Selection of mathematical model of on-board capacity energy storage as element of hybrid traction unit of motor car rolling stock

2.1. Ideal model

The “Ideal model” is based on an ideal capacitor $C$. The main advantage of this model is simplicity, but its use in research allows us only to obtain a basic understanding of the energy exchange processes between system objects and to form a certain conceptual solution for the use of energy storage devices (Fig. 4(a)).

2.2. Commercial model

The “Commercial model” is the most commonly used SC model, which consists of an ideal capacitor $C$ with equivalent series resistance $R_{ESR}$ (Fig. 4(b)). Equivalent resistance $R_{ESR}$ reproduces the omic losses of the SC in the process of its charge and discharge. This model combines simplicity and sufficient reliability of results in the study of energy storage systems, provided that the simulation of long-term storage of stored energy is not provided.

2.3. Simplified model

The “Commercial model” was supplemented by an equivalent resistor $R_{EPR}$ connected in parallel with a capacitor $C$ that simulates the process of self-discharge of the SC (Fig. 4(c)). This expands the scope of possible application of this model in the study of long-term energy storage projects. However, this model provides only a basic understanding of the energy exchange processes both in the SC and between the objects of the system as it ignores the important physical phenomena inherent in the SC, such as the process of charge redistribution, the variation of the capacitance depending on the voltage level at the terminals and the recovery process.

2.4. 1-branch model

The “1-branch model” has the advantages of the “Simplified Model” and details the behavior of the SC by taking into account the variation in capacitance in response to a change in voltage on the SC in the practical voltage range of the device (Fig. 4(d)). The variation of the SС capacitance is avoided by using a differential capacitor. The differential capacitor is a capacitor having a parallel connected constant capacitance $C_1$ and capacitance $C_V$, which depends on the voltage at the differential capacitor. Among the advantages of this model it is possible to distinguish accuracy of reproduction of SC behavior in short-term period of time during cycles of charge and discharge [17].

2.5. Zubieta-Bonert model

The “Zubieta-Bonert model” allows us to reproduce the behavior of the SC in sufficient detail for 30 minutes. [17] (Fig. 4(e)). It is represented by three parallel connected branches (“immediate branch”, “delayed branch”, “long-term branch”) with different time constants that simulate the work of the SC not only in moments of charge or discharge but also reproduce the recovery processes characteristic of the SC and charge redistribution. “Immediate branch” is a differential capacitor (parallel connected constant capacitance $C_1$ and capacitance $C_V$, which depends on the voltage on the differential capacitor) and resistor $R_1$. “Immediate branch” predominates in the reproduction of the behavior of the SC in the range of seconds in response to a charge or discharge [17, 18]. “Delayed branch” is a series-connected capacitor $C_2$ and resistor $R_2$. “Delayed branch” predominates in the reproduction of the behavior of the SC in the range of minutes [17, 18]. “Long-term branch” is a series-connected capacitor $C_3$ and resistor $R_3$. “Long-term branch” predominates in the reproduction of the behavior of the SC for more than 10 minutes [17, 18]. It should be added that the specificity of this type of mathematical models is to cover only a certain period of time of the SC, thus requiring a clearly defined time interval between the phases of the energy storage such as energy storage and return.

Fig. 4Equivalent circuit models of the SC: a) Ideal model; b) Commercial model; c) Simplified model; d) 1-branch model; e) Zubieta-Bonert model, where C – ideal capacitor; RESR – equivalent series resistance; REPR – equivalent parallel resistance; C1 - fixed capacitance differential capacitor; CV – voltage-dependent capacitance of the differential capacitor; R1, R2, R3 – resistance of branches Immediate branch, Delayed branch, Long-term branch accordingly; C2, C3 – capacitor of branches Delayed branch, Long-term branch accordingly

2.6. Stern model

“Stern model” is able to reproduce the change in the capacity of the SC, in accordance with the nonlinear dynamics of diffusion in it. The capacity of the SC is determined by two components – the Hermholtz capacity and the diffusion capacity. The Hermholtz capacity is a constant component of the SC capacity and depends only on its physical parameters.

The diffusion capacity varies according to the change in the concentration of ions in the electrolyte solution as a result of diffusion and electrostatic forces. The key feature of this model is a detailed reproduction of the processes of charge and discharge of the SC due to the mathematical description of the physicochemical processes occurring in it. As for the mathematical description, each model, in accordance with the attempt to detail the behaviour of the SC, has its own level of complexity. Table 1 shows the mathematical models of those mentioned above.

Table 1Mathematical models of SC

Name	Mathematical model
Ideal model	$U_C=U_{SC}\left(t_0\right)+\underset{t_0}{\overset{t}{\int }}\frac{i_{SC}\left(t\right)}{C}dt,U_{SC}=U_{С}+i_{SC}\left(t\right)\cdot R_{ESR}$
Commercial model	$\frac{d{U}_{С}}{dt}=\frac{1}{R_{ESR}C}\left(U_{SC}-U_C\right),U_{SC}=U_{С}+i_{SC}\left(t\right)\cdot R_{ESR}$
Simplified model	$\frac{d{U}_{С}}{dt}=-\frac{1}{R_{ESR}C}{U}_{С}+\frac{1}{C}{i}_{sc}\left(t\right),U_{SC}=U_{С}+i_{SC}\left(t\right)\cdot R_{ESR}$
1-branch model	$\begin{array}{l}\frac{d{U}_{{С}_{eq}}}{dt}=-\frac{1}{R_{ESR}\cdot {С}_{eq}}{U}_{{С}_{eq}}+\frac{1}{{С}_{eq}}{i}_{SC}\left(t\right),\\ {С}_{eq}=C_0+C_V,C_V=k\cdot U_{{С}_{eq}},\\ U_{SC}=U_{{С}_{eq}}+i_{SC}\left(t\right)R_{ESR}\end{array}$	$k$ – coefficient, the dependence of the capacitance of the SC relative to the voltage at its terminals
Zubieta-Bonert model	$\begin{array}{l}\frac{d{U}_{{С}_{eq}}}{dt}=\frac{1}{R_1{С}_{eq}}{U}_{{С}_{eq}}-\frac{1}{R_1{С}_{eq}}{U}_{SC}\\ {С}_{eq}=C_0+C_V,C_V=k{U}_{{С}_{eq}},\\ U_{SC}=U_{{С}_{eq}}+i_{SC}\left(t\right)R_1\end{array}$
Stern model	$\begin{array}{l}{U}_{SC}=\frac{N_s{Q}_{T}d}{N_p{N}_e\epsilon {\epsilon }_0{A}_i}+\frac{2N_p{N}_{e}RT}{F}\times \\ \times sin{h}^{-1}\left(\frac{Q_T}{N_p{N_e}^2{A}_i\sqrt{8RT\epsilon {\epsilon }_{0}c}}\right)-i_{SC}{R}_{ESR}\end{array}$	$A_i$ – interfacial area between electrodes and electrolyte $c$ – molar concentration $d$ – molecular radius $F$ – faraday constant $Q_T$ – electric charge $R$ – ideal gas constant $T$ – operating temperature $\epsilon$ – permittivity of material ${\epsilon }_0$ – permittivity of free space $N_e$ – number of layers of electrodes $N_p$ – number of parallel supercapacitors $N_s$ – number of series supercapacitors

However, for each of the above models, there are limitations concerning their ability to adequately reproduce the behaviour of the SC only for certain time intervals under certain conditions and for specific purposes. On the grounds of the analysis, Table 2 shows the main advantages and disadvantages of the models under consideration and the directions of possible application in conducting the studies.

In most cases, the time interval of the SC operating cycle provides for three phases – charge, discharge and energy storage. If the reproduction of the charge and discharge phases of the SC model does not need to be clarified in terms of the time interval of their course, then the duration of the energy storage phase plays an important role in the reliability of the simulation results. Firstly, this is due to the possibility of SC models to adequately reproduce the behaviour in this phase only in certain time frames, and secondly, partial or even complete neglect of some models of physical processes of SC in this phase. In general, the duration of the energy storage phase is determined by the concept of reuse of stored energy and spatial location of the drive. For the stationary placing, the range of possible duration of the energy storage phase will be formed by the train schedule. In the case of on-board placement, as shown by the results of studies [5, 14, 15], the appropriate use of stored energy is its focus on optimizing the peak of traction energy consumption, i.e. use in the time of the acceleration vehicle. Under these conditions, the storage phase of the charge will correspond to the stopping time of the rolling stock at the station (Fig. 5).

In this paper, we make a comparative assessment of the above six models for the possibility of their integration into the model of electric traction systems with on-board placement of the energy storage. During the study, a number of assumptions were made that the accuracy of reproduction of SC discharge models is identical to its charge within one cycle “charge-discharge” and the time of the energy storage phase will not exceed 3 minutes. A supercapacitor with a capacity of 50 Faraday of nominal voltage 2.7 V is considered as a test sample. The parameters given in [18] and the technical documentation of the test sample were taken as the basis of the model parameters. The parameters of certain mathematical models are given in Table 3.

The square root of the root mean square error (RMSE) and the mean absolute percentage error (MAPE) were used as indicators of the accuracy of reproducing the behaviour of the SC:

where $y_i$– the $i$-th value of the estimation parameter of the model; ${\widehat{y}}_i$ – the $i$-th true value of the evaluation parameter; $n$ – the number of measurements of the evaluation parameter.

Fig. 5Phases of operation of the energy storage in the case of its onboard placement while optimizing the peak of traction power consumption of the subway electric train

The evaluation parameter $y_i$ for which the comparison was performed is the data of the instantaneous value of the $U_{SC}$ voltage at the SC terminals, obtained during the simulation. The real value ${\widehat{y}}_i$ was the instantaneous value of the voltage at the outputs of the experimental sample $U_{SC}$, obtained during experimental studies (Figs. 6-8).

The phase of SC charge was carried out with a constant current I_SC at the level of 5 A in the operating voltage $U_{SC}$ range from 0.2 V (Cur1) to 2.5 V (Cur2) (Fig. 6). The value of 0.2 V is the relative complexity of the use of all stored energy in a short period of time. In this case, time of SC charge lasted |$\mathrm{\Delta }X$| 23.81 s.

Fig. 6Experimental results of the SC charge phase: USC – voltage at the terminals of the prototype SC; ISC – charge current of the prototype SC; CH1, CH2 – 1 and 2 oscillograph channels, respectively; Relaxation – redistribution of charge in the structure of the SC at the end of the charge phase; Cur1, Cur2 – 1 and 2 measurement cursor respectively; |ΔX| – absolute value of difference in time (along x axis) between Cur1 and Cur2; |ΔY| – absolute value of difference in amplitude signals between Cur1 and Cur2

The phase of SC discharge was carried out in the $U_{SC}$ range from 2.45 V (Cur1) to 1 V (Cur2) at a constant load with a resistance $R_{LOAD}$ of 0.5 Om (Fig. 7). In this case, time of SC discharge |$\mathrm{\Delta }X$| lasted 24.7 s.

Based on the experimental results of measuring the voltage drop values |$\mathrm{\Delta }Y$|, it was found that the internal series resistance of the $R_{ESR}$is 0.027 Om (Fig. 6, 7). It was also found the presence of variation in the value of $R_{ESR}$ in the range of 3-7 % due to changes in charge intensity, the number of continuous cycles “charge-discharge” or the occurrence of disturbances, whether a sharp drop in charge current (Fig. 6, 7; Table 3).

In the course of experimental studies, the presence of charge redistribution in the middle of the SC after the phases of charge (relaxation) and discharge (restoration) of the drive was confirmed (Fig. 6-8). In Fig. 8 shows the oscillograms of the change in the instantaneous voltage of the $U_{SC}$ at the terminals of the SC due to the processes of relaxation and restoration for the time interval |$\mathrm{\Delta }X$| 180 seconds at Cur1 and Cur2 measurement cursors. For the experimental results within the measurement cursors Cur1 and Cur2, the voltage change range |$\mathrm{\Delta }Y$| reached 7-8 %. According to the obtained results, we can state that the key factor that determines the instantaneous value of the SC voltage is mainly the stage of charge redistribution within the SC. Thus, it is somewhat incorrect to use self-discharge in models as a tool for reproducing the relaxation of the SC, because in the process there is no loss of accumulated energy but only its redistribution within the SC itself. Without a doubt, it is impossible to exclude the self-discharge of the SC, but the importance of its influence on the formation of the voltage level is not so great.

Fig. 7Experimental results of the discharge phase of the SC: USC – voltage at the terminals of the prototype SC; ISC – discharge current of the prototype SC; CH1, CH2 – 1 and 2 oscillograph channels, respectively; RLOAD – load resistance SC; Restoration – charge redistribution in the SC structure at the end of the discharge phase; Cur1, Cur2 – 1 and 2 measurement cursor respectively; |ΔX| – absolute | value of difference in time (along x axis) between Cur1 and Cur2; |ΔY| – absolute value of difference in amplitude signals between Cur1 and Cur2

Our experimental setup included a SC, a data acquisition system, a power supply, a current shunt, and a load (Fig. 9). The SC was used by SAMWHA DS series with a capacity of 50 F. A switching power supply with the function of limiting the output current was used as a source of constant power. Data collection was performed using a two-channel digital oscilloscope Instrustar ISDS 205A and software Record Recorder Multi VirAnalyzer. The sampling frequency of the sampling signal is 1 MHz. The load and current shunt are implemented on ceramic resistors with a scattering power of 5 watts.

The processing of the results of experimental data to isolate the based signal was carried out using the Signal Analyzer application of the Matlab software product. The comparative evaluation was carried out on the basis of the results of the simulation of the Matlab software product using the Simscape Electrical physical systems modeling package of the Simulink graphical simulation environment [19]. The “Stern model” was implemented by the “Supercapacitor” Matlab block using the predefined model parameters given in the block. As a numerical integration algorithm, a discrete time with a fixed sampling period of 0.001 s was used. Experimental data were presented in the form of a table function. When performing a comparative evaluation of mathematical models with experimental data in the examined dependent variable, the instantaneous value of stress at SC terminals was determined. As an independent variable for carrying out a comparative evaluation of the reproduction of the charge phase of SC, mathematical models are the charge current of SC at the level of 5 A. For the storage phase of stored energy, the time of the storage phase of stored energy within 3 minutes is selected as an independent variable. Figs. 10 and 11 show the results of the simulation of the charge and storage phases of each model. The parameters given in Table 3 were used as model parameters.

Fig. 8Experimental results of the energy storage phase of the SC: USC – voltage at the terminals of the prototype SC; CH1, CH2 – 1 and 2 oscillograph channels, respectively; Relaxation – redistribution of charge in the structure of the SC at the end of the charge phase; Restoration – charge redistribution in the SC structure at the end of the discharge phase; Cur1, Cur2 – 1 and 2 measurement cursor respectively; |ΔX| – absolute value of difference in time (along x axis) between Cur1 and Cur2; |ΔY| – absolute value of difference in amplitude signals between Cur1 and Cur2

Fig. 9Structural diagram of test platform of SC

Fig. 10Comparative evaluation of mathematical models of SC during the charge phase: USC – voltage at the terminals of the SC, error – models error of the SC

Fig. 11Comparative evaluation of mathematical models of SC during the energy storage phase: USC – voltage at the terminals of the SC, error – models error of the SC

In Fig. 11, we can observe the reproduction of the charge storage phase models. For all models except the “Zubieta-Bonert model”, the change in no-load voltage at the SC terminals is practically absent, because, as noted earlier, these models do not simulate the internal process of charge redistribution.

After analyzing the resulting simulation results and comparing them with experimental data, RMSE and MAPE accuracy measures were calculated and structured in Table 4. RMSE gives relatively great weight to significant errors, which will prevent further distortion of information about the value of the stored energy volume. MAPE has a more supervisory meaning and eliminates the problem of mutual compensation for positive and negative errors. Thus, the use of these accuracy indicators will reduce the likelihood of making a false decision. Determination of RMSE and MAPE was carried out for two separately separated phases of the SC cycle (charge, energy storage). The SC discharge phase was not considered as previously assumed. The accuracy of energy storage phase reproduction was considered within two time intervals of 1 and 3 minutes. This was done to give greater objectivity to the decision made when giving advantage to a particular model, either for hybridization of suburban rail electric transport or traction rolling stock of the subway. A longer storage phase was also determined with a time interval of 12 minutes. The time of 12 minutes corresponds to 95 % cases of travel time between stations by suburban railway transport in Ukraine.

A comparative evaluation of the RMSE and MAPE (Table 4) found the following. All models, except the “Ideal model”, reproduce the SC charge with high accuracy. For these models, the MAPE for the charge phase is in the range from 2.4 to 5.1 % (Table 4). According to Table 4, taking into account the presence of SC diffusion capacity in the models “1-branch model”, “Zubieta-Bonert” and “Stern model” allows to improve the adequacy of the reproduction of the charge phase almost 2 times compared to other models considered. Based on the accuracy indicators of RMSE and MAPE (Table 4), it can be argued that the reproduction of the charge redistribution process by the Zubieta-Bonert model in short-term energy storage phases up to 3 minutes does not significantly increase the adequacy of the model compared to other models, but only allows to achieve high accuracy. The main problem of the “Zubieta-Bonert” and 1-branch model models is their indirect determination of model parameters, based on the results of preliminary experimental studies of the original object.

From RMSE and MAPE (Table 4) it follows that “Simplified model” and “Stern model” provide the highest accuracy of reproduction of charge phases and energy storage in a time interval up to 3 minutes. Based on the accuracy of RMSE and MAPE (Table 4), it can be argued that the reproduction of the charge redistribution process model “Zubieta-Bonert” in the short phases of energy storage up to 3 minutes does not significantly increase the adequacy of the model compared to other models, but only achieves high accuracy in concepts with long-term energy storage. The main problem of the “Zubieta-Bonert” and “1-branch” models is their indirect determination of the model parameters, based on the results of previously conducted experimental studies of the original object. From RMSE and MAPE (Table 4) it follows that “Simplified model” and “Stern model” provide the highest accuracy of reproduction of charge phases and energy storage in a time interval up to 3 minutes. In this case, the majestic charge phase error (Fig. 6) for “Simplified model” and “Stern model” does not exceed 0.08 V and 0.05 V, respectively. Therefore, these models do not significantly distort the information about the amount of energy stored after the charge or discharge. For the energy storage phase after 180 s the magnitude of the error (Fig. 7) reaches 0.19 V and 0.21 V, respectively. At the same time, the results of experimental studies show that the transient process of changing the energy states of the SC after the completion of the charge or discharge phase has a similar nature and leads to a change in voltage at the terminals of the SC in a certain range (Fig. 8). This allows, using predefined constants or time functions, to increase the accuracy of models in the storage phase, without resorting to the complexity of a particular model. This will be sufficient for a closer understanding of the amount of “quickly available” energy and additional energy input due to the subsequent change in the energy state of the SC for recovery after the discharge phase.

If you look at the models in terms of availability of their parameters, the advantage of “Simplified model” is indisputable as the parameters of this model are in common use and are provided by each manufacturer. For the “Stern model” the situation is a bit more complicated, because the mathematical model operates with somewhat specific parameters. Thus, there are developments that allow us to model the SC on the grounds of pre-defined parameters of the model, but this does not give confidence in the reliability of the results without additional experimental studies. However, the possibility of mathematical description of these models as a control object is not significantly difficult. However, the mathematical description of the “Stern model” has a higher level of complexity due to the presence of a hyperbolic function in the mathematical model as opposed to the “Simplified model”.

So, “Simplified model” meets all our criteria and can be presented as a mathematical model of the SC object of the electric traction system under conditions of briefly long cycles of the energy storage. In our opinion, the accuracy of reproducing the SC behavior, taking into account the above steps to assess the energy state of the SC during the charge storage phase, will be sufficient to possess reliable information about the instantaneous value of the stored energy volume and simulate the course of traction-energy processes in the electric traction system of urban and suburban electric transport. However, this model will not allow to correctly simulate the operation of power accumulators of main trains. However, this model will not correctly simulate the operation of power storage of main trains in the case of long-term energy storage.