电能和热能存储设备外文文献

Contents lists available at ScienceDirect Journal of Energy Storage journal homepage Ragone plots and discharge efficiency power relations of electric and thermal energy storage devices Thomas Christen ABB Switzerland Ltd Segelhofstr 1K Baden D ttwil CH 5405 Switzerland A R T I C L E I N F O Keywords Energy storage Heat storage Ragone plot Efficiency power relations Endoreversible thermodynamics A B S T R A C T Ragone plots energy power relations and discharge efficiency power relations are important for characterizing energy storage ES devices as they contain the information on the maximum power and the available energy In this theoretical study these two characteristics and the losses per energy are derived in the framework of en doreversible thermodynamics for ideal electric and thermal ES systems with different dependencies of their intensive thermodynamic variables potentials temperatures of the reservoirs on the extensive state variables electric charge heat that quantify their state of charge While for battery and latent heat storage device the normalized Ragone plot is equal to the discharge efficiency power relation the two characteristics differ for electric capacitor and sensible heat storage device due to their intrinsically limited depth of discharge at constant power Due to the decrease of the exergy of heat at decreasing temperature the discharge efficiency of the ideal sensible heat storage device exhibits a local maximum at a finite power value 1 Introduction Driven by the progressing changes in the electric power infra structure and transport technologies worldwide growth in energy consumption and ecological sustainability requirements R compressed air various types of batteries flow batteries and fuel cells normal and hybrid supercapacitors flywheels and superconducting magnetic ES and thermal ES systems subdivided in sensible latent and thermo chemical ES technologies exist for a big variety of applications remote or grid independent power like for electric vehicles and backup power matching power demand requirements with supply like frequency regulation and power quality load leveling peak shaving buffering spinning reserve and black start with various different possible ben efits autonomy cost minimization loss reduction lifetime extension of other power system components contingency service and area control The optimization and the further development of ES devices and technologies is usually targeting the maximization of certain char acteristic quantities like the available energy power efficiency life time reliability and minimization of cost weight volume and waste This requires characterization recipes that can be used in a general technology independent way for the assessment comparison and benchmarking of different ES devices on equal footing 7 9 The present work focuses on two performance characteristics the relation between power and available energy the so called Ragone plot and the relation between power and efficiency The purpose is to investigate the connection between the two characteristics their properties for different types of ES technologies and their behavior for different de pendencies of their thermodynamic intensive variable on the state of charge For illustration electric and thermal ES will be considered with the intensive variable being voltage or more exact the electro che mical potential difference and temperature respectively Electric ES is generally required for many different applications and there are ongoing large global R Received in revised form 17 November 2019 Accepted 17 November 2019 E mail address thomas christen Journal of Energy Storage 27 2020 101084 Available online 20 December 2019 2352 152X 2019 Elsevier Ltd All rights reserved T dependence of the OCV on the state of charge of a real battery is gen erally not constant the energy released from a latent heat ES may generally contain a sensible heat part and there are sensible heat sto rage technologies running at constant temperature like e g in two tank heat storage 11 The present work with the simplified ideal ES models is conceptual and theoretical and the further application to real ES systems and other ES technologies like kinetic ES e g flywheels and superconducting magnetic ES is straightforward see e g Ref 14 but can be computationally cumbersome for realistic complex models In the next section Ragone plots and efficiency power relations are recalled In Section 3 they are derived in normalized form for electric ES The results were already discussed in the literature 14 15 how ever they are useful for comparison of thermal and non thermal ES furthermore they will be derived and expressed here in a slightly dif ferent but equivalent way than in Ref 14 In Section 4 the same will be done for latent and sensible heat ES All ES systems will be described in the framework of endoreversible systems see Fig 1 which consist of thermodynamic reservoirs reversible processes and irreversible links conductors or resistors 16 17 The results for the different devices will then be compared in Section 5 including irreversible dis charge losses in terms of the entropy production S 2 Ragone plots and efficiency power relations David Ragone emphasized already in his seminal publications 18 19 the usefulness of representing the properties of batteries for electric vehicles in the power energy plane PE or their densities because the performance characteristics as well as the application re quirements can be displayed in the same figure and because it makes evident the trade off between high power and large available energy The theory for calculating Ragone plots of batteries electrochemical capacitors and other ES devices has been described in detail in Refs 14 20 Energy power relations of ES systems can also be experimen tally determined see e g 21 for Li ion batteries however this work will not discuss experimental aspects further Another useful char acteristics of any kind of power device is the efficiency power P relation For ES both characteristics have in common that the device is discharged at constant work power P demanded by the load This is necessary since in applications P is prescribed by the energy consuming load and must thus be considered as a control parameter rather than an implicitly time dependent function of the intrinsic dynamic system variables state of charge voltage temperature current etc For ex ample a capacitor discharge with constant load resistance leads to an exponentially decaying load power which is certainly not useful for general application Of course in real applications P is typically not constant but there is rather an explicitly time dependent power load profile P t Although it might therefore seem academic to consider the available energy and the efficiency at maximum depth of discharge DoD and at constant P these conditions can serve as a simple well defined scenario to characterize different ES devices on equal footing besides the characteristic dimensional values of energy capacity max imum power or discharge time constant For modeling below endoreversible thermodynamics 16 17 will be used since it allows to describe different ES systems in a generic non equilibrium thermodynamics framework Its use for heat storage devices is rather common for instance the round trip efficiencies of thermal storage devices e g with pumped heat was investigated in this manner in Refs 11 22 An ES device with an initially stored energy E0and providing con stant power P to a load can do this only for a limited time t The graph of the maximum available energy as a function of P E PPtP 1 refers here to the Ragone plot which can also be expressed in nor malized form e P E P E 0 2 If the ES device is fully discharged at t Eq 2 is equal to the dis charge efficiency The round trip efficiency contains the charging effi ciency as an additional factor The charging efficiency can in principle be separately maximized without referring to a load profile in real applications load profiles are sometimes also prescribed for full load cycles e g in battery charging by recuperation of braking energy in electric vehicles Charging a thermal ES device consisting of a Carnot machine that provides the power to a heat capacity with a heat pump may lead to a round trip efficiency of unity in the reversible case 11 22 For the purpose of characterizing the discharge performance of an ES device the charging efficiency will not be considered in the following Depending on the system under consideration various loss contributions must be considered acting during discharge In the fol lowing we will include power losses outside the defined ES device into the output power P like power electronic losses in case of electric ES or generator losses in the case of thermal ES Only intrinsic losses in the ES device like those due to the internal battery capacitance re sistance or the heat transfer loss in the thermal ES will be included in the internal ES system losses In case of a limited depth of discharge to a finite residual energy E the energy to be stored in a charging cycle is reduced toEE 0 The efficiency becomes thus P E P EEP e P EE 1 00 3 In practice ES devices are often not fully discharged due to various reasons A trivial one is associated with the specific load profile like shallow load cycling of batteries a case which is not of interest in the present context of constant power discharge Another one may be the degradation of the ES device near the fully discharged state like in certain battery types or a lower temperature limit of a liquid sensible heat storage medium above ambient temperature due to its solidifica tion For each specific case such special conditions can be taken into account individually Here we assume maximum possible depth of discharge which can still be intrinsically limited by a maximum power condition as will be explained in Sects 3 2 and 4 2 Both normalized Ragone plot e P and discharge efficiency P Fig 1 Endoreversible models a Ideal battery capacitor with equivalent series resistance ESR initial open circuit voltage OCV U0 and electric current I connected to a load b Ideal latent sensible heat storage device consisting of a heat reservoir initially at temperature T1 a heat exchanger providing the heat current and a reversible Carnot process working between temperatures Tc and T2 In both cases a and b the load demands a constant power P T ChristenJournal of Energy Storage 27 2020 101084 2 are useful characteristics and which of them is relevant depends on the specific question of interest For energy cost considerations may be more relevant while e can be more important if the amount of energy obtained from a discharge is relevant 23 3 Electric energy storage devices Consider the leakage free ideal electric ES device illustrated in Fig 1 a Ideal means constant equivalent series resistance ESR R constant open circuit voltage U0 in case of the battery and constant capacitance C in case of the capacitor The state of charge of the ES device is generally defined here by the value s of the system s state variable s For electric ES it is the electric charge Q and the initial charge QQ t 0 0 is considered as the relevant reference charge For batteries this slightly differs from the usual definition of the state of charge Q Cbat where it is normalized to the battery capacity Cbat a parameter value that contains some ambiguity in practice The general procedure for calculating the Ragone plot starts with solving the or dinary differential equation modeling the discharge process for the state variable s in time dQ dt I 4 with initial condition Q tQ 0 0 The discharge current I and the load voltage drop U are related to the constant load power by PUI 3 1 Battery For the ideal battery the open circuit voltage U0is Q independent Hence U0 I and U are constant in time during discharge The initial energy of the battery is EQ U 000 The current I P is given by one of the two solutions of the quadratic equation IUI URIP 0 namely IUpR 11 2 0 where pP P maxwith maximum power PUR 4 max0 2 the relevant I P corresponds to the solution branch with the negative sign in front of the square root because for P 0 the load should be blocking I P 0 0 rather than short circuiting I U0 R associated with the positive sign The ES device can be discharged at constant power for a time t given by Q tQIt0 0 or tQ I 0 Putting the expressions together into Eq 1 leads to the energy obtained by the load i e the Ragone plot E P The normalized Ragone plot for the ideal battery as a function of the normalized power pP P maxfinally becomes e p p 11 2 5 where ppp 11 11 was used Because the battery is fully discharged at tt the Ragone plot is equivalent to the discharge efficiency power relation pe p The result is shown by the da shed curve in Fig 2 Obviously e 0 1because there are no leakage losses and e 1 1 2 which reflects impedance matching i e ESR equals load resistance at maximum power It is mentioned without further comment that if one includes charging of ES which can be easily done by considering negative P 0 in the present context a maximum power value analogous to the maximum discharge power does not exist 3 2 Capacitor The state of charge Q of a capacitor with capacitance C and ESR R Fig 1 a decays according to Eq 4 The initial energy is EQCCU 2 2 C00 2 0 2 where UQ C C 00 is the initial open circuit voltage of the capacitor with initial charge Q0 Solving UQ CRI for I inserting I in Eq 4 eliminating the load voltage with the help of UP I with constant P and multiplying by dQ dt leads to a quadratic equation in dQ dt with relevant solution dQ dt QP R Q 22 2 6 and with RC time constant RC The state of charge Q decays since obviously dQ dt 0 Because 6 must be real there is a lower limit for Q given by QC RP2 A residual energy remains on the capacitor EQCRCPE p 22 2 0 with pP P maxand PUR 4 Cmax 0 2 At the given power P the rest energy is unavailable this refers to the above mentioned limit of the depth of discharge The associated current value is IP R Separation of the variables t and Q in Eq 6 and in tegration from QQ0to QQ t gives t Q Q QQQ Q QQQ ln QQ t QQ 222 2 2 2 0 7 This solution describes the temporal decrease of the capacitor charge Q t The decaying voltages U0 t and U t the increasing current I t and the time dependent effective load resistance U I are then obtained by substituting Q t in the appropriate expressions At QQ I t and U t exhibit a turning point where the capacitor can no longer deliver the required power P 14 Putting Q tQP in Eq 7 leads to tt Q and multiplication by P provides eventually the Ragone plot 1 of the capacitor In terms of the dimensionless quantities eE E 0 and pP P max the final result reads e ppp p p p 1 2 1 11 2 ln 11 8 It holds that e 0 1and since at PPmaxthe equality QQoccurs at t0 one finds e p 1 0 The discharge efficiency power rela tion 3 becomes with EE p 0 p e p p p p p p p p 1 11 2 12 1 ln 11 9 with e 0 0 1 Furthermore p 1 1 2as for the battery because near PPmaxthe discharge occurs at practically constant ca pacitor voltage analogous to battery behavior The functions e p and p are shown in Fig 2 The limit case p 1 with 1 1 2can be seen from a Taylor expansion of the above results One obtains from e pp 1 with e 1 0that de dp 1 p1 From Eq 8 de dppp 1ln 11 2is obtained which confirms 1 1 2 Fig 2 Normalized Ragone plots e p and discharge efficiencies p for the ideal battery dashed and capacitor solid The triangle indicates the limit pP P 1 max where the capacitor acts as a constant voltage source like the battery due to the vanishing depth of discharge 15 T ChristenJournal of Energy Storage 27 2020 101084 3 4 Thermal energy storage devices Heat storage in the framework of Ragone plots has recently been discussed in detail in Ref 24 with the aim of deriving figures of merits for thermal storage materials Lumped circuit models were considered including time and internal space dependent temperature distributions We will not consider here internal space dependence but stay with the simple endoreversible model with average variables as illustrated in Fig 1 b with constant parameter values The model parameters describing the ideal ES device are the heat capacity Cth the specific latent heat L and the equivalent heat resistance Rth of the heat exchanger The values of Cthand L can be estimated relatively easily from well known system properties e g the product of the ac tive volume or mass and volumetric or specific material quantities like the specific heat sensible heat storage or the phase change en thalpy latent heat storage The heat resistance on the other hand may strongly vary between different technologies depending on the type of heat transfer or the design of the heat exchanger and its modeling is thus a challenging task in practice 12 13 In the present context nevertheless it is sufficient to assume given constant model parameter values Cth L and Rthof the ideal ES devices under con sideration Prior to a discussion of the discharge the Novikov model for the irreversible heat engine in the framework of endoreversible thermo dynamics is summarized see Fig 1 b The heat is stored in a finite size upper heat bath at temperature T1 which is connected via the heat exchanger to the Carnot heat engine embedded in the environment at lower temperature T2 A generalization of the present endoreversible model e g by considering other thermodynamic cycle processes than the Carnot process or adding other irreversible elements is feasible 16 The heat exchanger transports the heat from the storage reservoir at T1down to the upper temperature Tc T1 of the working fluid of the Carnot process according to the linear heat conduction law RTT thc1 The Carnot engine which provides the work power output P works reversibly between Tcand lower temperature T2of the working fluid The heat exchange resistance at the lower temperature side is neglected for simplic