3.1.1. Direct Estimation Methods
The direct SoC estimation approaches can be classified into the following five categories: open-circuit voltage (OCV), internal resistance (IR), impedance spectroscopy (IS), electromotive force (EMF) and CC. In the following sub-sections, a brief explanation of each method is given.
- (1)
Open Circuit Voltage
OCV is a conventional approach to evaluate the SoC by measuring the circuit voltage in an open circuit state and using SoC–OCV relationships. The SoC–OCV relationships are not the same for different batteries, they are influenced by battery capacity and electrode materials. For example, a lead-acid battery has a linear SoC and OCV relationship [
35]. Conversely, the LFP battery shows a quasi-flat SoC–OCV relationship region. Within this region, OCV variation is very small compared to SoC variation. Thus, misleading estimation of SoC might occur [
36] Although the SoC–OCV curve of lithium–ion batteries is relatively stable, it will change according to the charging/discharging rate, battery temperature, cell variation, and cycle life of the battery [
37]. To consider the aging mechanism, several adjustments to the OCV modeling curve are necessary, such as incremental capacity or differential voltage analysis [
38]. While this method demonstrates simplicity and high accuracy, it necessitates an extended resting period to attain the battery’s equilibrium state, a duration influenced by environmental conditions. Additionally, precise voltage measurements are essential due to the battery’s hysteresis characteristics.
- (2)
Internal Resistance
The IR method is based on the relationship between the internal resistance of the battery and SoC. This estimation approach evaluates the internal resistance measuring the charging/discharging current and the terminal voltage in the same short period [
39]. IR is computed as the ratio of the battery voltage and charging/discharging current following Ohm’s Law:
It is named Direct Current (DC) internal resistance [
40]. It is worth emphasizing that if the sampling period is shorter than 10 ms, only the ohmic internal resistance can be detected; for a longer time, the internal resistance assessment becomes more complicated [
41]. In general, the IR value is in the order of milliohms and its measure is also challenging due to the temperature and number of cycles’ influence. Furthermore, internal resistance changes slowly and is hard to observe for SoC estimation; thus, this approach is not suitable for online SoC estimation [
42].
- (3)
Impedance spectroscopy
IS is an experimental method to characterize electrochemical systems, such as batteries, and supercapacitors. During the measurement of an IS, a small alternate current (AC) flows through the battery, and the voltage, the response concerning amplitude and phase, is measured. The impedance of the system is determined by the complex division of AC voltage by AC. This sequence is repeated for a certain range of different frequencies, and the full range of frequency properties of the battery can be obtained. IS gives a precise impedance measurement in a wide band of frequencies, thus providing a unique tool for analysis of the dynamical behavior of batteries, which directly measures the nonlinearities as well as very slow dynamics [
20]. The IS method exhibits exceptional accuracy, quickly and non-destructively capturing the dynamic characteristics of batteries. Nonetheless, this method comes with a high cost, significant susceptibility to battery life and temperature sensitivity. Furthermore, an accuracy loss in SoC estimation is observed if the battery temperature changes greatly [
41]. Therefore, the impedance-based method is not sufficiently accurate to be implemented for vehicle applications [
42].
- (4)
Electromotive force
The electromotive force (EMF) voltage of a battery is the OCV or equilibrium voltage when a battery is in equilibrium or in an open circuit state for a long period [
43]. Therefore, EMF is useful to describe the relationship between the battery terminal voltage under equilibrium conditions and several parameters such as temperature and SoC [
44]. Through the OCV relaxation technique, the EMF can be calculated when the battery is charged or discharged and the current is subsequently interrupted. After a significant time has elapsed since the current interruption, the change in OCV is negligible and the EMF can be assessed as the OCV of the battery in its equilibrium condition (OCV = EMF) [
45]. The OCV relaxation process may take a lengthy time, especially if the battery is completely depleted in cold conditions or with an excessive charge/discharge current rate [
46]. Thus, this method is not suitable for online applications.
- (5)
Coulomb Counting
In the CC method, the charging or discharging current of the battery is considered by integrating the time to find the SoC. This method permits the inclusion of some internal battery effects such as self-discharge, capacity-loss, and discharging efficiency [
47]. The mathematical formulation [
48] of this approach is as follows:
where SoC(t
0) is the initial SoC value; T is the sampling period; C
N is the nominal battery capacity; η is the Coulomb efficiency; i(t) is the charging/discharging current; S
d is the self-discharge rate. The initial SoC value is the main concern because it will cause errors in the accuracy of the SoC estimation. When the initial SoC value is known, this method works more efficiently considering a short time-period. Although this method has been widely used in recent years, CC is not actually used as a sole tool for estimating SoC but rather is used in combination with other techniques, e.g., in [
49], the CC method is coupled with OCV to enhance the initial SoC estimation considering the effect of the internal resistance of the battery. Nonetheless, the easy implementation of the CC approach is not suitable for online SoC estimation [
50].
Table 2 summarizes the estimation errors via direct methods for further comparison.
3.1.2. Model-Based Estimation Methods
To address the direct methods’ limitations, in view of online battery parameter estimations, model-based techniques have been introduced. These methods connect through a battery model the measured battery signals (voltage, current and temperature) with the battery SoC. A high-fidelity battery model is required to capture the characteristics of the real-life battery and predict its dynamic behavior under varying operating conditions. In a BMS algorithm, the model uses signals as inputs to calculate the SoC and other battery states. The schematic battery state estimation diagram process is depicted in
Figure 4.
The most common battery models proposed are the electrochemical (EChM) and the equivalent circuit (ECM).
The ECM model is composed of an open-circuit voltage source connected with a combination of electric elements such as resistors, and capacitors to describe the battery behavior under a certain load [
55]. The simplest ECM, also known as the Rint model, is an ideal OCV voltage source connected in series with an internal ohmic resistance, both functions of SoC, SoH, and temperature [
56]. The internal resistance Rint value represents the voltage drop in the cell when it supplies/absorbs current under a certain load. This parameter defines battery performance and its SoH [
57]. Improved versions of this basic model are obtained by adding different resistor–capacitor (RC) branches to capture different time constants inherent in the battery system. In the Thevenin ECM or first-order ECM, a RC group is added to represent the voltage relaxation dynamics and describe the transient response during charging/discharging phases [
58]. A further improvement in the first-order ECM consists of adding a capacitor in series to the internal ohmic resistance to describe the OCV changing with the SoC value. This battery model is called Partnership for a New Generation of Vehicles (PNGV) [
59]. The PNGV model is an accurate nonlinear equivalent circuit model for transient response process simulation. It can be used with high currents and in charging/discharging severe conditions [
41]. Nevertheless, the relatively high complexity leads to an increase in computational effort and consequently lower real-time performance. The so-called second-order ECM is composed of two RC branches to take into account the slow and fast transient response caused by charge transfer and ion diffusion phenomena [
60]. The second-order ECM is widely used for online battery SoC estimation since it is computationally efficient and accurate. The increase in RC branches improves the model accuracy, however the computational effort increases as well [
61].
Figure 5 illustrates the typical representations of the aforementioned models.
The EChM models adopt differential equations to model battery physicochemical phenomena such as diffusion and electrochemical kinetics. Hence, the definition of the equations involves a specialized knowledge of electrochemistry [
62]. The most accurate EChM model is the pseudo two dimension (P2D). The key assumption for the P2D model is the one-dimensional dynamics of chemical reaction, neglecting the variations over the other two directions. Furthermore, to consider the intercalation/deintercalation of lithium on the solid matrix and the ion diffusion over a single direction the existence of small particles inside the electrolyte is assumed. As the main direction, the pseudo-radius of such particles is superimposed [
63]. The overall complexity of the model due to the high numbers of partial differential equations and the computational time-consuming process to solve them make the P2D not suitable for SoC online estimation [
60]. Several simplifications have been proposed for the P2D model for real-time application [
64,
65] at the cost of penalties in SoC estimation accuracy. In comparison, EChMs offer a key advantage by inherently incorporating the dependency of battery behavior on SoC and temperature. In contrast, electrical models necessitate the storage of their parameters as look-up tables across diverse SoC and temperature combinations to derive reliable SoC-OCV curves [
42]. Nevertheless, the merits of the ECM-based approach, such as low complexity and high accuracy, make it favorable for online SoC estimation.
Even though several approaches have been proposed to estimate directly SoC using a battery model [
9,
66], this section just reviews electrical and electrochemical models because they represent a prerequisite for battery state estimation approaches. Further advancements in SoC estimation via model-based approaches are obtained by coupling a battery model, generally first or second-order ECM, with an adaptive filter algorithm. Reducing the noise influence on the battery model by the filter can improve the accuracy and robustness of the battery SoC estimation [
40].
- (1)
Kalman Filter-based algorithms
The essence of the Kalman filter (KF) algorithm is to use a recursive formula to calculate the current state starting from a prior estimated state and current measurement signals to minimize the mean of the squared error [
60]. The filter then feeds back and recursively uses prior prediction to determine the new best guess at each time step [
67]. The self-correcting nature of the KF algorithm makes it suitable for SoC online estimation. A Kalman filter computes the states of the system by utilizing a process model, a measurement model, and a set of noisy measurements of the inputs and outputs of the system. While the process model contains all the information about the system dynamics, the measurement model relates the outputs of the system to its inputs and states. The process and measurement models predict the present state and correct the raw state estimation obtained from the process model [
68].
Depending on the equations’ linearization process, the KF algorithms can be grouped into two main categories: linear KF (LKF) and non-linear KF. Among the nonlinear KF algorithms, extended KF (EKF), sigma-point KF (SPKF) and cubature KF (CKF) can be mentioned. SPKF is further divided into central difference KF (CDKF) and unscented KF (UKF) [
60].
LKF is commonly used as a data fusion algorithm in several technical applications due to its robustness and acceptable computational cost to filter parameters from inaccurate observation [
69]. The basic idea of KF is to compare the measured terminal voltage with the modeling one, and the difference is fed back to update the predicted SoC through a gain matrix, as schematically depicted in
Figure 6. The algorithm works as an optimal state estimator with a self-correcting nature for real-time SoC estimation of the battery. It employs a recursive process estimating the unknown SoC by exploiting previous knowledge, system predictions and noisy measurement [
70]. LKF is composed of two equations: a process Equation (4), which is used to predict the current state x
k from the prior state x
k−1; a measurement Equation (5) useful for updating the current state to converge to the real value [
70]:
where x represents the system state; u is the control input; w is process noise to capture the uncertainties in the model; y is measurement input; v is measurement noise to capture the measurement error; meanwhile, A, B, C and D are the time-varying covariance matrixes that describe the dynamics of the system. Both measurement noise and process noise are defined as Gaussian errors. Yatsui and Bai [
67] presented a LKF-based SoC estimation method for lithium-ion batteries. Experimental results validate the effectiveness of KF during the online application reporting and also a low estimation of SoC errors. Dong et al. [
71] have developed a simplified linearized ECM to simulate the dynamic characteristics of a battery when the OCV is not linear to apply LKF for SoC estimation. Despite LKF being an efficient filtering algorithm proposed for tracking the state of linear systems in Gaussian noise environments, its performance is limited when it is applied to systems which exhibit hysteresis effects and strong nonlinearities during charging/discharging events [
72]. Thus, improved methods have been put forward to tackle this issue.
A widely used method for battery parameter/state estimation for non-linear systems is the extended Kalman filter (EKF) [
73]. To deal with the non-linear characteristics of battery models, EKF employs partial derivates and a first-order Taylor series to linearize the battery model. The linearization process occurs at each step. In particular, Equations (4) and (5) are modified as follows [
74]:
At each time step, matrices of f(x
k,u
k) and g(x
k,u
k) are linearized close to the operation point by the first order in the Taylor series and the remainder of the series are truncated. In the technical literature, several applications of EKF or an improved version of it have been proposed for SoC estimation. Jiang et al. [
75] proposed a battery SoC estimation approach via EKF. The experimental results reported showed an average SoC estimation error of 1% [
75]. In [
76], a comparison between a standard EKF and an improved EKF algorithm was proposed. Although the experimental findings demonstrated that both filters have good performance, the improved EKF showed a slightly better SoC estimation accuracy. Similarly, Sepasi et al. [
74] proposed an improved EKF variant including aging effects in the battery electrical model. The novel approach reported has shown a low computational burden with a good SoC estimation accuracy making it suitable for online implementation. In several works, EKF has also been used with DD algorithms to enhance the SoC estimation accuracy. For example, in [
77] EKF has been adopted with a neural network. In particular, NNs have been used to model the non-linear battery behavior, whereas the EKF has been adopted for SoC estimation. Similarly, an EKF data-driven approach was proposed in [
78]. The novel approach has produced an accurate SoC estimation within the 2% error.
Since the linearization process in EKF uses the first-order Taylor series, a linearization error may occur under highly non-linear conditions due to a lower accuracy of the first-order Taylor series. Furthermore, the accuracy of the EKF algorithm depends on battery model parameters and the prior knowledge of the system noise signals. The assumption of fixed measurement and process noise covariance matrices in EKF reduces the overall performance of SoC estimation [
79]. Thus, in practice, inappropriate initial noise information will make the approach fail in ensuring its performance. To overcome this issue, the adaptive updating of these matrices has been introduced through the adaptive EKF (AEKF) [
80]. As stated previously, misleading SoC estimation may occur for highly nonlinear models adopting EKF. To mitigate this problem, an improved version of KF has been proposed named SPKF. This algorithm is capable of linearizing the process up to the third order of a Taylor series expansion [
81]. Rather than using Taylor-series expansions to approximate the required covariance matrices, SPKF performs several functional evaluations whose results are used to compute an estimated covariance matrix [
82]. The algorithm selects a set of sigma points with weighted mean and covariance values exactly like the values of the mean and covariance of the model being developed [
35]. This approach presents comparable computer complexity compared with EKF. In addition, the original functions do not need to be differentiable and no derivate calculation is needed [
82]. Based on the weighing factor, the SPKF algorithm is classified into two categories: unscented Kalman filter (UKF) and central difference Kalman filter (CDFK) [
83]. The UKF estimates covariance with statistical methods rather than with a Taylor series. In particular, UKF applies an unscented transformation, which is a method for calculating the statistics of a random variable propagating through a nonlinear system [
81]. In CDKF, a Sterling interpolation formula is used to avoid derivative computing through polynomial approximation. This approach uses central difference instead of a first or second-order Taylor series expansion [
83].
- (2)
H∞ Filter
The H∞ filter (HIF) represents another viable solution to overcome the noise influence on the accuracy of the traditional EKF algorithm. This algorithm considers the time-varying element of battery parameters and does not require the details of process noise and measurement noise [
84]. HIF can restrict the effects of the uncertainty and perturbation of the system model and no specifications of the disturbances and model uncertainties are necessary. Accordingly, the battery SoC may be determined without needing the exact statistical features of the system and measurement errors. Despite its robustness and easy implementation, ageing, hysteresis and temperature effects could influence the accuracy of the model [
70]. In [
85], the HIF algorithm was used to estimate the SoC of lithium-ion batteries. The method was validated through real-time experimental battery data. Zhang et al. [
86] proposed a robust HIF to estimate the SoC of a lithium battery pack. The proposed method takes into account battery time-varying parameters with no prior knowledge of the process and measurement noise, respectively. Several UDD cycles have been performed to test the algorithm’s performance. In order to enhance the SoC estimation accuracy, HIF has been combined with DD techniques [
87,
88] and filter-based methods [
89].
- (3)
Particle Filter
A particle filter (PF) is a probability-based estimator that uses the Monte Carlo simulation technique to approximate the probability density function of a non-linear system with a set of random weighted particles without any explicit assumption about the form of the distribution [
90]. The weight represents the chance of the particles to be selected in the probability density function [
91]. When designing a particle filter, the main difficulty is to select the proper proposal distributions that can approximate the posterior distributions [
50]. Because PF is suitable for estimation for non-linear systems, such as battery models, it can be adopted as a SoC estimator algorithm. In [
91], a PF-based approach is proposed for estimating simultaneously in real-time the state of charge and internal temperature of a prismatic lithium-ion battery using a first-order ECM model. The PF-based solution was compared with the traditional EKF algorithm in terms of SoC and temperature estimation showing a faster convergence to the real values of SoC and internal temperature when compared to the EKF solution. A similar result is reported by Gao et al. [
92]. To deal with the computational cost limitations of the PF, it is necessary to select a number of particles that provide a good trade-off between the accuracy and reliability of the results [
91]. In several works [
93,
94], PF was implemented with other techniques to improve its efficiency at a cost of more complexity.
Table 3 summarizes the estimation errors for further comparison.
- (4)
Observer-based methods
An observer-based method is realized to provide state feedback on the estimated values of the state variables of a system based on external measurements [
95]. Several observer-based methods have been proposed for battery state estimation.
Sliding-mode observer (SMO) is an observer algorithm with the advantage of compensating the modelling errors caused by variation in the parameters of the circuit model and can help overcome some of the drawbacks that other model-based methods present [
55]. With respect of SoC estimation, Ning et al. [
96] used SMO to estimate the battery SoC based on a parameter adaptive battery model to reduce systematic errors. The proposed approach shows good estimation accuracy. In [
97], based on a second-order ECM, a novel SMO was proposed for SoC estimation. Different test cycles were performed to assess the robustness and estimation accuracy of the proposed algorithm.
The proportional integral observer (PIO) is an efficient algorithm to estimate the state of a system with unknown input disturbance [
95]. It has been observed that this approach has lower computational complexity but high precision without matrix operation, even though the original SoC is uncertain [
50]. Xu et al. proposed a PIO approach for SoC estimation based on a first-order ECM. To validate the proposed algorithm, a UDDS cycle is performed experimentally [
98]. In [
36], a dual-circuit observer based was proposed to estimate SoC. A PIO circuit path was used to deal with capacity error and initial error. Even though the initial SoC was unknown, the proposed approach yielded reasonable SoC accuracy.
Another observer approach is that of the non-linear observer (NLO). This approach is used to deal with linear systems and non-linear observation equations. An advantage of this method is that it does not need complicated matrix operations, thus the computation cost can be reduced [
99]. In [
100], an NLO is tuned through an optimization algorithm to enhance the observer robustness and SoC estimation accuracy. In [
99], a novel method for SoC estimation using a NLO is presented. The proposed approach is then compared with EKF and SMO showing a faster convergence and an improved SoC accuracy, respectively.
Table 4 summarizes observer-based SoC estimation errors.
3.1.3. Data-Driven
Data-driven (DD) approaches consider the battery as a black-box model. In this case, the battery is presumed to be an unknown system and the internal dynamics have been learned through a vast quantity of data [
50]. Specifically, the model considers online measurable parameters such as battery current, voltage and temperature as inputs and battery state of charge (SoC) as the output. The model utilizes intelligent algorithms to train on input and output data, establishing the relationship between them [
41].
Figure 7 schematically depicts the process. The main DD approaches exploited for SoC estimation, as shown in
Figure 3, include fuzzy logic (FL), neural network (NN), genetic algorithm (GA), and particle swarm optimization (PSO). These approaches, being free of capturing any physico-chemical mechanisms, have potential advantages such as flexibility and strong adaptability and being highly nonlinearly matching [
101]. However, the main disadvantage of these algorithms is their sensitivity to the quality of the training dataset and that they may easily encounter overfitting or underfitting problems. In addition, the on-board implementation of DD methods for online SoC estimation is currently challenging.
In the following sub-sections, the DD approaches previously listed are presented and discussed.
- (1)
Fuzzy logic
The FL method provides a powerful means of modeling nonlinear and complex systems [
47]. This methodology is regarded as a problem-solving approach that simplifies all input data, characterized by noise, vagueness, ambiguity and imprecision, through the application of objective rules to determine the real value of the input [
102]. In addition, FL does not require a precise mathematical model of the system, as it only uses the input data and identifies the parameters using the fuzzy rule base. Fuzzy methods are robust and tolerant to imprecise measurements and component variations with rules that are easily tunable. The basic idea of a fuzzy algorithm is to formulate human knowledge and reasoning as a collection of “If–Then–Else” rules tractable by a computer [
103]. The FL process can be divided into the following steps [
104]: First, the inputs are fuzzified, or otherwise are converted into fuzzy language and grouped into membership functions. In the rule step, the relationship between input and output variables is described and a database defines the membership functions for the input and output variables. Finally, the fuzzy output value is defuzzied and translated into a real analogue value output.
Figure 8 shows the basic steps of the FL approach.
At present, few studies use only FL to assess battery SoC [
105,
106]. Commonly, fuzzy algorithms are combined with other intelligent algorithms to enhance SoC estimation accuracy. For example, Burgos et al. [
107] proposed a novel fuzzy logic algorithm to predict the battery SoC. In particular, the fuzzy model characterized the relationship between the battery OCV, SoC and the discharge current. It was used in combination with an EKF to predict the battery ScC. In [
108], the authors proposed a method to estimate battery SoC involving fuzzy algorithms to process data obtained by IS and CC with an estimation error of about 5%. Similarly, Malkhandi et al. [
109] proposed a model for SoC estimation adopting CC and a learning system based on FL. Despite a reasonable accuracy in SoC estimation, the FL approach is expensive in terms of storage and computational effort requirements to determine the parameters of a complex and nonlinear system. Thus, this approach is not suitable for online SoC estimation.
- (2)
Neural network
NN is a mathematical tool with adaptability and self-learning skills able to form a complex nonlinear system. Commonly, a NN is formed from three layers [
102]: the input layer, the hidden layer and the output layer. The input layer transfers the data to the hidden layer. Generally, as input data discharge current, the terminal voltage and temperature are considered [
35]. The hidden layer provides the mathematical junction between input and output through its neurons [
62]. In the third layer, SoC is addressed as output. The NN main advantage is that it can be utilized without knowledge of the cell’s internal structure. Thus, NN is suitable for the SoC estimation of all kinds of batteries. Nevertheless, to ensure a reliable SoC estimation, a large quantity of training data and storage are required. In addition, non-negligible computational effort is necessary to manage the whole training process [
33]. Therefore, the NN estimation approach is challenging to implement on board. Chen et al. [
110] combined a NN with an EKF-based algorithm for the estimation of SoC. This hybrid approach provides an estimation of SoC accuracy within 1%. In [
111], an OCV-based method for SoC estimation using the dual neural network fusion battery model was proposed. A first NN is used to estimate battery parameters of first- and second-order models. A second NN is employed to assess SoC via OCV-based methods for both ECMs.
- (3)
Genetic algorithm
A genetic algorithm (GA) is an optimization technique where the variables of interest of the system to be optimized are characterized in the form of strings called chromosomes. GA simulates natural biological evolution according to the “fitness level” of the individuals which provides a large set of possible solutions to a given problem. Through the genetic operators (selection, crossover and mutation) and natural selection, improved generations are bred. By selection operator, parent solutions that have better fitness levels are more likely to reproduce which means better genes are more likely to dominate the next generation. Crossover combines the features of two parents at a certain crossover fraction to form new solutions by swapping corresponding segments of parent chromosomes. By randomly changing one or more genes at a low mutation rate, mutation introduces variability into the next generation that will stop GA converging at a local minimum [
112]. A GA can be used both for single and multi-objective optimization, with the search strategy that, through iterations, maximizes or minimizes a given function of a properly formulated problem [
113]. The idea of coupling a numerical model of a given engineering unit with an algorithm for decision-making has been proven to be an effective and cost-saving option to achieve the desired results, even without resorting to real counterparts and spending time in heavy experimental tests [
114]. Practically, in SoC estimation applications, the chromosome is a string containing battery parameters such as SoC. The algorithm after the creation of a random set of chromosomes, via an iterative process, finds the optimal solution. It is necessary to obtain the definition of a stop criterion via an objective function in order to select the best population in each iteration [
62]. Xu et al. [
115] combined CC and a first-order ECM to estimate the battery SoC. Their proposed approach used GA to optimize battery parameters. Similarly, in [
112], the GA was used to find the optimum parameters of an equivalent model of a LiFePO
4 battery pack to estimate battery SoC via UKF.
- (4)
Particle swarm optimization
PSO is a nature-inspired optimization technique. The inspiration came from the social behavior of groups of animals, such as schools of fish or flocks of birds. It was first presented by Kennedy and Eberhart [
116]. The algorithm first generates a random population, then the next population is generated based on an objective function to be optimized [
62]. PSO is simpler than the GA approach and has several advantages such as fewer parameters to be tuned, lower computational effort and higher degree of convergence. Nevertheless, it is time-consuming to properly tune the parameters [
117]. PSO is used in a wide range of industry applications [
118,
119,
120], and has been extensively used in battery parameter estimations [
121,
122]. Sun et al. [
123] adopted PSO to tune up a Thevenin ECM to identify the critical parameters useful to SoC estimation. The proposed algorithm can estimate battery SoC with negligible errors. Similarly, Ye et al. [
124] combined PSO and an adaptive algorithm to estimate battery parameters and SoC. DD-based algorithms’ SoC estimation errors are summarized in
Table 5 for further comparison.