**4. Algebraic Targeting Approach (Cascade Analysis)**

The algebraic targeting approach with the cascade analysis technique was originally developed for resource conservation networks [59], and has its roots in the Problem Table Algorithm and the Heat Cascade developed for Maximum Heat Recovery networks [21]. Cascade analysis for CCEP is an algebraic targeting approach for the identification and determination of the minimum amount of zeroand/or low-carbon sources to achieve emission limits. The method was introduced by Foo et al. [13] and could overcome the inaccuracy problems of the graphical approach.

Cascade analysis for determining the minimum amount of zero- or low-carbon sources consists of various steps [13], all of which are presented in detail by Foo et al. [13]. In the following section, the steps of the algorithm are demonstrated on an example which targets electricity demand in the production of 1 t of aluminum slugs.

In the first column of the cascade analysis table (see Tables 3–6), emission factors are sorted in ascending order in *k* intervals (*ck*). Their increments between successive rows (Δ*ck*) are presented in the second column of the cascade analysis table. The following columns represent energy demand (*Dj*) and energy source (*Si*) at the interval of the corresponding emission factor, where the number of intervals *k* is equal to the number of energy demand (*j*) and source types (*i*). The difference between supply and demand (*Si* − *Dj*) gives the net surplus or deficit of energy at each interval *k*. A cumulative energy surplus or deficit (*Fk* = Δ(*Si* − *Dj*)) is the sum of increments between successive rows, starting from

either a zero-carbon or a low-carbon source. In the next column, the CO2 emission load at interval *k* (*Ek*) is calculated as:

$$E\_k = F\_k \cdot \Delta c\_k \tag{1}$$

Next, the cumulative CO2 load (Δ*Ek*) between successive rows of *Ek* is calculated. If Δ*Ek* has a negative value, the CO2 limit has been exceeded, and the cascade is infeasible. In the last column, clean source demand at interval *k* (*FCS*,*k*) is shown, which is calculated as:

$$F\_{CS,k} = \frac{\Delta E\_k}{c\_k - c\_{CS}} \tag{2}$$

where *cCS* represents an emission factor or the clean (zero- or low-carbon) energy source.

In the following section, both cases will be shown, i.e., the minimum demand for the zero- and low-carbon energy sources. The cascade analysis approach was implemented in Excel by using the actual (not rounded) numbers for calculations.
