PyBatteryID – A Python Package for Data-Driven Battery Model Identification using LPV Framework

Input–output LPV representation

A brief overview of the battery model structure can be given as follows. Consider an nth-order LTI input–output representation for the battery overpotentials as given by

where y_k represents the model output at time instant k, u_k the model input at time instant k, and a_i,b_i the model parameters. The above LTI representation can be converted to an LPV representation by rendering the parameters a_i,b_i as functions of the scheduling variable p. Concretely,

where the notation (□⋄p)_k corresponds to having an arbitrary dependence on the variable p at different time instants, e.g., ${(a_1\diamond p)}_k=a_1(\mathrm{\dots },p_{k–1},p_k,p_{k+1},\mathrm{\dots })$. In the following, the so-called shifted-form input–output representation will be considered, which assumes a special p-dependence structure, that is, ${(a_j\diamond p)}_k=a_j(p_{k–j})$. Accordingly, the above input–output representation can be given by

where the parameter corresponding to the output or input term lagged by j time instants depends on the value of the scheduling variable also lagged by j time instants, that is, p_k–j. Such a choice of p-dependence has been motivated owing to the analytical convenience associated with the shifted-form input–output representation as explained in [9].

Definition of p-components

As mentioned earlier, the package allows SOC, current magnitude, current direction and temperature as the components of the variable p, that is, p≜[sδ|u|T]^T, where s represents the SOC, δ the current direction, |u| the current magnitude, and T the temperature. Note that the value of SOC s can be determined using a first-order difference equation as given by

where τ represents the model sampling time, Q the battery capacity, and u the battery current. Additionally, the current direction δ has also been defined using a first-order difference equation [9], as can be given by

where ε is a current-dependent parameter as given by

with ${\epsilon }_0,{\epsilon }_1\in [0,1]$ the hyperparameters. In effect, various current-direction trajectories can be generated by choosing certain values of ε₀ and ε₁, henceforth denoted by ${\delta }^{[{\epsilon }_0,{\epsilon }_1]}$.

Selection of suitable basis functions

Once the components of the variable p have been defined, the next step can be given by the specification of suitable basis functions for each p-component. For instance, the following basis functions for the SOC, current magnitude, current direction and temperature have been considered in our recent papers,

where 𝒢_s, 𝒢_|u|, 𝒢_δ and 𝒢_T represent the sets of basis functions for SOC, current magnitude, current direction and temperature, respectively; see our recent papers [9, 10] for the motivation behind choosing such functional forms for each p-component. Subsequently, a larger set of basis functions can be generated that includes individual basis functions specified above as well as their higher-order combinations. Formally, a set 𝒢^(l) can be defined representing all the considered basis functions as given by

where l represents the nonlinearity order specifying the order of basis-function combinations, and ${\mathcal{G}}^{(1)}=\{1\}\cup {\mathcal{G}}_s\cup {\mathcal{G}}_{|u|}\cup {\mathcal{G}}_{\delta }\cup {\mathcal{G}}_T$. Finally, the model parameters ${\{a_i\}}_{i=1}^n$, ${\{b_i\}}_{i=0}^n$ can be expanded using the set 𝒢^(l) as given by

where {a_ij} and {b_ij} represent the to-be-estimated free model parameters, g^(j) the jth element in 𝒢^(l), and r the number of elements in the set 𝒢^(l).

Defining the Battery Model Structure

The model structure can be defined using the ModelStructure class provided in the package that takes model sampling time and battery capacity as the inputs, as given by

Subsequently, the package requires the battery EMF function to be known a priori, which can then be used to calculate the battery overpotentials from the measured battery terminal voltage as given by y_k = V_k–V_EMF(s_k), where y represents the battery overpotential, V the battery terminal voltage, V_EMF the battery EMF, and s the battery SOC. Essentially, the EMF function can be defined using the add_emf_function provided by the ModelStructure class, which requires a Python dictionary consisting of a set of SOC values and the corresponding EMF values as given by

Additionally, the package allows the definition of a temperature-dependent EMF function V_EMF(s,T) using the first-order approximation, that is,

where the functions V_EMF(s,T_ref) and $\frac{\partial V_{\text{EMF}}}{\partial T}(s)$ can be obtained experimentally using the potentiometric method; see [10] for the detailed procedure. Now, the add_emf_function requires a dictionary of four elements to define a temperature-dependent EMF function, namely, (i) a set of SOC values, (ii) the corresponding EMF values, (iii) the value of the reference temperature T_ref at which the EMF values are obtained, and (iv) the values for the $\frac{\partial V_{\text{EMF}}}{\partial T}$ function corresponding to the SOC values given as the first element. Concretely,

The next step for the model identification procedure can be given by specifying the basis functions as given in (7), which are needed to explain the dependence of the model parameters on the p-variables. In this regard, the package allows the user to conveniently specify various functional forms of the p-variables as strings, which are automatically parsed by the package. Note that the characters s, i, d and T are reserved for the SOC s, current u, sign of the current sgn[u] and temperature T, respectively. The following functional forms are currently supported by the package,

• No functional transformation. In this case, the variables are specified as is, e.g., s, and T.

• Inverse. The string format given by 1/□ needs to be specified to invert a variable, e.g., 1/s, and 1/T.

• Logarithm. The logarithm of a variable can be specified using the string log[□], e.g., log[s].

• Low-pass filtering. A variable can also be low-pass filtered according to (5) using the string □[ε₀, ε₁]. Note that such an operation only makes sense for the current direction d, e.g., d[0.01,0.99].

• Exponential. For certain basis functions, e.g., 𝒢_|u| and 𝒢_T, it may be needed to perform multiplication, addition, taking the absolute value, or even raising the variable to a certain power before taking the exponential of the variable. In such cases, the string of the form exp[c*[|a*□+b|]^d] can be used, where a, b, c, d represent suitable numbers. For instance, the string exp[[0.00366*T+1]^-1] can be used to specify the basis function $e^{273.15/(T+273.15)}=e^{{(0.00366T+1)}^{–1}}$, where T represents the temperature in Celsius. Furthermore, if the exponent d equals 0.5, the string can also be specified as exp[c*sqrt[|a*□+b|]], e.g., exp[0.05*sqrt[|i|]].

Accordingly, the package provides the add_basis_functions function to specify the basis functions using a Python list of the strings described above, as can be given by

Modelling batteries with hysteresis. Multiple approaches can be adopted using the package to incorporate the hysteresis phenomenon in the model structure, for instance, by introducing a second input that represents maximum hysteresis overpotential as investigated in [8]. Alternatively, the hysteresis phenomenon can be considered as an extremely sluggish relaxation process compared to other battery chemistries, as done in [14], which can then be modelled by including suitable current-trajectories δ mimicking such sluggish relaxation as part of the basis functions without introducing the second input. Nevertheless, the latter approach has not been thoroughly investigated by the authors and will not be discussed further. Accordingly, the second input can be introduced in the model structure using the add_hysteresis_function function provided by the ModelStructure class, as given by

Furthermore, the package allows additional basis functions that are specific to the second input – the hysteresis overpotential, using another list of identifier strings as a second input to the add_basis_functions function. For instance,

Identifying the Battery Model

Following the definition of the battery model structure, the battery model of a certain model order n and nonlinearity order l can be identified using a predefined list of regression methods; see (1) and (7) for the definition of n and l, respectively. More precisely, a linear regression setting can be constructed to estimate the free model parameters in (8) as given by

where $Y\in {\mathbb{R}}^{N\times 1}$ represents the output vector containing the output measurements ${\{y_k\}}_{k=1}^N$, $\mathrm{\Phi }\in {\mathbb{R}}^{N\times M}$ the regression matrix containing the regression vectors ${\{{\varphi }_k\}}_{k=1}^N$ corresponding to each output measurement, and θ∈ℝ^M the unknown parameter vector. Note that the regression vector ${\varphi }_k:\mathbb{Z}\to {\mathbb{R}}^M$ contains all the candidate model terms which can be generated using the set 𝒢^(l) as given by

where

Note that ⊗ denotes the Kronecker product, $f_k:\mathbb{Z}\to {\mathbb{R}}^{2n+1}$ represents the vector containing the output and the input terms, $g_k:\mathbb{Z}\to {\mathbb{R}}^r$ the vector containing the basis functions, and M = r(2n+1). Furthermore, the parameter vector θ can be expressed in terms of the free model parameters as given by

where {a_ij}, {b_ij} correspond to the free model parameters.

To solve the regression problem in (10), the package allows the usage of either or both of the following methods, namely (i) the least absolute shrinkage and selection operator (LASSO), and (ii) the ridge regression method. Quite briefly, the LASSO and ridge regression methods can be summarised as follows

where ${\widehat{\theta }}_{\text{LASSO}}$ represents the LASSO estimate, and λ₁ the hyperparameter controlling the amount of regularization via the l₁-norm. Note that the LASSO attempts to induce sparsity in the parameter estimate ${\widehat{\theta }}_{\text{LASSO}}$ by setting some of its elements to exactly zero depending on the value of λ₁, thereby removing the corresponding candidate terms from the model equation [15]. Subsequently,

where ${\widehat{\theta }}_{\text{Ridge}}$ represents the final model estimate, and λ₂ the hyperparameter controlling the amount of regularization via the l₂-norm. In this regard, the package currently provides wrappers for the following algorithms,

• lasso.cvxopt: A LASSO algorithm provided along with the CVXOPT package, which uses λ₁ = 1 [16].

• lassocv.sklearn: A LASSO algorithm provided by the Scikit-learn package that uses cross-validation to find the optimal value of λ₁ [17].

• ridgecv.sklearn: A ridge regression algorithm provided by the Scikit-learn package that uses cross-validation to find the optimal value of λ₂ [17].

Now, the battery models can be identified using the function identify_model, which requires the following arguments, namely the (i) identification dataset, (ii) model order n, (iii) nonlinearity order l, and (iv) list of regression methods. Note that the desired algorithms for the regression methods need to be specified in the form of a list with correct sequence using the identifier strings (e.g., lasso.cvxopt). For instance, a battery model with n = 3 and l = 4 can be identified as follows

where the argument identification_dataset needs to be given using a Python dictionary as follows

Additionally, in this example, the argument optimizers specifies cross-validated LASSO and ridge regression methods to be performed sequentially, which ultimately results in (i) variable selection using LASSO, that is, the terms corresponding to zero values in ${\widehat{\theta }}_{\text{LASSO}}$ are automatically removed, and then (ii) re-estimation of the parameters corresponding to the remaining candidate terms; see [9] for the motivation behind augmenting ridge regression method with LASSO.

In the case of temperature-dependent battery model identification, the argument identification_dataset accepts an additional temperature_values key corresponding to the temperature measurements of the battery. Furthermore, as explained in [10], the temperature-dependent model identification may require multiple identification experiments, each resulting in a separate regression problem Y_i = Φ_iθ. In this regard, the package provides two strategies to combine multiple regression problems, namely (i) concatenate, and (ii) interleave; see [10] for more details. An example of temperature-dependent model identification can be given by

where identification_datasets represents a list of Python dictionaries corresponding to each identification experiment, that is, [{…}, {…}, …, {…}].

Simulating the Battery Model

The identified battery model can be simulated for a given current profile to obtain the output voltage using the simulate_model function as given by

1    voltage_output = simulate_model(model,
2        current_profile)

where model represents the identified battery model (an instance of the Model class), and current_profile a Python dictionary containing the current values, initial SOC and initial voltage values depending on the model order, as can be given by

Again, in case of a temperature-dependent battery model, an additional temperature_values key is required in the current_profile dictionary corresponding to the temperature measurements of the battery.

Various Utility Functions

The package provides various utility functions that can enhance user experience during the battery model identification procedure. In the following, such utility functions will be briefly described along with suitable example usages.

Generating identification input signal. The package allows the generation of a current profile that can be used as an input signal during the identification experiments; see [9] for details regarding the current profile design and the associated parameters. More precisely, the function generate_current_profile can be used as follows

Saving/Loading models to/from a file. The identified battery model can be saved to (respectively, loaded from) a file using the function load_model_from_file (respectively, save_model_to_file). For instance, an identified model model can be saved and then loaded from a file as given by

Analysing an experimental dataset. Occasionally, it may be important to check various details of an experimental dataset for a better understanding of the dataset. In this regard, the package provides a utility function analyze_dataset, which provides various details for a dataset, such as experimental time, extracted charge during the experiment, voltage range, SOC range and temperature range. An example usage can be given by

Other utility functions. Other worth-mentioning utility functions can be given by invert_voltage_function and print_model_details. The former function can be used to calculate the initial SOC for a given experimental dataset by inverting the EMF function using the initial voltage and temperature (if applicable) values as inputs. The latter function can print the model terms along with the model order and nonlinearity order of an identified battery model.