circuit model), we present how Bayesian Optimization can be contextualized to ... At every step of Bayesian Optimization, the next set of unknown parameters.
Parameters Estimation for State-of-Power (SOP) Prediction of Lithium-ion Batteries: A Bayesian Optimization Approach Ali Baheri
1
Introduction
In this brief, we start to investigate parameters estimation for lithium-ion battery model using Bayesian Optimization. First, we present the available approaches to model battery systems. Next, by introducing the mathematical model for the simplest case (i.e., equivalent circuit model), we present how Bayesian Optimization can be contextualized to estimate parameters of the lithium-ion battery. Bayesian Optimization has proven to be an effective strategy for finding the global optimum of an unknown, expensive to evaluate, and blackbox function within only a few function evaluations [1, 2, 3, 4, 5]. One popular approach is to model the unknown function as a Gaussian Process (GP) [6, 7], where Bayesian Optimization puts prior belief on an objective function to describe the overall structure of that function. At every step of Bayesian Optimization, the next set of unknown parameters (i.e., optimization variables) are selected to maximize some acquisition function, which characterizes (i) how much will be learned by this new set of unknown parameters and (ii) what the likely performance level will be at that next unknown parameters.
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Battery Model
Fig. 1 illustrates an overview of battery models along with associated modeling complexity [8]. Full order electrochemical model consists of several PDEs and ODEs in time and space resulting to difficult control and parameters estimation task. In such situations, Single Particle Model with Electrolyte Dynamics (SPMe) model provides a reduced order with good accuracy compared to the full order model. In this work, we illustrate how a powerful machine learning algorithm is employed to parameters estimation of simple model of a lithium-ion battery model.
2.1
Equivalent Circuit Model
Let's start with the simplest battery model called equivalent circuit model with hysteresis (See Figure 2). In this model, Rc and Rs denote the charge/discharge energy loss in the cell
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Future direction Scope of this work
Figure 1: Overview of battery models (adopted from [8]) and charge transfer resistance, respectively. Cd represents the charge transfer. Moreover, Vd and V presents the short term diffusion voltage of the cell and terminal voltage, The voltage hysteresis model is modeled by, ∂Vh = −ρ(ηi − vSd ) Vh ∂t
max
+ sign(i)Vh
(1)
where ρ is the hysteresis parameter, η is the coulomb efficiency, i is the instantaneous current applied to the battery, v is the self-discharge multiplier for hysteresis expression, Sd is the self-discharge rate, and Vh max is the maximum hysteresis voltage. A discrete-time battery model can be expressed as, s 1 0 0 0 − CηT max i(k) X(k + 1) = 0 γ 0 X(k) + Rc (1 − γ) 0 Vh max 0 0 H 0 (H − 1)sign(i) y(k) = V (k) = Voc SOC(k) − Vd (k) − Rs i(k) + Vh (k)
(3)
Voc (SOC) = c0 exp(−c1 SOC) + c2 SOC 3 + c3 SOC 2 + c4 SOC + c5
(4)
2
(2)
Vc Rc Rs
Cd
Voc (SOC)
+
V -
i Figure 2: Equivalent Circuit Model (ECM) represents the simplest form of a battery model. where X(k + 1) = [SOC(k + 1) vd (k + 1) vh (k + 1)]T is the states. y(k) is the measured (−
Ts
)
output, and Cmax is the maximum capacity of battery. Furthermore, γ = exp Rc Cd and H(i) = exp(−ρ|i|Ts ), and ci represent the coefficients used to parametrize the Voc (SOC) curve.
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Proposed Framework Estimated battery parameters
i
(ˆ ↵)
Estimated battery parameters
Bayesian Optimization
Battery model
(ˆ ↵)
k =k+1
i _
Vˆ ⌃ V
+
Battery model
_
Vˆ
Performance metric
⌃ V
Bayesian Optimization
+
Performance metric
Battery system
Battery system
Figure 3: Offline parameter estimation framework
Figure 4: Online parameter estimation framework
Using the simple battery model of Fig. 2, the ultimate goal is to identify unknown battery parameters: α = [Vc , Rc ] (5) where Vc represents the constant battery voltage in Volts and Rc denotes the battery polarization resistance in Ohms. Several methods have been proposed for state and pa3
rameter estimation of battery systems, however, the existing literature includes two main drawbacks. Firstly, most estimation algorithms are model-based methods. Secondly, high computational complexity prevents them from being implemented in real-time embedded systems. To address these issues, we introduce a new framework that treats the parameter estimation problem as an offline/ online learning optimal control problem. Fig 3 and 4 illustrate the idea behind parameter estimation using Bayesian Optimization. At each iteration of Bayesian Optimization, a battery terminal voltage, V , is measured under a specific current, i. The measured V is compared with the estimated voltage, Vˆ , obtained from the battery model based on the measured current using estimated parameters. The difference between V and Vˆ is then used to construct a performance metric. Next, a Bayesian Optimization algorithm is utilized to identify the next best estimated battery parameters. It has been proven that the process converges within only a few evaluations of the performance metric under some conditions [5]. Mathematically, if α is the vector of unknown battery parameters, then the optimal learning control problem takes the following form: Z tf minimize J αk = [V (t) − Vˆ (t)]2 dt. (6) ti
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Preliminary Results
To evaluate the effectiveness of the offline estimation framework, a simple 1.2V (6500mAh) battery is subjected to a discharge and charging experiment. The model input is the battery current and the model output, the battery terminal voltage, is calculated from the battery state-of-charge. Once a single experiment is run, the Bayesian Optimization algorithm is utilized to estimate unknown battery parameters within only a few evolutions of performance metric. The convergence of unknown battery parameters is shown in Fig. 5 and Fig. 6. One can conclude from these figures that unknown battery parameters converge after only about 12 iterations of Bayesian Optimization (i.e., after only about 12 rounds of system performance evaluation).
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Online Parameter Identification
Online estimation framework presents more applicable scenario. Because Bayesian Optimization is an iteration-based optimization algorithm, it is crucial to make a clear connection between the concept of discrete iteration (used in Bayesian Optimization), and continuous time over the course of simulation. To achieve this goal, several interesting open-ended questions should be answered:
4
10-3
Battery Resistance (Ohms)
Volatage (V)
2 1.5 1 0.5 0
5
10
15
1 0.8 0.6 0.4 0.2 0
20
Number of Iterations
5
10
15
20
Number of Iterations
Figure 5: Evolution of the battery voltage over the course of optimization
Figure 6: Evolution of the battery resistance over the course of optimization
• What is the shortest feasible iteration length, and how does this relate to concepts of persistent excitation? • When the level of excitation is different during different iterations, how can Bayesian optimization be amended to account for this excitation bias? • Can iteration lengths be adjusted to achieve a uniform/nearly uniform level of excitation during each iteration? • Can longer overlapping iterations be used to drive Bayesian optimization? Note that information would be duplicated with overlapping iterations.
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Future Work
The research plan is aimed at achieving a number of objectives. These include, but are not limited to: • Develop and implement an online estimation framework that leads to crystal clear answers for the aforementioned questions; • Replace the simple equivalent circuit model with high-fidelity models such as SPMeT under different current profiles; • Incorporate other battery parameters into the estimation frameworks; • Provide the proof of the convergence for the proposed frameworks.
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References [1] Ali Baheri, Shamir Bin-Karim, Alireza Bafandeh, and Christopher Vermillion. Realtime control using bayesian optimization: A case study in airborne wind energy systems. Control Engineering Practice, 69:131–140, 2017. [2] Roman Garnett, Michael A Osborne, and Stephen J Roberts. Bayesian optimization for sensor set selection. In Proceedings of the 9th ACM/IEEE International Conference on Information Processing in Sensor Networks, Stockholm, Sweden, 2010. [3] Ali Baheri and Chris Vermillion. Altitude optimization of airborne wind energy systems: A Bayesian Optimization approach. In American Control Conference, Seattle, US, 2017. [4] Ali Baheri, Praveen Ramaprabhu, and Chris Vermillion. Iterative in-situ 3D layout optimization of a reconfigurable ocean. In ASME 2017 Dynamic Systems and Control Conference, Tysons Corner, US, 2017. [5] Ali Baheri, Joe Deese, and Chris Vermillion. Combined plant and controller design using Bayesian optimization: A case study in airborne wind energy systems. In ASME 2017 Dynamic Systems and Control Conference, Tysons Corner, US, 2017. [6] Carl Edward Rasmussen and Christopher K. I. Williams. Gaussian processes for machine learning. Number ISBN 0-262-18253-X. The MIT Press, 2006. [7] Shamir Bin-Karim, Alireza Bafandeh, Ali Baheri, and Christopher Vermillion. Spatiotemporal optimization through gaussian process-based model predictive control: A case study in airborne wind energy. IEEE Transactions on Control Systems Technology, 2017. [8] Hector Eduardo Perez. Model Based Optimal Control, Estimation, and Validation of Lithium-Ion Batteries. PhD thesis, University of California, Berkeley, 2016.
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