Elsevier

Journal of Power Sources

Volume 91, Issue 2, December 2000, Pages 210-216
Journal of Power Sources

Theory of Ragone plots

https://doi.org/10.1016/S0378-7753(00)00474-2Get rights and content

Abstract

The general theory of Ragone plots for energy storage devices (ESD) is discussed. Ragone plots provide the available energy of an ESD for constant active power request. The qualitative form of Ragone plots strongly depends on the type of storage (battery, capacitor, SMES, flywheel, etc.). For example, the energy decreases as a function of power for capacitive ESD, but increases for inductive ESD. Analytical results for a representative set of ideal ESD (battery, capacitor, and SMES) are compared. Furthermore, the effect of leakage and of the specific loss type (Coulomb, Stokes, and Newton friction) is discussed for inductive ESD. Finally, we address the problem of how composite ESD should be treated, and illustrate it for a battery with inductance.

Introduction

Energy storage devices (ESD) are characterized by the energy and the power being available for a load [1], [2]. A prominent example is the comparison of conventional batteries and capacitors. While batteries have high energy densities (about 105J/kg specific energy) but only low power densities (below 100 W/kg specific power), capacitors have rather high power densities (about 106 W/kg) but low energy densities (about 100 J/kg). Batteries, capacitors, flywheels, superconducting magnetic energy storage devices (SMES), pressure storage devices, etc., are thus located in characteristic regions in the power–energy plane. Typical examples are shown in Fig. 1. These regions are related to specific applications by energy and power requirements. The boundaries of the regions are determined by internal losses and/or leakage, etc., of the various ESD. The characteristic time of an application is of the order of the energy-to-power ratio of the ESD. In the log–log plane of Fig. 1, the time corresponds to straight lines. Obviously, batteries are useful for long time applications (>100 s), while conventional capacitors are useful for short time applications (<0.01 s).

Since the efficiency of an ESD is usually dependent on the working point, a single device belongs to a whole curve in the energy–power plane (see inset of Fig. 1). These so-called Ragone plots , which are usually presented in a log–log plot, are standard in the battery community since a long time [1]. First, they provide the limit in the available power of a battery or a capacitor. Secondly, they provide the optimum region of working, which is given by the part of the curve where both energy and power are high. The aim of this paper is to present a unified discussion of the qualitative behavior of Ragone plots for different ESD. Here, we will focus on the specific curves rather than on the specific regions where these curves are located. It turns out also that the specific form of the Ragone curve depends on internal loss and leakage properties of the ESD. A typical qualitative behavior of a Ragone curve is sketched in the inset of Fig. 1. Consider for example a capacitor or a battery. The internal self discharge leads to a decrease of the energy that can be utilized, if the characteristic time of the application exceeds the self discharge time. This fact corresponds to a drop of the Ragone curve for sufficiently low power. On the other hand, the effective series resistance leads to a lower time limit and thus to a maximum power. It is clear that, irrespective of the type of ESD, there are always physical limits to minimum and maximum speed of discharge of an ESD. These limits are reflected in the low/high power behavior of the Ragone plot.

In the next section, we introduce the general class of ESD that will be investigated, and we propose a mathematical definition of the Ragone plot. In Section 3, we discuss two specific cases of potential ESD, which may be interpreted as ideal battery and capacitor. In Section 4, the Ragone plot of a purely inductive ESD is studied, which may be interpreted as a SMES or a flywheel. Furthermore, the effect of various types of friction forces in kinetic ESD is addressed. Section 5 provides a brief discussion of the stability problem of circuits containing a constant power load; as a particular example, we discuss the battery with a series inductance.

Section snippets

Ragone plots

Consider the general circuit of Fig. 2. For example, the ESD may consist of a voltage source, V(Q) , depending on the stored charge Q, an internal series resistor, R, and an internal inductance, L. Note that this ESD can describe many kinds of electrical power sources. For example, a current source delivering a current I0 can be described by L=0, and R, V→∞, with V/RI0. The ESD is connected to a load which draws constant (active) power P≥0. Of course, in general such a load is not related to a

Storage of potential energy

In this section, we consider ESD without inductance (L=0 ). We focus on the particular cases of an ideal battery and an ideal capacitor. ‘Ideal’ means that there is neither frequency dependence nor an intrinsic nonlinearity (e.g. faradaic contributions, pseudo-capacitance, etc.). Battery and capacitor differ in their charge dependence of V(Q) in Eq. (1).

Storage of kinetic energy

Inductive or kinetic ESD store energy exclusively as kinetic energy (coming from the L-term of Eq. (1)), i.e., V≡0. In contrast to storage of potential energy, where load-free losses (P→0) are related to a separate ‘leakage’ property, the kinetic ESD dissipates energy due to internal friction, related to R in Eq. (1). In practice, inductive ESD have thus very small friction forces or internal resistances. We will show that internal friction influences kinetic ESD only at low power, and that the

General cases

The previous discussion pretends that the Ragone curve of an arbitrary dynamic system can be simply derived analytically or numerically, be it of the form of Eq. (1), be it more general. For example, it is in principle straightforward to derive Ragone plots for a hybrid system consisting of two different batteries being parallel, or battery and capacitor, etc. In general, however, things are far more complicated. ‘General’ means that both potential energy and inductive energy appear. As an

Conclusion

We introduced a mathematical scheme for the calculation of the Ragone plots of arbitrary energy storage devices. One has to solve the dynamic problem of the circuit of Fig. 2 with a load drawing a constant power, and to determine the time t when the ESD fails to be able to provide the desired power P. The Ragone curve is then given by E(P)=Pt(P). It is important that in the case where there is more than one solution branch, E1(P), E2(P)…, the Ragone plot belongs to the maximum energy, E(P)=max

Acknowledgments

We thank Christian Ohler for many helpful discussions and fruitful stimulations.

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