Machine Learning Estimation of Battery Efficiency and Related Key Performance Indicators in Smart Energy Systems

Citation: Luque, J.; Tepe, B.; Larios, D.; León, C.; Hesse, H. Machine Learning Estimation of Battery Efficiency and Related Key Performance Indicators in Smart Energy Systems. Energies 2023,16, 5548. https://doi.org/10.3390/ en16145548 Academic Editor: Quanqing Yu Received: 21 June 2023 Revised: 16 July 2023 Accepted: 20 July 2023 Published: 22 July 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). energies Article Machine Learning Estimation of Battery Efficiency and Related Key Performance Indicators in Smart Energy Systems Joaquín Luque 1,*, Benedikt Tepe 2, Diego Larios 1, Carlos León1and Holger Hesse 3 1Department of Electronic Technology, University of Seville, 41004 Seville, Spain; [email protected] (D.L.); [email protected] (C.L.) 2Chair of Electrical Energy Storage Technology, Department of Energy and Process Engineering, TUM School of Engineering and Design, Technical University of Munich (TUM), 80333 Munich, Germany; [email protected] 3Department of Mechanical Engineering, Institute for Energy and Propulsion Technologies (IEAT), Kempten University of Applied Sciences, 87435 Kempten, Germany; holger[email protected] *Correspondence: [email protected] Abstract: Battery systems are extensively used in smart energy systems in many different applications, such as Frequency Containment Reserve or Self-Consumption Increase. The behavior of a battery in a particular operation scenario is usually summarized using different key performance indicators (KPIs). Some of these indicators such as efficiency indicate how much of the total electric power supplied to the battery is actually used. Other indicators, such as the number of charging-discharging cycles or the number of charging-discharging swaps, are of relevance for deriving the aging and degradation of a battery system. Obtaining these indicators is very time-demanding: either a set of lab experiments is run, or the battery system is simulated using a battery simulation model. This work instead proposes a machine learning (ML) estimation of battery performance indicators derived from time series input data. For this purpose, a random forest regressor has been trained using the real data of electricity grid frequency evolution, household power demand, and photovoltaic power generation. The results obtained in the research show that the required KPIs can be estimated rapidly with an average relative error of less than 10%. The article demonstrates that the machine learning approach is a suitable alternative to obtain a very fast rough approximation of the expected behavior of a battery system and can be scaled and adapted well for estimation queries of entire fleets of battery systems. Keywords: battery energy storage system; smart energy systems; machine learning; battery operation KPI; operation strategy 1. Introduction The number and applications of battery-powered devices have significantly increased over the last decades [ 1 ]. Mobile phones, laptops, electronic watches, portable radios, children’s toys, and many others are a few examples of daily-life battery-operated appliances. Additionally, battery systems are playing an increasingly prevalent role in the support of smart energy systems [ 2 , 3 ] by means of providing energy storage functionalities to make the electric grids more flexible, clean, and efficient. Smart energy systems are the application of Information and Communication Technologies (ICT), computer-based algorithms, and/or artificial intelligence (AI) methodologies to design, operate, and maintain a sustainable, efficient, and reliable energy supply infrastructure [ 4 ]. As modern smart energy systems incorporate more intelligent devices with their own computing and connectivity capabilities, they build up an Internet of Energy [ 5 ] where the centralized control is complemented with, and in certain cases even substituted by, a distributed control over cloud computing architectures [6]. Energies 2023,16, 5548. https://doi.org/10.3390/en16145548 https://www.mdpi.com/journal/energies Energies 2023,16, 5548 2 of 18 Frequency Containment Reserve (FCR) is one of these applications of battery storage systems (BSSs) to power networks. According to the European Union Network Code on System Operation [ 7 ], FCR denotes “the active power reserves available to contain system frequency after the occurrence of an imbalance”. These reserves are used whenever a deviation of nominal frequency occurs, either absorbing or supplying the imbalanced amount of energy. The main target of FCR is to provide a fast response (in seconds) to changes in the network frequency to contain and limit its deviation from the nominal value (50 Hz in Europe), while other energy storage or sources with more capacity but larger time responses can begin to operate and restore the normal operation state. Self-Consumption Increase (SCI) in the private sector is considered, besides FCR, one of the most prominent applications of BSSs to power systems [ 8 , 9 ]. In a typical SCI application, a household owner with a private photovoltaic (PV) installation tries to accommodate its power demand to the solar-dependent generation profile. When PV-generated power exceeds the demanded power, the surplus energy is stored in a battery system. On the other hand, when the electricity demand surpasses solar-dependent generation, the BSS supplies the lacking power. In this article, we focus on FCR and SCI only, although batteries can be employed in smart energy systems for a variety of other applications such as peak shaving [ 10 ], electric service reliability [ 11 ], spot market trading [ 12 ], and many others, as are summarized in [13]. The behavior and performance of a BSS in a certain application can be analyzed using different approaches. In the first place, a set of physical experiments on laboratorycontrolled or in-field conditions can be undertaken [ 14 ]. However, these tests require physical (and usually expensive) equipment, they must be run in real time (which is very time-consuming), and they lack enough flexibility to investigate many different scenarios and assess rare events. To overcome these problems, simulation-based analysis is a common practice for BSS assessment. For running these simulations, the most precise and detailed models consider the physicochemical equations ruling the BSS behavior [ 15 ]. Besides that, thermal, mechanical, and electrical models complement the theoretical description of all the subsystems that make up a complex BSS [16]. Physicochemical models have shown themselves to be very accurate, offer a deep and detailed insight into internal processes, and are very flexible in testing different experimental conditions. They are also about one order of magnitude faster than physical tests, which means that the most optimized models can be run in about one-tenth of the real-time duration of the simulated experiment [ 17 ]. However, although these results are a remarkable improvement over the physical test, they require a well-parametrized and battery-chemistry-specific model. Additionally, they require an enormous amount of computing resources, making them useless for long-term experiments. For instance, a one-year-length analysis would still require about one month to be simulated using this type of model. An extensively used alternative to physicochemical simulations is based on considering the battery cells as black box electric circuits which can be described using an equivalent electric model [ 18 ]. While they still offer very good estimations of the BSS behavior, the corresponding simulations can be run about two to four orders of magnitude faster than the real test. Then, for instance, a one-year-length experiment could be simulated in about one day. Therefore, due to this reduced computation time, many battery operation conditions can be tested and compared. Some examples of simulators based on equivalent electric models are summarized in [19]. Although electric circuit models are certainly suitable for many analyses, they already have severe limitations when many different BSSs and/or operation conditions have to be considered. To cope with these cases, several machine learning (ML) approaches have been proposed [ 20 – 22 ]. They use the inputs and outputs of previous experiments, either real or simulated, to build a data-driven ML model that, once completely trained, can Energies 2023,16, 5548 3 of 18 obtain remarkable estimations of the outputs corresponding to novel inputs. Training an ML model is usually a time-demanding process in human and computer resources which mainly depends on the complexity of the model and the size of the dataset available. On the other hand, the prediction step can be performed with the highest computing efficacy (about five to eight orders of magnitude faster than real tests). The comparison of different analysis methods of BSSs is sketched in Figure 1. Energies 2023, 16, x FOR PEER REVIEW 3 of 18 an ML model is usually a time-demanding process in human and computer resources which mainly depends on the complexity of the model and the size of the dataset available. On the other hand, the prediction step can be performed with the highest computing efficacy (about five to eight orders of magnitude faster than real tests). The comparison of different analysis methods of BSSs is sketched in Figure 1. Figure 1. Comparison of different analysis methods of battery cell and system behavior. The impact that a certain input (for instance, the frequency signal for FCR or the household load and PV generation for SCI) has on a BSS can be described in terms of a set of output signals. They typically measure the evolution of some physical magnitudes, such as the storage power provided for (drawn from) the BSS, its state of charge (SOC), or its state of health (SOH). It is a common practice to summarize this impact, that is, the set of output signals, using a handful of KPIs [23] to better analyze and compare the storage load profiles. Using KPIs is a way to summarize the behavior of a BSS in a handful of values. The KPIs selected in this research either emphasize the performance of the BSS, such as the battery efficiency (η), or they are precursory indicators of the expected battery degradation, such as the number of Full Equivalent Cycles (FEC) and the number of charging-discharging swaps (𝑛). This research is focused on the development of an ML methodology to estimate the impact, as it is summarized by a set of KPIs, of a given input signal in a BSS when it is used for FCR or SCI applications. This methodology does not require any knowledge about the physicochemical process inside the BSS, nor even an equivalent electric model. The ML approach is based on an agnostic model which is trained using datasets with the results of previous experiments. The main contribution of the research is to show the capabilities of the ML techniques to describe, in a very fast way, the behavior of a BSS under different operation conditions. The use of ML methods in this research has been proposed mainly for three reasons: they do not require an explicit set of equations to describe the BSS (agnostic model); they can be run in much less time (see Figure 1); and, finally, because ML is a well-established discipline that has proven successful in many different applications. The paper is organized as follows: in Section 2, the datasets employed and the proposed methodology are described; the main results are explained in Section 3, outlining the estimation errors of the different KPIs; Section 4 discusses these results and takes into consideration their dependence on design parameters such as the length of the input signals, their time resolution, or the number of input profiles available for training; and finally, the main conclusions of the research are presented in Section 5. Figure 1. Comparison of different analysis methods of battery cell and system behavior. The impact that a certain input (for instance, the frequency signal for FCR or the household load and PV generation for SCI) has on a BSS can be described in terms of a set of output signals. They typically measure the evolution of some physical magnitudes, such as the storage power provided for (drawn from) the BSS, its state of charge (SOC), or its state of health (SOH). It is a common practice to summarize this impact, that is, the set of output signals, using a handful of KPIs [ 23 ] to better analyze and compare the storage load profiles. Using KPIs is a way to summarize the behavior of a BSS in a handful of values. The KPIs selected in this research either emphasize the performance of the BSS, such as the battery efficiency ( η ), or they are precursory indicators of the expected battery degradation, such as the number of Full Equivalent Cycles (FEC) and the number of charging-discharging swaps (nswap). This research is focused on the development of an ML methodology to estimate the impact, as it is summarized by a set of KPIs, of a given input signal in a BSS when it is used for FCR or SCI applications. This methodology does not require any knowledge about the physicochemical process inside the BSS, nor even an equivalent electric model. The ML approach is based on an agnostic model which is trained using datasets with the results of previous experiments. The main contribution of the research is to show the capabilities of the ML techniques to describe, in a very fast way, the behavior of a BSS under different operation conditions. The use of ML methods in this research has been proposed mainly for three reasons: they do not require an explicit set of equations to describe the BSS (agnostic model); they can be run in much less time (see Figure 1); and, finally, because ML is a well-established discipline that has proven successful in many different applications. The paper is organized as follows: in Section 2, the datasets employed and the proposed methodology are described; the main results are explained in Section 3, outlining the estimation errors of the different KPIs; Section 4discusses these results and takes into consideration their dependence on design parameters such as the length of the input signals, their time resolution, or the number of input profiles available for training; and finally, the main conclusions of the research are presented in Section 5. Energies 2023,16, 5548 4 of 18 2. Materials and Methods 2.1. General Description This research explores ML methods to obtain BSS KPIs when they are excited with some input profile (one or several input signals). On one side, an input profile is used as the stimulus of an equivalent circuit model operating in a particular simulation scenario. As a result, the corresponding storage profile is obtained, containing two signals: the storage power supplied to the battery (positive when charging) and the SOC. Then, from the storage profile, several KPIs are extracted summarizing the overall impact of the input profile in the BSS. In this research, the Simulation Tool for Stationary Energy Storage Systems (SimSES) [19] is used for these purposes. On the other hand, each input profile is characterized using a set of features or attributes: that is, a reduced set of values that approximately describe this input profile. Then, the extracted features are used for an ML model to estimate the corresponding KPIs. This ML model is trained using the datasets described in the following subsection. To evaluate the results produced by the ML approach, the estimated KPIs are compared to the KPIs obtained by SimSES, which are considered as the true values. An overall description of the process is depicted in Figure 2. Energies 2023, 16, x FOR PEER REVIEW 4 of 18 2. Materials and Methods 2.1. General Description This research explores ML methods to obtain BSS KPIs when they are excited with some input profile (one or several input signals). On one side, an input profile is used as the stimulus of an equivalent circuit model operating in a particular simulation scenario. As a result, the corresponding storage profile is obtained, containing two signals: the storage power supplied to the battery (positive when charging) and the SOC. Then, from the storage profile, several KPIs are extracted summarizing the overall impact of the input profile in the BSS. In this research, the Simulation Tool for Stationary Energy Storage Systems (SimSES) [19] is used for these purposes. On the other hand, each input profile is characterized using a set of features or attributes: that is, a reduced set of values that approximately describe this input profile. Then, the extracted features are used for an ML model to estimate the corresponding KPIs. This ML model is trained using the datasets described in the following subsection. To evaluate the results produced by the ML approach, the estimated KPIs are compared to the KPIs obtained by SimSES, which are considered as the true values. An overall description of the process is depicted in Figure 2. Figure 2. General description of the research methodology. “SimSES” refers to a battery time series simulation tool incorporating an ECM battery model [16]. 2.2. Datasets For the FCR application, the evolution of the grid frequency for Continental Europe in a five-year period (2013–2017) is used as the input profile. This information, with a one-second resolution, is provided by the transmission system operator 50 hertz Transmission GmbH [24]. A typical example of the evolution of this signal over one hour is shown in Figure 3a. Figure 3. Example of input profiles: (a) Evolution of the frequency in Continental Europe for one hour; (b) Normalized values of the photovoltaic power generated in a typical sunny day, the electric power demanded by a household, and the difference between them (surplus). Figure 2. General description of the research methodology. “SimSES” refers to a battery time series simulation tool incorporating an ECM battery model [16]. 2.2. Datasets For the FCR application, the evolution of the grid frequency for Continental Europe in a five-year period (2013–2017) is used as the input profile. This information, with a onesecond resolution, is provided by the transmission system operator 50 hertz Transmission GmbH [ 24 ]. A typical example of the evolution of this signal over one hour is shown in Figure 3a. Energies 2023, 16, x FOR PEER REVIEW 4 of 18 2. Materials and Methods 2.1. General Description This research explores ML methods to obtain BSS KPIs when they are excited with some input profile (one or several input signals). On one side, an input profile is used as the stimulus of an equivalent circuit model operating in a particular simulation scenario. As a result, the corresponding storage profile is obtained, containing two signals: the storage power supplied to the battery (positive when charging) and the SOC. Then, from the storage profile, several KPIs are extracted summarizing the overall impact of the input profile in the BSS. In this research, the Simulation Tool for Stationary Energy Storage Systems (SimSES) [19] is used for these purposes. On the other hand, each input profile is characterized using a set of features or attributes: that is, a reduced set of values that approximately describe this input profile. Then, the extracted features are used for an ML model to estimate the corresponding KPIs. This ML model is trained using the datasets described in the following subsection. To evaluate the results produced by the ML approach, the estimated KPIs are compared to the KPIs obtained by SimSES, which are considered as the true values. An overall description of the process is depicted in Figure 2. Figure 2. General description of the research methodology. “SimSES” refers to a battery time series simulation tool incorporating an ECM battery model [16]. 2.2. Datasets For the FCR application, the evolution of the grid frequency for Continental Europe in a five-year period (2013–2017) is used as the input profile. This information, with a one-second resolution, is provided by the transmission system operator 50 hertz Transmission GmbH [24]. A typical example of the evolution of this signal over one hour is shown in Figure 3a. Figure 3. Example of input profiles: (a) Evolution of the frequency in Continental Europe for one hour; (b) Normalized values of the photovoltaic power generated in a typical sunny day, the electric power demanded by a household, and the difference between them (surplus). Figure 3. Example of input profiles: ( a ) Evolution of the frequency in Continental Europe for one hour; ( b ) Normalized values of the photovoltaic power generated in a typical sunny day, the electric power demanded by a household, and the difference between them (surplus). Energies 2023,16, 5548 5 of 18 For the SCI applications, two signals are required to define an input profile: first, the power demanded over one year for 74 households, information which has been published by the HTW Berlin [ 25 ]. Second, SCI requires us to know how much solar power is available. For this purpose, the power generated by a photovoltaic installation at the TU Munich has been recorded during a one-year (2009) experiment [ 26 ]. Both signals also have a resolution of one second and, to match their scales, they are normalized in the range of 0–100% of their respective maximum values. Then, the surplus of generation (normalized photovoltaic power minus normalized household demand) is used as a single combined input signal. An example of these two signals for a typical household on a sunny day and the corresponding surplus is shown in Figure 3b. Until now, the number of instances (examples of input profiles) in the dataset is limited: only one example for FCR and 74 for SCI. These numbers are insufficient to train an ML model. Therefore, to increase the number of instances, each input signal is split into segments of one-day length. In this way, up to 1825 (365 × 5) FCR and 27,010 (365 × 74) SCI segments are available. As each FCR segment is analyzed in three different simulation scenarios (two scenarios in the case of SCI), a total of 59,495 instances are available after data preprocessing. Simulation scenarios will be described in the following subsection, while the validity of splitting the input signals into segments of one-day length will be analyzed in the discussion section. The proper characterization of the input signals is also a key element in the development of these models [ 27 – 29 ]. Once a one-day input segment has been obtained, it is described using a feature extraction procedure. For this purpose, no specific applicationoriented feature set has been developed. Instead, an off-the-shelf set of general-purpose attributes is used. Up to 16 time-domain and 13 frequency-domain features are computed using several mathematical operations on the input segments. The resulting 29 features are described in [ 30 ]. Additionally, the information about the simulation scenario must be included for a fully defined input segment. This can be made by using a single column containing a category value (one out of five possibilities) or, alternatively, using five columns in a one-hot coding scheme. The latter is the solution implemented in this research. Finally, the resulting ML design matrix contains 59,495 rows and 34 columns. 2.3. Simulation Scenarios To analyze the performance of the ML approach in estimating the KPIs of BSS under different operating conditions, five simulation scenarios have been explored (see Table 1). On one side, two types of battery are considered: a Lithium-Iron-Phosphate (LFP) cathode and a Carbon-Graphite (C) anode; and a Nickel-Manganese-Cobalt-Oxide (NMC) cathode with C anode. The batteries used in FCR applications have a storage capacity of 1.6 MWh and a maximum power of 1.6 MW, while in SCI these parameters are 5 kWh and 5 kW, respectively. Table 1. Simulation scenarios. Scenario A-FCR B-FCR C-FCR D-SCI E-SCI Application FCR FCR FCR SCI SCI Battery type LFP:C LFP:C NMC:C LFP:C LFP:C Storage Capacity 1.6 MWh 1.6 MWh 1.6 MWh 5 kWh 5 kWh Maximum Power 1.6 MW 1.6 MW 1.6 MW 5 kW 5 kW Power Electronic Single unit 3 modular 3 modular Single unit Single unit Operation strategy FCR regulations Germany (status 2021) Greedy Feed-in damp On the other hand, the power electronic required in the BSS is usually modeled as a single unit. However, in certain highly demanding applications such as FCR, an alternative 3-unit modular power electronic system is also considered. Finally, three different operation strategies have been simulated. In the case of FCR applications, the simulation was executed according to the German regulatory framework [ 31 ] Energies 2023,16, 5548 6 of 18 where the BSS must provide a certain maximum prequalified power for a predefined time. On the other hand, SCI applications are operated using either greedy or a feed-in damp strategy. In the first case, the solar power surplus is directly devoted to charging the battery. However, to obtain a more stable power fed into the grid, a feed-in damp strategy [ 26 ] is also considered. 2.4. Key Performance Indicators The simulation process yields a storage profile defined by two signals: the storage power (SP) supplied to (or drawn from) the battery; and the SOC. These signals span over the same time interval as the input profile segment, in our case, for one day. Several variables and parameters can be defined from the SP and SOC. They are summarized in Table 2. Table 2. Variables and parameters describing the storage profile signals. Name Description Ebat Energy storage capacity of the battery Epos Positive energy (supplied to the battery); when SP >0 Eneg Negative energy (drawn from the battery); when SP <0 SOCstart SOC at the start of the signal SOCend SOC at the end of the signal Based on the previous definition, three KPIs are defined to characterize the impact of the input profiles on the BSS. The KPIs have been proposed in [ 23 ] and are summarized in Table 3. The importance of the KPI “efficiency” has a straightforward interpretation. On the other hand, the other two KPIs have a direct influence on the battery lifetime, as the aging process is due to the number and type of charge-discharge cycles (besides some other factors such as temperature and c-rate) [32], as well as calendric effects [33]. Table 3. Key Performance Indicators (KPIs). Name Description Expression ηEfficiency Eneg Epos−(SOCstart−SOCend)Ebat FEC Full Equivalent Cycles Epos Ebat nswap Number of swaps (charging-discharging) Defined in [23] From an ML point of view, the 3 KPIs are the target variables. The ML estimation can be addressed by modeling either a single multiple (3)-target problem or multiple (3) single-target problems. In the first case, the target information is defined in a target matrix containing 59,495 rows and three columns, while in the second case, three target vectors are used, each one containing 59,495 rows. Both approaches have been considered and compared in the following sections. 2.5. Regression Performance Metrics To compare different options in the modeling process and to select the best algorithms, a regression performance metric must be defined. For that purpose, the coefficient of determination is generally used [34] that, for the k-th KPI, is defined as R2 k=1− ∑n i=1y(i) k−ˆ y(i) k2 ∑n i=1y(i) k−yk2, (1) Energies 2023,16, 5548 7 of 18 where y(i) k and ˆ y(i) k are the respective actual and estimated values of that KPI for the i -th input profile and yk is the mean of the actual value of the KPI. The coefficient of determination can be in the range of R2 k∈[−∞, 1] . A naïve regressor which always predicts the mean value yk as the estimation ˆ y(i) k for any input profile would yield a coefficient of determination of R2 k= 0. As such, for models with a better prediction than the naïve regressor, the coefficient of determination will be in the range of R2 k∈[0, 1] . This result is a very convenient range to compare models and, therefore, R2 will be the metric used to tune and select the different ML models. Although R2 is a very convenient and extended metric, it is mainly algorithm-oriented and does not have an intuitive meaning. Seeking a more human-oriented way to assess the predictions produced by the final selected regressor, a first and more straightforward metric is the error between the actual and the estimated value for a KPI instance. More formally, the prediction error of the k -th KPI corresponding to the i -th input profile can be defined as e(i) k=ˆ y(i) k−y(i) k, (2) where y(i) k and ˆ y(i) k are the respective actual and estimated values for that KPI and input profile . In many cases, the important fact relies on how far the prediction from the actual value is, and not if their difference is positive or negative. For that reason, it is a common practice to measure the absolute values of the differences. Then, the absolute error of the k -th KPI corresponding to the i-th input profile can be defined as AE(i) k=  e(i) k  =  ˆ y(i) k−y(i) k  . (3) The errors for a certain KPI are expressed in the units of that KPI. Then, as these units may be different, it makes no sense to compare the errors for different KPIs. Moreover, even in the case that all the KPIs were in the same unit, they may be in different scales, making a comparison between KPIs meaningless. To tackle this problem, errors can be normalized using a reference value. For instance, a relative absolute error of the k -th KPI corresponding to the i-th input profile can be defined using the actual value as the reference, that is, rAE(i) k=     e(i) k y(i) k      =     ˆ y(i) k−y(i) k y(i) k      =     ˆ y(i) k y(i) k −1     . (4) The relative error so defined has the problem that a small absolute error relative to a very low actual value (near 0) yields a very large relative error with a significant distortion of the average error. Then, it is usually preferred to make the errors relative, not to a single actual value, but to an average (the mean) of these values. Then, the relative (to the mean) absolute error of the k-th KPI corresponding to the i-th input profile can be defined as RAE(i) k=AE(i) k |yk|=     e(i) k yk      =     ˆ y(i) k−y(i) k yk      . (5) For each KPI, the statistical distribution of these errors can be drawn in the form of histograms, probability density functions, or other techniques. In this research, the distribution of the errors will be depicted using boxplots, where the central values are computed using the median as it is more robust than the mean in the presence of outliers (values of errors rarely large). For the same robustness reason, boxplots indicate the statistical dispersion using the 25th and 75th percentile values and the difference between them, called the interquartile range (IQR). For a dataset containing many input profiles, it is useful to summarize the error values in a single metric, for instance, the average value of the boxplot. For that purpose, the Energies 2023,16, 5548 8 of 18 median of the absolute error ( MAE ) and the relative absolute error ( MRAE ) of the k -th KPI are defined as MAEk=median iAE(i) k=median i  ˆ y(i) k−y(i) k  . (6) MRAEk=median iRAE(i) k=median i      ˆ y(i) k−y(i) k yk      . (7) 2.6. Regression Models The estimation of the KPIs corresponding to a certain input profile is a good example of an ML regression problem where the target variables (the KPIs) are continuous. Although dozens of regression algorithms have been described in the literature, this paper focuses on three of them: the linear regressor with Tikhonov’s regularization (also known as ridge regressor) [ 35 ], the random forest regressor (an ensemble of decision trees) [ 36 ], and a feed-forward neural network [37]. Each of the three models contains many parameters and several hyperparameters. To select their values, a hold-out technique has been applied [ 38 ]; that is, the dataset (design matrix and target matrix or vector) has been split into three parts, each one containing randomly selected instances: the training (60%), validation (20%), and testing (20%) datasets. First, the training dataset is used to train the models: that is, to select the values of their parameters that minimize an error-related cost function. Second, a validation dataset is used to determine the hyperparameters of the models (including the selection of the best model). Finally, the testing dataset is used to generalize the selected and fitted model: that is, to give an estimation of the error that should be expected when the model is excited with new instances not previously seen. 3. Results 3.1. Optimal Models The optimal models are obtained through a validation process which selects the best hyperparameters of the models. An example of this process is shown in Figure 4, where the regression performance (coefficient of determination R2 ) is drawn for different values of the regularization hyperparameter ( λ ) for a single-target model. In this case, a value of λ=10−3has been selected as one that yields the best regression performance. Energies 2023, 16, x FOR PEER REVIEW 8 of 18 median of the absolute error (𝑀𝐴𝐸) and the relative absolute error (𝑀𝑅𝐴𝐸) of the 𝑘-th KPI are defined as 𝑀𝐴𝐸=median  𝐴 𝐸 󰇛󰇜=median 𝑦󰇛󰇜−𝑦󰇛󰇜. (6) 𝑀𝑅𝐴𝐸=median 𝑅𝐴𝐸 󰇛󰇜=median 𝑦󰇛󰇜−𝑦󰇛󰇜 𝑦. (7) 2.6. Regression Models The estimation of the KPIs corresponding to a certain input profile is a good example of an ML regression problem where the target variables (the KPIs) are continuous. Although dozens of regression algorithms have been described in the literature, this paper focuses on three of them: the linear regressor with Tikhonov’s regularization (also known as ridge regressor) [35], the random forest regressor (an ensemble of decision trees) [36], and a feed-forward neural network [37]. Each of the three models contains many parameters and several hyperparameters. To select their values, a hold-out technique has been applied [38]; that is, the dataset (design matrix and target matrix or vector) has been split into three parts, each one containing randomly selected instances: the training (60%), validation (20%), and testing (20%) datasets. First, the training dataset is used to train the models: that is, to select the values of their parameters that minimize an error-related cost function. Second, a validation dataset is used to determine the hyperparameters of the models (including the selection of the best model). Finally, the testing dataset is used to generalize the selected and fitted model: that is, to give an estimation of the error that should be expected when the model is excited with new instances not previously seen. 3. Results 3.1. Optimal Models The optimal models are obtained through a validation process which selects the best hyperparameters of the models. An example of this process is shown in Figure 4, where the regression performance (coefficient of determination 𝑅) is drawn for different values of the regularization hyperparameter (𝜆) for a single-target model. In this case, a value of 𝜆=10 has been selected as one that yields the best regression performance. Figure 4. Selection of the regularization parameter. The value 𝜆=10  is selected for the three regression models. Figure 4. Selection of the regularization parameter. The value λ=10−3 is selected for the three regression models. Energies 2023,16, 5548 9 of 18 Using a similar validation process, 100 decision trees have been selected for the random forest regressor. Analogously, an architecture of two hidden layers with 200 nodes each has been the choice for the neural network regressor. Using these hyperparameters, the R2 metric has been computed for the predictions obtained using the validation dataset and the six single-target and multiple-target regression algorithms. The results are summarized in Table 4, where the best results for each KPI are in bold font. The random forest regressor overperforms the other two models in all cases. The best results are obtained using a single-target approach for the FEC and nswap KPIs, and a multi-target model for the efficiency ( η ). These will be the optimal models used in the remaining sections. Table 4. Performance of different regression models ( R2 score). Optimal models in boldface and green background. KPI Single Target Multi-Target Ridge R. Forest N. Network Ridge R. Forest N. Network η0.011 0.122 −0.064 0.016 0.570 −0.034 FEC 0.811 0.927 0.918 0.866 0.919 0.902 ηswap 0.954 0.985 0.965 0.953 0.980 0.962 3.2. Generalization of Predictions Applying the design matrix of the instances in the testing dataset to the optimal regressors, a generalization of the expected errors in the prediction of unseen data can be obtained. In Figure 5, the optimal prediction for each KPI is depicted versus its actual value. As the number of instances in the testing dataset is high (more than ten thousand), a random sample (5%) of these results is plotted to avoid an overcrowded, too-dense, and unreadable graph. The main diagonal (dotted line) in the graph represents a perfect prediction ( ˆ y(i) k=y(i) k ). As most of the dots are close to this diagonal line, a good result should be expected for the error metrics. Energies 2023, 16, x FOR PEER REVIEW 9 of 18 Using a similar validation process, 100 decision trees have been selected for the random forest regressor. Analogously, an architecture of two hidden layers with 200 nodes each has been the choice for the neural network regressor. Using these hyperparameters, the 𝑅 metric has been computed for the predictions obtained using the validation dataset and the six single-target and multiple-target regression algorithms. The results are summarized in Table 4, where the best results for each KPI are in bold font. The random forest regressor overperforms the other two models in all cases. The best results are obtained using a single-target approach for the 𝐹𝐸𝐶 and 𝑛 KPIs, and a multi-target model for the efficiency (𝜂). These will be the optimal models used in the remaining sections. Table 4. Performance of different regression models (𝑅  score). Optimal models in boldface and green background. KPI Single Target Multi-Target Ridge R. Forest N. Network Ridge R. Forest N. Network 𝜂 0.011 0.122 −0.064 0.016 𝟎.𝟓𝟕𝟎 −0.034 𝐹𝐸𝐶 0.811 𝟎.𝟗𝟐𝟕 0.918 0.866 0.919 0.902 𝑛  0.954 𝟎.𝟗𝟖𝟓 0.965 0.953 0.980 0.962 3.2. Generalization of Predictions Applying the design matrix of the instances in the testing dataset to the optimal regressors, a generalization of the expected errors in the prediction of unseen data can be obtained. In Figure 5, the optimal prediction for each KPI is depicted versus its actual value. As the number of instances in the testing dataset is high (more than ten thousand), a random sample (5%) of these results is plotted to avoid an overcrowded, too-dense, and unreadable graph. The main diagonal (dotted line) in the graph represents a perfect prediction (𝑦󰇛󰇜=𝑦󰇛󰇜). As most of the dots are close to this diagonal line, a good result should be expected for the error metrics. Figure 5. Generalization results: estimated values of the KPIs using the optimal predictor versus their actual value. Results obtained for a random 5% of the testing dataset. The generalization errors can be summarized using the median absolute error (𝑀𝐴𝐸) and the median relative absolute error (𝑀𝑅𝐴𝐸) metrics, defined in Section 2.5. The values obtained are described in Table 5, where the selected optimal regressor for each KPI and the corresponding 𝑅 score are also indicated. The 𝑀𝑅𝐴𝐸𝑠 are below 8% for all KPIs under investigation. Table 5. Generalization errors of the optimal models (𝑀𝑅𝐴𝐸 metric and 𝑅  score). KPI 𝑴𝑨𝑬 𝑴𝑹𝑨𝑬 𝑹𝟐 Score Optimal Regressor Algorithm Targets 𝜂 0.056 7.38% 0.570 R. forest Multi-target 𝐹𝐸𝐶 14.42 5.42% 0.919 R. forest Single target 𝑛 28.04 7.72% 0.980 R. forest Single target Figure 5. Generalization results: estimated values of the KPIs using the optimal predictor versus their actual value. Results obtained for a random 5% of the testing dataset. The generalization errors can be summarized using the median absolute error ( MAE ) and the median relative absolute error ( MRAE ) metrics, defined in Section 2.5. The values obtained are described in Table 5, where the selected optimal regressor for each KPI and the corresponding R2 score are also indicated. The MRAEs are below 8% for all KPIs under investigation. Energies 2023,16, 5548 16 of 18 the functioning of the battery have been predicted with an average relative error (MRAE) of less than 10%. These average errors may differ depending on the KPI and the battery operation scenario, ranging from 2% in the better case to 16% in the worst case. Although the precisions estimating the battery KPIs are not very high, the ML approach proposed in this article offers the advantage of obtaining a prediction in a relatively short time (a few seconds), a remarkable result compared to the computing effort required when using simulations based on electric-equivalent circuits (several hours). The precision obtained in this research highly depends on the number of training examples available. As the learning curves have shown, the selected ML algorithm (a random forest regressor) has not reached its maximum performance and better predictions should be expected with a larger training dataset. The predicting models are valid for the same conditions employed to obtain the datasets. If the application, the operation strategy, and/or the parameters of the battery system change, a new ML model has to be trained from the corresponding dataset. From the previous paragraphs, it can be stated that the methods proposed in this research have two main limitations: they can only be used in cases where similar examples have been previously analyzed, either in lab tests or by equation-based simulations; and, on the other hand, the estimation of KPIs through ML techniques does include an estimation error (10% on average in our study). ML techniques are a very powerful tool to have at hand, but they do not replace an in-depth analysis and domain knowledge-based discussion of results altogether. As applied herein, they do not eliminate the need for precise physical-model-based simulations or lab-controlled experimentation, but they can offer a first and very fast rough approximation of the expected behavior of a BSS. The results presented in this article, mainly those concerning to FEC and nswap , indicate that ML techniques can probably be extended to estimate the SOH of a battery after a given input profile, a question that the authors will research in the future. Author Contributions: Conceptualization, H.H.; methodology, J.L.; software, J.L. and B.T.; validation, H.H., D.L. and C.L.; resources, H.H. and C.L.; writing—original draft preparation, J.L. and D.L.; writing—review and editing, B.T., H.H. and C.L. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Data Availability Statement: Publicly available datasets were analyzed in this study. This data can be found here: https://www.50hertz.com/de/Transparenz/Kennzahlen/Regelenergie/ArchivNetzf requenz (accessed on 19 July 2023); https://solar.htw-berlin.de/elektrische-lastprofile-fuer-wohng ebaeude/ (accessed on 19 July 2023). 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