*4.1. Minimum Demand for the Zero-Carbon Source*

First, the demand for the zero-carbon source was determined to achieve the specified emission limits. The data used in the study are as shown in Table 1. Cascade analysis was applied to one aluminum product, i.e., aluminum slugs, and is shown in Table 3.

The analysis in Table 3 shows an infeasible cascade for a minimum zero-carbon energy source. A feasible cascade should have a non-negative value for the cumulative CO2 load in the last interval. A feasible cascade is shown in Table 4, where almost all the steps remain the same as before. The exception is that the largest negative value of *FCS*,*<sup>k</sup>* is considered as the quantity of the zero-carbon energy source, and is taken as an absolute positive value.


**Table 3.** Cascade analysis for zero-carbon energy source where not all CO2 loads are non-negative.

**Table 4.** Minimum zero-carbon energy solution applying cascade analysis.


\* Pinch Point.

The negative value of *FCS*,*<sup>k</sup>* in the last interval from Table 3 (−2.137 MWh) is the amount of required zero-carbon energy source *Si* in the first interval in Table 4 (where emission factor *ck* is zero). In this way, the cascade is adjusted to become feasible. From Table 4, it can also be seen that the excess energy from the fossil source is the same as the minimum amount of zero-carbon energy (2.137 MWh; last interval of *Fk*). The Pinch Point is obtained at interval *k*, where Δ*Ek* is equal to zero, which is at 1.015 t CO2/MWh.
