*3.4. Electric Drive*

The electric drive, just like the internal combustion engines, was modeled using experimental static maps. Laboratory tests allowed for obtaining the torque and efficiency characteristics of the traction electric drive (electric machine + inverter) for two voltage levels, namely 650 V and 450 V, which covered the bulk of the prototype supercapacitor operating range. Between these voltages, the characteristics were interpolated, and were extrapolated below 450 V. The main parameters of the electric drive obtained from the experimental data are summarized in Table 1.


**Table 1.** The main performance parameters of the traction electric drive.

Figure 5 shows the efficiency map and the maximum torque characteristics of the electric drive in motor mode.

Given the electric machine torque (commanded by the control system) and the shaft speed (calculated by the transmission model), as well as the supercapacitor voltage *uSC* calculated via its model, one can obtain the electric drive current *iem*,*dc* at the DC side:

$$\dot{q}\_{cm,dc} = \frac{T\_{cm} \cdot \omega\_{cm}}{\mu\_{SC} \cdot \eta\_{cm}^{s\_{\rm{ST}}(T\_{cm})}},\tag{4}$$

where η *sgn*(*Tem*) *em* is the electric drive efficiency, taking into account the torque direction (positive for the motor mode). The current calculated with this formula is used by the SC model as the input signal.

**Figure 5.** Efficiency map and torque characteristics of the electric drive (motor mode).

### *3.5. Energy Storage System*

The following considerations were taken into account when choosing the approach to mathematical modeling of the supercapacitor. Chemical processes within the SC lie beyond the scope of the research. At the current stage, temperature effects are also neglected under the assumption that the powertrain operates within a moderate range of temperatures, which imposes no substantial effect upon the supercapacitor characteristics. The operating parameters of the SC relevant for the powertrain modeling in this study were the current, voltage, efficiency, and energy content. These considerations suggest using equivalent circuit modeling, which is a widely employed approach for the analysis of ESS not as electrochemical systems but rather electrical components of the powertrain. References [25,31,32] describe an equivalent circuit for the supercapacitor in the form of a "ladder" with each "step" consisting of a capacitor and a resistance. Depending on the required model fidelity, one can use different numbers of "steps," which in the simplest case, would be a single one. The comparison performed in this work with the involvement of experimental data has shown that a reasonable trade-off between model accuracy and complexity is provided by a circuit consisting of three "steps," as shown in Figure 6.

**Figure 6.** Equivalent circuit for a supercapacitor.

In this scheme, *r*1, *r*2, and *r*<sup>3</sup> are the internal resistances; *c*1, *c*2, and *c*<sup>3</sup> are the capacitances; and *uSC* and *iSC* are the voltage and current at the supercapacitor electric terminals.

From the equivalent circuit, one can derive the following equation system using Kirchhoff's current and voltage laws:

$$\begin{cases} \begin{aligned} \boldsymbol{u}\_{\rm SC} &= \boldsymbol{u}\_{1} = \frac{\int \frac{i\_{\rm SC} \, dt - c\_{2} \boldsymbol{u}\_{2} - c\_{3} \boldsymbol{u}\_{3}}{c\_{1}}}{\int \boldsymbol{i}\_{2} &= \frac{\boldsymbol{u}\_{1} - \boldsymbol{u}\_{2} - (\boldsymbol{u}\_{2} - \boldsymbol{u}\_{3}) \boldsymbol{r}\_{2}/r\_{3}}{c\_{2}r\_{2}}}, \\\ \boldsymbol{i}\_{3} &= \frac{\boldsymbol{u}\_{2} - \boldsymbol{u}\_{3}}{c\_{3}r\_{3}}, \end{aligned} \end{cases} \tag{5}$$

where *u*1, *u*2, and *u*<sup>3</sup> are the voltages within the sub-circuits corresponding to each "step" of the "ladder."

The initial conditions can be obtained for this model from the assumption of equality between the voltages of all the capacitors, which takes place when no load is applied and transient processes have ended: *u*<sup>1</sup> = *u*<sup>2</sup> = *u*<sup>3</sup> = *uSC*,*init*. Substituting this condition into the first equation of the system (Equation (5)) yields the initial condition for the integral term:

$$\int i\_{\rm SC} dt = \mu\_{\rm SC,init} (c\_1 + c\_2 + c\_3). \tag{6}$$

The term *c*<sup>1</sup> + *c*<sup>2</sup> + *c*<sup>3</sup> is the total capacitance of the "ladder." Identification of the model parameters using experimental data showed that the best accuracy for voltage calculation was obtained when the sum *c*<sup>1</sup> + *c*<sup>2</sup> + *c*<sup>3</sup> was equal to the rated capacitance *cSC* of the modeled SC. Furthermore, this provided a correct calculation of the supercapacitor energy content with no need for ad hoc corrections.

Figure 7 shows the results of modeling that simulated two experiments involving the supercapacitor installed within the powertrain of the hybrid city bus mentioned in Section 2. The bus was driven in the pure electric mode through velocity patterns consisting of accelerations and decelerations. The top plots demonstrate the supercapacitor current logged during the experiments (positive for charging). The current was used as the input signal for the equivalent circuit model, which responded with the calculated voltage. The latter is shown in the bottom plots (denoted as "Model") along with the measured voltage ("Experiment"). The identification of the model parameters was performed using the criterion of the minimum root mean square error (RMSE) of voltage calculation. The parameter set satisfying this criterion is listed in Table 2. The RMSEs obtained with these parameters amounted 0.7–3% in about a dozen experiments.

**Figure 7.** Comparison of the supercapacitor simulation results with the experimental data.

**Table 2.** Supercapacitor equivalent circuit parameters.


The equivalent circuit includes the internal resistances, which not only provide a voltage drop, but also introduce the dissipation of energy, and therefore, allow for calculating the capacitor efficiency. To do this, it is necessary to determine the power dissipated at the resistances.

The current through the resistance *r*<sup>1</sup> is equal to that at the SC terminals, i.e., *i*<sup>1</sup> = *iSC*. While deriving the system (Equation (5)), one can find that the current through the second resistance is equal to . *<sup>u</sup>*2*c*<sup>2</sup> <sup>+</sup> . *<sup>u</sup>*3*c*3. The current in the third "step" is equal to . *u*3*c*3. Therefore, the total power loss is:

$$P\_{\rm SC,loss} = r\_1 i\_{\rm SC}^2 + r\_2 \left(\dot{u}\_2 c\_2 + \dot{u}\_3 c\_3\right)^2 + r\_3 \left(\dot{u}\_3 c\_3\right)^2. \tag{7}$$

The supercapacitor net power *PSC* = *iSC*·*uSC*. The total power drawn from the SC or received by it is a sum of the net power and the loss power, taking into account the signs thereof. These considerations result in the following expression for the SC efficiency:

$$\eta\_{\rm SC} = \left(\frac{|P\_{\rm SC}|}{|P\_{\rm SC}| - \text{sgn}(P\_{\rm SC}) \cdot P\_{\rm SC,loss}}\right)^{-\text{sgn}(P\_{\rm SC})}.\tag{8}$$

Note that *PSC* is positive for charging.

Figure 8 demonstrates the simulation results for yet another one of the mentioned field tests, in which the supercapacitor was discharged with the maximum continuous current and then charged back to the initial voltage. Three upper plots show the voltage, current, and net power of the SC, respectively. The fourth plot demonstrates the components of power dissipation associated with the three internal resistances of the equivalent circuit. One can see the clear differences in the dynamics of these components. The first term is the major contributor to the power dissipation. The dynamics of this term is defined by the dynamics of supercapacitor loading (i.e., dynamics of the *iSC* current). The second and third terms constitute processes with slower dynamics and lower amplitudes. The SC efficiency (the bottom plot) diminished along with the voltage, while the loading current remained constant, as well as the power dissipation at the resistance *r*1. However, the power losses at the two other resistances grew due to dynamics of the currents at the capacitors *c*<sup>2</sup> and *c*3. Analysis of the obtained results suggests that when using the considered supercapacitor as the ESS in a hybrid powertrain, one should avoid a deep discharge (i.e., below 450 V), either using the control strategy or via supercapacitor sizing or, most likely, both.

**Figure 8.** Model analysis of the power losses and efficiency of the supercapacitor.

It is worth noticing the interval of transition between discharging and charging (in the vicinity of the 14th second), where the power dissipation at *r*<sup>1</sup> drops down to zero while the powers at *r*<sup>2</sup> and *r*<sup>3</sup> continue to attenuate at the rates defined by the parameters of the corresponding "steps" of the equivalent circuit. After the load removal, the capacitors *c*1, *c*2, and *c*<sup>3</sup> exchange energy between each other until their voltages become equal. This process is accompanied by the dissipation of energy at the resistances *r*<sup>2</sup> and *r*3. If no external load is present during these periods of voltage stabilization, the efficiency of the SC becomes negative. However, in the estimation of the SC mean efficiency, such energy losses may be assigned to the preceding operating interval, which allows for avoiding an unnecessary introduction of the negative efficiency.

An alternative way of determining the supercapacitor efficiency (the average for a certain operating period) stems from the fact that the equivalent circuit model incorporates the power dissipation, and the supercapacitor voltage calculated by it reflects the actual state of energy (at least under no-load conditions). Therefore, one can integrate the net power *PSC* and multiply it by the average efficiency, which is calculated to bring the integrated energy level to that of the equivalent circuit model by the end of the operating cycle. This can be expressed as the following equation:

$$
\eta\_{SC}^{\text{sgn}(i\_{SC})} \int\_0^{t-\text{cnd}} P\_{SC} dt = 0.5 c\_{SC} u\_{SC, t-\text{cnd}'}^2 \tag{9}
$$

where #η*SC* is the average efficiency of the SC during the given period, *<sup>t</sup>* <sup>−</sup> *end* is the end time of the period, and *uSC*,*t*−*end* is the SC voltage at the end of the period.
