A Double Resistive–Capacitive Approach for the Analysis of a Hybrid Battery–Ultracapacitor Integration Study

Abstract

The development of energy storage systems is significant for solving problems related to climate change. A hybrid energy storage system (HESS), combining batteries with ultracapacitors, may be a feasible way to improve the efficiency of electric vehicles and renewable energy applications. However, most existing research requires comprehensive modelling of HESS components under different operating conditions, hindering optimisation and real-world application. This study proposes a novel approach to analysing the set of differential equations of a substitute model of HESS and validates a model-based approach to investigate the performance of an HESS composed of a Valve-Regulated Lead Acid (VRLA) Absorbent Glass Mat (AGM) battery and a Maxwell ultracapacitor in a parallel configuration. Consequently, the set of differential equations describing the HESS dynamics is provided. The dynamics of this system are modelled with a double resistive–capacitive (2-RC) scheme using data from Hybrid Pulse Power Characterisation (HPPC) and pseudo-random cycles. Parameters are identified using the Levenberg–Marquardt algorithm. The model’s accuracy is analysed, estimated and verified using Mean Square Errors (MSEs) and Normalised Root Mean Square Errors (NRMSEs) in the range of a State of Charge (SoC) from 0.1 to 0.9. Limitations of the proposed models are also discussed. Finally, the main advantages of HESSs are highlighted in terms of energy and open-circuit voltage (OCV) characteristics.

1. Introduction

To combat climate change, 196 countries emitting 99.75% of the world’s greenhouse gases (such as carbon dioxide—CO₂) signed the Paris Agreement on 4 November 2016 [1]. The signatories agreed to limit the global temperature increase to 1.5 °C and to reduce greenhouse gas emissions. The European Green Deal [2] aims to achieve these goals for net-zero greenhouse gas emissions by 2050. Transport is responsible for approximately 25% of the EU’s greenhouse gas emissions—71% of which are from road transport (Figure 1).

These goals can be achieved by developing sustainable and intelligent mobility solutions [3], such as deploying at least 30 million zero-emission electric vehicles in the EU by 2030. These vehicles will be supported by three million public charging points, including ultra-fast charging infrastructure compliant with the DC CHAdeMO 3.0 (ChaoJi) standard [4,5], delivering up to 900 kW of power according to IEC 61851-23:2014 [6] and the Alternative Fuels Infrastructure Regulation (AFIR) [7].

In addition to achieving climate neutrality, one of the most critical aspects of the targets is the sustainable development of renewable energy. The countries in the European Union have set specific targets for the overall share of renewable energy in final energy consumption, aiming to reach 32% by 2030 [8] or even up to 45% [9]. These targets include using renewable energy for heating, cooling, transport and electricity generation.

Additionally, EU regulations require member states to develop infrastructure and technologies to store heat and electricity simultaneously. This includes stabilising the power system [9]; promoting prosumer systems [10]; expanding alternative fuel infrastructure for vehicles, especially electric vehicles [7]; and integrating renewable energy sources into the primary power grid [9]. Given the limited predictability and daily variability of renewable energy production, sustainable development must prioritise the development of energy storage such as batteries (BESs) and ultracapacitors (UCs) [11]. These advances would increase the effective annual utilisation of renewable energy installations, thereby increasing the share of alternative, clean energy sources in the overall energy balance.

Electrochemical batteries [12] and ultracapacitors [13] are critical components in the charging stations developed by Gemamex Motion Co., with power ratings of up to 250 kW [14]. Moreover, they are integral to the drive systems of Chariot Motors’ battery electric buses [14] and ultracapacitor buses [15].

In addition, ultracapacitors and batteries are critical for starting large cogeneration systems based on internal combustion engines. Furthermore, ultracapacitors and batteries are used in Heavy-Duty Vehicles (HDVs), including those with conventional powertrains [16], such as trucks and all-electric vehicles [17,18,19,20,21,22,23,24,25,26,27,28].

When used under challenging conditions such as pulsed loads or low ambient temperatures, batteries experience significant performance degradation [29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44]. This includes increased internal resistance (especially electrolyte resistance), decreased electromotive force and reduced terminal voltage. Additionally, when starting internal combustion engines under these conditions, mechanical resistance increases due to the higher density of engine oil. These factors reduce the power and energy available from the battery, resulting in higher loads and a diminished service life [13].

Higher-capacity starter batteries are used to overcome these challenges and increase the specific energy density and power, or multiple batteries are often connected in parallel, increasing the cost. An alternative solution is a hybrid energy storage system (HESS) based on the parallel combination of an electrochemical battery and an ultracapacitor, whose equivalent circuit model is analysed in this paper.

Advances in electrochemical battery technology have significantly reduced the problems associated with starting internal combustion engines under standard operating conditions. Today’s starter batteries are affordable, efficient and relatively durable. However, high starting currents cause deep voltage drops at the battery terminals under challenging conditions, reducing starter performance and making it difficult to start internal combustion engines. Such conditions include low ambient temperatures, long periods of engine inactivity or high stationary battery loads, such as in vehicles with start–stop systems [22,23]. In start–stop systems, frequent engine starts and short battery charging periods accelerate battery discharge, reducing usable capacity and shortening battery life [22,23]. Other drawbacks include self-discharge and difficulty detecting impending battery discharge, often resulting in sudden, total discharge without warning [28,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62]. These limitations can be mitigated by connecting a battery and an ultracapacitor in parallel, as described in Section 2.

Section 3 describes the test bench and details the employed load cycles, including the Hybrid Pulse Power Characterisation (HPPC) and a pseudo-random cycle. Section 4 introduces the equivalent circuit model for the battery–ultracapacitor system along with its mathematical representation. Section 5 explains the assumptions made for the parameter identification and error analysis, and Section 6 presents the model and experimental data results. The conclusions and recommendations are given in Section 7.

2. General Characteristics of HESSs Based on BES and UC Components

This section presents the open-circuit voltage (OCV) characteristics as a function of the State of Charge (SoC) for a battery (OCV_BES), an ultracapacitor (OCV_UC) and a hybrid energy storage system (HESS) based on the combination of a battery and ultracapacitor (OCV_{UC + BES}). Additionally, energy characteristics for the battery (E_BES), ultracapacitor (E_UC) and HESS (E_{UC + BES}) are also presented.

2.1. General Characteristics of BESs

The energy-versus-SoC plot for the example battery shown in Figure 2 has a flat, slightly sloping shape. This characteristic is advantageous, as it maximises the energy stored around the operating point. The energy stored in the battery can be defined as the integral of the OCV_BES with respect to the stored charge Q. This is because Q is proportional to the SoC. Therefore, the energy stored in the battery can be expressed as follows:

$$E_{BES}=\int OCVd{Q}_{|SoC=Q{Q}_{nom}^{-1}\Rightarrow dQ=Q_{nom}dSoC}=Q_{nom}\int OCVdSoC$$

(1)

Since the OCV is nearly constant over the entire SoC range, the stored energy increases almost linearly with the increasing SoC. However, the slight variation in OCV with SoC means that even a small measurement error in the OCV can result in a significant error in SoC estimation (dOCV/dSoC ≈ 0 implies dSoC→∞). When estimating the OCV under load, the following relationship can be used:

$$OCV=U-Ri$$

(2)

It is important to note that SoC estimation errors are significant because dynamic processes such as voltage relaxation are not considered. More accurate methods, such as electrochemical impedance spectroscopy [63] and Kalman filtering [12,64,65], estimate the SoC by simultaneously measuring voltage and charge Q simultaneously, where Q is determined as the integral of the load current. Despite the superiority of adaptive methods in terms of better SoC estimation accuracy, they are not immune to error [66].

2.2. General Characteristics of UCs

As demonstrated in Figure 3, the energy stored in an ultracapacitor is equivalent to the integral of the OCV to the stored charge Q. Given that Q is proportional to the SoC and the OCV increases almost linearly with increases in the SoC, a quadratic function of the SoC approximates the stored energy. The stored energy equation therefore is given by

$$\begin{array}{l}{E}_{UC}=Q_{nom}\int OCVdSoC=Q_{nom}\int Q{C}^{-1}dSoC\cong \\ Q_{nom}{C}^{-1}\int SoC\cdot Q_{nom}dSoC={\displaystyle \frac{1}{2}}So{C}^2{Q}_{nom}{U}_{nom}={\displaystyle \frac{1}{2}}So{C}^{2}C{U}_{nom}^2\end{array}$$

(3)

In contradistinction to batteries, ultracapacitors have the capacity to store energy even at low voltages (close to 0 V) [13,62]. Nevertheless, the total energy stored is less than that of batteries. The linear increase in the OCV_UC with the SoC is advantageous for estimating the SoC and assessing the stored energy. In the simplest case, the SoC of an ultracapacitor can be evaluated by measuring the OCV and using the function OCV = f(SoC), as demonstrated in Figure 3. It should be noted that the OCV varies significantly with the SoC; even substantial OCV measurement errors will result in minimal SoC estimation errors, expressed as follows:

$$dOCV/dSoC\approx U_{nom}\Rightarrow dOCV\approx U_{nom}dSoC$$

(4)

As with batteries, the OCV_UC of an ultracapacitor under load can be estimated using Equation (4). However, significant SoC estimation errors remain significant due to dynamic processes such as voltage relaxation. To achieve more accurate SoC estimation, even under load, the utilisation of tools such as Kalman filters [12,67,68,69,70] or predictive methods [71,72] based on recursive neural networks [39,40,73,74] is recommended.

2.3. HESSs Based on BESs and UCs

The characteristics of ultracapacitors and batteries are often complementary, especially when connected in parallel, thus creating a system that takes advantage of both energy storage solutions (see Figure 4a). A parallel configuration of ultracapacitors allows the system to simultaneously achieve both high energy density (battery effect) and high power density (ultracapacitor effect) [75,76]. This configuration maintains performance even at low ambient temperatures (ultracapacitor effect) and low SoC levels, as discussed in [12,13,21,77]. The relationship between the energy stored in the battery (E_BES), the ultracapacitor (E_UC) and the HESS (E_UC+BES) as a function of voltage is presented in Figure 4a.

These graphs demonstrate the advantages of combining these components in parallel. On the one hand, the system provides substantial energy in the higher voltage range (supplied by the battery). At the same time, it retains the ability to store energy at lower voltages (ultracapacitor effect). With an acceptable large ultracapacitor capacity, the stored energy may be sufficient to start an internal combustion engine (ICE) [75], even when the battery is discharged (i.e., at very low system voltages). Battery discharge can occur due to low ambient temperatures or high currents from electrical equipment onboard.

As illustrated in Figure 4b, the OCV_UC+BES of the ultracapacitor–battery system exhibits a substantial curvature for SoC < 0.2, indicating a notable relationship between the two parameters. A low SoC is the cause of a more intense battery OCV drop. This behaviour enables earlier detection of an approaching SoC = 0, reducing the risk of sudden power unavailability and improving system reliability.

3. Description of the Test Stand

A special test bench was developed to test the battery and the ultracapacitor. The technical specifications of the battery and ultracapacitor are presented in

Table 1. Technical data of the BES and UC components.

Battery (VRLA AGM)		Ultracapacitor (Maxwell BMOD0058 E016 B02)
Parameter	Value	Parameter	Value
Nominal voltage	6 V	Maximum rated voltage	16 V
Minimum voltage	4.5 V	Minimum voltage	0 V
Maximum voltage	7.5 V	Maximum voltage	17 V
Capacity	12 Ah	Capacity	58 F
Internal resistance	10 mΩ	Initial ESR	22 mΩ
Maximum discharge current	135 A	Continuous/peak discharge current	23 A 190 A
Dimensions (L × W × H)	151 mm × 50 mm × 94 mm	Dimensions (L × W × H)	226.5 mm × 49.5 mm × 75.9 mm
Service life	6–9 years	Service life	10 years
Weight	1.9 kg	Weight	0.63 kg

.

The selected components are reference objects for 6 V low-voltage installation tests. The measurement track included the following components:

• PC with LabVIEW 2018 software: Used to develop an application for recording selected experimental data. The application cooperated with National Instruments measurement modules:
  • NI 9206: Voltage acquisition at battery terminals.
  • NI 9213: Temperature measurement at terminals, battery housing and ambient temperature using K-type thermocouples.
  • NI 9401: TTL digital I/O module for controlling relays in charge and load (discharge) circuits.
• TA018 PICO current clamps: Acquired current values for both components.

The test stand (Figure 5) allowed for the programming of charge and discharge cycles for the energy storage components. Charge and load current values were controlled via an Ethernet connection to a programmable power supply (TTI CPX400DP) and programmable load (TTI LD400P).

The measurement track was protected from damage by pre-set current and temperature limits. If these limits were exceeded, the system automatically disconnected components to prevent damage, especially during long-term operational testing. In addition, a notification system kept the user informed via a TCP/IP network (e.g., a “warning status” when temperature or current thresholds were exceeded), and the programme had reporting capabilities for mobile devices.

The battery and the ultracapacitor (Figure 5) were connected in a passive topology, i.e., physically connected in parallel.

Adopted Load Cycles

The Hybrid Pulse Power Characterisation (HPPC) cycle was performed at the beginning of the study (Figure 6a). The open-circuit voltage (OCV) was determined from the HPPC cycle, as shown in Figure 6b. The U_ocv value ranged from 5.89 V (SoC = 0) to 6.55 V (SoC = 1).

A pseudo-random cycle (Figure 6c) was designed to run the experiment at constant values of SoC = const., transitioning from full battery charge (SoC = 1) to full battery discharge (SoC = 0) in steps of dSoC = 10%. However, due to strongly nonlinear processes at SoC = 1 (instantaneous overload) and SoC = 0 (total discharge), these limits were excluded when identifying the battery parameters.

4. Mathematical Models of the BES-UC System Based on a Replacement Double RC Loop Diagram

This section presents an equivalent circuit of the battery–ultracapacitor module, using a passive parallel connection between the battery and the ultracapacitor (Figure 7). The equivalent diagram of the battery–ultracapacitor module can be divided into two parts: the first, through which the current i_uc flows—this part is related to the ultracapacitor—and the second, through which the current i_b flows—this part is related to the battery.

Based on the analysis of the current distribution and according to Kirchhoff’s first law, it can be written for Figure 7 as follows:

$$i=i_b+i_{uc}\Rightarrow i_b=i-i_{uc}$$

(5)

Respectively, the voltage changes for the battery and the ultracapacitor can be written as a system of equations:

$$\left\{\begin{array}{l}u=U_t=U_{OCV}-U_1-U_2-U_0\\ u=u_c-u_L-u_{ESR}=u_c-L{\displaystyle \frac{d{i}_{uc}}{dt}}-i_{uc}ESR\end{array}\right.$$

(6)

In Equation (6), the voltages U₁, U₂ and U₀ are unknown. They can be determined from the following system of equations:

$$\left\{\begin{array}{l}{\dot{U}}_1=-{\left(R_1{C}_1\right)}^{-1}{U}_1+C_1^{-1}{I}_b\Rightarrow U_1=R_1{C}_1\left(C_1^{-1}{I}_b-{\dot{U}}_1\right)\\ {\dot{U}}_2=-{\left(R_2{C}_2\right)}^{-1}{U}_2+C_2^{-1}{I}_b\Rightarrow U_2=R_2{C}_2\left(C_2^{-1}{I}_b-{\dot{U}}_2\right)\\ 0=-R_0^{-1}{U}_0+I_b\Rightarrow U_0=I_b{R}_0\end{array}\right.$$

(7)

Another unknown parameter in dependency (7) is the voltage, u_c, which can be calculated based on the diagram analysis from Figure 7:

$$\left\{\begin{array}{l}{u}_c=u_{c1}+u_{c2}\\ {\dot{u}}_{c_1}=-{\left(R_1{C}_{1uc}\right)}^{-1}{u}_{c1}+C_{1uc}^{-1}{i}_{uc}\Rightarrow u_{c1}=R_1{C}_{1uc}\left(C_{1uc}^{-1}{i}_{uc}-{\dot{u}}_{c_1}\right)\\ {\dot{u}}_{c2}=-{\left(R_2{C}_{2uc}\right)}^{-1}{u}_{c2}+C_{2uc}^{-1}{i}_{uc}\Rightarrow u_{c2}=R_2{C}_{2uc}\left(C_{2uc}^{-1}{i}_{uc}-{\dot{u}}_{c2}\right)\end{array}\right.$$

(8)

We know that for the ultracapacitor, the relationship between current i_uc and voltage uc can be written as

$$i_{uc}=C{\displaystyle \frac{d{u}_c}{dt}}=C_{1uc}{\displaystyle \frac{d{u}_{c1}}{dt}}+C_{2uc}{\displaystyle \frac{d{u}_{c2}}{dt}}=C_{1uc}{\dot{u}}_{c1}+C_{2uc}{\dot{u}}_{c2}$$

(9)

Inserting Equation (5) into Equation (6) results in

$$u-R_1{i}_{uc}-{\displaystyle \frac{R_2{C}_2}{C_1}}{i}_{uc}-R_0{i}_{uc}=U_{OCV}-R_{1}i+R_1{C}_1{\dot{u}}_1-{\displaystyle \frac{R_2{C}_2}{C_1}}i+R_2{C}_2{\dot{u}}_2-R_{0}i$$

(10)

By inserting u_1c and u_c2 from Equation (8) into Equation (6), we obtain

$$\begin{gathered} u=u_{c1}+u_{c2}-u_L-u_{ESR}=u_c-L{\displaystyle \frac{d{i}_{uc}}{dt}}-i_{uc}ESR=R_1{C}_{1uc}\left(C_{1uc}^{-1}{i}_{uc}-{\dot{u}}_{c_1}\right)+ \\ {R}_2{C}_{2uc}\left(C_{2uc}^{-1}{i}_{uc}-{\dot{u}}_{c2}\right)-L{\displaystyle \frac{d{i}_{uc}}{dt}}-i_{uc}ESR \end{gathered}$$

(11)

Replacing i_uc in Equation (12) gives

$$\begin{array}{c}u=R_1{C}_{1uc}\left(C_{1uc}^{-1}\left(C_{1uc}{\dot{u}}_{c1}+C_{2uc}{\dot{u}}_{c2}\right)-{\dot{u}}_{c_1}\right)\hfill \\ +R_2{C}_{2uc}\left(C_{2uc}^{-1}\left(C_{1uc}{\dot{u}}_{c1}+C_{2uc}{\dot{u}}_{c2}\right)-{\dot{u}}_{c2}\right)+\hfill \\ -L\left(C_{1uc}{\ddot{u}}_{c1}+C_{2uc}{\ddot{u}}_{c2}\right)-\left(C_{1uc}{\dot{u}}_{c1}+C_{2uc}{\dot{u}}_{c2}\right)ESR\hfill \end{array}$$

(12)

Solving Equation (10) with Equation (12), we obtain the final form of the relationship in the battery–ultracapacitor module with two RC loops:

$$\begin{array}{c}{U}_{OCV}-R_{1}i+R_1{i}_{uc}+R_1{C}_1{\dot{u}}_1-{\displaystyle \frac{R_2{C}_2}{C_1}}i+{\displaystyle \frac{R_2{C}_2}{C_1}}{i}_{uc}+R_2{C}_2{\dot{u}}_2-R_{0}i+R_0{i}_{uc}=\hfill \\ u+L\left(C_{1uc}{\ddot{u}}_{c1}+C_{2uc}{\ddot{u}}_{c2}\right)+\left(C_{1uc}{\dot{u}}_{c1}+C_{2uc}{\dot{u}}_{c2}\right)ESR+\hfill \\ -R_1{C}_{1uc}\left(C_{1uc}^{-1}\left(C_{1uc}{\dot{u}}_{c1}+C_{2uc}{\dot{u}}_{c2}\right)-{\dot{u}}_{c_1}\right)\hfill \\ -R_2{C}_{2uc}\left(C_{2uc}^{-1}\left(C_{1uc}{\dot{u}}_{c1}+C_{2uc}{\dot{u}}_{c2}\right)-{\dot{u}}_{c2}\right)\hfill \end{array}$$

(13)

Reordering relationship (13), we obtain

$$\begin{gathered} U_{OCV}-\left(R_1+C_1^{-1}{C}_2{R}_2-R_0\right) i+R_1{C}_1{\dot{u}}_1+R_2{C}_2{\dot{u}}_2= \\ u+\left(R_1+C_2^{-1}{C}_1{R}_2+R_0\right) i_{uc}+L\left(C_{1uc}{\ddot{u}}_{c1}+C_{2uc}{\ddot{u}}_{c2}\right) \\ +ESR\left(C_{1uc}{\dot{u}}_{c1}+C_{2uc}{\dot{u}}_{c2}\right) \end{gathered}$$

(14)

Equations (5)–(14) were implemented in MATLAB and Simulink programmes to identify the parameters of the battery–ultracapacitor module model in Section 5.

5. The Process of BES-UC Parameter Identification and Error Analysis

5.1. Identification of BES-UC Model Parameters

This subsection outlines the methods for identifying parameters of the components of the parallel-connected battery–ultracapacitor module, an equivalent circuit with two RC loops.

In the system of differential equations that serves as the mathematical model of the hybrid energy storage system (Figure 8) with a parallel connection of a BES and UC, each element consists of two RC loops (2-RC), a series ESR resistor and an electromotive force source. The complete model includes six battery parameters (R_{1_BES}, R_{2_BES}, C_{1_BES}, C_{2_BES}, R_{0_BES} and U_{OCV_BES}) and six ultracapacitor parameters (R_{1_UC}, R_{2_UC}, C_{1_UC}, C_{2_UC}, ESR_{_UC} and EMF_{_UC}). Differential equations describing the system are automatically generated by the MATLAB 2020a SimScape software (a system fragment is shown in Figure 8) but can be provided manually.

The input signal to the loop model is the total current signal obtained from experimental research. This is similar to the method for identifying individual battery and ultracapacitor parameters. Since the battery and the ultracapacitor currents were measured independently during the bench tests, their parameters were identified separately. This approach reduced the number of identified parameters from six to twelve, significantly speeding up the identification process.

By solving the system of differential equations, the voltage waveform across the common terminals of the battery and ultracapacitor (connected in parallel) is estimated as a function of time. The voltage waveform depends on the values assigned to the battery parameters (R_{1_BES}, R_{2_BES}, C_{1_BES}, C_{2_BES}, R_{0_BES} and U_{OCV_BES}) and the ultracapacitor parameters (R_{1_UC}, R_{2_UC}, C_{1_UC}, C_{2_UC}, ESR_{_UC} and EMF_{_UC}). The identification process minimises the difference between the simulated and measured voltage waveform obtained during laboratory tests.

The identification task is to find a set of six battery parameters (R_{1_BES}, R_{2_BES}, C_{1_BES}, C_{2_BES}, R_{0_BES} and U_{OCV_BES}) and six ultracapacitor parameters (R_{1_UC}, R_{2_UC}, C_{1_UC}, C_{2_UC}, ESR_{_UC} and EMF_{_UC}), such that the simulated voltage waveforms and the measured voltage waveforms during testing differ as little as possible. The measure of the identification error is the integral of the squared error e² = (y_E − y(Ξ,x,u))², where y(Ξ,x,u) is the calculated function and y_E is the measured value from the tests. The identification process was performed using a “nonlinear data-fitting” procedure that minimised the value of the integral J of the squared error e² by iteratively modifying the values of the parameter vector Ξ = [R₁, R₂, C₁, C₂, ESR/R₀, EMF/U_OCV]. This was achieved by applying increments ΔΞ = [ΔR₁, ΔR₂, ΔC₁, ΔC₂, ΔESR/ΔR₀, ΔEMF/ΔU_OCV] (for the battery and ultracapacitor, respectively) using the values of the gradient (∇J) and Hessian (H) values, as described by Equations (15)–(17).

$$J={\int }_{t_0}^{t_{end}}{e}^{2}dt={\int }_{t_0}^{t_{end}}{\left(y_E-y(\Xi ,x,u)\right)}^{2}dt$$

(15)

Knowing that

$${\displaystyle \frac{\partial J}{\partial \Xi }}=-{\int }_{t_0}^{t_{end}}2e{\displaystyle \frac{\partial y(\Xi ,x,u)}{\partial \Xi }}dt={\int }_{t_0}^{t_{end}}{\left(y_E-y(\Xi ,x,u)\right)}^{2}dt$$

(16)

The Jacobian function J reaches a minimum under the following condition:

$$\begin{gathered} J=\mathit{min}\iff {\displaystyle \frac{J(\Xi +\Delta \Xi )-J\left(\Xi \right)}{\Delta \Xi }}={\displaystyle \frac{\partial J}{\partial \Xi }}\Delta \Xi +{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\partial }^{2}J}{\partial {\Xi }^2}}\Delta {\Xi }^2=0\Rightarrow \Delta \Xi =-2{\displaystyle \frac{\frac{\partial J}{\partial \Xi }}{\frac{{\partial }^{2}J}{\partial {\Xi }^2}}} \\ =-2{\displaystyle \frac{\nabla J}{H}} \end{gathered}$$

(17)

To modify the parameter values Ξ = [R₁, R₂, C₁, C₂, ESR/R₀, EMF/U_OCV] (for the battery and ultracapacitor, respectively), the Levenberg–Marquardt error backpropagation method was used. This method uses appropriately weighted gradient approximations of the minimised function (∇J) and its Hessian (H), with the Jacobian being recalculated at the end of each iteration. The Jacobian was determined based on the obtained error change Δe, related to the parameter vector change ΔΞ.

The adaptation of parameter vector values Ξ = [R₁, R₂, C₁, C₂, ESR/R₀, EMF/U_OCV], through successive increments ΔΞ = [ΔR₁, ΔR₂, ΔC₁, ΔC₂, ΔESR/ΔR₀, ΔEMF/ΔU_OCV], was carried out iteratively until the stationary value of the integral J of the square error e² was achieved. This indicated that a local minimum had been reached. Stationarity was defined as changes in the relative value |ΔJ/J| < ε_ps, where ε_ps = 1 × 10⁻¹⁰ was assumed in the performed tests.

The correct assumption of the initial values for the parameters Ξ₀ = [R₁₀, R₂₀, C₁₀, C₂₀, ESR₀/R₀, EMF₀/U_OCV0] was crucial. Converging to a sub-optimal local solution in gradient-based methods is risky rather than using the global optimum. Therefore, the initial parameter values were set to typical orders of magnitude for the tested battery and ultracapacitor. The following initial conditions were adopted:

• Battery: Ξ_{0_BES} = [R_{10_BES} = 1.0 × 10, R_{20_BES} = 1.0 × 10⁻², C_{10_BES} = 1.0 × 10, C_{20_BES} = 1.0, R_{0_BES} = 1.0 × 10⁻², U_{OCV0_BES} = 6.0];
• Ultracapacitor: Ξ_{0_UC} = [R_{10_UC} = 1.0, R_{20_UC} = 1.0 × 10⁻³, C_{10_UC} = 1.0 × 10⁻³, C_{20_UC} = 1.0 × 10, ESR_{0_UC} = 1.0 × 10⁻³, EMF_{0_UC}/U_{OCV0_UC} = 6.0].

Additionally, all parameter values were constrained to be positive during the identification process.

The identified parameters of the 2RC/DP model of the VRLA battery operating in a hybrid module with an ultracapacitor were expressed as polynomial functions of the SoC (0.1 ≤ SoC ≤ 0.9). The general forms are as follows:

$$\begin{gathered} {R_{int}}_{BES}\left(SoC\right)=A_{RintBES}So{C}^6+B_{RintBES}So{C}^5+C_{RintBES}So{C}^4+D_{RintBES}So{C}^3 \\ +E_{RintBES}So{C}^2+F_{RintBES}SoC+G_{RintBES} \\ {R}_{1BES}\left(SoC\right)=A_{R1BES}So{C}^6+B_{R1BES}So{C}^5+C_{R1BES}So{C}^4+D_{R1BES}So{C}^3 \\ +E_{R1BES}So{C}^2+F_{R1BES}SoC+G_{R1BES} \\ {R}_{2BES}\left(SoC\right)=A_{R2BES}So{C}^7+B_{R2BES}So{C}^6+C_{R2BES}So{C}^5+D_{R2BES}So{C}^4 \\ +E_{R2BES}So{C}^3+F_{R2BES}So{C}^2+G_{R2BES}SoC+H_{R2BES} \\ {C}_{1BES}\left(SoC\right)=A_{C1BES}So{C}^5+B_{C1BES}So{C}^4+D_{C1BES}So{C}^3+E_{C1BES}So{C}^2 \\ +F_{C1BES}SoC+G_{C1BES} \\ {C}_{2BES}\left(SoC\right)=A_{C2BES}So{C}^5+B_{C2BES}So{C}^4+D_{C2BES}So{C}^3+E_{C2BES}So{C}^2 \\ +F_{C2BES}SoC+G_{C2BES} \\ {U}_{OCVBES}\left(SoC\right)=A_{U_{OCVBES}}So{C}^2+B_{U_{OCVBES}}SoC+D_{U_{OCVBES}} \end{gathered}$$

(18)

Table 2. Values of the polynomial coefficients for the VRLA battery (BES) operating in a hybrid module with an ultracapacitor, derived from Equation (18) for the 2RC/DP model.

Parameter Name	Rint BES	R1 BES	R2 BES	C1 BES	C2 BES	UOCV BES
A	−11	4.2 × 10−3	26	1.8 × 10−4	1.3 × 10−4	1.1
B	34	−1.3 × 10−4	−84	−4.1 × 10−4	−3.1 × 10−4	−1.7
C	−43	1.4 × 10−4	1.1 × 10−2	3.4 × 10−4	2.8 × 10−4	5.7
D	26	−8.1 × 10−3	−74	−1.2 × 10−4	−9.1 × 10−3	–
E	−7.5	2.4 × 10−3	27	1.6 × 10−3	1.2 × 10−3	–
F	0.91	−3.7 × 10−2	−5.5	72	–46	–
G	0.033	55	0.53	–	–	–
H	–	–	0.01	–	–	–

presents the values of the polynomial coefficients for the identified parameters, R_1BES(SoC), R_2BES(SoC), C_1BES(SoC), C_2BES(SoC), ESR_BES(SoC) and U_OCVBES(SoC), as shown in Equation (18).

Figure 9 shows graphs of the identified parameters of the battery: (a) U_{OCV BES} = f(SoC), (b) R_{int BES} = f(SoC), (c) R_{1 BES} = f(SoC), (d) R_{2 BES} = f(SoC), (e) C_{1 BES} = f(SoC) and (f) C_{2 BES} = f(SoC).

The identified parameters of the 2RC/DP model of the ultracapacitor in a hybrid module with a battery over the SoC range from 0.1 to 0.9 are described by polynomial functions in the following form:

$$\begin{gathered} ES{R}_{UC}\left(SoC\right)=A_{ESRUC}So{C}^6+B_{ESRUC}So{C}^5+C_{ESRUC}So{C}^4+D_{ESRUC}So{C}^3 \\ +E_{ESRUC}So{C}^2+F_{ESRUC}SoC+G_{ESRUC} \\ {R}_{1\mathrm{UC}}\left(SoC\right)=A_{1\mathrm{UC}}So{C}^5+B_{1\mathrm{UC}}So{C}^4+C_{1\mathrm{UC}}So{C}^3+D_{1\mathrm{UC}}So{C}^2+E_{1\mathrm{UC}}SoC+F_{1\mathrm{UC}} \\ {R}_{2\mathrm{UC}}\left(SoC\right)=A_{R2\mathrm{UC}}So{C}^5+B_{R2\mathrm{UC}}So{C}^4+D_{R2\mathrm{UC}}So{C}^3+E_{R2\mathrm{UC}}So{C}^2+F_{R2\mathrm{UC}}SoC \\ +G_{R2\mathrm{UC}} \\ {C}_{1\mathrm{UC}}\left(SoC\right)=A_{C1\mathrm{UC}}So{C}^6+B_{C1\mathrm{UC}}So{C}^5+D_{C1\mathrm{UC}}So{C}^4+E_{C1\mathrm{UC}}So{C}^3+F_{C1\mathrm{UC}}So{C}^2 \\ +G_{C1\mathrm{UC}}SoC+H_{C1\mathrm{UC}} \\ {C}_{2\mathrm{UC}}\left(SoC\right)=A_{C2\mathrm{UC}}So{C}^5+B_{C2\mathrm{UC}}So{C}^4+D_{C2\mathrm{UC}}So{C}^3+E_{C2\mathrm{UC}}So{C}^2+F_{C2\mathrm{UC}}SoC \\ +G_{C2\mathrm{UC}} \\ {U}_{OCV\mathrm{UC}}\left(SoC\right)=A_{U_{OCV\mathrm{UC}}}So{C}^2+B_{U_{OCV\mathrm{UC}}}SoC+D_{U_{OCVUC}} \end{gathered}$$

(19)

Table 3. Values of the polynomial coefficients for an ultracapacitor operating in a hybrid module with a VRLA battery, as derived from Equation (19) for the DP model.

Parameter Name	ESRUC	R1UC	R2UC	C1UC	C2UC	UOCV UC
A	−5.2	1 × 10−3	8.6	−5 × 10−4	−8.4 × 10−4	0.87
B	15	−3.9 × 10−3	−24	1.6 × 10−5	2.7 × 10−5	−1.3
C	−17	4.8 × 10−3	24	−2.2 × 10−5	−3.4 × 10−5	1.3
D	8.3	−2.3 × 10−3	−12	1.5 × 10−5	2 × 10−5	5.8
E	−1.8	4.6 × 10−2	2.7	−5.6 × 10−4	−5.8 × 10−4	–
F	0.12	−6.2	−0.27	1.2 × 10−4	7.4 × 10−3	–
G	0.018	–	0.027	−1.2 × 10−3	−3.3 × 10−2	–
H	–	–	–	1.2 × 10−2	–	–

presents the values of the polynomial coefficients of the identified parameters, R_{1 UC}(SoC), R_{2 UC}(SoC), C_{1 UC}(SoC), C_{2 UC}(SoC), ESR_UC(SoC) and U_{OCV UC}(SoC), as shown in Equation (19).

Figure 10 shows graphs of the identified parameters of the ultracapacitor: (a) U_{OCV UC} = f(SoC), (b) ESR_UC = f(SoC), (c) R_{1 UC} = f(SoC), (d) R_{2 UC} = f(SoC), (e) C_{1 UC} = f(SoC) and (f) C_{2 UC} = f(SoC).

5.2. Error Analysis

This section describes the indicators used to assess the identification accuracy of parameter identification. The BES voltage estimation error is defined as follows:

$$e_{rr}\left(k\right)=y_{DP}\left(j\right)-y\left(j\right)$$

(20)

where y(j) = U_{z experiment}(j) is the experimental voltage, y_DP(j) = U_{z model DP}(j) is the model voltage and j is the number of samples.

The Mean Square Error (MSE) is calculated as follows:

$$MSE={\displaystyle \frac{1}{N}}\sum _{k=1}^N\left[e_{rr}\right(j){]}^2$$

(21)

The quality of a single resistive–capacitive model (DP) with two RC loops for the battery and the ultracapacitor was evaluated using the MSE, while the goodness of fit between the model-estimated data from the model and the experimental data were evaluated using the Normalised Root Mean Square Error (NRMSE) against 1 [77]:

$$NRMSE(j)=1-{\displaystyle \frac{‖e_{rr}\left(j\right)‖}{‖y\left(j\right)-\overline{y}‖}}$$

(22)

where $\overline{y}$ is the average value of the modelled signal. The NRSME value ranges from –∞ to 1, where a value close to 1 indicates a good fit.

The summary tables in Section 6 present the MSE and NRMSE values for the considered SoC values during pseudo-random cycles for the hybrid BES-UC systems.

6. Comparison of Results Obtained from the Experiment and the Model for the HESS

This section presents validation results of voltage waveforms in the hybrid system, incorporating the battery (BES) and an ultracapacitor (UC) using the 2-RC model. The results cover SoC values from 0.1 to 0.9. Selected results for SoC = 0.1, 0.5 and 0.9 are shown in the following figures:

✓

SoC = 0.1: Figure 11a (BES) and Figure 11b (UC);

✓

SoC = 0.5: Figure 12a (BES) and Figure 12b (UC);

✓

SoC = 0.9: Figure 13a (BES) and Figure 13b (UC).

All of the examples are shown for a pseudo-random pulse loading cycle.

The maximum voltage error for the battery and ultracapacitor occurred at SoC = 0.9:

✓

For the battery, the maximum error was 0.15 V (Figure 13a);

✓

For the ultracapacitor, the maximum voltage error was 0.19 V (Figure 13b).

In other cases, for both the battery and the ultracapacitor, within the SoC range from 0.1 to 0.8, the voltage error did not exceed 0.12 V. The results obtained are considered satisfactory.

Table 4. Comparison of Results Obtained from the Experiment and the Model presented in papers [47,54,55,57,78] and HESS presented in this paper.

Parameter	BES in [47]	BES in [54]	BES in [55]	BES in [57]	HESS in [78]	This Paper
BES equivalent scheme	1-RC (Thevenin), PNGV, 2-RC (DP),	PNGV	2-RC (DP)	2-RC (DP)	1-RC	2-RC (DP)
UC equivalent scheme	n.a	n.a.	n.a.	n.a.	1-RC	2-RC (DP)
HESS equivalent scheme	n.a.	n.a.	n.a.	n.a.	1-RC	2-RC (DP)
Parameter identification method	Genetic algorithm	Kalman filter algorithm	Levenberg–Marquardt algorithm	Extended Kalman filter algorithm	EKF, UKF, RLS	Levenberg–Marquardt algorithm
Error indicator name	V2	Error	RMSE	Error	RMSE, MAE	MSE, NRMSE
Error indicator accuracy	1-RC, PNGV: Medium-high 2-RC (DP): High	PNGV: medium-high	2-RC (DP): high	2-RC (DP): high	High: MAE, Medium-high: RMSE	High for MSE, Medium-high for NRMSE
Experimental validation test	HPPC, DST	Dynamic test	Pulse test	Normalised FTP-75 test	DST, UDDS	HPPC, Pseudo-random
Usability in practical applications	Yes	Yes	Yes	Yes	Yes	Yes
Scalability	Yes	Yes	Yes	Yes	Yes	Yes
Objects in practical applications	EV	EV	EV	EV	EV	EV, RES integration

includes a comparison of the results obtained from the experiment and the model presented in papers [47,54,55,57,78] and the HESS presented in this paper.

Table 5. Error indicator values for HESS components (BES-UC) using the 2-RC/DP model.

State of Charge (SoC)	DP ModelBES–UC
	MSEDP–UC	NRMSEDP–UC	MSEDP–BES	NRMSEDP–BES
0.1	0.0006	0.7386	0.0002	0.8978
0.2	0.0009	0.6232	0.0001	0.9111
0.3	0.0016	0.4810	0.0001	0.9113
0.4	0.0017	0.4468	0.0002	0.9046
0.5	0.0004	0.7245	0.0002	0.8931
0.6	0.0010	0.5730	0.0002	0.8778
0.7	0.0007	0.6518	0.0003	0.8623
0.8	0.0007	0.6686	0.0004	0.8416
0.9	0.0024	0.4611	0.0011	0.7597

summarises the MSE and NRMSE values for SoC values ranging from 0.1 to 0.9. The MSE values are low:

-

For the battery, they do not exceed 1.1 × 10⁻³ (at SoC = 0.9).

-

For the ultracapacitor, they do not exceed 2.4 × 10⁻³ (at SoC = 0.9).

The lowest NRMSE value for the battery is 0.7597 at SoC = 0.9, while for the ultracapacitor, it is 0.4468 at SoC = 0.4. Adding more RC loops could increase the model’s accuracy, as shown in [47,48,49,50,51,52,53,54,55,79] and in Table 4. However, this comes at the cost of significantly longer parameter identification times. The approach presented in this paper represents a compromise between achieving high accuracy in estimating unmeasurable parameters, minimising parameter identification time and maintaining the accuracy of the identified parameters.

For a 1-RC loop, it is necessary to identify four parameters for the battery (R_{1_BES}, C_{1_BES}, R_{0_BES} and U_{OCV_BES}) and four for the ultracapacitor (R_{1 UC}, C_{1 UC}, ESR_UC and U_{OCV UC}), for a total of eight parameters. For a 2-RC loop (as used in this work), twelve parameters need to be identified for both components. For a 3-RC loop, the number increases to sixteen parameters, while for a 4-RC loop, it increases to twenty parameters.

As shown in [79], using more than two 2-RC loops does not significantly improve the accuracy of voltage estimation. However, it significantly increases the time required to identify the model parameters, even by a factor of several times.

7. Conclusions and Recommendations

This study presents experimental and modelling research on energy storage using a battery–ultracapacitor module in a passive parallel connection. These energy storage systems have potential applications in various transport and infrastructure segments, including electric vehicles, plug-in hybrid vehicles, machines, wheelchairs and mobile robots, considering scalability.

Table 6. Summary of applications and recommendations for BES, UC and HESS energy storage technologies.

Role		Application	Description/Recommendation	HESS	BES	UC
Transportation and Infrastructure	Electromobility operation	Flexible operation of primary energy sources and RES	Limiting the value of pulsating loads of primary energy sources (e.g., ICE, FC) and RES. Reducing ICE start-up time in extreme ambient conditions (e.g., low temperature) by increasing volumetric power density.	+		+
		EV recharging process	Reduction of charging time through the use of higher charging power. Reduction of the risk of thermal instability. Increase in the reliability of charging at low temperatures by providing energy from the low-voltage system to start the high-voltage system in the drive system of an electric vehicle. Extension of battery durability.	+		+
	Sustainable development of EV infrastructure	Sustainable development of charging points	Addressing areas with low population density and lack of electrification by providing access to autonomous RES-based charging points. Ensuring the sustainable development of charging points along communication routes (e.g., TEN-T). Facilitating bi-directional energy flow at charging points in urban areas.	+	+	+

summarises the potential applications of BES, UC and HESS energy storage technologies and recommendations for their use in the transportation and infrastructure sectors.

The main advantages, disadvantages and limitations of the HESS study based on the 2-RC model are presented in

Table 7. Advantages and limitations of the 2-RC model-based HESS study.

Aspect	Pros	Cons
Hybrid energy storage system	Demonstrates efficient integration of battery and ultracapacitor for energy density and power density.	Higher cost and increased system mass due to additional ultracapacitor components.
2-RC circuit model	Provides high accuracy for voltage and parameter estimation.	Accuracy decreases slightly at SoC extremes (e.g., SoC = 0.1 or 0.9).
Parameter identification	Effective use of the Levenberg–Marquardt algorithm ensures rapid and accurate parameter optimisation.	Requires careful selection of initial parameters to avoid local minima during optimisation.
Error metrics	Low MSE (<0.0024) and reasonable NRMSE values indicate good model reliability.	Lower NRMSE for ultracapacitors compared to batteries suggests scope for further refinement.
Experimental validation	Real-world testing using pseudo-random and HPPC cycles validates the model’s accuracy.	Requires specialised equipment and significant computational resources for parameter fitting.
Practical application	Applicable for EVs, renewable energy systems and other transport applications.	Increased complexity requires advanced battery management systems for real-world use.
Sustainability impact	Supports renewable energy integration and EV infrastructure development.	Limited immediate scalability for cost-sensitive or lightweight applications.

. The HESS module has several advantages in mobile applications, including limiting currents, smoothing voltage variations in the system, and achieving high energy density (battery effect) and high power density (ultracapacitor effect), which are included in Table 7. However, the main disadvantages of such a module in mobile applications are higher cost, increased system mass and the need to monitor the voltage of each component in the system through the BMS.

Another significant advantage of hybrid energy storage in stationary applications is its potential for the start-up of massive gas engine-powered systems, particularly those transported by gas engine systems used at temporary events such as concerts. These stores can also be used with wind turbines and other renewable energy sources to increase flexibility. Their use in zero-emission buildings and off-grid installations is recommended, considering households’ and prosumer farms’ daily energy demand cycles equipped with micro-renewable energy systems. Hybrid energy storage can contribute to developing sustainable infrastructure for electric vehicles using only renewable energy. This could accelerate their development, particularly on intercity routes and in low-density areas.

This work highlights the usefulness of a novel analysis method using models based on replacement diagrams with a 2-RC loop for modelling HESSs in the form of a battery and an ultracapacitor connected in a passive parallel topology. The MSE values for the battery and the ultracapacitor did not exceed 0.0024, while the NRMSE values were not lower than 0.7597 for the battery and 0.4468 for the ultracapacitor, respectively, at SoC = 0.9. These results confirm that the developed simulation models based on equivalent schemes provide highly accurate results and can be used for model-based estimation of non-measurable parameters for the battery–ultracapacitor module. The obtained results can assist in the component selection process for HESSs in EV propulsion systems and distributed generation devices.

To foster future perspectives and the study’s limitations, we suggest the following:

-

The exploration of advanced battery management systems (BMSs) for real-time monitoring and balancing is recommended.

-

Integrating hybrid energy storage with smart grids is recommended to optimise energy flow and improve grid stability.

-

Exploration of the 2-RC loop modelling process is recommended to ensure the maintenance of accuracy whilst reducing computational intensity for real-time applications.

-

The research should be extended beyond VRLA-AGM batteries and Maxwell ultracapacitors to enhance the generalisability of findings to other energy storage technologies.