*4.2. Minimum Demand for the Low-Carbon Source*

The algebraic targeting approach is further extended to determine the amount of the low-carbon energy source to achieve the specified emission limits. All steps in the cascade analysis remain the same as for the zero-carbon example, except that *Si* for the renewable energy source is now the sum of the previous amount of renewable energy source (1.425 MWh) plus the minimum amount of zero-carbon energy obtained above (2.137 MWh, see Table 4). *FCS*,*<sup>k</sup>* was obtained by considering a *cCS* of 0.038 (see also Table 1). The cascade analysis determining the minimum amount of the low-carbon energy source is shown in Table 5.


**Table 5.** Cascade analysis for the low-carbon energy source where not all CO2 loads are non-negative.


\* undef—result is undefined because of division by zero.

**Table 6.** Minimum low-carbon energy solution applying cascade analysis.

\* Pinch Point.

The cascade analysis in Table 5 gives an infeasible solution for the low-carbon source owing to the negative value of Δ*Ek* in the last interval. A feasible solution to the cascade analysis is shown in Table 6, where again, the most negative value of *FCS*,*<sup>k</sup>* (−0.083 MWh) is added to the low-energy source (*Si* for renewable energy source with *ck* of 0.038). This is the amount (in absolute numbers) of the required addition to the low-carbon supply shown in Table 5 (3.562 MWh, increased to 3.645).

As obtained also by the graphical approach, the amount of excess energy from fossil sources is increased to 2.22 MWh. The Pinch Point is again obtained at 1.015 t CO2/MWh.

In the following section, the optimization-based approach is used to minimize the cost of electricity sources while achieving CO2 emission targets. Two products are considered for optimization, i.e., aluminum slugs and aluminum evaporator panels.
