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OCV-SOC Parameter Estimation for Li-ion Battery

## Solution Approach

Submit a project report that has the following sectionsÃ¯Â¿Â½
Introduction - Problem definition

Solution approach Results and discussions Conclusions The following rubric will be used to mark the report
(20) Organization of the report (is it typed? how neat? does it have a table of content? references? figure/table numbers? are plots in figures appropriately labeled)

Ã¯Â¿Â½
(20) Literature review (are existing papers discussed and cited:) (25) Correctness of derivations (25) Correctness of results shown (10) Diseussions albout the resultsÃ¯Â¿Â½
The open cireuit voltage (OCv) of a Liion battery has a non-linenr relationship to its state of charge (SOC). Figure 1 shows a sample OCV-SOC curve of a Li-ion battery. Obtaining the paramn eters of the OCV-SOC curve involves offline data collection and optimization at cach temperature. Below. the OCV characterization appronch is summarized [1]. OCV-soC characterization is essentially a curve fitting problem: given different measurements of the {0CV, SOC} pairs spanning the entire SOC range (i.e., SOC E [0, 1|), the gonl is to fitÃ¯Â¿Â½
a curve that can aceurately represent the OCV-SOC measurements using as few parameters asÃ¯Â¿Â½
pOsible.

Ã¯Â¿Â½

The firststep in OCV-SOC modeling is the selection of a function that is best suitable to capture the OCV-SOC representation of a partieular battery chenistry For this, many diferent functions from straight lines to polynomials of varyving order and components - were tried in the past (see [1 for a review of such existing models). One of the well adopted OCV-sOC models is known as combined model which uses the following function for OCV-SOC characterization:

where o, ...K are the parameters of the combined model, s E [0, 1]) denoted the SOC and Vos) denotesthe OCV corresponding to that SOC. The range of OCV depends on the type of battery: for a single li-ion cell this range is OCv = V<(Ã¯Â¿Â½) e 34.2) for SOC =se |0,1] 1. In [), a slightlymore complex model that was shown to reduce modeling errors at lower SOC regions was introduced. The resulting model, referred hereafter as the combined+3 model uses the following function for OCV-SOC characterization.

where ko, k1,. , kz are the parameters of the combined-+3 model.

The next step in OCV-SOC modeling is to estimate the modelparameters, i.e., ko. k,...,ky in the case of combined+3 model. By collecting (OCV, SOC) data points spanning the entire range of OCV and SOC, the model parameter vector k = [ko, k1....,kg|can be linearly estimated using the least square approach.Figure 1: OCV vs. SOC curve of a Li-ion battery. This particular curve is obtained from a Samaung EB575152 battery. The state of charge (SOC) is indicated as a ratio of remaining charge and battery capacity

## Rubric for Report

Figure 2 shows the equivalent cireuit of a battery when it is slowly charged/discharged with a constant rate. First, we define the SOC at a given time asÃ¯Â¿Â½

where the notation, that reads "defined as", is used to assign a new variable name with slightly different context, e.g., the value s at time k is defined as Ã¯Â¿Â½lk) in (3). The true SOC at time k can be recursively computed using the Coulomb counting eqjuation

Ã¯Â¿Â½

where A is the time difference between two measurements, i[k] is the current through the battery and Cbatt is the battery capacity in Ampere hour (Ah). The Coulomb counting appronch is susceptible to the following five sourees of noise (3]: current measurement error, battery capacity uncertainty, initial SOC uncertainty, Coulomb integration approximation error, and the timing oscillator error. During the OCV characterization process, the following measures are taken to minimize these errors: OCV characterization data is collected in laboratory setting using high precision measurement systems, such as the Arbin tester |2] -this

Figure 2: Equivalent circuit model of a battery during slow charge/discharge. It must be noted that the above equivalent cireuit model is suitable when the battery experiences constant current of very low amplitude. This model allows ius to estimate the OCV-\$OC curve.

Ã¯Â¿Â½reduces the current measurement and timing oscillator errors; the charge discharge current is kept constant this nuliies the integration error; the OCV characterization current profile is selected inma way that initial SOC and the battery capacity can be accurately measured betore the parameters are obtained. Now, let us consider the measured voltage across the battery terminalsÃ¯Â¿Â½

Ã¯Â¿Â½

ZV {k} =v{k} + nv{k}

Ã¯Â¿Â½

where vlk] is the true voltage acros the battery terminals, and n,k| is the voltage meaxurement noise which is modeled as white Gaussian with standarddevintion (s.d.) o During the OCV experiment ie., when the battery is being slowly charged/discharged, the terminal voltage can beÃ¯Â¿Â½
written asÃ¯Â¿Â½

where hk] is the hysteresis or voltage "pul" which is a funetion of curent and SOC of the battery [4. Since the OCV test is performed at a very low current, we assume that the hysteresis is proportional to the current only |2], ie.Ã¯Â¿Â½

Ã¯Â¿Â½

Finally, the OCV-SOC modeling can be summarized as follows:

Ã¯Â¿Â½
Summary:OCV-\$OCModelingat Temperature T

1. Fully charge the battery at Tmax

Ã¯Â¿Â½
2. Bring the battery to temperature T

Ã¯Â¿Â½
3. Collect v[k], ik] during steps 4) and 5)Ã¯Â¿Â½

4. Slow-discharge the battery at C/30 rate until empty

Ã¯Â¿Â½
5. Slow-charge the battery at C/30 rate until fullÃ¯Â¿Â½

6. Compute battery capacity at TÃ¯Â¿Â½

7. Compute SOC s|k] using Coulomb counting through (4)

Ã¯Â¿Â½
8. Estimate the model parameters using (18)Ã¯Â¿Â½

Remark 1Ã¯Â¿Â½
Fig. 4 shows a linear scaling approach that can be used to avoid substituting s = 0 and s = 1 in equations, such as (2), that might lead to mumerical instability. The scaling approach maps the sOC domainse |0,1) to s e 0+6,1 - el in a linear fashion as described in Fig. 4. That way sÃ¯Â¿Â½
is prevented from reaching 0 or 1 and provides stability in computations 5). Before computing theÃ¯Â¿Â½
OCV, the SOC is scaled asÃ¯Â¿Â½
s'= (1-2e)s +e (20)Ã¯Â¿Â½
where the value of e needs to be selected based on the model; it was reported in 5] that e = 0.175Ã¯Â¿Â½
gives optimal results in Combined model and its variants.Ã¯Â¿Â½
Question 1Ã¯Â¿Â½
Re-derive the OCV parameter estimation approach to the following three models:Ã¯Â¿Â½
Linear model.

Use the data provided as an attachment in the file 'data.xls' to obtain the OCV-SOC parametersÃ¯Â¿Â½
of the following four models discuSsed so far:Ã¯Â¿Â½
Linear model Polynomial model (select the polynomial order)
Combined modelÃ¯Â¿Â½