*2.3. Dichotomy Method*

As already mentioned in Section 2.2, the Simulink model receives both the battery capacity (as variable) and a peak threshold (as data input). To find out whether or not that threshold will be met, all we have to do is run the simulation and check the maximum load power Max(Pload). On the one hand, if the threshold is too low, the system will be unreliable (Max(Pload) > PThreshold) due to insufficient battery capacity, whereas, on the other hand, if the threshold is too high (Max(Pload) ≤ PThreshold) the system will be reliable but the battery is overdimensioned. Consequently, for each load profile and a given battery capacity, there is only one threshold that minimizes the load power (See Figure 5). To find the solution for our optimization problem we deployed the 'dichotomy method'. In the next paragraph, follows a short description of the algorithm.

**Figure 5.** Flow chart—Dichotomy method: (**a**) Pseudocode (left), (**b**) Midpoint evolution (right).

Dichotomy method (Figure 5):


#### *2.4. Definition of Performance Metrics*

Before continuing with the presentation of the simulation results, first, we need to give the definitions of our performance metrics, based on which we evaluated the peak shaving potential of the users. In our approach, we would rather associate the word 'potential' explicitly to energetic assessments. The extent to which these can be translated into economic terms (e.g., revenues, expenses, ROIs) depends certainly on the tariff structure under consideration as well as the cost for the battery storage system. Although, as shown in Section 3, we do provide some insights specifically for Belgium, preferably, each reader ought to make his own reflections.

*Peak reduction (%):* It is the percentual difference between the initial peak power and the final peak power after peak shaving:

$$A\_{\text{peak red}} = \frac{P\_{\text{max i}} - P\_{\text{max f}}}{P\_{\text{max i}}} \cdot 100\tag{9}$$

where Apeak red is the peak reduction, Pmax i is the initial peak power, Pmax f is the final peak power after peak shaving.

*Peak reduction-to-capacity:* It is the difference between the initial peak power and the final peak power after peak shaving divided by the battery capacity. This metric can serve us as a rough estimation of the profitability of the installation if we can express the revenue and costs linearly proportional to the peak reduction and battery capacity respectively.

$$R\_{\text{peak red}-to-cap} = \frac{P\_{\text{max i}} - P\_{\text{max f}}}{C\_{\text{bat}}} \tag{10}$$

where Rpeak red−to−cap is the ratio peak reduction-to-capacity, Pmax i is the initial peak power, Pmax f is the final peak power after peak shaving, Cbat is the battery capacity.

*SoC active time (%):* It is the average percentage of time per year that the battery is deployed for peak shaving. This metric can be very useful, especially when our intention is to combine peak shaving with other services (e.g., increasing the self-sufficiency of PV, ancillary services, Time-of-Use (ToU) prices).

$$\begin{aligned} \text{SoC}\_{\text{act time}} &= \sum\_{\mathbf{i}=1}^{1096 \cdot 96} \mathbf{i} \cdot \frac{100}{1096 \times 96} \\ \mathbf{i} &= \begin{cases} 1, \text{ } | \text{P}\_{\text{bat}} | > 0 \\ 0, \text{ } \text{P}\_{\text{bat}} &= 0 \end{cases} \end{aligned} \tag{11}$$

where SoCact time is the SoC active time, Pbat is the battery power, i is the quarter index of the simulation, 1096 × 96 is the total number of quarters within the 3 years period (1st January 2014–31st December 2016).

*Battery utilization (cycles*/*year):* It is the average total energy discharged by the battery within a year divided by the battery capacity. This metric can be used to assess how fast the battery reaches the end of its lifetime. Particularly for peak shaving applications, it is desirable that the battery be utilized as low as possible since our cost savings are exclusively dependent on the power component (cost in function of kW). Conversely, when the aim is to increase the self-sufficiency of the installation (PV or wind), the battery utilization should be as high as possible, since our cost savings are mainly dependent on the energy component (cost in function of kWh).

$$\mathbf{U}\_{\text{bat}} = \frac{\mathbf{E}\_{\text{dis,tot}}}{\mathbf{C}\_{\text{bat}} \cdot \mathbf{3}} \tag{12}$$

where Ubat is the battery utilization, Edis tot is the total discharged energy within the 3 years period, Cbat is the battery capacity.

*Consumption increase (%):* It is the percentage of energy consumption increase due to efficiency losses of the battery storage system. In addition to the initial capital expenditures for the battery, the additional energy consumption should be taken into account as operating cost.

$$A\_{\rm incr} = \frac{E\_{\rm load \, f} - E\_{\rm load \, i}}{E\_{\rm load \, i}} \cdot 100 \tag{13}$$

where Aincr is the consumption increase, Eload f and Eload i is the total energy consumed within the 3-year period after and before peak shaving, respectively.
