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Energy Storage and Loss in Fractional-Order Systems
Abstract
As fractional-order systems are becoming more widely accepted and their usage is increasing, it is important to understand their energy storage and loss properties. Fractional-order operators can be implemented using a distributed state representation, which has been shown to be equivalent to the Riemann–Liouville representation. In this paper, the distributed state for a fractional-order integrator is represented using an infinite resistor–capacitor network such that the energy storage and loss properties can be readily determined. This derivation is repeated for fractional-order derivatives using an infinite resistor–inductor network. An analytical example is included to verify the results for a half-order integrator. Approximation methods are included.
... , i.e., if the system is really at rest. This definition of energy has been validated by a physical interpretation [32,35]. Fractional energy has been used to derive system stability based on the Lyapunov method [33,34]. ...
... The infinite state fundamentals have already been presented in several conference and journal papers (see for example [19,[22][23][24][25][33][34][35]). A two volume monograph [32] provides a "state of the art" of this methodology and its application to different problems arising in fractional system theory. ...
... The optimal excitation ) ( t u has to satisfy the two Euler conditions (35) and (36) with the boundary conditions ...
Optimal control of fractional order systems is a long established domain of fractional calculus. Nevertheless, it relies on equations expressed in terms of pseudo-state variables which raise fundamental questions. So in order remedy these problems, the authors propose in this paper a new and original approach to fractional optimal control based on a frequency distributed representation of fractional differential equations called the infinite state approach, associated with an original formulation of fractional energy, which is intended to really control the internal system state. In the first step, the fractional calculus of variations is revisited to express appropriate Euler Lagrange equations. Then, the quadratic optimal control of fractional linear systems is formulated. Thanks to a frequency discretization technique, the previous theoretical equations are converted into an equivalent large dimension integer order system which permits the implementation of a feasible optimal solution. A numerical example illustrates the validity of this new approach.
... A source of inspiration for the construction of Lyapunov functionals for mechanical systems is the total mechanical energy. For the case of a springpot, an interpretation as an infinite arrangement of springs and dashpots, see [13,30,37], leads to a potential energy term which was also derived in [11,42,43] for an electrical system. It leads to an energy Lyapunov functional with expressions based on the infinite state representation (also known as diffusive representation) of fractional integrators, which were introduced by Montseny [28], Matignon [26] and have been elaborated by Trigeassou et al. [42][43][44][45][46]. ...
... where the elongation q of the springpot element changes according to (11) depending on the force λ acting on it, where c > 0 and α ∈ (0, 1) are constant. For a springpot, a unit step force input −λ(t) = (t) (12) with the Heaviside step function leads to a timedependent elongation output of the form which shows a behavior "in-between" a spring (α = 0) and a dashpot (α = 1), see Fig. 2. A mechanical interpretation of springpots as an infinite arrangement of springs and dashpots is given in [13,15,30,37] and a related potential energy E, which is useful for the direct method of Lyapunov, can be formulated as ...
... with z(η, t) as in (3), see [13,15]. In [11,40], the energy storage of a fractional element in an electrical circuit was derived, which through the mechanical-electrical analogy, is equivalent to the potential energy (14). ...
In this paper, we introduce a generalization of Lyapunov’s direct method for dynamical systems with fractional damping. Hereto, we embed such systems within the fundamental theory of functional differential equations with infinite delay and use the associated stability concept and known theorems regarding Lyapunov functionals including a generalized invariance principle. The formulation of Lyapunov functionals in the case of fractional damping is derived from a mechanical interpretation of the fractional derivative in infinite state representation. The method is applied on a single degree-of-freedom oscillator first, and the developed Lyapunov functionals are subsequently generalized for the finite-dimensional case. This opens the way to a stability analysis of nonlinear (controlled) systems with fractional damping. An important result of the paper is the solution of a tracking control problem with fractional and nonlinear damping. For this problem, the classical concepts of convergence and incremental stability are generalized to systems with fractional-order derivatives of state variables. The application of the related method is illustrated on a fractionally damped two degree-of-freedom oscillator with regularized Coulomb friction and non-collocated control.
... To close the discussion on the work of Lorenzo and Hartley, in the context of the ideas discussed in the Introduction it is necessary to mention that similar studies related to energies of fractional-order elements and transfer functions have been developed in the areas of control and electrical engineering [15][16][17], all of which are based on the Riemann-Liouville fractional derivatives and integrals. However, as declared above, here we restrict ourselves to mechanical systems only and focus on problems beyond the simple power-law memory formulations of the force-displacement relationship. ...
... For m = 0, we have W power−law (m = 0) = 1 k λ Γ(1) Γ(λ+1) t λ , which coincides with (15). For m = 1, the result (38) reduces to (20), namely, W power−law (m = 1) = 1 ...
The energies of the classical Maxwell mechanical model of viscoelastic behavior have been studied as a template with a variety of relaxation kernels in light of a causal formulation of the force-displacement relationship. The starting point uses the Lorenzo-Hartley model with the time-fractional Riemann-Liouville derivative. This approach has been reformulated based on critical analysis, allowing for the application of a variety of relaxation (memory) functions mainly based on the Mittag-Leffler family, in order to meet the need for broader modeling of viscoelastic behavior. The examples provided include cases of the types of forces used by Lorenzo and Hartley as well as a new family of force approximations such as a general power-law ramp, polynomials, and the Prony series.
... Consequently, the fractional integrator is modelled by an infinite dimensional distributed differential system. This methodology, called Infinite State (IS) approach [20] has proved its efficiency for the initialization [25], Lyapunov stability [26] and fractional energy definition [27,28] of fractional differential systems. But basically, it is an efficient tool for FDE simulation [29,30]. ...
... In other words, the state vector X k of the sampled system (27) has an infinite dimension. The simplicity of the I n GL (q −1 ) simulation technique has to be paid by an important cost due to this infinite number of terms: the number of elementary arithmetic operations (additions, multiplications) and memory length (to memorize past values x k−i ) increase considerably with time. ...
Two algorithms for the simulation of fractional differential equations are analyzed and compared numerically. The well-known Grünwald–Letnikov (GL) approach is interpreted in terms of an equivalent fractional integrator methodology. The infinite state (IS) integrator is time discretized in order to compare its discrete impulse response to the GL one. Analysis demonstrates that the two approaches, though deeply different in their principles, are numerically equivalent. Drawbacks of GL algorithm are highlighted—infinite memory requirement, memory truncation, difficulties to take into account initial conditions—whereas the IS algorithm performs a good compromise precision/computation time, associated to initialization ability.
... Based on the fractional energies, Lyapunov functions are proposed and stability conditions of fractional systems involving implicit fractional derivatives are derived respectively by the dissipation function [24,25] and the energy balance approach [26,27]. The energy storage properties of fractional integrator and differentiator in fractional circuit systems have been investigated in [28][29][30]. Particularly in [29], the fractional energy formulation by the infinite-state approach has been validated and the conventional pseudo-energy formulations based on pseudo state variables has been invalidated. Moreover, energy aspects of fractional damping forces described by the fractional derivative of displacement in mechanical elements have been considered in [31,32], in which the effect on the energy input and energy return, as well as the history or initialization effect on energy response has been presented. ...
... The energy storage properties of fractional integrator and differentiator in fractional circuit systems have been investigated in [28][29][30]. Particularly in [29], the fractional energy formulation by the infinite-state approach has been validated and the conventional pseudo-energy formulations based on pseudo state variables has been invalidated. Moreover, energy aspects of fractional damping forces described by the fractional derivative of displacement in mechanical elements have been considered in [31,32], in which the effect on the energy input and energy return, as well as the history or initialization effect on energy response has been presented. ...
This paper addresses the total mechanical energy and equivalent differential equation of motion for single degree of freedom fractional oscillators. Based on the energy storage and dissipation properties of the Caputo fractional derivatives, the expression for total mechanical energy in the single degree of freedom fractional oscillators is firstly presented. The energy regeneration due to the external exciting force and the energy loss due to the fractional damping force during the vibratory motion are analyzed. Furthermore, based on the mean energy dissipation and storage in the fractional damping element in steady-state vibration, two new concepts, namely mean equivalent viscous damping and mean equivalent stiffness are suggested and the above coefficient values are evaluated. By this way, the fractional differential equations of motion for single-degree-of-freedom fractional oscillators are equivalently transformed into integer-order ordinary differential equations.
... The fractional exponent a of the CPE has been widely used in literature [9e11] to study the influence of the electrode porosity, surface inhomogeneity, distributed reactivity, roughness and geometry on UC behavior. Energy storage, losses and efficiency in fractional-order circuits and systems have also been correlated with C f and a [12,13]. ...
... UC fractional-order model. for the calculation of power and energy based on charge-discharge profile [12,13]. For a UC charged with a constant current i, the power at time t, is given as, ...
The usage of ultracapacitors for development of energy storage devices and alternative power sources is increasing at a very rapid rate. However accuracy in selection of ultracapacitor model parameter plays key role in the design of such devices, especially in applications involving wide operating frequency. Ultracapacitors are known to exhibit fractional dynamics and the model parameters vary significantly with frequency. This paper proposes a piecewise modelling and parameter estimation approach for ultracapacitors using a hybrid optimization and fuzzy clustering approach. The proposed modelling technique has been applied over impedance frequency response data acquired from a commercially available ultracapacitor. The model is able to represent the experimental data over different operating points with reduced number of model parameters. Comparative numerical simulations have been carried out to validate the benefits of the proposed approach. The estimated parameters revealed the disparity in the frequency dependent behavior of ultracapacitors and standard electrolytic capacitors.
... The energy storage properties of fractional integrator and differentiator in fractional circuit systems have been investigated in Refs. [15][16][17]. Particularly in Ref. [16], the fractional energy formulation by the infinite-state approach has been validated, and the conventional pseudo-energy formulations based on pseudostate variables has been invalidated. Moreover, energy aspects of fractional damping forces described by the fractional derivative of displacement in mechanical elements have been investigated in Refs. ...
... [15][16][17]. Particularly in Ref. [16], the fractional energy formulation by the infinite-state approach has been validated, and the conventional pseudo-energy formulations based on pseudostate variables has been invalidated. Moreover, energy aspects of fractional damping forces described by the fractional derivative of displacement in mechanical elements have been investigated in Refs. ...
This paper addresses the Lyapunov functions and sliding mode control design for two degrees-of-freedom (2DOF) and multidegrees-of-freedom (MDOF) fractional oscillators. First, differential equations of motion for 2DOF fractional oscillators are established by adopting the fractional Kelvin-Voigt constitute relation for viscoelastic materials. Second, a Lyapunov function candidate for 2DOF fractional oscillators is suggested, which includes the potential energy stored in fractional derivatives. Third, the differential equations of motion for 2DOF fractional oscillators are transformed into noncommensurate fractional state equations with six dimensions by introducing state variables with physical significance. Sliding mode control design and adaptive sliding mode control design are proposed based on the noncommensurate fractional state equations. Furthermore, the above results are generalized to MDOF fractional oscillators. Finally, numerical simulations are carried out to validate the above control designs.
... Remark 1: Both Riemann-Liouville's and Caputo's derivative definition can be viewed as special cases of proposed frequency distributed model. Though some problems of indirect Lyapunov method with frequency distributed model have not been solved [18], [19], it indeed provides a effective way to deal with the stability of fractional order systems and has been widely utilized. ...
... Subscribe (19) into the (17) and (18), and one can obtain the following equations ...
... An alternative methodology, based on the infinite state approach, has been proposed to analyze Lyapunov stability of both linear and non linear FDEs [19,22,23,25]. Its main feature is an original definition of fractional energy [4,19,22]. LMI conditions have been derived for linear commensurate order FDEs. Non commensurate fractional systems correspond to fractional differential systems with different fractional orders for each derivative. ...
... Moreover, energy decrease is interpreted as internal Joule losses. A similar approach to fractional energy has already been proposed in [4]; moreover, it has been validated by its application to infinite RC line energy. ...
Lyapunov stability of linear noncommensurate order fractional systems is treated in this paper. The proposed methodology is based on the concept of fractional energy stored in inductor and capacitor components, where natural decrease of the stored energy is caused by internal Joule losses. The Lyapunov function is expressed as the sum of the different reversible fractional energies, whereas its derivative is interpreted in terms of internal and external Joule losses. Stability conditions are derived from the energy balance principle, adapted to the fractional case. Examples are taken from electrical systems, but this methodology applies also directly to mechanical and electromechanical systems.
... The case of a single rectangular charge and discharge pulse (where charging and discharging currents are both constant, but not necessarily equal) has been analyzed by Hartley et al. [23], [24]. By setting the charge and discharge currents and times to maximize energy efficiency, one obtains an efficiency directly related to the fractional order α: ...
Ageing of rechargeable batteries is routinely characterized in the frequency domain by electrochemical impedance spectroscopy, but the technique requires laboratory measurements to be made on a time scale of days. However, the normal cycling of a battery as it is used
in situ
provides equivalent information in the time domain, though extracting robust frequency information from a time series is challenging. In this work, we explore, in the time domain, the relationship between instantaneous voltage-current phase difference and cycle efficiency. Moreover, we demonstrate that phase measures can be used to identify battery ageing. We have cycled a 250 mA h Nickel-Cobalt cell several hundred times and used Hilbert Transforms to identify phase difference between voltage and current. This phase difference becomes closer to zero as the battery ages, commensurate with a drop in energy cycle efficiency. In another experiment, we applied a synthetic current profile mimicking behaviour of an electric car cell, to a 3.2 A h LiNiMnCoO
<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub>
cell, for ~100 days. For this more complicated profile with a wide range of frequency content, we used wavelet analysis to identify changes in phase difference and impedance as the battery aged. For this cell, drop in cycle efficiency was associated with a rise in internal resistance. The results imply that time-series analysis of
in situ
measurements of voltage and current, when applied with equivalent circuit models and underlying theory, can identify markers of battery ageing.
... If 0 ≤ α 1, the bounding conditions of α will corresponds to the discrete conventional elements: the resistor at α = 0 and the ideal capacitor at α = 1, as illustrated in Figure 1B. As α goes to 0, (Z i ) convergence to 0, and thus the fractional element looks like that a pure resistor, whereas as α goes to 1, (Z r ) converges to 0 and hence, the fractional element serves as a pure capacitor, (Oustaloup et al., 2000;Krishna et al., 2011;Hartley et al., 2015;Trigeassou and Maamri, 2020). Figure 2B represents the schematic diagram for a FOC along with the ideal resistor and capacitor. ...
The blood flow dynamics in human arteries with hypertension disease is modeled using fractional calculus. The mathematical model is constructed using five-element lumped parameter arterial Windkessel representation. Fractional-order capacitors are used to represent the elastic properties of both proximal large arteries and distal small arteries measured from the heart aortic root. The proposed fractional model offers high flexibility in characterizing the arterial complex tree network. The results illustrate the validity of the new model and the physiological interpretability of the fractional differentiation order through a set of validation using human hypertensive patients. In addition, the results show that the fractional-order modeling approach yield a great potential to improve the understanding of the structural and functional changes in the large and small arteries due to hypertension disease.
... The fractional order elements can be approximated via an infinite number of parallel series-RC branches. 68,69 In the next two subsections, we use this fact to analyze the optimal charging problem using the traditional tools of multivariate calculus of variations. ...
In this paper, we discuss the optimal charging and discharging of supercapacitors to maximize the delivered energy by deploying the fractional and multivariate calculus of variations. We prove mathematically that the constant current is the optimal charging and discharging method under R s -CPE model of supercapacitors. The charging and round-trip efficiencies have been mathematically analyzed for constant current charging and discharging. © 2020 The Electrochemical Society ("ECS"). Published on behalf of ECS by IOP Publishing Limited.
... This definition of the Lyapunov function corresponds to fractional energy. It was first proposed in [34]; then it received a physical interpretation in [35]. Its derivative can be expressed as ...
A typical phenomenon of the fractional order system is presented to describe the initial value problem from a brand-new perspective in this paper. Several simulation examples are given to introduce the named aberration phenomenon, which reflects the complexity and the importance of the initial value problem. Then, generalizations on the infinite dimensional property and the long memory property are proposed to reveal the nature of the phenomenon. As a result, the relationship between the pseudo state-space model and the infinite dimensional exact state-space model is demonstrated. It shows the inborn defects of the initial values of the fractional order system. Afterward, the pre-initial process and the initialization function are studied. Finally, specific methods to estimate exact state-space models and fit initialization functions are proposed.
... In view of this initialization theory, the initialized function is an infinite dimensional distributed initial condition z(x, t 0 ), which summarizes all the past behavior for t < t 0 . Furthermore, energy storage and loss in fractional-order systems have been evaluated based on the diffusive model [26]. Besides, several techniques for estimating the initialization functions have been proposed in Refs. ...
Fractional calculus is viewed as a novel and powerful tool to describe the stress and strain relations in viscoelastic materials. Consequently, the motions of engineering structures incorporated with viscoelastic dampers can be described by fractional-order differential equations. To deal with the fractional differential equations, initialization for fractional derivatives and integrals is considered to be a fundamental and unavoidable problem. However, this issue has been an open problem for a long time and controversy persists. The initialization function approach and the infinite state approach are two effective ways in initialization for fractional derivatives and integrals. By comparing the above two methods, this technical brief presents equivalence and unification of the Riemann-Liouville fractional integrals and the diffusive representation. First, the equivalence is proved in zero initialization case where both of the initialization function and the distributed initial condition are zero. Then, by means of initialized fractional integration, equivalence and unification in the case of arbitrary initialization are addressed. Connections between the initialization function and the distributed initial condition are derived. Besides, the infinite dimensional distributed initial condition is determined by means of input function during historic period.
... The authors in [23] proposed the concept of "indirect Lyapunov method", which derived the stability of a fractional order system by proving the stability of the corresponding infinite-dimensional integer order system. Moreover, the energy problem [24,25] and initialization problem [26] for fractional order systems have also been well interpreted based on the proposed model. ...
In this paper, we first verify that fractional order systems using Caputo’s or Riemann–Liouville’s derivative can be represented by the continuous frequency distributed model with initial value carefully allocated. Then, the relation of the stability between the fractional order system and its corresponding integer order system is discussed and it is proven that stability of integer order system implies the stability of its corresponding fractional order system under some mild conditions. Moreover, we extend the stability theorems to the finite-dimensional case since fractional order systems are always implemented by approximation. Some illustrative examples are finally provided to show the usage and effectiveness of the proposed stability theorems.
... Particularly in [29] , the fractional energy formulation by the infinite-state approach has been validated and the conventional pseudo-energy formulations based on pseudo state variables has been invalidated . Moreover, energy aspects of fractional damping forces described by the fractional derivative of displacement in mechanical elements have been considered in [31, 32], in which the effect on the energy input and energy return, as well as the history or initialization effect on energy response has been presented. On the basis of the recently established fractional energy definitions for fractional operators, our main objective in this paper is to deal with the total mechanical energy of a single degree of freedom fractional oscillator. ...
This paper addresses the total mechanical energy of a single degree of freedom fractional oscillator. Based on the energy storage and dissipation properties of the Caputo fractional derivatives, the expression for total mechanical energy in the single degree of freedom fractional oscillator is firstly presented. The energy regeneration due to the external exciting force and the energy loss due to the fractional damping force during the vibratory motion are analyzed. Furthermore, based on the mean energy dissipation of the fractional damping element in steady-state vibration, a new concept of mean equivalent viscous damping is suggested and the value of the damping coefficient is evaluated.
... [15], the impedances of commercial supercapacitors from various vendors were experimentally validated against a fractional-order model consisting of an equivalent resistance R 0 in series with a constant phase element (CPE, which is an element with impedance Z(s)¼1/(C a s a ) (0 < a < 1) and constant phase angle q¼Àap/2) and have shown a perfect fit with the dispersion coefficient falling to values as low as a¼0.5 or less. However, to the authors' best knowledge, very few attempts have been made to derive the expressions for the electrical power and energy based on fractionalorder impedances, such as the work of Hartley et al. [16]. This resulted in simple reliance on conventional relations for the estimation of these characteristics. ...
Characterizing and modeling electrical energy storage devices is essential for their proper integration in larger systems. However, basic circuit elements, i.e. resistors, inductors, and capacitors, are not well-suited to explain their complex frequency-dependent behaviors. Instead, fractional-order models, which are based on non-integer-order differential equations in the time-domain and include for instance the constant phase element (CPE), are mathematically more fit to this end. Here, the electrical power and energy of fractional-order capacitance and inductance are derived in both steady-state and transient conditions, and verified using a number of commercial supercapacitors and fractional-order coils. A generalized expression for the energy stored in a supercapacitor/fractional-order inductor is derived and found to depend on the capacitance/inductance and the dispersion coefficient of the device, as well as on the properties of the applied voltage waveform.
The dependence on renewable energy to solve the major energy issues related to global warming and shortage of energy resources is increasing drastically. This has led to high level awareness of proper energy storage and management. In this regard, supercapacitors have evolved as an efficient energy storage solution and hence successfully employed in several applications. This is attributed to its high‐power density, superior performance, and extended maintenance‐free lifetime. Also, it is known for low Equivalent Series Resistance (ESR), that means, only a small amount of energy is lost while its charging/discharging even at high current. As a result, it has developed a lot of inquisitiveness among the research community of recent past for further exploration of different aspects of supecapacitors, ranging from material science, modeling and characterizations, engineering applications etc. This review article aims at putting forth an exclusive study of their characterization of supercapcitor for different charging methods and applications. Further, the prime focus is given to the efficiency estimation of supercapacitors which is very essential as it denotes the amount of energy loss or the utilizable State of Charge (SOC). The analysis has been carried out based on different charging methods and applications, which is essential for improving overall system reliability and performance. The purpose is to organize all important research carried out in this domain till date and to stimulate further insights for effective energy storage and management.
This manuscript revisits the assertion of Berthier et al. that competing fractional-element equivalent-circuit models of battery cells are indistinguishable in the presence of noise. Starting with Westerhoff’s general equivalent-circuit model (ECM) of 2016 and an idealized impedance spectrum of a lithium battery, a contemporary set of possible ECMs are chosen. Their distinguishability is investigated. Using the extended frequency range recommended by Mauracher & Karden in 1997 or Hasan & Scott in 2019, an approach is presented that permits selection of the appropriate model, even in the presence of noise. For the given frequency range, models with up to five elements or eight real parameters are studied. Fitting to measured data with straightforward numerical methods and choosing the most primitive appropriate sub-circuit is shown to reproduce the data. Typically a three-element, five-parameter ECM is shown to model real, measured data with precision comparable to individual sample error.
Recently, interest in fractional‐order inductors (FOIs) has increased since they allow for accurate and robust models of dynamical systems to be designed. However, practical implementation of these models has not been possible due to the lack of single FOI realizations. To address this challenge, in this work, we propose a simple‐to‐realize fractional‐order inductor design with variable constant phase angle (CPA). The design relies on characteristics of transverse electromagnetic (TEM) mode propagating on a coaxial structure filled with conductive material, more specifically NaCl‐water solution and flour‐based mixtures. The CPA of the resulting FOI can be tuned by changing the conductivity of the dough mixture. Analysis of the proposed FOI design show that the CPA can vary in a range from 0° to 90°. Two of these CPA values are verified against experiments in the frequency band changing from 1 to 10 MHz. In this work, we propose a simple‐to‐realize fractional‐order inductor design with variable constant phase angle (CPA). The design relies on characteristics of transverse electromagnetic (TEM) mode propagating on a coaxial structure filled with conductive material. Analysis of the proposed FOI design shows that the CPA can vary in a range from 0° to 90°, with two of these CPA values verified against experiments in the frequency band changing from 1 to 10 MHz.
At present, the three-phase power system has been widely used in the power system of all countries in the world, and it is the most extensive production system. The three-phase power system is the basis for the production, transmission, and utilization of electric energy. And the structure of the system has been highly standardized to meet the needs of industrialized production.
This paper investigates the inverse Lyapunov theorem for linear time invariant fractional order systems. It is proved that given any stable linear time invariant fractional order system, there exists a positive definite functional with respect to the system state, and the first order time derivative of that functional is negative definite. A systematic procedure to construct such Lyapunov candidates is provided in terms of some Lyapunov functional equations.
Lyapunov stability of linear commensurate order fractional systems is revisited with the energy balance principle. This methodology is based on the concept of fractional energy stored in inductor and capacitor components, where natural decrease of the stored energy is caused by internal Joule losses. Previous stability results are interpreted, thanks to an equivalent fictitious fractional RLC circuit. Energy balance is used to analyze the usual Lyapunov function and to provide a physical interpretation to the weighting positive matrix. Moreover, the classical linear matrix inequality (LMI) condition is interpreted in terms of internal and external Joule losses.
The performance of electrical devices, depending on the processes of the electrolytes, have been fully described and the physical origin of each parameter is well established. However, the influence of the irregularity of the electrodes was not a subject of study and only recently, this problem became relevant in the viewpoint of fractional calculus. This paper describes an electrolytic process in the perspective of fractional order capacitors. In this line of thought, are developed several experiments for measuring the electrical impedance of the devices. The results are analyzed through the frequency response, revealing capacitances of fractional order that can constitute an alternative to the classical integer order elements.
This paper deals with a fractional order state space model for the lithium-ion battery and its time domain system identification method. Currently the equivalent circuit models are the most popular model which was frequently used to simulate the performance of the battery. But as we know, the equivalent circuit model is based on the integer differential equations, and the accuracy is limited. And the real processes are usually of fractional order as opposed to the ideal integral order models. So here we propose a lithium-ion battery fractional order state space model, and compare it with the equivalent circuit models, to see which model fit with the experiment results best. Then the hybrid pulse power characterization (HPPC) test has been implemented in the lithium-ion battery during varied state-of-charge (SOC). Based on the Levenberg-Marquardt algorithm, the parameters for each model have been obtained using the time-domain test data. Experimental results show that the proposed lithium-ion fractional order state space model has a better fitness than the classical equivalent circuit models. Meanwhile, five other cycles are adopt here to validate the prediction error of the two models, and final results indicate that the fractional model has better generalization ability.
A new modeling approach was developed for galvanic devices including batteries and fuel cells. The hybrid modeling approach reduces the complexity of the First Principles method and adds a physical basis to the empirical method. It provides a practical technique for device users to obtain a physics-based, accurate, chemistry and device independent model of the device. Practical uses of the model include device characteristics analysis, system and circuit design that enables a more efficient and intelligent utilization of galvanic devices.
This paper considers the energy aspects of fractional-order elements defined by the equation: force is proportional to the fractional-order derivative of displacement, with order varying from zero to two. In contrast to the typically conservative assumption of classical physics that leads to the potential and kinetic energy expressions, a number of important nonconservative differences are exposed. Firstly, the considerations must be time-based rather than displacement or momentum based variables. Time based equations for energy behavior of fractional elements are presented and example applications are considered. The effect of fractional order on the energy input and energy return of these systems is shown. Importantly, it is shown that the history, or initialization, has a significant effect on energy response. Finally, compact expressions for the work or energy, are developed.
Proper initialization of fractional-order operators has been an ongoing problem, particularly in the application of Laplace transforms with correct initialization terms. In the last few years, a history-function-based initialization along with its corresponding Laplace transform has been presented. Alternatively, an infinite-dimensional state-space representation along with its corresponding Laplace transform has also been presented. The purpose of this paper is to demonstrate that these two approaches to the initialization problem for fractional-order operators are equivalent and that the associated Laplace transforms yield the correct initialization terms and can be used in the solution of fractional-order differential equations.
In this paper, fractional-order electrical elements are considered as energy storage devices. They are studied by comparing the energy available from the element to do future external work, relative to the energy input into the element in the past. A standard circuit realization is used to represent the fractional-order element connected to an inductor with a given initial current. This circuit realization is used to determine the energy returned by both capacitive and inductive fractional-order elements of order between zero and one. Plots of the energy stored versus time are provided. The major conclusion is that fractional-order elements tend to rapidly dissipate much of their input energy leaving less energy for doing work in the future.
This paper considers the energy aspects of fractional elements defined by the equation Display FormulaFt=kλDtλ0xt. In contrast to the typically conservative assumption of classical physics that lead to the potential and kinetic energy expressions, a number of important non-conservative differences are exposed. Firstly, the considerations must be time-based rather than displacement or momentum based variables. Time based equations for energy behavior of fractional elements are presented and example applications are considered. The effect of fractional order on the energy requirements and energy return of these systems is shown. Importantly, it is shown that the history, or initialization, has a strong effect on energy requirements. Finally, compact expressions for the work or energy, are developed.
The objective of this paper is to demonstrate that the Infinite State Approach, used for fractional order system modeling, initialization and transient prediction is the generalization of integer order system theory. The main feature of this classical theory is the integer order integrator, according to Lord Kelvin's principle. So, fractional order system theory has to be based on the fractional order integrator, characterized by an infinite dimension frequency distributed state. As a consequence Fractional Differential Systems or Equations generalize Ordinary Differential Equation properties with an infinite dimension state vector. Moreover, Caputo and Riemann-Liouville fractional derivatives, analyzed through their associated fractional integrators, are no longer convenient tools for the analysis of fractional systems.
Fractional order differentiation is generally considered as the basis of fractional calculus, but the real basis is in fact fractional order integration and particularly the fractional integrator, because definition and properties of fractional differentiation and of fractional differential systems rely essentially on fractional integration. We present the frequency distributed model of the fractional integrator and its finite dimension approximation. The simulation of FDSs, based on fractional integrators, leads to the definition of FDS internal state variables, which are the state variables of the fractional integrators, as a generalization of the integer order case.The initial condition problem has been an open problem for a long time in fractional calculus. We demonstrate that the frequency distributed model of the fractional integrator provides a solution to this problem through the knowledge of its internal state. Beyond the solution of this fundamental problem, mastery of the integrator internal state allows the analysis and prediction of fractional differential system transients. Moreover, the finite dimension approximation of the fractional integrator provides an efficient technique for practical simulation of FDSs and analysis of their transients, with a particular insight into the interpretation of initial conditions, as illustrated by numerical simulations.Laplace transform equations and initial conditions of the Caputo and the Riemann–Liouville derivatives are used to formulate the free responses of FDEs. Because usual equations are wrong, the corresponding free responses do not fit with real transients. We demonstrate that revised equations, including the initial state vector of the fractional integrator (used to perform differentiation) provide corrected free responses which match with real transients, as exhibited by numerical simulations.
In this paper we propose using a non-linear least squares fitting process to estimate the impedance parameters of a fractional order model of supercapacitors from their collected step response datasets, without requiring direct measurement of the impedance or frequency response. Experimentally estimated parameters from 1 F supercapacitors verify the proposed time domain method showing less than 2% relative error between the simulated response (using the extracted fractional parameters) and the experimental step response, over the entire dataset.
Four practical sinusoidal oscillators are studied in the general form where fractional-order energy storage elements are considered. A fractional-order element is one whose complex impedance is given by Z = a(jω)±α, where a is a constant and α is not necessarily an integer. As a result, these oscillators are described by sets of fractional-order differential equations. The integer-order oscillation condition and oscillation frequency formulae are verified as special cases. Numerical and PSpice simulation results are given. Experimental results are also reported for a selected Wien-bridge oscillator. Copyright © 2007 John Wiley & Sons, Ltd.
With EV and HEV developments, battery monitoring systems have to meet the new requirements of car industry. This paper deals with one of them, the battery ability to start a vehicle, also called battery crankability. A fractional order model obtained by system identification is used to estimate the crankability of lead-acid batteries. Fractional order modelling permits an accurate simulation of the battery electrical behaviour with a low number of parameters. It is demonstrated that battery available power is correlated to the battery crankability and its resistance. Moreover, the high-frequency gain of the fractional model can be used to evaluate the battery resistance. Then, a battery crankability estimator using the battery resistance is proposed. Finally, this technique is validated with various battery experimental data measured on test rigs and vehicles.
Lyapunov stability of fractional differential equations is addressed in this paper. The key concept is the frequency distributed fractional integrator model, which is the basis for a global state space model of FDEs. Two approaches are presented: the direct one is intuitive but it leads to a large dimension parametric problem while the indirect one, which is based on the continuous frequency distribution, leads to a parsimonious solution. Two examples, with linear and nonlinear FDEs, exhibit the main features of this new methodology.
Using the method of the Laplace transform, we consider fractional
oscillations. They are obtained by the time-clock randomization of ordinary
harmonic vibrations. In contrast to sine and cosine, the functions describing
the fractional oscillations exhibit a finite number of damped oscillations with
an algebraic decay. Their fractional differential equation is derived.
The concept of ''diffusive representation'' was previously introduced in the aim of transforming non standard convolutive causal operators such as fractional integrodifferential ones, into infinite-dimension dissipative classical input-output dynamic systems. The existence of an explicit dissipative semigroup makes then possible the use of classical tools of functional and numerical analysis of PDE, energy methods, control, filtering, etc., generally ill-fitted to the standard convolutive formulations, namely when long memory dynamics are present. The aim of the paper is to present in a synthetic statement, first the general frame of diffusive representations, and secondly some of the essential characteristics and properties of this new tool. Simple and concrete examples are given. More significant applications (essentially in the fractional context) will be found in the referenced papers.
This paper considers various aspects of the initial value problem for fractional order differential equations. The main contribution of this paper is to use the solutions to known spatially distributed systems to demonstrate that fractional differintegral operators require an initial condition term that is time-varying due to past distributed storage of information.
Our control philosophy is to charge the NiH2 cell in such a way that the damage incurred during the charging period is minimized, thus extending its cycle life. This requires nonlinear dynamic model of NiH2 cell and a damage rate model. We must do this first. This control philosophy is generally considered damage mitigating control or life-extending control. This presentation covers how NiH2 cells function, electrode behavior, an essentialized model, damage mechanisms for NiH2 batteries, battery continuum damage modeling, and battery life models. The presentation includes graphs and a chart illustrating how charging a NiH2 battery with different voltages and currents affects damages the battery and affects its life. The presentation concludes with diagrams of control system architectures for tracking battery recharging.
A new linear capacitor model is proposed. It is based on Curie's
empirical law of 1889 which states that the current through a capacitor
is i(t)=U<sub>0</sub>/(h<sub>1</sub>t<sup>n</sup>), where h<sub>1</sub>
and n are constants, U<sub>0</sub> is the dc voltage applied at t=0, and
0<n<1. It implies that the insulation resistance is
R<sub>i</sub>(t)=h<sub>1</sub>t<sup>n</sup>, that is, it increases
almost in proportion to time since n nearly equals 1.0. For a general
input voltage u(t) the current is i(t)=Cd<sup>n</sup>u(t)/dt<sup>n</sup>
where use is made of the fractional derivative, defined by means of its
Laplace transform. The model gives rise to a capacitor impedance
Z(iω=1/[(iω)<sup>n</sup>C], with a loss tangent that is
independent of frequency. The model has other properties: the capacitor
`remembers' voltages it has been subjected to earlier, dielectric
absorption is an example of this. Capacitor problems require solving
integral equations. The model is dynamic, i.e. electrostatic processes
are simply slow dynamic processes. The model is applied to several
problems that cannot be treated with conventional theory
The Space of Neutrally Stable Linear Fractional Oscillators of the Fractional Meta-Trigonometry
- C F Lorenzo
- R Malti
- T T Hartley
Kinetics of Rapid Electrode Reactions