Estimation of Best Linear Approximation from Varying ...

Estimation of Best Linear Approximation from Varying Operating Conditions for the Identification of a Li-ion Battery Model ? Rishi Relan ∗ Koen Tiels ∗ Jean-Marc Timmermans ∗∗ Johan Schoukens ∗ ∗ Department ELEC Department ETEC-MOBI Vrije Universiteit Brussel, 1050 Brussels, Belgium (e-mail: [email protected]) ∗∗

Abstract: The short term dynamic response of the battery varies with varying operating conditions. Hence, even before proceeding towards the modelling step, it is important to fully characterise and understand the dynamic behaviour of the battery at varying operating conditions. In this paper, a data-driven methodology for characterising the battery’s short term electrical response at varying operating conditions e.g. at different levels of SoC and different temperature levels is discussed. Furthermore, a novel way to estimate the best linear approximation from the data acquired at these operating conditions with varying levels of noise and nonlinear distortions is proposed. Keywords: System identification; Multiple experiments; Nonlinear models; Li-ion battery; Best linear approximation. 1. INTRODUCTION Nowadays due to the environmental concerns, there is an increasing demand for cleaner energy supply and energy efficient systems. Availability, certainty and efficiency of rechargeable electro-chemical energy systems, persuade us to consider them as alternative energy source in different applications such as electric and hybrid vehicles, heavy transportation systems, renewable energy systems and smart grids (Lund et al. (2012)). Furthermore, the success of the next generation systems, from consumer electronics, aerospace systems to the modern transportation systems rely on their energy consumption, efficiency and lifetime (both calendar life and cycling life). One of the most crucial components and still one of the weakest links in such systems is the main energy storage component i.e. the battery or battery pack (Franco (2015)). For instance,the heart of the any vehicle has moved from its engine to the battery. Lithium ion (Li-ion) batteries are attracting significant and growing interest because their high energy and high power density render them an excellent option for energy storage, particularly in hybrid and electric vehicles, as well as an ideal candidate for a wide variety of other applications. Hence, our understanding regarding the behaviour of different types of energy storage systems under different operating conditions has to be extended. 1.1 General Remarks In order to have a reliable performance of a lithium ion battery system, an autonomous management system is ? The corresponding author ([email protected]).

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needed to monitor and control the status of each cell (Rahimi-Eichi et al. (2013)). In order to efficiently control the state of a battery, it should be characterised carefully under many operating conditions. In this regard, it is important to identify all aspects of the battery, considering both the linear and nonlinear behaviours. Furthermore an accurate and comprehensive model is needed to describe its behaviour and predict the output based on arbitrary dynamic or static input load profiles. Many papers describe the characterisation of a battery based on conventional test procedures such as capacity test, hybrid pulse and dynamic performance tests and investigated battery specification mentioned in datasheet such as energy and power density, energy efficiency, discharge capacity and internal impedance (Mulder et al. (2011)). 1.2 Modelling of Battery In the field of battery modelling many different battery models exist (Cuma and Koroglu (2015)) for both the short and long term behaviour of battery cells. These models can be classified in the following categories: a) electro-chemical models, b) analytical and stochastic models of a cell, c) impedance based models, d) equivalent circuit models, e) empirical and semi-empirical models, Depending on the application and specifications, a choice between different classes of models can be made. In the short term electrical modelling of the batteries and the estimation of the associated parameters, most of the research effort relies on hybrid pulse power (HPPC) tests, and are based on a first or second order equivalent circuit (He et al. (2012)). The most common procedure is to consider the battery either as a linear system with variable parameters, or model the battery straightaway as a nonlinear system. Hardly, there

has been an effort to study the behaviour of the battery under different operating conditions. In this paper, we are mainly concerned with characterisation and modelling of the short term electrical behaviour of the battery cell. The short term behaviour of a cell consists of the voltage response, the usable capacity and state of charge (SoC) determination; in other words how a cell behaves over one cycle. For instance, the short term behaviour of a cell can be used to optimize the energy efficiency of the system and reduce heating of the cells. Short-term electrical behaviour of the battery cell varies due to the manufacturing variations as well as due to the different cell chemistries. Along with that, different loading conditions (charging/discharging profiles) and temperature play an important role in the determination of the cell response. 1.3 Practical Challenges in Data Acquisition One of the main challenges which is encountered in datadriven modelling process of batteries is the experiment design and data acquisition. For example, there are many situations that can lead to a series of sub-records of data with equal (Markovsky and Pintelon (2015)) or unequal lengths (Schoukens et al. (2012)). A first illustration is a long experiment where some parts in the data have extremely poor quality due to a sensor failure or due to very large disturbances coming from other processes. Eliminating these bad parts results in a series of broken sub-records of the data. In other experiments, it might be impossible to measure for a very long time without interruption; only a series of shorter tests can be performed. Finally, we can consider battery as a system whose dynamic response varies slowly due to changing operational conditions, e.g., a varying temperature, pressure etc. In that case, a series of sub-records that are collected under similar conditions can be grouped. If, for one of these reasons, a set of shorter sub-records is available, it turns out that it is advantageous to process them all at once, considering the data to come from one single experiment. Therefore in this paper we concentrate on developing a methodology to characterise and handle data from multiple experiments within the system identification framework. 1.4 Contribution and Organization of the Paper This article focuses on Li-ion battery nonlinear characterisation under different operating conditions. The characterisation is done over a 2-D grid of SoC and temperatures making use of various RMS levels of high current multisine input current signals. Furthermore, this work investigates the linear modelling of a battery cell in the nonlinear settings i.e. considering both the effect of disturbing noise and NL. As the first step, nonlinear distortions (NL) caused by high current, low SoC levels or temperature are quantified. Similarly, the noise disturbance caused by environmental and measurement equipment is analysed and modelled. Quantifying both the NL and disturbing noise at varying operating conditions, enable us to extract Best Linear Approximation (BLA) of the system. Finally, an efficient way to calculate a common BLA from all the operating conditions is presented. The structure of the paper is the following: Section 2 briefly introduces the multisine excitation signals. Section

2.1 describes very briefly the reasoning behind, how the properties of specially designed multisine excitation signals can be utilized for the detection and quantification of nonlinearities and noise in a dynamic system. Next, in Section 3, a procedure to identify the BLA of battery’s short term electrical dynamics at varying operating conditions is explained. Furthermore, it is explained how this identification method can be extended to calculate a common BLA from all the operating conditions in Section 3.1. The measurement setup and experiment design are described in Section 4. Section 5 presents the results and discussion. Finally the conclusions are stated in Section 6. 2. MULTISINE AS THE PERTURBATION SIGNAL Before proceeding to model the battery short term dynamics, it is important to characterise the battery electrical response under varying operational conditions of temperature and SoC in terms of the level as well as kind of nonlinearities. Using a broadband (Al Nazer et al. (2013)) excitation signal it is possible to extract efficiently maximum information about the behaviour in the band of excitation. Multisine excitation signals offer various advantages over Gaussian noise signals in extracting information from dynamical systems (Bayard (1993); Rivera et al. (2009)). A detailed information on the use of multisine signals for system characterisation and identification can be found in (Pintelon and Schoukens (2012); Relan et al. (2016b); Schoukens et al. (2016)). Assumption 1. The signal u is a random phase multisine (Pintelon and Schoukens (2012); Schoukens et al. (2016)) The frequency domain representation of the multisine signal is the sum of the Fourier transforms of the individual sines and is given by: X 1 Ums (jω) = √ A(k)δ(ωk − ωke )ejϕk (1) π Nk k∈±K exc

where δ(•) is the Dirac delta function, Nk the number of excited frequencies and ϕk ∼ U[0, 2π[ are the phases. The amplitudes of the multisine components A(k) ≥ 0 can be chosen arbitrarily, depending on the application. In addition, some frequencies are not excited i.e. A(kn.exc ) = 0. The output spectrum at these unexcited frequencies (detection lines) contains valuable information about the presence of non-linearities and/or the time variations of the system. In the next subsection, we will briefly explain how the properties of a multisine signals can be exploited to gain better insight into the dynamics of the system. 2.1 Nonparametric data analysis For this purpose, a random odd multisine (Schoukens et al., 2016) current signal is used to excite the battery at different operating points in terms of SoC and temperature. In this paper, characterisation of the battery’s short term electrical response is performed at between 0% − 100% SoC (in steps of 2% for the SoC between 0%−10% and 90%−100%; whereas in steps of 20% for the SoC between 10% − 90%). The analysis was repeated at different operating temperatures e.g. at 5°C, 14°C, 25°C, 35°C and at 40°C using a nonparametric characterisation technique proposed by (Schoukens et al. (2016); Relan et al. (2016a,b)).

3. BEST LINEAR APPROXIMATION Definition 1. The BLA of a nonlinear system is defined as the model G belonging to the set of linear models G, such that
GBLA (q) = arg min Eu |y(t) − G(q)u(t)|2 (3) G(q)∈G

where the E is taken over the random phase realisations of the input.

Fig. 2. Time domain representation of the problem

Fig. 1. Response PISPO: Total output is the sum of linear contributions (at excited lines), even NL (at even lines), odd NL (at odd lines) and noise (at all lines, not displayed here).

Assumption 2. For this particular analysis, it is assumed that the battery discharge capacity and the corresponding SoC levels can be estimated rather accurately. Assumption 3. The battery can be modelled as a weakly nonlinear periodic-in-same-period-out (PISPO) system described by a Volterra series (see Pintelon and Schoukens (2012) for more details). Remark 1. A nonlinear system is called PISPO if the steady state response to a periodic input is also a periodic signal with the same period as the input (with preservation of the period length). This includes systems with saturation and discontinuous nonlinearities, but it excludes systems with period multiplication, chaotic behaviour, subharmonics, and hysteresis, see (Schoukens et al. (1998); Pintelon and Schoukens (2012); Schoukens et al. (2016)) for a more formal definition. Since the nonlinear system (here battery) is operating in open-loop, the output Discrete Fourier Transform (DFT) spectrum of each period p where p = 1, 2, 3, ..., P , of the steady state response (with known periodic input) to an odd random phase multisine with random harmonic grid is given by: Y [p] (k) = Y0 (k) + V [p] (k) + YS (k)

(2)

The total response of the system (see Fig. 1) is the sum of linear (Y0 (k)), stochastic nonlinear (even & odd) contributions YS (k), and V [p] is the noise term (Pintelon and Schoukens (2012); Relan et al. (2016b); Schoukens et al. (2016)). The information about the effect of NL and the need for differentiating between odd and even frequencies during the nonparametric test is essential in order to get an idea about the contributions of NL to the FRF of the battery dynamics under varying conditions.

Set Up Here, we focus for simplicity on the nonparametric estimation of a discrete time single-input-single-output (SISO) model GBLA (q) of the BLA of a nonlinear system, which is excited with signals belonging to the Riemann equivalence class of asymptotically normally distributed excitation signals (Pintelon and Schoukens (2012)), see Fig. 2. For an infinitely long data record t = −∞, ..., N −1, the input-output relation of the nonlinear system can be written as: y(t) = GBLA (q)u0 (t) + ys (t) + H0 (q)e(t). (4) with q −1 the backward shift operator (q −1 x(t) = x(t − 1)) and ys (t) are the stochastic nonlinear contributions. The exact input u0 (t) is assumed to be known, while the output is disturbed with additive noise v(t), then y(t) = y0 (t) + v(t). The noise v(t) is assumed to be filtered white noise, v(t) = H0 (q)e(t), where H0 (q) is the noise model. For a finite record length t = 0, ..., N − 1, as it is in practical applications, this equation has to be extended with the initial conditions, or in other words, the transient effects of the dynamic plant and noise system tG , tH : y(t) = GBLA (q)u0 (t) + ys (t) + H0 (q)e(t) + tG (t) + tH (t). (5) An exact frequency domain formulation of (5) is obtained as: Y (k) = GBLA (ωk )U0 (k) + Ys (k) + H0 (ωk )E(k) + TG (ωk ) + TH (ωk ) (6) where the index k points to the frequency kfs /N , with fs the sampling frequency, and ωk = ej2πkfs /N . The finite record length requires the use of transient terms in (5), and it turns out that the leakage errors of the DFT are modelled by very similar terms in the frequency domain (Pintelon and Schoukens (2012)). It is most important for the rest of this paper to understand that (6) is an exact relation, where leakage effects are modelled by the transient terms (Pintelon et al. (1997); Pintelon and Schoukens (2012)). All these terms tG (t), tH (t), TG (k), TH (k) are described by rational forms in q −1 (time domain) or z −1 (frequency domain), hence they are smooth functions of the frequency. Within the above described set up, the nonparametric BLA can be calculated using either the Fast or the Robust method explained in (Pintelon and Schoukens (2012)), or the Local Polynomial Method (LPM), which makes an optimal use

of the smooth behaviour of GBLA and TG to significantly reduce the leakage errors (Schoukens et al. (2009a)). This results in superior properties compared with the classical windowing methods and provides a good estimation of 2 the BLA as well as its variance (σBLA ) (Schoukens et al. (2009a)). An alternative to the LPM method to capture transient effect is TRansient Impulse response Modelling Method (TRIMM) proposed by (Hägg et al. (2016)). 3.1 BLA from Multiple Experiments Suppose we carry out M independent experiments at different settings of SoC and for each of these settings J independent experiments at different settings of temperature, then the nonparametric BLA of each setting of SoC and temperature can be calculated individually using the Fast, Robust or the LPM nonparametric identification procedure described in (Pintelon and Schoukens (2012); Schoukens et al. (2016)). During the data acquisition at different temperatures, the set of excited and non-excited frequency lines were different from one set of temperatures i.e. [5°C, 14°C & 35°C] to other set of temperatures [25°C, & 40°C]. The BLA obtained from each set of experiments is rather smooth in nature within the band of interest and differs only by a scaling factor (see Section 5 below). Hence before calculating a common BLA (CBLA ) for all datasets, the smooth nature of the FRF’s is exploited and the FRF of each experiment is interpolated over a set of common frequency points within the band of excitation. After the linear interpolation step, the CBLA of the system under consideration can be calculated from the set of individual BLAs by calculating the statistical mean (calculated at each frequency line k in the set of frequency lines) of all the nonparametric BLAs. In a similar way the variance of CBLA can be obainted by calculating the sample variance of the (obtained in the previous step) individual BLAs: M 1 X BLAAvg,i (k) = BLAir (k) M r=1 M

BLAV ar,i (k) =

1 X |BLAir (k) − BLAAvg,i (k)|2 M − 1 r=1

J 1X CBLA (k) = BLAAvg,i (k) J i=1 2 σC (k) = BLA

J 1X BLAV ar,i (k) J i=1

(7)

3.2 Parametric BLA In this step, a parametric model is fitted to the nonparametric BLA. The parametric model can be used to better understand the system behaviour using the polezero representation. Thus, using the nonparametric FRF 2 estimate (CBLA ) and its variance (σC ), which is found BLA in the previous step (Pintelon and Schoukens (2012); Schoukens et al. (2009b)), we estimate a parametric model of our system by solving a nonlinear weighted least squares (NLWLS) problem. This model (discrete-time) describes the system as a rational transfer function. The model considered here is a rational function in the backward shift operator q −1 :

b0 + b1 q −1 + b2 q −2 + ...... + bnb q −nb ˆC G (q, θtf ) = , BLA a0 + a1 q −1 + a2 q −2 + ...... + ana q −na (8) The parameter vector θtf ∈ R(nb +na +2)×1 contains the parameters [a0 , a1 , . . . , ana , b0 , b1 , . . . , bnb ]T . Since one parameter can be chosen freely because of the scaling invariance of the transfer function, only nb + na + 1 independent parameters need to be estimated by minimizing the following NLWLS cost function: F X ˆC |CBLA (jωk ) − G (q, θtf )|2 BLA Vtf (θtf ) = , (9) 2 σC (jωk ) BLA k=1

The order of the parametric model in (8) can for example be determined using a signal theoretic measure such as the minimum description length (MDL) criterion (see page no. 439 of Pintelon and Schoukens (2012)). This NLWLS framework also guarantees the lowest possible uncertainty on the model parameters, i.e. the efficiency of the estimates (Pintelon and Schoukens (2012)). Thereafter, a balanced ˆBLA , CˆBLA , D ˆ BLA ) state space realization Gss = (AˆBLA , B ˆ for the stable portion of the linear system GBLA (q, θtf ) can be calculated (MathWorks (2013)), where the subscript ss stands for the state space. For stable systems, this is an equivalent realization for which the controllability and observability Gramians are equal and diagonal (Moore (1981); Laub et al. (1987)). In the next section, we describe briefly the measurement setup and experiment design for the acquisition of the data at various operating conditions. 4. MEASUREMENT SETUP A high energy density Li-ion Polymer Battery (EIGePLB-C020, Li(NiCoMn)) with the following electrical characteristics: nominal voltage 3.65V , nominal capacity 20Ah, AC impedance (1 KHz) < 3mΩ along with the PEC battery tester SBT0550 with 24 channels is used for the data acquisition. The tests are performed on a pre-conditioned battery inside a temperature controlled chamber at different temperatures. An odd-random phase multisine signal is used as an input excitation signal. The band of excitation is kept between 1Hz–5Hz, because the dynamic range of interest of the battery for HEV’s and EV’s applications is covered well within this band of excitation. It also takes into consideration the limitations of the battery tester in terms of the sampling frequency. The excitation signal has a period of 5000 samples and the sample frequency fs is set to 50Hz resulting in a frequency resolution of fo = 0.01Hz. The input is zero mean with a RMS value of 5, 10 and 20A. A random realization of the phases of the multisine signal with 7 periods is acquired at different levels of SoC and temperatures. For the test, the battery is first charged using a constant C3 rate, where C is the rated capacity, to the maximum charge voltage of 4.1V using the constant current-constant voltage method. Then, after a relaxation period of 30 minutes, it is discharged to the desired SoC level Ah-based and considering the actual discharge capacity at the respective temperatures until the end of discharge voltage 3.0V of the cell. After each discharge a rest period of 60 minutes is applied before the multisine tests are performed. It is made sure that the synchronization is maintained between the signal generation and acquisition side.

5. RESULTS This section presents the results of the nonparametric analysis and BLA estimation performed at varying operating conditions of SoC and temperature.

35°C and 40°C). Predominently even NL were observed at most operating points but mostly at lower temperatures and lower SoC levels odd NL also became significant. 5.2 Temperature Effect on BLA

5.1 Nonparametric Analysis

Fig. 6. BLA at (6% SoC, 10A RMS, 25°C, 40°C) Fig. 3. Voltage response at (6% SoC, 10A RMS, 40 °C)

Fig. 7. BLA at (6% SoC, 10A RMS, 5°C, 14°C, 35°C)

Fig. 4. Voltage Response at (6% SoC, 10A RMS, 5 °C)

Figures 6 and 7 show the effect of temperature on the nonparametric BLA calculated using two different realizations of a multisine signal. It can been clearly seen that the BLA and variance of the BLA changes at different operating conditions only as a scaling factor. This observation and the smooth nature of BLA is further exploited to calculate the CBLA . Figure 8 shows the CBLA calculated using (7).

Fig. 5. Voltage response at (30% SoC, 10A RMS, 40 °C) Figures 3, 4 and 5 show the result of nonparametric analysis performed at specific settings of SoC and temperatures using 10 A RMS multisine input current profile. It can be clearly observed that the level of NL changes w.r.t the operating conditions. It can be seen from the Figure 5 that the battery behaves linearly at (30% SoC, 40°C), whereas at (6% SoC, 40°C) even NL dominate and the level of odd NL is at the level of noise (see Fig.3). At (6% SoC, 5°C) both even as well as odd NL grow significantly both within and outside the band of interest (see Fig.4). Similarly the level of noise is higher as compared to the other settings of SoC and temperature. A similar analysis was performed at other settings of SoC (not displayed here due to limitation in space) and at 5 different temperatures (5°C, 14°C, 25°C,

Fig. 8. CBLA and individual BLA’s at (10% SoC, 5°C) and (4% SoC, 40°C) The CBLA was calculated using the data acquired at all operating condition, but to avoid the overcrowding of the Figure 8 only the BLA’s at two extreme operating conditions i.e. at (10% SoC, 5°C) and (4% SoC, 40°C) are shown. It can be seen that CBLA is a reasonable approximation of the individual BLA’s estimated at extreme operating conditions. The final values of the parameters of the linear model (CBLA ) can be used as an initialization for the datadriven identification of a nonlinear model (Relan et al. (2016a)), which is valid at varying operating conditions

or the individual BLAs can be used to develop black-box linear time-varying or parameter-varying models. 6. CONCLUSION This paper proposed a data-driven methodology to develop the BLA of the battery’s short term electrical dynamics from the data acquired using multiple experiments at varying operating conditions. The proposed framework paves the way for handling data records of arbitrary lengths from multiple experiments. This gives a practical advantage when performing long experiments is not feasible or rather expensive. The data of extremely poor quality can also be handled. This whole process can be carried out in relatively short measurement time, which is the need of many industrial processes. The next part of this research ˆC is to utilize the CBLA or G to identify a Polynomial BLA Nonlinear State-Space model of the battery’s short term dynamics (Relan et al. (2016a)) valid at multiple operating conditions. ACKNOWLEDGEMENTS This work was supported in part by the IWT-SBO BATTLE 639, FWO-Vlaanderen, by the Flemish Government (Methusalem grant), Inter university Poles of Attraction (IAP VII) Program, and by the ERC advanced grant SNLSID, under contract 320378.

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