) and reference potential ϕ2 = 0 is specified at the top of current collector of negative electrode.2.4 Parameters used in the modelIn the simulations, it was assumed that electrolyte is asymmetric (not binary). H 2SO 4 is an example for asymmetrical electrolyte. It dissociates into two H + cations and one SO 42−. So, concentration change at both electrodes is not the same as that in the case of symmetrical electrolyte, further both the ions have different diffusivities and ionic sizes. Diffusivity of electrolyte in bulk medium is given by (Newman and Thomas-Alyea, 2004)Diffusivity of i th ion in any porous mediumwhere ε is porosity of given subdomain.Diffusivity of salt in bulk electrolytewhere εs = 1 for bulk electrolyte.Diffusivity of salt in porous medium (Newman and Thomas-Alyea, 2004),where ε is porosity of the medium.Conductivity of electrolyte in porous electrode is given as (Bruggeman empirical relation),Table 1: Parameters used in simulating the modelWhere κp0, the conductivity of electrolyte in bulk medium in the context of dilute solution theory (Newman and Thomas-Alyea, 2004), is given by3. Results and discussionThe models discussed in the previous sections are simulated using COMSOL Multiphysics (v4.2a), with multiphysics (PDE coefficient) and AC/DC (electric current (ec)) modules.3.1 1D resultsThe domain considered for 1D modeling spans 28 units. Thickness of the porous electrode on one side of the current collector represents one unit (L). This geometry was meshed to 3500 elements. Further refinement of the mesh did not change the simulation results. Solver used is ’Direct( PARDISO)’ with flexibility to take time steps on its own. Ends of the domain are connected with periodic boundary condition (PBC) to maintain continuity of fluxes and variables.Using the concentration dependent model, the galvanostatic discharge curve (V cell vs t) obtained at 1 A/m2and 3M H2SO4solution is shown in figure 4. The SC is initially charged to 1V. Figure 4 shows that there is sudden potential drop followed by a slow linear decrease. The reason for this behavior is the resistance offered to ions in the electrolyte. This discharge curve profile is in qualitative agreement with the actual SC with assumptions used for simulations. These result are consistent withThe amount of charge recovered for 1 A/cm2 is less than given by equation 35 due to the resistance offered by the electrolyte, it can be recovered though after the capacitor is allowed to relax.Figure 4 also shows anode and cathode potentials at 1 A/cm2discharge current and 3M electrolyte concentration. These potentials are computed by introducing a reference point in the center of SC, i.e. in between the inner faced electrodes.These potentials are expressed aswhere V a is anode potential, V c is cathode potential and V cell is cell potential. ϕ1,+, ϕ1,−are potentials of positive and negative electrodes respectively. ϕ2,r is potential of reference point. After t=1.3, the cell was left to relax after discharge (see Figure 4). During this period, the unrecovered charge helps in raising the potential.The potential difference (ϕ1−ϕ2) between the electrode and the electrolyte at an interface isan indication of the extent of charge present at the latter. These profiles at 1 A/cm2, 3M are shown in Figure 5. Initially, ϕ1−ϕ2profiles are flat. The non uniform potential difference profiles shown for t=0.02s represents the spatially non uniform charge distribution at the interface in the electrode. The figure shows that ϕ1−ϕ2profiles approach zero as the SC discharges. This kind of non uniform discharge of electrode is not observed at low currents, say 0.1 A/cm2 because the electrolyte resistance does not dominate, and therefore does not accumulate leading to full recovery of charge stored in SC.Figure 4: Cell voltage, anodic and cathodic potential profiles during discharge and relaxation at 1 A/cm2, 3M.The concentration profiles during discharge at1 A/cm2 and 3M initial concentration are shown in Figure 6. The aqueous electrolyte used in SC dissociates into H+ and SO42− . Cations, H+ have nine times higher diffusivity than anions, SO42−. The electroneutrality constraint tells that at any point in spaceTo maintain this condition, H+ ions released at the negative electrode approach the positive electrode faster than the approach of SO42− ions released on positive electrode to negative electrode. This manifests through difference in peak heights of salt concentration (C s) at both electrodes as shown in Figure 6.Figure 5: Potential difference profiles across the double layer in porous electrode at 1 A/cm2, 3M.3.2 2D resultsThe transport parameters required in the model are assumed to be isotropic. COMSOL Multiphysics is used. The dimensions of geometry of SC is taken is 20*12 after scaling the space coordinate with L, the thickness of the porous electrode coated on one side of current collector. The electrodes of size 2*8 are placed at the top of the supercapacitor at coordinates (5,12) and (15,12) (Figure 1). This geometry is meshed into 15406 triangular elements, 840 edge elements and 29 vertex points. The solver used to solve the system of equations are ”Direct(PARDISO)” and the relative tolerance is taken as 10−5. Figure 7 shows the comparison of galvanostatic discharge curves of SC at 1 A/cm2 and 3M initial electrolyte concentration. The result obtained for 2D are close to those obtained with 1D as shown in Figure 7. Though they are not exact, differences are minimum.Figure 6: Concentration variation profiles during discharge at 1 A/cm2, 3MThe assumption made in 1D, i.e. L s2, the space between outer faced electrodes is double of the inner space (L s1) is need to be extended toaccount for more resistance.Figure 7: Comparison of discharge profiles from 2D and 1D simulations at 1 A/cm 2Figure 8: Gradient of electrolyte potential streamlines at 1 A/cm 2, 3MFigure 8 shows the streamlines of gradients of ionic flux of x-component, ϕ2x and y-component, ϕ2y . The figure shows that streamlines of potential gradients of inner faced and outer faced electrodes are quiet independent of each other. The major assumption made in 1D model which considered electric field between the outer faced and the inner faced electrodes separately draws support from the 2-D results shown in the figure. The phenomenon of potential recovery after discharge is the same as that for 1D. Figure 7 shows recovery in potential after discharge at 1 A/cm 2 and 3M.4. ConclusionIn this work, a 1D concentration dependent model is developed for laboratory scale two electrode aqueous electrolyte double layercapacitor, that doesn’t involve any Faradaic processes. This model, based on the transport approach, is built from the first principles of ionic movement. Using the isotropic transport properties, 2D model equations were also developed. The 2D model does not require the assumption of periodicity made in the 1D model and is more realistic. A comparison of both 1D and 2D model prediction shows that the approximation made in developing 1D model are valid.The simulation results show that both the models explain recovery of potential during relaxation after dis(charge) and its dependence on current and concentration, and the failure of SC during charging at high currents and low concentrations of electrolyte. The models also explain galvanostatic discharge curves, relative utilization of charge, process of double layer formation and its disappearance.8. References1.Burke, A. (2000) Ultracapacitors: why, how, and where is the technology. Journal of power sources 91(1), 37–50.2. Johnson, A. and Newman, J. (1971) Desalting by means of porous carbon electrodes. Journal of the Electrochemical Society 118, 510.3. Lin, C., Popov, B. and Ploehn, H. (2002) Modeling the effects of electrode composition and pore structure on the performance of electrochemical capacitors. Journal of the Electrochemical Society 149, A167.4. Newman, J. and Thomas-Alyea, K. (2004) Electrochemical systems . Wiley-Interscience.5. Srinivasan, V. and Weidner, J. (1999) Mathematicalmodeling of electrochemical capacitors. Journal of the Electrochemical Society 146, 1650–1658.。

第24卷第2期电力科学与工程Vol.24,No.22008年3月Electric Power Science and Engineering Mar.,20081收稿日期：2007-10-21.基金项目：国家自然科学基金资助项目（50677018）；华北电力大学重大项目预研基金（20041303）.作者简介：汪娟华(1978－),女,华北电力大学电力系统保护与动态安全监控教育部重点实验室硕士研究生.超级电容器储能系统统一模型的研究汪娟华，范伟，张建成，阮军鹏（华北电力大学电力系统保护与动态安全监控教育部重点实验室，河北保定071003）摘要：从超级电容器储能系统的运行机理出发，设计了含双向DC-AC-DC 变换器的超级电容器储能系统主电路结构，并建立了其统一模型。

仿真结果证明了所建统一模型的正确性和有效性,并表明超级电容器储能系统提高了分布式发电系统的运行稳定性。

关键词：电容器储能系统；模型；双向DC-AC-DC 变换器中图分类号：TM761文献标识码：A0引言随着电力系统的发展，分布式发电技术越来越受到人们的重视[1]。

储能系统作为分布式发电系统必要的能量缓冲环节，其作用越来越重要。

除此之外，储能系统对提高电力系统的运行稳定性也具有非常重要的作用。

超级电容器储能系统利用多组超级电容器（称为超级电容器组件阵列）将能量以电场能的形式储存起来，当能量紧急缺乏或需要时，再将存储的能量通过控制系统释放出来，准确快速地补偿系统所需的有功和无功，从而实现电能的平衡、稳定控制[2]。

超级电容器具有非常突出的优点，其具有如下的特性：（1）其功率密度高，为7000~18000W/g ，可以在短时迅速放出能量；（2）循环寿命长，高低温性能好，效率高，其循环寿命大于500000次，效率高于95%；（3）超级电容器的充放电速度快，充放电效率高，其充电方式比起其他的储能系统简单，控制也相对容易；（4）超级电容器的材料几乎没有毒性，环境友好，而且在使用中无需维护[3，4]。

清洁能源与新能源混合电动车用超级电容能量源建模强国斌,李忠学,陈　杰(上海交通大学机械与动力工程学院,上海　200030)摘　要:阐述了超级电容的一种基于试验数据的建模方法。

结合试验研究超级电容容量特性和内阻特性,全面分析了对建模有影响的超级电容各种功率特性,继而得出超级电容数学模型,最后用M AT LAB /Simulink 做出仿真模型进行仿真示例。

关键词:超级电容;能量源;仿真;建模;MA T LAB /Sim ulink中图分类号:T M 93 文献标识码:A 文章编号:1005-7439(2005)02-0058-04Modeling of Ultracapacitor Energy Storage Used in Hybrid Electric VehicleQIANG Guo -bin ,LI Zhon -xue ,C HEN Jie(Scho ol of M echanical Eng ineering ,Shang hai Jiao T ong U niv ersity ,Shanghai 200030,China )A bstract :It introduces a method of modeling ultr aca pacito r based on data by ex periments.F ir st ,capacity pr operty and resistance property a re studied by experiments.T hen ,all the impor tant pro per ties o f ultracapacitor are analyzed to make the mathematic model.In the end ,a simulation model is set up by M A T L AB /Simulink for a demonstra tion.Keywords :U ltr acapacito r ;Energ y sto rag e ;Simula tion ;M odel ;M A T LA B /Simulink 超级电容是20世纪60年代发展起来的一种新型储能单元,具有功率密度大、充电时间短、使用寿命长、充放电效率高等优异特性,它与二次电池联用作为电动汽车的动力系统,已被认为是今后解决电动汽车发展的最优途径之一。

光电导纳米超级电容器模拟建模优化方案总结摘要：随着科技的迅猛发展，超级电容器作为一种新型的储能设备，受到了越来越多的关注。

其中，光电导纳米超级电容器以其高能量密度和长寿命的特点，成为研究的热点之一。

本文从模拟建模和优化角度，对光电导纳米超级电容器进行了研究和总结。

文章首先介绍了光电导纳米超级电容器的原理和结构，然后详细阐述了模拟建模的过程和方法，并针对模拟建模中存在的问题提出了一些优化方案。

最后，通过实验验证了优化方案的有效性，并对未来的研究方向进行了展望。

1. 引言光电导纳米超级电容器是一种基于纳米材料的新型储能设备，其在能量密度和循环寿命方面具有明显优势。

然而，由于其特殊的结构和材料特性，光电导纳米超级电容器的模拟建模和优化仍然是一个挑战。

本文旨在通过建模和优化方案的研究，提高光电导纳米超级电容器的性能和可靠性。

2. 光电导纳米超级电容器的原理和结构光电导纳米超级电容器是通过光能转化为电能进行储能的一种设备。

其主要结构由电极、电解质和纳米材料组成。

电极通常使用碳纳米管或二维材料，而电解质则是具有高离子导电性和低电阻率的材料。

纳米材料作为电容储能介质，具有高比表面积和良好的电化学性能。

3. 模拟建模的过程和方法为了研究光电导纳米超级电容器的性能，模拟建模是必不可少的工作。

首先，通过将光电导纳米超级电容器的结构离散化为网格，建立数值模型。

然后，根据电场和电流的分布情况，使用有限元方法或有限差分方法，解析求解电荷传递和扩散的方程。

最后，通过改变电极和纳米材料的结构参数，分析模型的输出，评估光电导纳米超级电容器的性能。

4. 模拟建模中存在的问题及解决方案模拟建模中存在一些问题，如网格生成困难、边界条件的选择和非线性关系的处理等。

针对这些问题，本文提出了一些解决方案。

首先，采用自适应网格生成方法，根据电场和电流分布的变化，生成合适的网格。

其次，选择合适的边界条件，比如固定电势或固定电流，以准确模拟实际场景。

2010年2月第36卷第2期北京航空航天大学学报Journal of Beijing University of Aer onautics and A str onautics February 2010Vol .36　No 12　收稿日期:2009202223　基金项目:台达环境与教育基金资助项目　作者简介:盖晓东(1978-),男,山东莱阳人,博士生,f orrestgxd@.改进的超级电容建模方法及应用盖晓东 杨世彦 雷　磊 席玲玲(哈尔滨工业大学电气工程与自动化学院,哈尔滨150001) 摘 要:提出了一种改进的基于物理2端行为特性的超级电容建模方法,模型包含3部分分支电路,在即时分支电路里,采用了一个电压受即时电路端电压控制的电压源和一时间常数恒定的电容串联来模拟超级电容器的即时特性,同时给出了等效电路的参数确定方法.仿真结果表明,该模型可以精确描述超级电容充放电过程的非线性及充放电之后的电压自恢复特性,具有很好的静态特性和动态特性.该模型简单实用,已成功应用于均衡电路参数优化设计中,并在其它采用超级电容作为能量源的电力电子装置中有很好的应用前景.关　键　词:超级电容器;等效电路;建模方法;非线性中图分类号:T M 9106文献标识码:A 文章编号:100125965(2010)022*******Advanced u ltra cap ac it o r m o de li ng m e tho d and app li ca ti o n sGai Xiaodong Yang Shiyan Lei Lei Xi L ingling(Depart m ent of Electrical Engineering,Harbin I nstitute of Technol ogy,Harbin 150001,China )Ab s tra c t:An ultra capacit or (UC )modeling method based advanced physical ter m inal behavi or was p r o 2posed .This model with perfect stable and dyna m ic characteristics according t o the actual physical p r operties was consisted with three parts of the branch circuits,a voltage contr olled voltage s ource serried by a constant capacitance was used in real 2ti m e branch circuit t o si m ulate real 2ti m e characteristics of super 2capacit ors .The method t o deter m ine model para meters was als o p r oposed .Si m ulati ons results show that this model can well re 2flect the nonlinear charge and discharge behavi or and de monstrate self voltage recovery behavi or after charge and discharge .This model has been successfully app lied t o the op ti m um design of equalizati on circuit and it is si m p le enough t o be easily app lied for si m ulati on of power electr onics app licati ons involving UC p ractically .Ke y wo rd s:ultra 2capacit or;equivalent circuit;modeling method;nonlinear 超级电容器作为电压源型电能存储元件,具有功率密度高、充电时间短、可靠性高等优点,由超级电容构成的串联储能电源组广泛应用于电动车辆、不间断电源、分布式发电等相关领域.作为串联储能电源组的关键技术之一[1],储能单体间的能量均衡电路必须具备动态特性好、效率高和均衡速度快的特点,设计一只参数可靠、性能优良的能量均衡电路往往要通过反复的电路仿真和参数优化过程来实现,关键的一环是要有精确的超级电容器等效电路模型.人们提出了多种基于物理特性的超级电容模型[2-6],传输线模型、多阶梯形模型是其中代表.这类模型的优点是参数具有特定的物理意义,但用其描述充放电特性仍有局限性.1　超级电容器建模选取10组电容器进行充放电实验,如图1所示,由充放电特性曲线可以看到:1)超级电容在充放电开始瞬间分别有电压陡升和陡降现象,表明超级电容器件内有一定内阻.2)超级电容器在充放电结束后,电压有自恢复过程,自恢复过程表现出很大的时间常数,表明在充放电过程和结束后其内部电荷有重新分布、均匀化的过程.这种重新分布均匀化可以用时间常数较大的RC 分支来表现,超级电容器应该有多个时间常数不同的RC 分支.a 充电特性b 放电特性图1　超级电容特性1.1　改进的物理2端行为特性模型建立的超级电容器物理2端行为等效电路模型应遵循如下原则:1)所建的模型结构和参数应具有实际的物理意义,模型应尽量简单.2)所建的模型应具有相当的精度,在一定的时间范围内(30m in )能够真实表现超级电容的特性曲线.3)模型中参数应该能够通过测量实际超级电容器件端部特征分析得到.电容特性的两大影响因素是电压和时间.实际的超级电容器有电压自恢复过程,很难用单一理想可变电容来模拟超级电容器,本文结合超级电容的实际物理机理和器件端行为特性提出了改进的三分支等效电路模型,包含即时充电分支电路(C 0和电压控制电压源(VCVS,Voltage Con 2tr olled Voltage Sourse ))、自再分布分支电路(R 1,R 2,C D ,C L )和泄放分支电路(R L )3部分,见图2.1.2　参数确定1)R S0,R S1,C S1.图2　改进的物理2端行为特性模型在恒流充放电开始瞬间,电容C S1可视为短路,电压的跳变完全由R S0上压降产生,电阻R S0等于恒流充放电瞬间电压跳变幅值与施加电流之比,即R S0=ΔuI(1)其中,Δu 是电压跳变幅值;I 是充电电流大小.串联分支R S1,C S1的作用是用来描述电容单元大电流充放电过程电荷受分布过程中扩散速度限制而引起的局部短时间电荷聚集的“物理极化”现象,主要影响充放电瞬间的暂态过程,在较长时间的充放电和自放电、自恢复过程中可以忽略不计.利用图3所示的简化电路和式(2)计算R S1和R S2.图3　确定参数R S1,C S1的等效电路u =(R S0+R L )i 1u =(R S0+R L +R S1)i 2(2)其中,i 1为放电过程初始电流峰值;i 2为稳态放电电流大小;R L 为负载电阻.极化内阻是放电电流的函数,负载电阻越大,放电电流越小,“极化”内阻越大.这是由于在电容器内部瞬间电荷积聚密度相同的情况下,电流越大电极上电荷越容易释放,瞬间积聚越少,由此引起的“极化”电阻也越小.试验表明电流尖峰持续时间非常短,即C S1充电过程非常短暂,其值可通过仿真步长大小进行估算,令其值引起的充放电过程小于一个步长即可.2)C 0,f[v 1(t )].超级电容容量值是电压和时间的函数,因此即时充放电分支电路可用可变电容C (t )表达,即C (t )=f [u (t ),t ],而对于可变电容C (t ),可用一只固定电容和电压控制电压源的串联来描述,如371　第2期 盖晓东等:改进的超级电容建模方法及应用图4所示,其等效电路关系如式(3)所示.图4　求解C (t )等效电路C (t )=C 0[1-f (v 1(t ))](3) 超级电容充电快速,C (t )所处的即时分支电路相较于其他两分支电路具有很小的时间常数,由此可假定电荷在充电开始阶段全部存储在该可变电容中.即时分支电路参数可以通过一个完整的大恒定电流充放电循环确定,在此充电过程中测得的电容就可认为是该时刻的C (t )值.实验数据表明C (t )大体与超级电容两端电压呈线性关系,即:C(t )=A +B ・v (t )=A [1+B /A ・v (t )](4) 这样可以确定C 0的值应等于A,而v 1(t )就是端电压u (t ),而函数f [v 1(t )]就等于-B /A ・v (t ),而A 和B 则可通过试验确定.3)C D ,C L .自再分布电路是由短时分支电路和长时分支电路构成,短时分支相对长时分支同样具有相对较短的时间常数.短时分支决定了超级电容充放电结束后10m in 时间内表现出的端行为特性,而长时分支则用来描述30m in 内超级电容表现出的端行为特性.在充电阶段所有的电荷都被注入到即时分支电路中,充电结束后,自再分布分支电路开始对超级电容特性产生影响.10m in 内部分电荷从C 0转到C D ,此后的20m in 内部分电荷从C D 转到C L .这一转移过程可通过图5所示的简化电路表示,由式(5)和式(6),通过测量u 1(t )和u 2(t ),确定参数C D 和C L 值.u 1(t )=u C (t 0)C (t 0)+u C 1C DC (t 0)+CD +C 1(u C (t 0)-u C 1)C (t 0)+CD etτ1(5)u 2(t )=u C (t 0)C (t 0)+u C 2C LC (t 0)+C L+C 2(u C (t 0)-u C 2)C (t 0)+C Letτ2(6)u C (t 0),u C 1,u C 2为C (t ),C D ,C L 自放电阶段初始电压;τ1=R 1C (t 0)C D C (t 0)+C D为短时分支电路时间常数;τ2=(R 1+R 2)C 0C LC (t 0)+C L为长时分支电路时间常数.图5　求解C D ,C L 等效电路2　模型仿真及应用本文通过对容量标称30kF 超级电容各种不同充电条件下的响应特性进行仿真,以验证该改进的物理2端行为特性超级电容模型.其等效电路模型参数如表1所示.表1　等效电路模型参数名称数值名称数值R S0/m Ω0.058C S1/F 40R S1/m Ω0.77C 0/F 11160R 1/m Ω12.9C 1/F 11945.3R 2/m Ω27.13C 2/F 5321.7R L /kΩ200f[v 1(t )]-0.7u (t ) 图6是超级电容等效电路在100A 充放电电流条件下的端电压特性,可以清楚地看到超级电容器在充放电结束后端电压的自恢复过程,同时可以看到仿真结果和试验结果差别不大,模型精确度高.a 100A 恒流充电曲线b 100A 恒流放电曲线图6　模型端电压特性仿真结果图7是在100A 恒流充电600s 后静止,2000s 内超级电容等效电路模型中C 0,C D 和C L471北京航空航天大学学报 2010年　等各部分元件相应的电流响应波形.图7　模型中各元件电流仿真波形 通常采用基于buck 2boost 开关变换结构的能量均衡电路来均衡超级电容单体间端电压的不平衡.能量均衡电路均衡过程也通过PSP I CE 仿真得到,如图8所示.在该过程中,为缩短仿真时间,超级电容等效电路模型中的电路参数都按一定比例缩小.a 均衡电路b 仿真结果图8　超级电容模型在均衡电路中的仿真应用 C R 1和C R 2分别表示真实的超级电容单体,在均衡电路参数优化设计中,C R 1和C R 2采用改进的物理2端行为特性超级电容模型代替.利用该模型进行的优化设计案例中,均衡电路中的电感确定为200μH.3　结　论本文提出了一种能用于能量均衡电路仿真和设计的超级电容模型.采用该方法获得的超级电容器模型能够成功地应用于指导能量均衡电路参数的优化设计.1)本文提出的超级电容模型中,包含3部分分支电路,在即时分支电路里,采用了一个电压受即时电路端电压控制的电压源和一个时间常数恒定的电容串联来模拟超级电容器的即时特性,采用自再分布分支电路来描述此超级电容充放电结束后的电压自恢复过程.2)本文给出了确定模型中参数值的方法,仿真结果表明,实际超级电容器充放电行为特性和利用改进的物理2端行为特性超级电容模型得到的充放电特性有很好的一致性,该模型不但有良好的静态行为特性,同时还具有良好的动态响应特性.参考文献(References )[1]Luis Zubieta,R ichard Bonert .Characterizati on of double 2layercapacit ors for power electr onics app licati ons[J ].I EEE Trans I nd App licati ons,2000,36(1):199-204[2]Nel m s R M,Cahela D R,B ruce J.Tatarchuk .Modeling double 2layer capacit or behavi or using ladder circuits[J ].I EEE Transac 2ti ons on Aer os pace and Electr onic Syste m 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