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Battery management systems (BMS) must estimate the state-of-charge (SOC) of the battery accurately to prolong its lifetime and ensure a reliable operation. Since batteries have a wide range of applications, the SOC estimation requirements and methods vary from an application to another. This paper compares two SOC estimation methods, namely extended Kalman filters (EKF) and artificial neural networks (ANN). EKF is a nonlinear optimal estimator that is used to estimate the inner state of a nonlinear dynamic system using a state-space model. On the other hand, ANN is a mathematical model that consists of interconnected artificial neurons inspired by biological neural networks and is used to predict the output of a dynamic system based on some historical data of that system. A pulse-discharge test was performed on a commercial lithium-ion (Li-ion) battery cell in order to collect data to evaluate those methods. Results are presented and compared.

Cost, cycle-life, reliability, and safety are major concerns in all batteries. Due to their high cost in general, batteries must be carefully charged and discharged in order to prolong their cycle-life and thus minimize their maintenance and replacement costs. The battery must also be operated within its design limits to ensure a reliable and safe operation. That is, overcharging batteries produces excessive heat that could damage the battery, shorten its lifetime, and further, result in a fire hazard, where over-discharging batteries could result in an irreversible damage or reduced service-time [

Unlike a fuel gauge, the SOC is an immeasurable quantity that is rather estimated. Nonetheless, there are several methods that can be used to estimate the SOC ranging from simple to sophisticated methods. Those methods can be classified into two types: direct and indirect. Some examples of direct methods include current-integration or “coulomb counting”, which is one of the most common methods used to estimate the SOC. Although current-integration method can be implemented in real-time and can be very accurate, it has one major limitation due to its open-loop nature. That is, with a very small error in the current measurement, the estimated SOC can easily diverge from the true SOC value after few cycles due to accumulated error in the current measurement unless a reset mechanism is periodically used [

On the other hand, there are several indirect methods that are used to estimate the SOC. Those methods can be very accurate and reliable in general. Among those methods are extended Kalman filter (EKF) and artificial neural network (ANN). EKF methods employ advanced battery cell models and require a relatively high computation capability. In contrast, ANN methods don’t rely on any electrical, physical, chemical, or thermal model and they require less computation capability compared to EKF methods. However, ANN methods require huge data to train the network in order to calculate the weight functions for the network [

To evaluate and compare the capabilities of the presented methods, some battery cell data were gathered from a commercial 1.1 Ah, 3.6 V Li-ion cell. To ensure that the gathered data represent the steady-state and transient battery responses, a pulse-discharge test was performed over the entire capacity range of the cell (from 100% to 0% SOC). In this test, the reference (true) SOC value is obtained by integrating the current with time (coulomb- counting). The battery was fully charged using constant current/constant voltage (CC/CV) profile, then it was discharged using a 1-Amp pulsed current. The pulse and rest durations were set at 6 and 4 minutes, respectively. Since the test comprises only one cycle, the SOC calculated using coulomb-counting is believed to be 100% accurate and thus it will be used later as a reference. The test was performed at room temperature. Since the SOC is strongly dependent on temperature, the test must be repeated at all possible temperatures to evaluate the methods at any temperature. However, the temperature effect is beyond the scope of this paper and thus the methods were evaluated only at room temperature. Also, the calculated SOC in this test and the estimated SOC in the next two sections are found assuming a constant battery total capacity. In order to further generalize the analysis, the test must be performed at different battery conditions, i.e. different battery ageing levels, which also was not considered in this paper. The voltage and SOC are shown in

Kalman filter (KF) is an established technique that is used as an optimal estimator to estimate the inner state of a dynamic linear system. For nonlinear systems, a linearized version called extended Kalman filter or EKF is rather used to estimate a system’s state. Basically, KF or EKF is a recursive set of equations that consists of two

steps: prediction and correction. In the prediction step, the state of the system is estimated using the previous available measurement, while in the correction step, the estimated state is updated using the current measurement.

If the current is assumed to be positive during charging and negative during discharging, then, the equation of the state-of-charge,

where

which is defined as the number of ampere-hours obtained from a fully charged battery at a

To estimate the SOC, an EKF algorithm is constructed as given in the following equations [

・ Firstly, the model is expressed in state-space representation

where

(process uncertainty) with a covariance matrix

・ Secondly, the filter is initialized

where

・ Thirdly, for

1) Compute the following partial derivative matrices:

2) Perform the time update of the state estimate and estimation error covariance:

3) Compute the following partial derivatives:

4) Measurement update:

To accurately estimate the SOC using EKF (the battery is a nonlinear time-variant dynamic system), an accurate battery model must be available. Some existing battery models are briefly presented [

Shepherd Model:

Unnewehr Model:

Nernst Model:

In equations (3)-(5):

Linear Model: Another battery model is the linear equivalent-circuit model shown in

where

RC Model: The last battery model discussed in this paper is the RC equivalent-circuit model. This model consists of an RC network as well as an internal resistance as shown in

where, again,

In (7), t is the transient time when the cell turns from the charge or discharge state to the idle state and vice versa. For both the linear and RC models, the relationship between the SOC and OCV is found by charging and discharging the cell at a very low current while monitoring the SOC. The OCV-SOC pairs are stored in a lookup table. The exact value of the models’ parameters in equations (3)-(7) that give the least square error can be found using least square estimation [

ANN is a subfield of artificial intelligence. It is a mathematical tool that is inspired by the way the human brain processes information [

ANNs are highly stable and robust due to their massive parallel structure. ANNs consist of inputs and outputs and are made of neurons interconnected with each other. The weights in

The weights of an ANN are calculated in the training phase by minimizing the loss function (usually a quadratic function of the output error). One common technique that is widely used in training ANNs is the back-propagation, which is a supervised learning method that is commonly used in the training phase to calcu-

late the weights of a neural network. Back-propagation uses a steepest-descent technique based on the computation of the gradient of the loss function with respect to the network parameters [

To evaluate the performance of ANNs in estimating the SOC, two common network architectures were selected: the first is nonlinear input-output (NIO) feed-forward network, and the second is nonlinear autoregressive with exogenous input (NARX) feed-back network. For both networks, Levenberg-Marquardt back-propagation algorithm was used in training.

NIO feed-forward network predicts one time series given the current and past values of an input time series as shown in

The output equation of this network is given in (8).

NARX feed-back network, on the other hand, predicts one time series given the current and past values of an input time series as well as the past and current values of another time series, called external or exogenous time series (output). A block diagram of this network is shown on

The output equation of this network is given in (9).

In (8) and (9):

For both networks, the delay parameter indicates the delay in the input/output time series and it can be selected by trial and error until the desired accuracy is obtained. The number of neurons in the hidden layer was set at 10. The value of the delay parameter and the number of neurons in the hidden layer were purposefully varied in order to find the effect of the selected values on the network performance.

The results of estimating the SOC for both the NIO and NARX models are shown in

In order to obtain practical results, a zero-mean process noise with an absolute peak value of 10 mA and a zero-mean measurement noise with an absolute peak value of 10 mV were added to the current and voltage readings. The results in

It is obvious in

newehr and Nernst models because they are more advanced since they use an OCV-SOC lookup table. The RC model outperformed all the other models because it handles both the internal voltage drop and transient (capacitance) voltage of the battery.

In fact, one advantage of EKFs is that they can still be accurate even if the initial guess was incorrect. To verify that, the initial SOC was purposefully changed from 100% (true value) to 80% (wrong value), and it was noticed that every time the algorithm converged to the true value after a relatively short time (within 5 minutes). Another advantage of EKFs is that they can also provide accurate performance even when the process and measurement have noise. To verify this, a white noise (Gaussian) was added to both the current and voltage. All the EKF algorithms, regardless what battery model they used, were able to achieve almost the same results before the process and measurement were contaminated.

In practice, however, the process and measurement noise is not necessarily white. If so, conventional EKFs will not perform well. Thus, EKFs can be a good option only and only if an accurate battery and noise models are used; otherwise, EKFs will not be accurate and will be prone to stability issues.

On the other hand, the presented ANNs showed different performances. NARX networks obviously outperformed NIO networks due to their feedback loop. In the presented NARX network, the feedback loop was used in the training phase to calculate the weight functions. The feedback resulted in a more accurate performance as can be seen by comparing the NARX and NIO models. However, although the calculated weight functions for both the NIO and NARX models are optimal in the sense that they gave the best possible performance, the weight functions must be able to adapt to changes in the ambient conditions or battery parameters.

One remark on the presented ANNs is the relationship between the delay parameter

Model | RMSE | Regression |
---|---|---|

Shepherd | 0.1739 | 0.95478 |

Unnewehr | 0.1178 | 0.95155 |

Nernst | 0.0672 | 0.98321 |

Linear | 0.0284 | 0.99530 |

RC | 0.0202 | 0.99788 |

NIO (d = 2) | 0.0579 | 0.97694 |

NIO (d = 4) | 0.0433 | 0.98742 |

NIO (d = 6) | 0.0379 | 0.99176 |

NIO (d = 8) | 0.0421 | 0.98995 |

NIO (d = 10) | 0.5889 | 0.98860 |

NARX (d = 2) | 0.0074 | 0.99990 |

NARX (d = 4) | 0.0128 | 0.99961 |

NARX (d = 6) | 0.0193 | 0.99920 |

NARX (d = 8) | 0.0253 | 0.99867 |

NARX (d = 10) | 0.0311 | 0.99807 |

choice when no output data is available. It was shown that as

Another remark on the ANNs models is that there was almost no relationship between the network accuracy and the number of neurons in the hidden layer. It was noticed that when the number of neurons in the hidden layer is less than 5, the accuracy is slightly lower than when the number of neurons is above 5. However, the performance of the network didn’t change at all when the number of neurons in the hidden layer was changed randomly between 5 and 100 for both NIO and NARX networks.

To quantitatively compare the presented methods, the root mean square error (RMSE) and regression were calculated and summarized in

The regression in

where

Two advanced SOC estimation methods were analyzed, compared and evaluated. The two methods, EKFs and ANNs, can perform well under certain conditions. That is, EKFs can perform well if an accurate battery and noise models are used, where ANNs can perform well if they were trained with an artificial experience (training) that is very close or identical to the real experience. However, even with ideal battery model or training, the actual EKF or ANN SOC estimator will not be ideal due to the variation between battery cells in their internal parameters even if they were from the same manufacturer. Other factors that must be taken into account when designing an SOC estimator are the dependence of the battery on temperature and the change in the battery parameters (i.e. capacity, internal resistance, etc.) through the battery lifetime. Thus, adaptability is a key issue for highly reliable SOC estimators through the battery lifetime.

This research was supported in part by the United Arab Emirates University under startup research grant # G00001287.