# Comparison of NARX and Dual Polarization Models for Estimation of the VRLA Battery Charging/Discharging Dynamics in Pulse Cycle

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{4}) battery voltage using SOC and current load signal, for the electric vehicle. In Reference [45] there is reported SOC and SOH estimation based on dynamically driven NARX recurrent networks designed for both state of charge (SOC) and State of Health (SOH) estimation for LiFePO

_{4}and lithium titanate (LTO) batteries. Authors of Reference [114] describe the research on SOC/SOH monitoring of Li-ion batteries with the hybrid method, combining Electrochemical Impedance Spectroscopy (EIS) models and R–ANNs. Another hybrid method is presented in Reference [117] where wavelet-neural-network-based NARX battery model combined with particle filter estimator is used for the State-Of-Energy (SOE) estimation of the LiFePO

_{4}battery. An interesting approach is discussed in Reference [118] where dual neural network fusion model is used to estimate the relation between Open Circuit Voltage (OCV) and SOC.

## 2. Test Stand Description

_{2}; negative electrode material: Pb.

_{term}—the temperature at the terminals, T

_{body}—body temperature, T

_{ambient}—ambient temperature, see Figure 1a,b). The test stand also featured a NI 9263 analog output module for control over current load generating unit, which allowed for the generation of pulse load profile. Finally, a NI 9401 digital TTL input/output module was used for relay switching between charge and load circuits. Total electric power of a single load circuit reached around 1 kWe. Use of two custom load generation units allowed for the increase of total electric power to about 2 kWe. Currently, use of up to six load generation units is possible, which can operate together or independently creating independent measurement tracks. The presented test stand is currently being expanded to six independent, parallel measurement tracks for electrochemical batteries and ultracapacitors, which can be tested independently or in one of two hybrid configurations: Parallel without universal bidirectional DC-DC converter such as described in Reference [119] (without ability to control the load distribution between battery and ultracapacitor) and with two-way, universal bidirectional DC/DC converters (possible control overload distribution).

## 3. Battery Parameters Estimation Based on NARX Model and DP Physical Model

#### 3.1. Dual Polarization (DP) Model Description and Parameters Identification

_{OCV}(SOC) (U

_{OCV}, measured in the work cycle during relaxation stage for the given SOC value when no charging/discharging occurs or is determined during identification); R

_{0}(SOC)—internal resistance (ohmic resistance); activation polarization resistance R

_{1}(SOC); R

_{2}(SOC)—density polarization resistance. The diagram also features capacitances C

_{1}(SOC) and C

_{2}(SOC), which describe the transitional states during supplying power to the battery or from the battery as well as the activation and density polarizations. Currents I

_{b}

_{1}and I

_{b}

_{2}are the electric currents flowing through C

_{1}and C

_{2}. U

_{1}and U

_{2}are the voltage drops occurring in the first and second loop.

_{bat}to nominal capacity Q

_{n}described as:

_{0}is the initial state of charge for time t = t

_{0}, η

_{c}is the columbic efficiency, I

_{b}is the battery charging (–)/discharging (+) current at time t.

_{min}and EMF

_{max}values in Equation (3) can be determined on the basis of experiments from the following equation:

_{0}is the internal resistance of the battery.

_{ch}= I

_{dch}= 1 C the value of R

_{0}can be calculated as follows:

_{ch}is the charging voltage and U

_{dch}is the discharging voltage.

#### 3.1.1. Parameters Identification of DP Model

_{0}connected in series and electromotive force source, a total of six parameters are present (R

_{1}, R

_{2}, C

_{1}, C

_{2}, R

_{0}, U

_{OCV}) that require identification. In the DP model project, the differential equations are generated automatically by Matlab Simscape software (a fragment of the project is presented in Figure 4) or they can be designated using traditional methods.

_{1}, R

_{2}, C

_{1}, C

_{2}, R

_{0}, U

_{OCV}) the calculated voltage will differ from the voltage acquired during measurements.

_{1}, R

_{2}, C

_{1}, C

_{2}, R

_{0}, U

_{OCV}) so that the calculated voltage values differ as little as possible. The measure of identification error is the value of integral of squared error e

^{2}= (y* − y(Θ, x, u))

^{2}between the value of calculated function y(Θ, x, u) and obtained from measurements y*. The identification process is conducted with use of “nonlinear data-fitting” procedure, which minimizes the value of integral J, presented in dependency (6), of the squared error e

^{2}by iterative adaptation of ΔΘ value of parameters (R

_{1}, R

_{2}, C

_{1}, C

_{2}, R

_{0}, U

_{OCV}), ΔΘ = [ΔR

_{1}, ΔR

_{2}, ΔC

_{1}, ΔC

_{2}, ΔR

_{0}, ΔU

_{OCV}] with use of $\nabla J$ gradient and hessian H.

_{1}, R

_{2}, C

_{1}, C

_{2}, R

_{0}, U

_{OCV}] the Levenberg–Marquardt error back propagation method was used, which uses properly weighted approximations of gradient of the minimized function $\nabla J$ and its hessian H with use of Jacobian calculated at end of each iteration based on the obtained change of error Δe, related to vector of change of parameters ΔΘ.

^{2}is obtained, meaning a minimum was reached. Stationarity is reached when the change of the relative value |ΔJ/J| < ε. In the research the value assumed was ε = 1 × 10

^{9}.

_{0}= [R

_{10}, R

_{20}, C

_{10}, C

_{20}, R

_{0}, U

_{OCV}

_{0}], as in gradient methods a risk of founding a suboptimal local solution instead of the optimal solution exists. In the conducted research the assumed values were equal to order of magnitude of typical values for the given battery type Θ

_{0}= [R

_{10}= 1 × 10

^{1}, R

_{20}= 1 × 10

^{−2}, C

_{10}= 1 × 10

^{1}, C

_{20}= 1.0, R

_{0}= 1 × 10

^{−2}, U

_{OCV}

_{0}= 6.0].

_{1}(SOC), R

_{2}(SOC), C

_{1}(SOC), C

_{2}(SOC), R

_{0}(SOC) and U

_{OCV}(SOC) presented in dependency (9).

#### 3.2. NARX Model Description

_{term}(t), current load I

_{load}(t), ambient temperature T

_{ambient}(t), the battery body temperature T

_{body}(t) and the temperature at battery terminals T

_{term}(t).

_{termNN}(k) is the NARX estimated output (voltage) signal at step k, n

_{u}is the number of delayed steps of output (poles) signal, n

_{I}, n

_{a}, n

_{b}, n

_{t}are the number of delayed steps of input signals (current load, ambient temperature, battery body temperature, temperature at battery terminals, respectively).

#### 3.2.1. NARX-ANN Architecture

- U
_{termNN}(k − 1), ..., U_{termNN}(k − n_{u}): Previous values of outputs on which the actual output depends. These inputs are generated in the recurrent loop with tapped delay lines. - I
_{load}(k), ..., I_{load}(k − n_{I}): Previous and delayed values of the current load signal. - T
_{ambient}(k), ..., T_{ambient}(k − n_{a}): Previous, delayed values of the ambient temperature. - T
_{body}(k), ..., T_{body}(k − n_{b}): Previous, delayed values of the battery body temperature. - T
_{term}(k), ..., T_{term}(k − n_{t}): Previous, delayed temperature at battery terminals.

_{u}= 1, n

_{I}= 1, n

_{a}= 1, n

_{b}= 1, n

_{t}= 1.

_{i}is the i-th input to the network (with delayed inputs), ${b}^{(1)}{}_{(n)}$ is the threshold offset of the n-th neuron in the 1st layer. The second term of Equation (12) describes the recurrent feedback loop with tapped delay line from output to input, in case when n

_{u}= 1.

#### 3.2.2. NARX-ANN Training

_{term}(k), y

_{NN}(k) = U

_{termNN}(k), k is the sample number.

_{u}= 1, n

_{I}= n

_{a}= n

_{b}= n

_{t}= 2 shall give satisfactory estimation results. In each NARX-ANN model there was one nonlinear hidden layer, and one linear output layer (see Figure 6).

^{−6}.

- Training dataset: Subset used updating the NARX-ANN weights and biases during network training.
- Validation dataset: Subset used to validate the NARX-ANN in every iteration during training. The performance function value on the validation set was monitored during the training. The rise of the error on the validation subset was an indication that the network begins to overfit the data. When the performance function for validation error increased for a specified number of iterations, the training stopped, and the weights and biases at the minimum of the validation error were chosen for the network.
- Testing dataset: Subset not used in the training, used to compare different models during evaluation.

## 4. Validation of Results Acquired from Measurements, DP and NARX Models

^{−3}(DP model for SOC = 0.8) and 1 × 10

^{−4}(NARX model for SOC = 0.2). The NRSME value for NARX model in SOC range from 0.2 to 0.8 is above 0.93 (the lowest for SOC = 0.5, NRSE

_{NARX}= 0.9371), which indicates a well-identified model.

_{DP}is above 0.77 (with the lowest value NRMSE

_{DP}= 0.7737 for SOC = 0.8), the model was identified correctly. The lowest value of NRMSE

_{DP}for SOC = 0.8 is mainly a result of the battery temporarily exceeding the maximum voltage of 6.8 V (the hyperbolic character of voltage increase at battery terminals).

## 5. Conclusions

- Increase in the accuracy of identified parameters with the use of models with additional RC loops (three and four RC loop pairs), which will greatly increase accuracy particularly for SOC >0.8 and SOC <0.2. Another approach, instead of increasing the number of RC loop pairs and also a compromise between the number of estimated parameters and the accuracy, is to use constant phase elements (CPE) [135,136,137,138].

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Acronyms | |

AGM | Absorbent Glass Mat |

BMS | Battery management system |

CPE | Constant Phase Elements |

D | Tapped Delay Line |

DP | Dual polarization |

HPPC | Hybrid Pulse Power Characterization |

MSE | Mean Squared Error [–] |

NARX | Nonlinear AutoRegressive eXogenous |

NRMSE | Normalized Root Mean Square Error |

OCV | Open Circuit Voltage |

R | ANN–Recurrent Artificial Neural Network |

SOA | Safe Operating Area |

SOC | State of Charge |

SOE | State of Energy |

SOH | State of Health |

VRLA | Valve Regulated Lead Acid |

Greek symbols, subscripts, superscripts and abbreviations | |

${b}^{(m)}{}_{(n)}$ | The threshold offset of the n-th neuron in layer m [–] |

C_{1}, C_{2} | Capacitances [F] |

e | Error |

EMF | Electromotive force [V] |

H | Hessian |

I_{load} | Loading current from control unit [A] |

I_{ch}/I_{dch} | Charging/discharging current [A] |

I_{b1}, I_{b2} | Currents flowing through C_{1} and C_{2} [A] |

J | Jacobian |

k | Sample number [–] |

N | The number of data samples [–] |

n | The number of neurons in nonlinear layer [–] |

n_{I}, n_{a}, n_{b}, n_{t} | Number of delayed steps of inputs [–] |

n_{u} | Number of delayed steps of output (poles) [–] |

$nW$ | Number of neural network weights [–] |

p | The number of neuron inputs in nonlinear hidden layer [–] |

R_{0} | Ohmic resistance [Ω] |

R_{1} | Activation polarization resistance [Ω] |

R_{2} | Density polarization resistance [Ω] |

SOC_{0} | The initial State of Charge for time t = t_{0} [–] |

Q_{bat} | Effective capacity [Ah] |

Q_{n} | Nominal capacity [Ah] |

T_{ambient} | Ambient temperature [K] |

T_{body} | Body temperature [K] |

T_{term} | Temperature at terminals [K] |

U_{1}, U_{2} | The voltage drops occurring in the first and second loop [V] |

U_{ch} | The charging voltage [V] |

U_{dch} | The discharging voltage [V] |

U_{OCV} | Open Circuit Voltage [V] |

U_{termNN} | NARX model output at step k [V] |

$W={[{w}_{1},\dots ,{w}_{nW}]}^{T}$ | The weight vector [–] |

${w}^{(m)}{}_{ij}$ | The weight of the i-th input to j-th neuron in layer m [–] |

$\nabla J$ | Gradient |

$\nabla {E}_{NN}({W}_{(i)})$ | The gradient vector |

η_{c} | The columbic efficiency [–] |

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**Figure 2.**Plots of the: (

**a**) Current in pulse cycle; (

**b**) battery body temperature in pulse cycle; (

**c**) temperature at the battery terminals in pulse cycle; (

**d**) ambient temperature.

**Figure 5.**Graphs of the identified parameters of DP model: (

**a**) U

_{OCV}; (

**b**) R

_{0}; (

**c**); R

_{1}; (

**d**) R

_{2}; (

**e**) C

_{1}; (

**f**) C

_{2}for State of Charge (SOC) ranging from 0.2 to 0.8 in pulse work cycle.

**Figure 6.**The parallel (recurrent) structure Nonlinear AutoRegressive eXogenous Arificial Neural Network (NARX-ANN) model with two neurons in hidden layer, and one neuron in the output layer (D-Tapped Delay Line).

**Figure 7.**The performance of NARX-ANN models during training, respectively for: (

**a**) SOC = 0.2; (

**b**) SOC = 0.3; (

**c**) SOC = 0.4; (

**d**) SOC = 0.5; (

**e**) SOC = 0.6; (

**f**) SOC = 0.7 and (

**g**) SOC = 0.8 in pulse work cycle.

**Figure 8.**The linear regression plots of NARX-ANN model for training, validation and testing subsets, (

**a**) SOC = 0.2; (

**b**) SOC = 0.3; (

**c**) SOC = 0.4; (

**d**) SOC = 0.5; (

**e**) SOC = 0.6; (

**f**) SOC = 0.7 and (

**g**) SOC = 0.8 in pulse work cycle.

**Figure 9.**Error histograms of NARX-ANN models after training, respectively for: (

**a**) SOC = 0.2; (

**b**) SOC = 0.3; (

**c**) SOC = 0.4; (

**d**) SOC = 0.5; (

**e**) SOC = 0.6; (

**f**) SOC = 0.7 and (

**g**) SOC = 0.8 in pulse work cycle.

**Figure 10.**Graphs of voltage at battery terminals. Comparison between NARX and DP models and measurement results, with presented error values, respectively for: (

**a**) SOC = 0.2; (

**b**) SOC = 0.3; (

**c**) SOC = 0.4; (

**d**) SOC = 0.5; (

**e**) SOC = 0.6; (

**f**) SOC = 0.7 and (

**g**) SOC = 0.8 in pulse work cycle.

Coefficient Name | R_{0} | R_{1} | R_{2} | C_{1} | C_{2} | U_{OCV} |
---|---|---|---|---|---|---|

A | −42 | −3.9 × 10^{−5} | 110 | 6 × 10^{8} | −8.9 × 10^{4} | 14 |

B | 140 | 3.1 × 10^{5} | −320 | −1.4 × 10^{9} | 2.3 × 10^{5} | −27 |

C | −200 | – | 390 | 1 × 10^{9} | −2.2 × 10^{5} | 17 |

D | 140 | – | −250 | 3.6 × 10^{8} | 1.1 × 10^{5} | −3.3 |

E | −57 | – | 92 | 1.8 × 10^{8} | −2.3 × 10^{4} | 5.8 |

F | 11 | – | −18 | –3.8 × 10^{7} | 2 × 10^{3} | – |

G | −0.85 | – | 1.4 | 3.5 × 10^{6} | – | – |

State of Charge (SOC) Data Set Value | NARX Model | DP Model | ||
---|---|---|---|---|

MSE_{NARX} | NRMSE_{NARX} | MSE_{DP} | NRMSE_{DP} | |

0.2 | 9.6263 × 10^{−5} | 0.9453 | 0.2009 × 10^{−3} | 0.9210 |

0.3 | 4.9337 × 10^{−5} | 0.9546 | 0.2673 × 10^{−3} | 0.8986 |

0.4 | 3.3201 × 10^{−5} | 0.9427 | 0.0695 × 10^{−5} | 0.9192 |

0.5 | 2.9521 × 10^{−5} | 0.9371 | 0.0542 × 10^{−3} | 0.9147 |

0.6 | 2.4470 × 10^{−5} | 0.9376 | 0.1323 × 10^{−3} | 0.8549 |

0.7 | 2.1085 × 10^{−5} | 0.9407 | 0.0528 × 10^{−3} | 0.9062 |

0.8 | 4.2477 × 10^{−5} | 0.9489 | 0.9224 × 10^{−3} | 0.7737 |

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Chmielewski, A.; Możaryn, J.; Piórkowski, P.; Bogdziński, K. Comparison of NARX and Dual Polarization Models for Estimation of the VRLA Battery Charging/Discharging Dynamics in Pulse Cycle. *Energies* **2018**, *11*, 3160.
https://doi.org/10.3390/en11113160

**AMA Style**

Chmielewski A, Możaryn J, Piórkowski P, Bogdziński K. Comparison of NARX and Dual Polarization Models for Estimation of the VRLA Battery Charging/Discharging Dynamics in Pulse Cycle. *Energies*. 2018; 11(11):3160.
https://doi.org/10.3390/en11113160

**Chicago/Turabian Style**

Chmielewski, Adrian, Jakub Możaryn, Piotr Piórkowski, and Krzysztof Bogdziński. 2018. "Comparison of NARX and Dual Polarization Models for Estimation of the VRLA Battery Charging/Discharging Dynamics in Pulse Cycle" *Energies* 11, no. 11: 3160.
https://doi.org/10.3390/en11113160