*3.2. Operation Limits and Calculation of Power Losses and Performance of the SCESS*

As mentioned above, in order to decide which storage system is best suited to a certain application, it is very important to consider its energy efficiency. In general, storage systems are complex non-linear systems and it is not easy to know their performance over the entire operating range. One of the parameters on which efficiency depends is the state of charge (SoC), that is, the operating point at which the system is working. In the case of SCs, efficiency is a function of current and voltage (SoC = f (V)) at each moment. In the present application the ESS, formed by supercapacitors, is connected to a DC bus through a DC/DC converter. Therefore, to calculate the overall efficiency it is necessary to also take into account the performance of this converter. Another parameter on which

the efficiency of the ESS depends is the duty cycle. It is necessary to take into account the frequency with which the SCs are charged/discharged, because, among other parameters (temperature), the value of the series resistance (ESR) depends on this frequency. Therefore, when a performance value is provided, the associated duty cycle must be specified. In the sizing of the storage system for the present application, the overall performance of the set composed by the electronic converter and the SCs for each operating point has been considered. The process followed to calculate these losses is explained in the next points.

(**a**) (**b**)

**Figure 3.** (**a**) Image of one supercapacitor energy storage system (SCESS) module during a voltage balancing process between cells and (**b**) image of one of the eight drawers that make up the SCs cabinet upon which this study is based.

As has already been commented, the SCESS is connected to a DC bus through a power converter. Another converter, connected between the electrical grid and the DC bus, keeps the voltage value of this bus at 950 V. This DC/DC converter regulates the charge/discharge current according to the received power command. The maximum power and energy available for each SC cabinet (unit module) is 125 kW and 0.5 kWh respectively. These values correspond to the following operating limits for each cell that makes up the SC cabinet:


The nominal voltage of each cell was 2.7 V (absolute maximum voltage is 2.85 V), but the maximum operating value was set at 2.65 V for two reasons. The higher the voltage in the SCs, the more accelerated their ageing. Reducing the maximum voltage by 0.05 V greatly lengthened the lifetime of the SCs. On the other hand, the ESS was formed by a series connection of cells and the distribution of the total voltage between all of them was not perfect. To ensure that the voltage in any cell was higher than 2.7 V, 2.65 V was set as the maximum value. The total working voltage range of each cabinet will be 382-636 V.

These limits imply an input and output voltage ratio in the DC/DC converter of 1/3-2/3, which did not significantly restrain its performance [43].

#### 3.2.1. Power Losses in the Equivalent Series Resistance (ESR)

In order to calculate the performance of the storage system, in addition to taking into account the efficiency of the storage system (ESR losses) and the converter, the losses in the connection plates, in the cables and in the voltage balancing system were calculated [36]. In this case, the variation of the ESR and capacitance with the temperature in the cell was not considered, since each cabinet had a cooling turbine in the upper part that maintains the temperature of the cells in the working range of 25-40 ◦C. In this temperature range the variation of both parameters (ESR and C) was very small. Regarding the variation of capacitance with frequency, it was not studied in this paper, only analyzing the variation with voltage.

To calculate the power losses in the ESR, Equation (3) must be taken into account:

$$P\_{ESR}(t) = \mathcal{U}\_{\text{SC}}(t) \cdot i\_{\text{SC}}(t) \tag{3}$$

*PESR(t)*: Instantaneous power losses;

*USC(t):* SC voltage;

*iSC(t):* Instantaneous current through the SCs.

If the current is considered as a periodic function of period T, it can be written as a Fourier series:

$$i\_{\rm SC}(t) = \sum\_{i=0}^{+\infty} I\_{\rm SC}(i) \cdot \sin(iwt + \vartheta\_i) \tag{4}$$

*ISC(i)*: Harmonic component of the current; *w*: Angular frequency.

The voltage across the ESR can be expressed as:

$$\mathcal{U}\_{ESR}(t) = \sum\_{i=0}^{+\infty} ESR(iw) \cdot I\_{SC}(i) \cdot \sin(iwt + \theta\_i) \tag{5}$$

*UESR (i)*: Voltage across ESR

*ESR (iw)*: ESR value for each frequency

If Equations (4) and (5) are multiplied, the expression for the instantaneous power losses in the ESR is obtained. Applying the Lagrange identity and the orthogonal property of sine and cosine the total power losses can be expressed as [40]:

$$P\_{ESR}(t) = I\_{RMS}^2 \cdot \left[ \sum\_{i=0}^{+\infty} ESR(iw) \cdot \frac{I\_{SC(i)}^2}{I\_{rms}^2} \right] = I\_{RMS}^2 \cdot ESR(eq) \tag{6}$$

*IRMS:* r.m.s value of the current; *ESR (eq)*: Total ESR.

To calculate the equivalent resistance in series for each frequency, it is necessary to complete an analysis of the SCs in the frequency domain. This analysis is based on experimental tests and on the design of a simulation model whose frequency response is adjusted to the results obtained experimentally. A complete analysis of the frequency response of the cell model (BCAP3000) on which the present study is based is presented in [44]. In this way, using Equation (6) the efficiency (*ηESR*) of the SCESS cabinet can be calculated including the losses in the ESR for a given duty cycle (frequency) as per:

$$\eta\_{ESR} = \frac{\mathcal{U}I\_{\mathcal{SC}} \cdot I\_{rms}}{\mathcal{U}\_{\mathcal{SC}} \cdot I\_{rms} + P\_{ESR}} \tag{7}$$
