*3.2. Electric Test Bench*

To compare the hydraulic accumulator with the ultracapacitor set, an electric test bench was designed and constructed. A schematic representation of the test bench is presented in Figure 10.

**Figure 10.** Schematic representation of the test bench used for ultracapacitors.

The ultracapacitors and the rheostats are shown in Figures 11 and 12, respectively.

**Figure 11.** Ultracapacitors used in the electric test bench.

**Figure 12.** Resistor bank used in the electric test bench.

This test bench was designed to measure the flow of energy through the ultracapacitors during charging and discharging. The resistance or load was made using a bank of resistors connected in parallel, which made the value of the resistance variable and easy to adjust manually. The ultracapacitor arrangement had six cells connected in series in a balancing board and six of these boards connected in parallel, so the total number of ultracapacitors used was 36. This number was selected based on calculations for the energy storage capacity for hydraulic accumulators and ultracapacitors. The selection of 36 ultracapacitors made the energy storage capacity comparable between the two systems. Switches S1 to S6 were used to activate the boards connected in parallel.

As mentioned previously, the number of ultracapacitors used in the test bench was based on the theoretical calculations for energy storage capacity in both systems. Energy stored in a hydraulic accumulator can be calculated with the following equation:

$$E\_{\text{acc}} = -\int\_{v\_{\text{g}}}^{v\_f} p dv \tag{4}$$

In Equation (4), *Eacc* is the total energy storage capacity of the hydraulic accumulator, *p* is the pressure, *v*<sup>0</sup> is the initial volume, and *vf* is the final volume. The charging and discharging process of the accumulator can be assumed as adiabatic, and the polytropic index of nitrogen can be assumed as 1.4, according to Rabie [20], so the relationship between pressure and the volume in the hydraulic accumulator can be expressed as follows:

$$pv^n = p\_0 v\_0^n \tag{5}$$

Plugging Equation (5) into Equation (4) to obtain Equation (6):

$$E\_{\rm acc} = \int\_{V\_{\rm o}}^{V\_f} p\_0 v\_0^n v^{-n} dv$$

$$E\_{\rm acc} = p\_0 v\_0^n \frac{v^{1-n}}{1-n} \Big|\_{v\_0}^{v\_f}$$

$$E\_{\rm acc} = \frac{p\_0 v\_0^n}{1-n} \Big[v\_f^{1-n} - v\_0^{1-n}\Big] \tag{6}$$

The final compressed volume can be expressed as a function of the maximum pressure in the accumulator.

$$p\_{\max}v\_f^n = p\_0v\_0^n$$

$$w\_f = \left(\frac{p\_0}{p\_{\max}}\right)^{1/n}v\_0\tag{7}$$

The final equation for energy in the accumulator can be obtained by plugging Equation (7) into Equation (6). Equation (8) is the expression for the energy in the hydraulic accumulator:

$$E\_{\rm acc} = \frac{p\_0 v\_0}{n - 1} \left[ \left( \frac{p\_0}{p\_{\rm max}} \right)^{\frac{1 - n}{n}} - 1 \right] \tag{8}$$

In Equation (8), *p*<sup>0</sup> is the precharge pressure in the hydraulic accumulator, *v*<sup>0</sup> is the initial gas volume, *pmax* is the maximum pressure, and *n* is the ideal gas constant. The values used for the calculation of energy capacity in the hydraulic accumulator are shown in Table 7.

**Table 7.** Estimated energy capacity of the hydraulic accumulator.


The energy storage capacity of the ultracapacitor arrangement needs to be approximately equal to the energy estimated from Equation (8) for the systems to be comparable. The energy in an ultracapacitor can be calculated with Equation (9):

$$E\_{\rm ult} = \frac{1}{2} C V\_{\rm ut}^2 \tag{9}$$

In Equation (9), *C* is the capacitance and *Vut* is the voltage of the ultracapacitors. The capacitance and the voltage of the arrangement can be calculated as function of the number of cells in series (*NC*) and the number of boards in parallel (*NB*) with Equations (10) and (11):

$$\mathbf{C} = \frac{N\_B}{N\_C} \mathbf{C}\_{cell} \tag{10}$$

$$\mathcal{V}\_{\rm ut} = \mathcal{N}\_{\mathbb{C}} \mathcal{V}\_{\rm cell} \tag{11}$$

The total energy in the ultracapacitor arrangement can be calculated with Equation (12):

$$E\_{\rm ult} = \frac{1}{2} \mathbb{C}\_{\rm cell} V\_{\rm cell} \mathrm{l}^2(\mathrm{N}\_{\rm B} \mathrm{N}\_{\rm C}) \tag{12}$$

The energy capacity of the ultracapacitors is close to the energy capacity in an accumulator with six cells connected in series in a single board and six boards connected in parallel. The estimated energy storage capacity of the ultracapacitor arrangement is shown in Table 8.

**Table 8.** Estimated energy capacity of the ultracapacitors.


As mentioned previously, this test bench was designed to measure the flow of energy through the ultracapacitors while charging and discharging. During charging mode, switch SB was on (see Figure 10), while switch SR was off. The current in the circuit depended on the value of the resistance. During discharge mode, switch SB was turned off and switch SR was turned on, which allowed the current to flow from the ultracapacitors to the rheostats, where the energy stored was dissipated as heat. During the experiments, the measured variables included voltage across the ultracapacitors and the current flowing through them. With these variables, the instantaneous power could be calculated, and the energy stored in the ultracapacitors could be estimated. The results for a charge and discharge experiment are shown in Figure 13 to illustrate the output.

Sixty experiments (30 for charge and 30 for discharge) like the one described in this section were made to calculate the efficiency and performance of the ultracapacitor arrangement. In the first round of experiments, just one board with six cells was connected, and five different values of resistance were tested. For the second round of experiments, two boards were connected and five different values of resistance were tested, this process was carried out until six boards were connected and tested with five different values of resistance. The same procedure was applied for discharge. The details for the tests performed in the electric testbench are summarized in Table 9; the number of boards connected, the equivalent capacitance (C), and the equivalent resistance (R) are presented in the table. A detailed explanation of the results is included in the next sections.

**Figure 13.** Results for charging (**a**) and discharging (**b**).


**Table 9.** Information of the tests performed in the ultracapacitor test bench.
