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**What are the different types of Battery Models available?**

30th May, 2021

You can find a variety of battery models with diverse objectives and difficulties that have been published in technical literature in the last years [Ref 1-6]. Different types of battery models are:

1. Mathematical and electrochemical model

2. Thermal model

3. Electrical models:

a) Thevenin-Based Electrical Model

b) Rint Electrical Model

c) Runtime-Based Electrical Model

d) Impedance-Based Electrical Model

e) Shepherd Model

f) Generic Library Model

## 1. Mathematical and Electrochemical Models:

These models have essentially been developed to describe fundamental mechanisms. The first models were published by Fuller, Doyle, and Newman [Ref 7,8]. They represent an important contribution to physical cell design and the quantification of macroscopic variables such as battery voltage and current, as well as microscopic such as concentration distribution and galvanostatic charge or discharge. These models require a variety of cell parameters and complex numerical computational methods. Regardless of the accuracy achievable with these models, they are unsuitable in a simulation environment in which the electrical terminal behavior of the battery and the state of charge is to be determined with reasonable computing times.

## 2. Thermal Models:

Newman and Pals [Ref 9,10] published their initial studies on thermal models in 1995. Thermal models have gained importance in the past several years, not least due to the growing market of lithium-ion in hybrid and electric vehicles [11]. These models are also characterized by numerous parameters and complex calculations. The battery system under examination is placed in an air-conditioned area in which the ambient temperature is controlled to a constant value. But, an evaluation of the cell temperature cannot be conducted in this manner. However, it is assumed in this study that the cell temperature does not affect the model behavior. So, only electrical models will be examined in more detail in the following sections.

## 3. Electrical models:

In contrast to other battery models, electrical models are intuitive and simple to use. All electrical models consist of equivalent circuits composed of passive components such as resistors and capacitors, perhaps inductors, and a voltage source. So, they are mostly right for use in circuit simulators. The precision achievable with these models about voltage & current characteristics, as well as state of charge, is enough for many applications. Next, a brief portrayal of the common models will be given.

### a) Thevenin-Based Electrical Model:

The simplest model, as shown in Figure 1(a), contains a series resistor RS, an RC network (Rt, Ct) to describe the basic charge transfer mechanism, and an open circuit voltage dependent on the SOC [VOCV(SOC)] [Ref 3]. But the simple model has limited accuracy. An improvement for the simulation of Li-ion cells can be achieved by adding a 2nd RC network (Figure 1 b) [Ref 12,13]. The first RC network represents short-term transient behavior (Rt,s, Ct,s), and the second, long-term transient behavior (Rt,l, Ct,l). In [Ref 14–17], a dependence of the network elements on the SOC was proposed to reach further accuracy. In detail, the dependence on the SOC is described by the listed set of equations:

V_{OCV}(SOC) = k_{0} + k_{1}·SOC + k_{2}·SOC_{2} + k_{3}·SOC_{3} + k_{4}·e ^{k5·SOC }^{(1)}

^{R}_{S}(SOC) = R_{s0} + k_{5}·e ^{k6·SOC}^{ }^{(2)}

^{R}_{t,s}(SOC) = R_{t,s0 }+ k_{7}·e ^{k8·SOC }^{(3)}

^{C}_{t,s}(SOC) = C_{t,s0 }+ k_{9}·e ^{k10·SOC }^{(4)}

^{R}_{t,l}(SOC) = R_{t,l0} + k_{11}·e ^{k12·SOC }^{(5)}

^{C}_{t,l}(SOC) = C_{t,l0} + k_{13}·e ^{k14·SOC }^{(6)}

The coefficients ki, i = 1 · · · 14 depend on the particular cell type and are subjects of measurements. In [Ref 18] an additional RC network is proposed, to define finally short-term, mid-term, and long-term transient performance. But this makes the calculation of the related resistances and capacities (Rx, Cx) much more complex.

Figure 1 (a) Thevenin-based model, 1 RC network (b) Thevenin-based model, 2 RC networks

### b) Rint Electrical Model:

As shown in Figure 2(a), the Rint model comprises a voltage source 𝑉_{OCV} representing the open-circuit voltage and an internal resistor 𝑅_{int}. Both network elements depend on the SOC. Moreover, the internal resistor can depend on the two operating manners of charging/discharging. This enables the model shown in Figure 2(b) to be specified with 𝑉_{OCV }(𝑆𝑂𝐶), 𝑅_{int }(𝑆𝑂𝐶, 𝑐ℎ𝑎𝑟𝑔𝑒), and 𝑅_{int }(𝑆𝑂𝐶, 𝑑𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒).

Figure 2 (a) Rint model (b) Rint model with individual internal resistors

### c) Runtime Based Electrical Model:

Figure 3 displays a runtime based model given in [Ref 14]. It comprises the growth of the original model introduced in [Ref 19] by an electrical network based on the Thevenin model. The left portion of the network holds the capacitor 𝐶_{C}, the self-discharge resistor 𝑅_{dis}, and the current-controlled current source 𝐼_{bat}, integral to the original runtime model. This portion models the capacity, the SOC, and the lifetime of the cell, while the right portion simulates the transient response. A voltage-regulated voltage source 𝑉_{OC }(V_{SOC}) is used to link the SOC to the open-circuit voltage. The full capacity capacitor 𝐶_{C} is presented to define the total charge stored in the battery. Influences of runtime and battery lifetime can be simulated with the help of this model. 𝐶_{C} is defined as a function of the nominal capacity of the battery 𝑄_{nom} and the correction factors 𝑓_{1}(𝐶𝑦𝑐𝑙𝑒) and 𝑓_{2}(𝑇𝑒𝑚𝑝), which depend on the cycle number and the cell temperature:

𝐶_{C} = 𝑄_{nom} ∙ 𝑓_{1}(𝐶𝑦𝑐𝑙𝑒) ∙ 𝑓_{2}(𝑇𝑒𝑚𝑝) (7)

Figure 3 Runtime Based Model

### d) Impedance Based Electrical Model:

The impedance model shown in Figure 4 is based on electrochemical impedance spectroscopy to model an AC-equivalent impedance 𝑍_{AC} in the frequency domain [Ref 20,21]. But, fitting 𝑍_{AC} to the impedance spectrum is very difficult and challenging. A full description of the method can be found in [Ref 22].

Figure 4 Impedance Based Model

### e) Shepherd Model:

The electrical circuit network in Figure 5 (a) is known as Shepherd’s Model, which was first referenced in [Ref 23] and improved by subsequent work, e.g., [Ref 24,25]. In the original Shepherd model, the open circuit voltage 𝑉_{OC} is given as:

VOC = V0 + (K Q) i/(Q-it) (8)

𝑉_{0} is the constant battery voltage, 𝐾 is the polarization resistance coefficient, 𝑄 is the battery capacity, 𝑖 is the dynamic battery current at the time 𝑡, 𝑖𝑡 is the discharge capacity. The usually used modified model is illustrated in Figure 5 (b) [Ref 11,26]. Here the battery current is also filtered by a low-pass (LP). To find open circuit voltage, a distinction is made between charging and discharging:

VOC,charge = V0-(K Q) ⅈ*/(it-0.1Q) - (K Q) it/(Q-it) + A e^(-Bit) (9)

VOC,dis = V0 - (K Q)(it+ⅈ* )/(Q-it) + A e^(-Bit) (10)

Coefficient A denotes the voltage amplitude in the exponential zone of the discharge curve, whereas coefficient B stipulates the time constant inverse in this region, and K is the polarization voltage. For a better understanding, Figure 6 shows the simulated discharge curve of a Li-ion cell. As indicated, the curve can be divided into three subintervals: Section 1 consists of the exponential zone starting at the full voltage V_{full}, section 2 describes the linear zone, and section 3 finally comprises the nonlinear zone up to the cut-off voltage V_{cut−off}. Using this curve, the coefficients A, B, and K in Equations (9) and (10) can be given as:

A = Vfull + Vtop (11)

B = (3/Qtop) (12)

The coefficient B is given in Equation 12, assuming that the end-value in the exponential region is reached after about 3-time constants. The coefficient K can be equated by Eq 10 for any point on the discharge curve [Ref 25]. It should be noted that the coefficient K in Equations (9) and (10) is multiplied by both the filtered battery current i∗ (A) and the discharge capacity it (Ah). The units of K must be Ω or V/Ah. So, the representation of the open circuit voltage using one polarization coefficient is not favorable.

Figure 5 (a) Original Shepherd Model (b) Modified Shepherd Model

Figure 6 Simulated discharge curve of Li-ion cell (1C rate) [Ref 27]

### f) Generic Library Model:

Figure 7 Powersimtech Battery Model

To meet the demand for a model that can define dynamic behaviour with enough accuracy, and that can also be applied as easily as possible in an electronic circuit simulator, this study examines the generic Li-ion model from the library of the software package PSIM. Figure 7 depicts a schematic of the model. The load is connected to the positive and negative pin, while the upper pin gives the actual SOC during simulation. For the user, the model appears as a mask sub-circuit, which is parameterized by the cell features [Ref 28]. The required cell voltage points (V_{full}, V_{exp}, V_{nom,} and V_{cut−off}) and capacities (Q_{full}, Q_{exp}, Q_{nom,} and Q_{max}) used from the discharge curve shown in Figure 6. The series resistance R_{S} can be derived from the corresponding datasheet. Also, a voltage derating factor and a capacity derating factor are available to adapt simulated curves. The default value of both factors is set to unity. When a cell is parameterized, a battery can be defined by the number of cells connected in series and parallel.

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