Risk Measures for Particle-Filtering-Based State-of ... - IEEE Xplore

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Risk Measures for Particle-Filtering-Based State-of-Charge Prognosis in Lithium-Ion Batteries Marcos E. Orchard, Member, IEEE, Pablo Hevia-Koch, Bin Zhang, Senior Member, IEEE, and Liang Tang, Member, IEEE

Abstract—This paper presents a class of risk measures to be used as damage indicators within particle filtering (PF)-based real-time prognosis algorithms, with application to the case of state-of-charge prediction in lithium-ion batteries. The proposed risk measure not only incorporates the risk of battery failure but also is a measure for the confidence on the prognosis algorithm itself. In addition, a novel simplified PF-based prognostic method is proposed to estimate the battery discharge time, while providing a computationally inexpensive solution. Computing times for both the novel prognosis routine and the associated risk measure are fast enough to allow their implementation in real-time applications, such as decision-making systems or path-planning algorithms. Index Terms—Lithium-ion (Li-ion) battery, risk management, state-of-charge (SoC) prognosis.

I. I NTRODUCTION

R

ISK MANAGEMENT can be defined as the study and mitigation of rare events that have potentially devastating outcomes. For this reason, in the particular case of failure prognosis of nonlinear dynamic systems (i.e., the prediction of the remaining useful life (RUL) of pieces of equipment), risk management is deeply entangled with concepts of isolation and characterization of the uncertainty sources that affect both the accuracy and precision of long-term predictions. Although models that consider stochastic variables in their formulation allow the representation of uncertainty related to observed phenomena, identification of their parameters is not straightforward due to a likelihood function that needs to be defined as an intractable integral. Furthermore, it is not always possible to model the process of interest solely on the basis of a fixed set of parameters. Thus, it is necessary to consider a scheme that allows not only model state estimation based on the observed process but also adaptive system identification.

Manuscript received May 4, 2012; revised August 13, 2012; accepted September 20, 2012. Date of publication October 11, 2012; date of current version June 6, 2013. This work was supported by Conicyt through Project Fondecyt 1110070. M. E. Orchard is with the Department of Electrical Engineering and the Advanced Mining Technology Center, Universidad de Chile, Santiago 837-0451, Chile (e-mail: [email protected]). P. Hevia-Koch is with the Department of Electrical Engineering, Universidad de Chile, Santiago 837-0451, Chile (e-mail: [email protected]). B. Zhang is with the Department of Electrical Engineering, University of South Carolina, Columbia, SC 29208 USA (e-mail: [email protected]). L. Tang is with Impact Technologies, LLC, Rochester, NY 14623 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2012.2224079

If uncertainty sources are assumed to be Gaussian and the structure of the process dynamic model is linear, then an optimal solution can be implemented through the use of Kalman filters. Unfortunately, not always these assumptions may hold, and thus, the implementation of numerical techniques (to find suitable sets of parameters for a given modeling structure, and hence to estimate unobserved components) has caught the attention of several scientific disciplines due to the high computational power, and the increasing storage capacity, which has been developed throughout the last decade. For this very reason, Bayesian state estimation techniques such as particle filtering (PF) [1]–[7] have become a key component of failure prognosis frameworks, providing a strong mathematical foundation to represent, and even manage, uncertainty in long-term predictions. Indeed, PF-based prognostic frameworks [8]–[11] have been established as the de facto state of the art in failure prognosis, mainly due to their capability to combine information available from system measurements and analytic/empirical models. However, most of the research effort does not focus on providing a consistent quantification of the risk within the faulty system that could consider information from different uncertainty sources such as errors in the state transition model, in the predicted load profiles, or even in the implementation of the filtering algorithm itself. In the case of energy storage devices (ESDs), such as Li-ion batteries, the concept of failure prognosis is of particular relevance since the notion of RUL has a direct relationship with terms that permeate our daily life, such as “autonomy” or “battery replacements” [10], [12]. ESDs have played a significant role in the development of novel and more efficient communication, transportation, and mobile systems. They represent not only the means to manage energy resources (securing the availability of electric supply for time-varying power demand, even when the system is isolated) but also an important constraint in terms of the maximum autonomy that any of those systems may attain. In recent years, the criticality of this role has increased as a result of the exponential growth of the industry of hybrid and electric vehicles, cell phones, laptop computers, and autonomous ground and unmanned aerial vehicles, among other electronic devices [15]. Regardless of the main purpose for which these technologies are being developed, it is a fact that end users expect at least three main features from the ESDs that enable them: 1) the ESD should provide a reasonable level of autonomy to the system (state of charge (SoC) [12]–[15]); 2) the ESD should require a brief period of time to accumulate the necessary amount of energy that guarantees autonomy; and 3) the ESD should allow to be reused

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ORCHARD et al.: RISK MEASURES FOR PF-BASED STATE-OF-CHARGE PROGNOSIS IN Li-ION BATTERIES

for a large number of operating cycles. It is for this reason that the specific problem of generating a risk measure that could provide information on the system autonomy has been considered in this paper as an excellent case study and an interesting research topic. The structure of this paper is as follows. Section II provides the theoretical background for Bayesian state estimation and prognosis, considering the general state estimation problem, and the use of particle filters for state prognosis. Section III introduces the concept of risk measures, gives a formal definition of a risk measure, and shows some desirable properties for a risk functional. Section IV focuses on the SoC estimation and prognosis problem, presenting a novel (and simplified) method for prognosis of the discharge time that is validated using actual Li-ion battery data. Section V proposes a risk measure for the particular problem of SoC estimation/prognosis and shows its performance when using the aforementioned data set. Finally, Section VI presents some insights about future work, main conclusions, and final remarks. II. PF-BASED P ROGNOSIS Consider a sequence of probability distributions {πt (x0:t )}t≥1, where it is assumed that πt (x0:t ) can be evaluated pointwise up to a normalizing constant. Sequential Monte Carlo (SMC) methods, also referred to as particle filters, are a class of algorithms designed to approximately obtain samples sequentially from {πt }, i.e., to generate a collection of N  1 weighted (i) (i) random samples {wt , x0:t }i=1,...,N satisfying [3], [4] N  i=1

(i)



(i)





wt φt x0:t −−−−→ N →∞

φt (x0:t )πt (x0:t )dxt

(1)

where φt is any πt integrable function. In the particular case of the Bayesian filtering problem, the target distribution πt (x0:t ) = p(x0:t |y1:t ) is the posterior probability density function (pdf) of x0:t , given a realization of noisy observations Y1:t = y1:t . Let a set of N paths {x0:t−1 (i)}i=1,...,N be available at time t − 1. Furthermore, let these paths distribute according to qt−1 (x0:t−1 ), also referred to as the importance density function at time t − 1. Then, the objective is to efficiently (i) obtain a set of N new paths (particles) {ˆ x0:t }i=1,...,N distributed according to π(ˆ x0:t ) [3]. For this purpose, the current (i) paths x0:t−1 are extended by using the kernel qt (ˆ x0:t |x0:t−1 ) = ˆ0:t = (x0:t−1 , x ˆt ). The δ(ˆ x0:t−1 − x0:t−1 ) · q(xt |x0:t−1 ), i.e., x importance sampling procedure generates consistent estimates for the expectations for any function, by approximating the following with the empirical distribution [3]: π ˆtN (x0:t ) =

N 

(i)δ(x0:t −ˆ x0:t )

w0:t

(2)

i=1

 (i) (i) (i) where w0:t ∝ w0:t (ˆ x0:t ) and N i=1 w0:t = 1. The most basic SMC implementation—the sequential importance sampling particle filter—computes the value of the

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(i)

particle weights w0:t , by setting the importance density function x0:t |x0:t−1 ) = equal to the a priori pdf for the state, i.e., qt (ˆ p(ˆ xt |xt−1 ). In that manner, the weights for the newly generated particles are evaluated from the likelihood of new observations. The efficiency of the procedure improves as the variance of the importance weights is minimized. The choice of the importance density function is critical for the performance of the particle filter scheme, and hence, it should be considered in the filter design. Prognosis, and thus the generation of long-term predictions, is a problem that goes beyond the scope of filtering applications since it involves future time horizons. Hence, if PF-based algorithms are to be used, it is necessary to propose a procedure with the capability to project the current particle population in time in the absence of new observations. Any adaptive prognosis scheme requires the existence of at least one feature providing a measure of the severity of the fault condition under analysis (fault dimension). If many features are available, they can always be combined to generate a single signal. In this sense, it is always possible to describe the evolution in time of the fault dimension through a nonlinear state equation [8]. By using the aforementioned state equation to represent the evolution of the fault dimension in time, it is possible to generate long-term predictions using kernel functions to reconstruct the estimate of the state pdf in future time instants, as it is shown in the following: x1:t+k−1 ) ≈ pˆ(xt+k |ˆ

N 

   (i) (i) (i) wt+k−1 K xt+k −E xt+k |ˆ xt+k−1

i=1

(3) where K(·) is a kernel density function, which may correspond to the process noise pdf, a Gaussian kernel, or a rescaled version of the Epanechnikov kernel [5], [8]. The resulting predicted state pdf contains critical information about the evolution of the fault dimension over time. One way to represent that information is through the computation of statistics (expectations and 95% confidence intervals) and of either the time of failure (TOF) or the RUL of the faulty system [8]–[10]. The TOF pdf depends on both long-term predictions and empirical knowledge about critical conditions for the system. This empirical knowledge is usually incorporated in the form of thresholds for main fault indicators. Therefore, the probability of failure at any future time instant t = tof (namely, the TOF pdf) is given by P {TOF = tof} =

N 

  (i) (i) P Failure|X = x ˆtof · wtof .

(4)

i=1

The conditional probability of failure in (4) may be defined via the determination of hazard zones [8], either using historical data or knowledge from process operators. The simplest case is where the concept of “failure” implies the moment when the fault feature crosses a given threshold. In that case, the probability of failure, conditional to the state, is equal to one if the state is exactly on the manifold that defines the threshold value.

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III. R ISK M EASURES : D EFINITION AND P ROPERTIES Risk can be defined as the potential that a chosen action (including the choice of inaction) will lead to a loss or undesirable outcome. In this sense, risk measures represent means for the quantification and assessment of uncertainty, allowing the incorporation of such type of information in the structure of models for system optimization purposes. When introduced in a system that makes use of prognostic algorithms, risk measures become a powerful tool that not only can give a measure of the uncertainty of the system but also can serve as a measure of the uncertainty in the prognostic algorithm itself. In classical optimization, a typical problem has the form min c0 (x) over all x ∈ S,

s.t. ci (x) ≤ 0;

i = {1, . . . , m}

where S is a subset of Rn composed vectors (xi , . . . , xn ) and ci : S → R. This formulation, however, is not free of challenges when facing environments with large-grain uncertainty. This is caused because some of the parameters to be optimized provide incomplete information about the system, a fact that may not be properly considered within the model. This difficulty can be represented in the formulation by assuming that, instead of ci (x), we have ci (x, ω), where ω belongs to a set Ω representing the future states of knowledge. In this way, choosing an x ∈ S no longer produces a specific ci (x) number but a collection of functions on Ω. In this sense, let X be random variables identified with functions from Ω to R that belong to the linear space L2 , introduced relative to a probability measure P on Ω. For quantifying risk, a value R(x) is assigned to each X ∈ L2 . Then, from a mathematical point of view, a risk measure is a functional R that goes from the linear space L2 to (−∞, ∞). In a context of failure prognosis for a faulty dynamic nonlinear system, X can be interpreted as the probability distribution of a fault condition. Several approaches have been proposed for constructing risk measures, including guessing the future, worst case analysis, relying on expectations, using standard deviation units as safety margins, specifying probabilities of compliance, and also using dynamic and stochastic programming [16]. In general, there are no “correct answers” in terms of defining a universal risk measure, given the different fields where risk measures are used. Nonetheless, there exist some properties of axiomatic behavior that are desirable for a good quantifier of risk. Although the work done by Artzner et al. [17], [18] and Delbaen [19] has been mostly focused on financial applications, they provide solid answers for properties that they identified as providing coherency. A functional R : L2 → (−∞, ∞] will be called a coherent measure of risk in the extended sense if it satisfies the following. R1) R(C) = C for all constants C. R2) R((1 − λ)X + λX ) ≤ (1 − λ)R(X) + λR(X ) for λ ∈ (0, 1) (“convexity”). R3) R(X) ≤ R(X ) when X ≤ X (“monotonicity”). R4) R(X) ≤ 0 when X k − X2 → 0 with R(X k ) ≤ 0 (“closedness”). It will also be called a coherent measure of risk in the basic sense if it also satisfies the following. R5) R(λX) = λR(X) for λ > 0 (“positive homogeneity”).

It is important to note that, sometimes, R1) is written in the more complicated form: R(X + C) = R(X) + C, which is tailored to a banking concept but follows automatically from R1) and R2), shown in [16] and [18]. Most systems require decisions to be made at some point in time. Unfortunately, complete information is often unavailable, and every decision must incorporate the uncertainty about what is going to happen in an optimal manner. Given this context, a risk measure is a powerful tool to be utilized in the decisionmaking process.

IV. S IMPLIFIED PF-BASED A PPROACH FOR BATTERY SoC P ROGNOSIS A. PF-Based SoC Estimator The SoC provides an indicator of the system autonomy that directly depends on the remaining battery energy and the mission profile, a critical piece of information for the design of path-planning/control strategies in autonomous vehicles. It is for this reason that the implementation of SoC estimation and prognostic algorithms [12]–[15], [20]–[26] is considered as the first step toward online characterization of both the end-of-discharge time and RUL of Li-ion batteries. One of the main difficulties in SoC estimation is that it cannot be measured directly, and thus, its value must be inferred from the observation of other variables such as the battery current, voltage, temperature, state-of-health degradation, and selfdischarge phenomena [15], [21]–[23]. Indeed, the utilization of more complex electrochemical models has been only suitable for offline studies, mainly because of the following: 1) These models require a large number of variables to represent the battery internal structure; 2) these models assume extremely accurate measurements; and 3) these models have an elevated computational cost [13], [15]. Other options for SoC monitoring include the open-circuit voltage (OCV) method. This approach has the advantage of providing a direct relationship between battery SoC and voltage measurements (the higher the OCV, the higher the SoC [24]). Unfortunately, the implementation of this test requires a large resting period for the battery, which also inhibits its use for online applications [13], [15], [23], [24]. Similarly, the “electrochemical impedance spectroscopy” (EIS) [14], [15] is a noninvasive method that intends to provide a complete characterization of the battery internal equivalent circuit. However, the implementation of an EIS test requires the acquisition of costly equipment (generally found only at laboratory test sites), which severely limits its widespread use in practice [26]. It is for this reason that current research efforts for SoC estimation and prognostic algorithms have focused on approaches that are mostly based on empirical models that incorporate only critical phenomenological aspects of the process, i.e., the relationship between currents, voltages, and temperatures of Li-ion cells. Among these methods, it is worth mentioning those that are based on fuzzy logic, neural networks, and Bayesian approaches (such as the extended Kalman filter and SMC methods).

ORCHARD et al.: RISK MEASURES FOR PF-BASED STATE-OF-CHARGE PROGNOSIS IN Li-ION BATTERIES

On the one hand, fuzzy logic models have been used for the SoC estimation either through the identification of equivalent circuit for the battery from EIS data or directly from voltage and current measurements [20]. Given that EIS data have proved to be very noisy in practice [12], [26], only the latter case represents a reasonable method for online SoC estimation and uncertainty characterization. However, even in that case, the problem of SoC prediction (related to battery prognosis) is still unresolved and mainly treated as a curve regression problem (which is insufficient for purposes of risk characterization). Neural networks have also been used to build a nonlinear relationship between battery measurements and the evolution of SoC in time [13], [15], [22]. These methods, however, do not provide an adequate representation for uncertainty in nonlinear systems, and thus, neither can they be used for risk quantification purposes. On the other hand, suboptimal Bayesian methods have proved particularly effective in the task of simultaneously incorporating information from noisy measurements and characterizing the sources of uncertainty [12], [13], [23], [25], [26]. In fact, experience has demonstrated that Bayesian state estimators are particularly well suited for real-time estimation problems associated to dynamic state models [8]–[12], [26]. In addition, these methods also provide a concrete characterization of uncertainty sources in both the filtering and prediction stages, a piece of information that is required for the generation of a risk measure associated to SoC prognosis. Bayesian estimators require a state-space model for the dynamic system, and prognostic modules based on a state-space formulation for the dynamic system are very sensitive to the initial condition of the state vector. For this reason, the implementation of accurate online SoC estimators is absolutely relevant for the development of real-time predictors capable of quantifying the feasibility (as well as the cost) of a particular vehicle trajectory. Depending on the validity of linear or Gaussianity assumptions, either an extended Kalman filter or a PF [6] approach may be needed. To evaluate which is the best option for this particular problem, it is first necessary to define a state model that represents adequately the dynamics associated to the evolution of the battery SoC, for a given usage profile. This research has considered for this purpose the problem of battery end-of-charge prognosis in a four-wheel ground robot. In this scenario, different discharge profiles can be verified due to terrain conditions (hills and surface changes) and other factors, while the robot autonomously has to perform a predetermined mission. Given that the robot is currently configured only for use on 2-D uniform terrain, it is necessary to simulate the environment through a variable load that has been attached to the battery. This variable load is made up of three resistors (6.23, 12.5, and 25 Ω), each wired in parallel to the battery to increase current draw. Each resistor can be activated via a relay controlled by the onboard computer’s data acquisition card. It provides eight different loading scenarios progressing linearly in magnitude. The onboard computer has a map of simulated terrain, and when the robot crosses into an area of higher simulated difficulty on the map, the onboard computer activates a larger loading scenario using the variable load. This allows for many simulated terrains while keeping the robot in a safe

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Fig. 1. Li-ion battery discharge curve. Battery voltage, measured in volts. Data illustrate the typical voltage drop that is observed in a complete discharge cycle.

Fig. 2. Li-ion battery discharge curve. Battery current, measured in amperes. Data show the uncertain nature of the discharge profile when using the battery on a mobile mechanical system.

uniformly flat environment. Online data consist of voltage and current measurements (with the corresponding timestamp), for a lithium iron phosphate (LiFePO4 ) battery (12.8 V, 2.4 Ah, and 14-A maximum discharge current). This experimental setup implies that, for full speed (700 mm/s), the current drained from the battery ranges between 1.6025 and 5.4738 A (depending on the value of the equivalent resistor that is connected in parallel to the battery) and, at 10% speed (70 mm/s), the current drained from the battery ranges between 0.6006 and 4.3971 A. As the battery voltage drops, maximum values of the drain current may increase. Figs. 1 and 2 show the measured data for the battery voltage (in volts) and current (in amperes), respectively, in an experiment where the energy accumulator was used until it discharged almost completely.

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Data show a clear correlation between the voltage drop and the current level on a particular time instant. This relationship clearly relates with the value of the battery internal resistance. However, it is also clear that the voltage drop in the battery increases as the remnant energy decreases. Both elements need to be included in a dynamic state model for the system. The discrete-time model for the battery operation requires defining necessarily one state as the battery SoC. In addition, it is important to consider adaptation of the model to a particular battery under supervision, through the definition of a state associated to an unknown model parameter. In this sense, the model definition considers the battery current level i(k) (in amperes) and the sample time δT (k) (in seconds) as input variables and the battery voltage v(k) (in volts) as system output. The states will be defined as x1 (k) (unknown model parameter) and x2 (k) (SoC, i.e., remnant battery energy measured in 10−5 VA · s. Equations (5) and (6) show the state model and the observation equation, respectively x1 (k + 1) = x1 (k) + ω1 (k) (5) x2 (k + 1) = x2 (k) − v(k)i(k)δT (k) · 10−5 + ω2 (k) x2 (k) + η(k) (6) v(k) = V0 − x1 (k)i(k) − e with V0 = 13 and  = 5.5687. Process noises ω1 and ω2 represent uncertainty on the a priori state estimates, and  is a constant that characterizes the battery voltage drop in terms of the remaining SoC. It is important to note that process noise (at least, noise ω2 ) is correlated with observation noise η, since uncertainty on battery SoC depends on the uncertainty of voltage measurements. This fact will be considered when designing the prognostic module. The state x1 in (6) represents the instantaneous value for the battery internal resistance. It is well known that this value depends on other environmental factors (e.g., temperature). As the experimental setup did not include temperature probes, then the filtering stage must infer the effect of the external temperature (and other unmeasured perturbations) into the state estimate x1 , based solely on voltage and current measurements. A more complex dynamic model could, for example, propose a structure that describes the evolution of the state x1 as an explicit function of temperature. Considering that the state model is nonlinear and that it is not possible to assume that all noises are independent and Gaussian, neither the Kalman filter nor its extended version can ensure optimality of the estimator [8]. For this reason, a PF state estimator was implemented to obtain an approximated value of the battery SoC in real-time operation. Given that the estimator only requires one-step-ahead predictions, then the routine met the specified standards for execution time of the algorithm. Fig. 3 shows the performance of the PF-based estimator [8], [9] for the discharge data set shown in both Figs. 1 and 2, in terms of its capability to predict the system output (battery voltage), always considering that the sample time and the battery current measurements are inputs to the estimator module. Although the PF-based estimator fulfills its task adequately in terms of predicting the output system, it is always important to analyze the state estimates to check for consistency and convergence of the solution. Figs. 4 and 5 show in particular

Fig. 3. PF-based estimator performance. (Coarse trace) Predicted versus (light gray fine trace) actual battery voltage.

Fig. 4. PF-based estimator performance. Estimate of the unknown model parameter x1 .

Fig. 5.

PF-based estimator performance. Battery SoC estimate (x2 ).

ORCHARD et al.: RISK MEASURES FOR PF-BASED STATE-OF-CHARGE PROGNOSIS IN Li-ION BATTERIES

the estimates of the unknown model parameter and the battery SoC, one of the main outputs of the estimator since it serves as initial condition for the prognostic module. Similar performance of the estimator was observed when other discharge data sets were analyzed, except for the fact that the unknown model parameter x1 (k) converged to a slightly different value. Both experiments help to validate both the estimation approach and the dynamic model that relates the battery SoC, the battery current level, and the observed voltage drop. The next step is to design an appropriate prognostic algorithm, capable of using the state model (5) and (6) for purposes of estimating the remaining time of operation of the robot, assuming that the future load profile (battery current, measured in amperes) is known. B. Battery SoC Predictor As it has been mentioned before, the next step in the design of battery SoC prognostic approaches is the implementation of prediction algorithms capable of both estimating the remaining battery discharge time and providing the corresponding uncertainty bounds for that estimate. Given that the definition of the problem involves the generation of uncertainty bounds for longterm predictions, one may think that a PF-based prognostic algorithm would be the most adequate solution [8], [12], [26]. The main challenge in this case, however, is not associated to the complexity of the nonlinear state equation defined by (5) and (6) but in the fact that the path-planning procedure requires executing the prediction routine for at least a hundred different profiles before deciding which is the optimal decision in terms of the remaining battery charge. Direct implementation of the PF-based prognostic framework proposed in [8], and later on used in [12] and [26], implies an execution time of 0.235 s per long-term prediction on an Intel(R) Core(TM) i7 CPU (2.93 GHz, 8-GB RAM, and 64-b operating system). That would imply almost 23.5 s of calculations just to gather information about the remaining SoC estimate that is associated to each one of the 100 possible routes under evaluation (not to mention that the path-planning algorithm still requires to consider other sources of information before making a decision). Real-time operation under those conditions is simply impossible, given that the algorithm would have to limit the number of particles so drastically that the implementation would lose sense as a suboptimal estimator in the mean-square sense. For this reason, this research effort proposes a novel PFbased prognostic routine. This approach considers the PF-based state estimate only as an initial condition for the prognostic algorithm and uses a simplified dynamic model to perform the long-term prediction analysis. One of the simplifications that have been considered in the aforementioned model corresponds to a Taylor approximation for the exponential term in the observation equation (6), which depends on the second state



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Fig. 6. Future load profile assumed for the SoC prognosis problem.

of the system (battery SoC). Thus, it will be considered that ex2 (k) ≈ 1 +  · x2 (k), which implies that the measurement equation can be written as v(k) ≈ V0 − x1 (k)i(k) − (1 + x2 (k)) .

(7)

Also, the unknown model parameter x1 will be assumed constant (x1 = α) for all future time instants, which leaves only a linear dynamic predictive model for the state x2 , which depends on the battery current level that the robot will require from the battery in the future, as given in (8), shown at the bottom of the page. It is assumed that the loading profile to the end of mission is divided into two stages, where tp represents the time at which the prognostic algorithm is executed. The first stage t ∈ [t, tp + t1 ] corresponds to a time period where the load profile is well known, while the second one t ∈ [tp + t1 , tp + t2 ] assumes a different load profile that is more uncertain. On each of those time periods, the future battery current level is assumed to be constant; see Fig. 6. This information must be considered when designing the prognostic algorithm, so that its execution time can be reduced to a minimum. This section focuses and describes the rationale behind the calculations that provide with approximate estimate for the battery remaining discharge time, and its uncertainty bounds, when the time of discharge is smaller than tp + t1 . The results for the other case can be generalized by considering the same equations but considering tp + t1 as the time origin. Let us assume that the future battery current level is given by i(tp + n) = I, and let R be a constant value defined by R = (1 +  · I · δT · 10−5 ). From (8), it is possible to find the discharge time (namely, the TOF for the battery in the context of one discharge cycle) by solving (9), where RUL = TOF − tp 0=R

RUL δT

x2 (tp ) − (V0 − αI − 1)IδT 10−5

R

RUL δT

−1 . (9) R−1

In (9), the value of the sampling time δT is computed as the average sample time of the acquired data. Equation (9) can be solved by applying the logarithm function (see the black dotted line in Fig. 7, which depicts the solution for the predicted SoC). As a matter of fact, the execution time associated to this computation is sufficiently small to be considered in the definitive implementation of the real-time prognostic module.

x1 (tp + n + 1) = x1 (tp + n) = x1 (tp ) = α x2 (tp + n + 1) = 1 +  i(tp + n)δT 10−5 x2 (tp + n) − (V0 − αi(tp + n) − 1) i(tp + n)δT 10−5 + K

(8)

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Fig. 7. Simplified algorithm for estimation of TOF standard deviation.

Regarding the computation of uncertainty bounds for the estimate of the time of discharge, it is important to note that the standard deviation σTOF of this estimate is closely related to the standard deviation of SoC long-term predictions. In that sense, the design proposes to approximate σTOF (note that σTOF = σRUL ) by calculating the difference between the expected time of discharge and the moment tp + μ · δT when the predicted −1σ curve for the state x2 hits the discharge point, i.e., when x2 (tp + μδT ) − σx2 (tp + μδT ) = 0.

(10)

Considering all of the aforementioned information, then the standard deviation of the estimate for time of discharge can be theoretically computed by solving μ −1 σx2 (μ) = Rμ x2 (tp ) − (V0 − αI − 1)IδT 10−5 RR−1 (11) σTOF ≈ |TOF − μδT |. An analytic expression for σx2 (μ) can be obtained when considering the fact that the uncertainty for state x2 , according to (5), has two separate sources: 1) process noise ω2 and 2) voltage measurement uncertainty, amplified by the battery current level (measured in amperes). Thus, assuming the simplified model (8) for long-term predictions, independent noise sources for future time instants i(tp + n) = I, and σx22 (tp ) = σ02 , we have σx22 (tp + n + 1) = σx22 (tp + n) + σ02 i2 (tp + n) = (n + 1) · σ02 I 2 .

(12)

A prognosis algorithm that delivers an estimate for σTOF by substituting (12) into (11) and solving for μ can be eventually implemented at this point. The problem, however, is that the procedure to solve the resulting nonlinear equation for μ still requires too much time to be used within an online optimization routine for usage planning (0.11 s per long-term prediction on an Intel(R) Core(TM) i7 CPU with 2.93 GHz, 8-GB RAM, and 64-b operating system). For this reason, one last algorithm simplification is made with the purpose of obtaining a fast estimate for the lower bound of the TOF confidence interval; see Fig. 7. This last simplification basically considers that a crude approximation for the −1σ curve of the state x2 can be obtained (see the light gray dotted line in Fig. 7) by subtracting σx2 (tp + n) from a line that links the estimated initial battery SoC, E(tp ), and the estimated discharge time tp + RUL (see the dark gray

Fig. 8. Output comparison between (coarse trace) the PF-based estimator and the simplified prognostic algorithm. Dotted lines show the expectation and 95% confidence limits of the predicted SoC.

continuous line in Fig. 7). As a result, μ (and, thus, σTOF ) can be computed as the solution of a quadratic equation A · μ2 + B · μ + C = 0

(13)

where A = (E(tp )/RUL) · δT , B = −σ0 I, and C = −E(tp ). This simplified prognostic algorithm allowed an execution time smaller than 0.014 s per load profile on a Intel(R) Core(TM) i7 CPU (2.93 GHz, 8-GB RAM, and 64-b operating system), assuming that all future battery discharge profiles follow the concept shown in Fig. 6. The proposed method allows a reduction of almost 94.04% in the execution time of the prognostic algorithm with respect to the approaches proposed in [8], [12], and [26], and more importantly, it enables the implementation of path-planning routines for ground robots that can recalculate the optimal trajectory every 2 s. The obtained results for the algorithm implementation on actual battery discharge data are shown in Fig. 8, where the coarse dotted trace represents the SoC predicted expectation and the fine (light gray) dotted lines illustrate the bounds for the ±1σ confidence interval for prediction. It is important to emphasize that the precision of the proposed prognostic allows generating 66% confidence intervals that represent only 13.38% of the total prediction window (σTOF = 93.2 s, while the ground truth RUL value is 1393.6 s) and estimates with only 4.03% error (56.1 s). V. R ISK M EASURE FOR BATTERY SoC P ROGNOSIS A. Definition of a Risk Measure for SoC Prognosis One of the most important uses of SoC estimation and prognosis is to aid in the process of decision making. In the context of Li-ion batteries, SoC estimation helps in the process of determining the operation cycle of the battery, providing information about the moment when the battery recharge must take place. This concept is of paramount importance, for example, in path-planning problems for electric vehicles, where the SoC must be considered as one of the constraints of any optimization problem.

ORCHARD et al.: RISK MEASURES FOR PF-BASED STATE-OF-CHARGE PROGNOSIS IN Li-ION BATTERIES

To propose a risk measure associated to the results obtained from SoC estimation/prognosis algorithms, we must first define which is the critical information that should be considered. To achieve this, it is necessary to clarify the failure modes that are expected in a system that depends on Li-ion batteries for its operation and their eventual costs. In the process of estimating the battery RUL (for a given discharge cycle), based on the output of SoC prognosis frameworks, there are two possible scenarios: either to overestimate or to underestimate the RUL. Although underestimating the RUL conveys losses of efficiency and entails costs associated with not utilizing the battery full capacity, the most costly error is, without a doubt, to overestimate the RUL. Overestimating the RUL of a battery implies having the battery charge depleted before it is expected, and thus, the system using the battery could become nonoperational before it was intended. It follows that one of the main sources to consider in a proper risk measure is the RUL estimate. In particular, it is proposed to consider the expected value of the RUL PF-based estimate μRUL and its associated standard deviation σRUL . Another important source of information is the estimated future load profile for the battery, measured in terms of the electric current (in amperes). Although the estimated average of the future electric current is taken into account by the proposed SoC prognostic algorithm, other sources of information (such as the standard deviation of the future profile) are not considered a priori. The standard deviation of the load profile provides an insight on how uncertain is the system future operation, therefore acting as a confidence index for the SoC prognosis result. Based on the methods proposed in [16], risk will be measured on the basis of the average RUL, its standard deviation σRUL , and the standard deviation of the future electric current profile σI as a security margin, i.e., μRUL + σRUL + σI . As shown in [16], this approach for defining a risk measure is well balanced and maintains some of the most important properties for coherence. It is important to note that, for maintaining the usual conception that positive values of a risk measure indicate negative benefits (a nondesirable situation), the proposed risk measure depends on μ−1 RUL . In order to counteract the explosive nature of this term, the structure utilizes its natural logarithm. Based on all of these considerations, the following risk measure could be used:

β R = α ln (14) + λσRUL + γσI μRUL where α, β, λ, and γ are normalizing constants that define the weight of each term in the risk measure. Even though the proposed risk measure currently considers most of the important factors in SoC estimation and prognosis, it does not take into account the level of confidence applicable to the prognosis algorithm in itself. In general, assuming that the RUL prediction is error free, one expects the RUL estimation to evolve as a line with a slope equal to minus one because, for each second that passes, the RUL estimation should decrease by 1 s (i.e., TOF is assumed constant). Following this

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line of thought, the more the RUL estimate slope deviates from this ideal line, the less consistent the algorithm is (and, thus, the less it can be trusted). It is important to note that this only holds true if the internal parameters of the model are time invariant. A simple way of measuring the aforementioned deviation is by means of the correlation coefficient. In this sense, the proposed risk measure is

β + λσRUL + γσI + δσx1 (RN + 1) (15) R = α ln μRUL where RN is the correlation coefficient between a vector that contains the last N computed PF-based RUL estimates and another that records the moment at which each estimate was generated, σx1 is the standard deviation associated to the estimation of the internal parameter x1 , and δ is an associated normalizing constant. Since RN ∈ [−1, 1], it is proposed to use the term (RN + 1) to ensure that the best case scenario (null risk) is represented by the condition RN = −1, thus facilitating a mathematical interpretation of the risk measure. The inclusion of σx1 as a weighting factor for the term (RN = −1) is inspired by the fact that, in the ideal case (i.e., if the unknown model parameter is time invariant and, thus, most of the consistency problems in RUL estimates would be caused only by implementation issues within the prognostic algorithm), σx1 reaches its minimum possible value σω1 (therefore, risk is also minimized in this case). If the unknown model parameter is time varying, then σx1 increases, amplifying the risk associated to the lack of consistency in the prognostic routine. B. Performance of the Proposed Risk Measure for SoC Prognosis In order to evaluate the performance of the proposed risk measure, the RUL is estimated for each time t using the proposed PF-based method, and the risk measure is computed in every step, using the results of the prognostic approach presented in Section IV-B (i.e., RUL expectation and standard deviation, the future load mean and standard deviation, and the estimated internal parameter x1 ). In this way, it is possible to see the evolution of the risk measure in time and how it changes as the battery drains and the RUL estimation (i.e., time before battery discharge) diminishes. Fig. 9 shows the evolution of the risk measure in time. The proposed risk measure that is shown in Fig. 9 uses coefficients α = 1, β = 10 000, λ = 0.005, γ = 1.2, and δ = 1.6, chosen through simple inspection in order to generate an appropriate response. From Fig. 9, it is possible to see that, as the time increases, the risk measure also increases, with a notorious growth in the final seconds of operation (when the SoC estimate approaches the null charge threshold). As a result of the good quality of the PF-based SoC estimation algorithm, it is possible to observe that, for most of the time, the signal-to-noise ratio for the proposed risk measure is adequate for decision-making purposes, except maybe for t ∈ [1300, 1500], which is a time period close to the battery discharge time and a moment where it is highly recommended to stop the system operation.

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Fig. 9. Evolution in time of proposed risk measure. Horizontal dotted line represents the threshold of 4.8 risk units that ensures 60 s until complete battery discharge (see vertical dotted lines).

The demonstrated performance of the proposed risk measure in time and its real-time operation possibilities allow us to use it in the context of real-time decision-making systems. In this sense, and extrapolating this problem to the arena of automated electric vehicles, it is perfectly feasible to use this measure and risk threshold value to define the point when the vehicle must stop its normal operation and return to the charging station. Consider Fig. 9, for example, where a threshold of 4.8 risk units provides the vehicle operator with approximately 60 s to return to the charging dock before the battery drains completely. This safety margin can be adjusted via the election of smaller values for the aforesaid threshold. VI. C ONCLUSION This paper has presented a PF-based real-time prognosis algorithm for the estimation of Li-ion battery discharge time, as well as a novel risk measure that includes information about the risk of battery failure and a measure of confidence for the prognostic algorithm itself. On the one hand, the proposed prognosis approach combines a PF-based state estimator and a simplified approach to the prediction problem that allows us to find an analytic expression for the discharge time (which, in this case, corresponds to the system TOF), as well as for the standard deviation of the battery RUL σRUL , providing the means to generate a confident estimate of the battery SoC in approximately 0.014 s on a Intel(R) Core(TM) i7 CPU (2.93 GHz, 8-GB RAM, and 64-b operating system), for a given future operation profile. On the other hand, the proposed risk measure was able to generate a confidence index for the system, integrating information about the battery discharge risk and the actual confidence on the prognosis algorithm performance. Computing times for the PF-based prognosis approach, and the associated risk measure, are fast enough to allow a real-time implementation in decision-making systems (such as path-planning algorithms), where several possible battery load profiles have to be evaluated in just a few seconds.

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ORCHARD et al.: RISK MEASURES FOR PF-BASED STATE-OF-CHARGE PROGNOSIS IN Li-ION BATTERIES

Marcos E. Orchard (M’06) received the B.S. degree in electrical engineering and the Certification in civil industrial engineering with major in electrical engineering from Pontifical Catholic University of Chile, Santiago, Chile, in 1999 and 2001, respectively, and the M.S. and Ph.D. degrees in electrical and computer engineering from the Georgia Institute of Technology, Atlanta, in 2005 and 2007, respectively. He was with the Intelligent Control Systems Laboratory, Georgia Institute of Technology. He is currently an Assistant Professor with the Department of Electrical Engineering, Universidad de Chile, Santiago, where he is also with the Advanced Mining Technology Center. His current research interest is the design, implementation, and testing of real-time frameworks for fault diagnosis and failure prognosis, with applications to battery management systems, mining industry, and finance. His fields of expertise include statistical process monitoring, parametric/nonparametric modeling, and system identification. His research work at the Georgia Institute of Technology was the foundation of novel real-time fault diagnosis and failure prognosis approaches based on particle filtering algorithms. He has published more than 50 papers in his areas of expertise.

Pablo Hevia-Koch received the B.S. degree in electrical engineering and the Certification in civil electrical engineering in 2012 from Universidad de Chile, Santiago, Chile. He is currently a Research Assistant with the Department of Electrical Engineering, Universidad de Chile. His research interests include the development of simultaneous localization and mapping (SLAM) modules for service robots and implementation of robotic operating system middleware.

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Bin Zhang (S’03–M’06–SM’08) received the B.E. and M.E. degrees from Nanjing University of Science and Technology, Nanjing, China, in 1993 and 1999, respectively, and the Ph.D. degree from Nanyang Technological University, Singapore, in 2007. He is currently with the Department of Electrical Engineering, University of South Carolina, Columbia. Prior to that, he was with Georgia Institute of Technology, Atlanta; Impact Technologies, LLC, Rochester, NY; and R&D, General Motors, Detroit, MI. His interests are prognostics and health management, intelligent systems, and their applications. He has published over 80 technical papers.

Liang Tang (M’04) received the Ph.D. degree in control theory and engineering from Shanghai Jiao Tong University, Shanghai, China, in 1999. He is currently a Team Lead for controls and prediction with Impact Technologies, LLC, Rochester, NY, with over 15 years of experience in signal processing, intelligent control, fault diagnostics, prognostics, and integrated vehicle health management. Prior to working with Impact Technologies, LLC, he was with the R&D Center, Ericsson, in 1999–2003, working on wireless communication systems, and was a Postdoctoral Research Fellow with Georgia Institute of Technology, Atlanta, in 2003–2004, working on unmanned aerial vehicle (UAV) fault diagnosis and fault tolerant control. He has been the Principal Investigator of multiple National Aeronautics and Space Administration and Department of Defense programs in which he led efforts in developing fault diagnosis system for jet engine, guidance and navigation system for UAV, and fault accommodation techniques for autonomous vehicles. He has published over 50 papers in his areas of expertise.