An effective battery prognostics method is fundamental for any application in which batteries have a critical role, such as in unmanned aerial vehicles. Given the batteries' variable nature, effectively predicting their End of Discharge or End of Life can become a difficult task. Therefore, developing an accurate and efficient model becomes a key step of this problem. The framework provided by traditional modeling techniques usually leads to inaccurate results, so newer state-of-the-art methodologies are needed to successfully build a model from a dataset. This paper compares the accuracy and time performance of three existing methods: a maximum likelihood optimal Support Vector Machine, a Bayesian Relevance Vector Machine, and a Fuzzy Inference System. Through this research, we aim to implement a real-time battery prognostics system in an Unmanned Aerial Vehicle. The three methods are used to model a Lithium-ion (Li-ion) battery's discharge curve while accounting for the State of Health of the battery for the estimation of voltage. This paper compares the accuracy and time performance of a maximum likelihood optimal Support Vector Machine, a Bayesian Relevance Vector Machine, and a Fuzzy Inference System for the modeling of Lithium-ion (Li-ion) batteries' discharge curve. Moreover, the model accounts for the State of Health of the battery for the estimation of voltage. We show that the three methodologies are valid for the modeling of the discharge curve with similar accuracy values. The Relevance Vector Machine proves to be the most computationally efficient method.
The electrical power system of an unmanned aerial vehicle (UAV) is one of the most critical subsystems in such aircraft. With advanced air mobility (AAM) poised as one of the future paradigms of civil aviation, these systems have been identified as key technologies for the successful integration of AAM ^{[1]}. Electrical Power systems, batteries, and emerging energy dense solutions are highlighted as technologies to be further developed and investigated for both safety as well as redundancy in the In-time Aviation Safety Management System report ^{[2, 3]}. Utilizing emerging artificial intelligence (AI) strategies to perform traditional
An electrical power system is formed by several components, batteries being the most critical ^{[4]}. A failure in a battery can result in catastrophic failure of the entire vehicle. Therefore, it is essential to have reliable prognostics for a battery's end of discharge (EOD) and end of life (EOL). Further, it is also interesting to assess the confidence of the resulting prediction. The first step to solve such a problem is to have a reliable method to model the state of charge – voltage (SOC-V) curve depending on the SOH of the battery.
One of the problems we encounter when working with batteries is their variable nature. Their performance is strongly affected by environmental conditions, as well as its prior use cycles ^{[5]}. Therefore, modeling using traditional methods is a difficult task. One of the possible alternatives is to use data-driven methods, which utilize machine learning algorithms to establish battery degradation models ^{[6, 7]}. This approach allows the battery state estimation without a deep prior knowledge about the internal characteristics of the battery ^{[8]}. Examples of these methods are relevance vector machines (RVM), support vector machines (SVM), and Fuzzy Inference Systems (FIS). Model-based methods build a set of rules that model the behavior of the system ^{[8]}. Their main disadvantage is the need for a deep knowledge about the system and a lengthy amount of time to build the model. These methods often rely on internal parameters, which are inaccessible once a battery has been manufactured ^{[9]}. Therefore, they are not well suited for UAV applications.
Previous work proved an RVM and particle filter (PF) algorithm to be successful in the prediction of the remaining useful life (RUL), both for the state of charge (SOC) and SOH ^{[10-12]}. These methodologies have also been tested for their application in UAVs ^{[13]}. Other research has shown that the combination of RVM and SVM with sample entropy has been provided as a valid framework for battery prognostics ^{[14]}. Regarding fuzzy systems, previous research employed a fuzzy neural network for the estimation of the SOC with lithium iron phosphate batteries ^{[15]}. Work has also been done with respect to the SOH assessment using a FIS to combine the SOH assessment obtained from capacity measurements and internal resistance values ^{[16]}. Related to this matter, another fuzzy granulation methodology was tested to obtain a minimum and maximum boundary in the estimation of SOH by looking at the charging cycles ^{[17]}. Most of the work done with SVM regarding battery prognostics is aimed at the control of batteries ^{[18, 19]} or grid-scale battery storage models ^{[20, 21]}. Publications for this topic include SVM applied to SOH estimation ^{[22]} in combination with PF ^{[23]}, fuzzy entropy ^{[24]}, or incremental capacity analysis ^{[25, 26]}.
In this work we are comparing RVM, a Mamdani FIS and a maximum likelihood optimal SVM for the modeling of the SOC-V curves in batteries. We generate the battery discharge models with each methodology and then compare their accuracy and computational requirements. As mentioned above, this curve depends on the SOH: a battery with a lower SOH will show a faster decaying SOC-V curve. For convenience, we substitute the SOC with the state of discharge (SOD), which we define as
In this section we present the SVM, RVM and FIS methodologies which were evaluated to solve the regression problem for a given dataset of battery discharge cycles. The general statement of the problem is as follows: obtain a function
To calculate a regression with SVR, we map the input data into a high-dimensional feature space by using a kernel function
where
The success of SVR depends on the choice of the loss function, which represents the noise model of the dataset. This means that one must have some
The optimization problem to solve is
subject to
where
Transformed using Lagrange multipliers, the dual-form problem is
where
that depends exclusively on the Lagrange multipliers
Equation (6) can be efficiently solved by an interior point optimization algorithm ^{[29]}. Once the values of
and the offset
where
RVM provides a probabilistic approach to the regression problem ^{[32]}. The method is similar to SVR in that it maps the inputs to a high-dimensional feature space and then computes a linear combination of weights with certain kernel functions
where
For a pair of input and target
where
Fuzzy logic was proposed in 1965 by Zadeh ^{[35]}. It distinguishes from other methodologies in two key concepts. The first is the use of linguistic variables, i.e., variables whose content are words instead of numbers. This concept allows for a granulation of the input and output data. The second concept is the use of if-then linguistic rules ^{[36]}. Overall, the usage of variables close to natural language provides a comprehensible and explainable approach for humans ^{[37]}. Moreover, explainability can become an essential matter in airborne systems ^{[38]}. Other AI algorithms are essentially black boxes with complex decision systems. This can be an issue for certification organizations, since the inability to understand the insights of the decision-making process can reduce the trust that end users have on any system. Therefore, a FIS introduces a significant advantage with respect to other AI methodologies.
The FIS used for this work is a Mamdani-type FIS. In early trials of this work a Takagi-Sugeno algorithm was tested, but its training took longer and its performance was worse than Mamdani's dut to the difficulty choosing initial model parameters and the genetic algorithm (GA) tuning processing used for the FIS, so it was discarded. A Mamdani algorithm is characterized by fuzzy if-then rules with linguistic variables both in the input and output variables. For example, a rule takes the form of
where
The membership functions make some rules fire that yield degrees of membership in the output membership functions. In order to
where
This section compares the battery discharge models obtained with the three algorithms. All of the models consist of 2 inputs (SOD and discharge cycle) and 1 output (battery voltage). The data utilized comes from the well-known Li-ion battery dataset provided by the NASA Ames Research Center ^{[41]}. The entire process is depicted in
Data flowchart of the process.
The parameters for the RVM algorithm were directly obtained from previous work ^{[13]}. The radii used for the radial basis functions (RBF) are
The user specified parameters in the SVM methodology can have a great impact on the results ^{[42]}. The initial parameters for the
The rule base and the initial membership function values of the FIS are manually set based on expert knowledge and previous experience with the topic. The membership functions, as shown in Equation (12), are all triangular. Further, with the aim of simplifying the learning process, all the triangles are isosceles. These triangles are handcrafted based on prior knowledge as an initial approximation and then further optimized using a GA ^{[44, 45]}. We use 3 membership functions for each input variable, and 5 membership functions for the output value. In the GA learning process, we encode the centers and base widths of all the triangular membership functions, giving a total of 22 genes per chromosome. The error function compares and minimizes the absolute value of the difference between the labeled values and the results obtained with the FIS, i.e.,
The NASA Ames Research Center dataset provides 3 different battery discharge cycle datasets named:
Comparison between SVM regression and datapoints. The surface represents the SVM regression. The datapoints used as Support Vectors are highlighted in green. SVM: support vector machin; SOD: state of discharge. The cycle is the number of times the battery has been charged and discharged throughout its lifetime.
Comparison between RVM regression and datapoints. The surface represents the RVM regression. RVM: relevance vector machine; SOD: state of discharge. The cycle is the number of times the battery has been charged and discharged throughout its lifetime.
Comparison between FIS regression and datapoints. The surface represents the FIS regression. FIS: Fuzzy Inference System; SOD: state of discharge. The cycle is the number of times the battery has been charged and discharged throughout its lifetime.
Comparison of results with dataset
^{a} |
|||||
0.0711 | 0.0551 | 0.0466 | 386.2 | 141.7 | |
RVM | 0.0708 | 0.0676 | 10142 | 50.2 | 1.2 |
Fuzzy-GA | 0.0894 | 0.0560 | - | - | 98.3 |
Comparison of results with dataset
Training time (min) | |||||
^{a}See footnotesin |
|||||
0.1136 | 0.0898 | 0.0604 | 749.4 | 137.6 | |
RVM | 0.1307 | 0.1317 | 194961 | 63.0 | 1.2 |
Fuzzy-GA | 0.1068 | 0.0626 | - | - | 100.4 |
Comparison of results with dataset
Training time (min) | |||||
^{a}See footnotesin |
|||||
0.1195 | 0.0662 | 0.0647 | 1137.6 | 150.4 | |
RVM | 0.1107 | 0.0859 | 1051199 | 74.2 | 1.1 |
Fuzzy-GA | 0.1234 | 0.0585 | - | - | 97.7 |
and Mean Absolute Error (MAE) defined as
There is no clear best method in terms of accuracy. The results vary across datasets and they depend on what error metric we consider. Looking at the results obtained from
The values of
In terms of computation time, RVM is the most efficient method by a big difference. Its training takes about a minute to complete, while
This study compares the accuracy and computational cost when modeling a battery discharge cycle with
The results presented show that the 3 methodologies could be used to solve this problem at varying computational costs. The
We did not anticipate these results, specially those obtained with the FIS. The
As previously stated, Matlab 2016a was used to obtain the results from
Further, one could also argue that the GA itself could be further optimized for faster performance or to yield more optimal results.
Another limitation of the results shown comes from the dataset itself. The data was obtained in controlled experiments in a lab. Therefore, this data does not account for inaccuracies, measurement errors or external noise, as with with UAVs subject to external factors. Future work will compare these same methodologies using battery discharge data obtained through hardware analysis using an Orion Jr. 2 Battery Management System as payload onboard a UAV. This will allow us to evaluate the results with new raw datasets coming directly from the target of our research.
AAM is one of the future paradigms of civil aviation and the use of AI offers an opportunity to fundamentally substitute, alter, or augment the traditional pilot functions. Batteries have been identified as critical subsystems in an electric UAV because their variable nature and dependence on prior usage makes their modeling difficult. This paper investigates the application of SVM, RVM, and FIS to model a battery's discharge curve. We show how to apply the three methodologies while accounting for the SOH of the battery in the moment of the discharge. The results prove that the three methodologies provide useful frameworks to solve the problem. SVM shows the best accuracy in most cases, but at a high computational cost. RVM can provide results similar in accuracy at a much lower computational cost. The FIS is able to improve the MAE of the other methodologies, with an intermediate computational cost. In future steps of this research we will implement the outline methodologies on-board an electric UAV and apply them in a real-time manner.
Concept development: Martin J, Ouwerkerk JN, Lamping AP, Cohen K
Manuscript drafting: Martin J
Manuscript edition and review: Ouwerkerk JN, Lamping AP, Cohen K
The dataset used for this work was created by the NASA Ames Research Center and is publicly available at https://c3.ndc.nasa.gov/dashlink/resources/133.
None.
All authors declared that there are no conflicts of interest.
Not applicable.
Not applicable.
© The Author(s) 2022.
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