**Positive variable**


Equation (3) is calculated for each source *i*, and implies that the sum of specific electricity source *i* consumed for all *j* products (*xij*) should be greater than or equal to the availability of that electricity source *i* (*ai*). Equation (4) is calculated for each demand for energy *j* and requires that the amount of electricity from all the sources satisfy the demand for electricity to produce each product *j* (*bj*). CO2 emissions originating from electricity consumption for the production of each product *j* (*Ej*) are

calculated with Equation (5), where the amounts of electricity consumed *xij* are multiplied by the emission factors for energy source *i* (*Fi*). Equation (6) calculates the fraction of electricity source *i* (*wij*) used to produce each product *j*. Finally, Equation (7) represents the objective, which is to minimize the cost of the electricity supply mix, which is calculated by multiplying the amount of electricity consumed (*xij*) by the price of each electricity source *i* (*Pi*).

Three different cases are studied:


$$\sum\_{i} x\_{i\bar{j}} F\_{i\bar{i}} \le E\_{j\bar{\prime}} \; \forall j \in J \tag{8}$$

• *Case 2:* This case is similar to Case 1, but here, it is assumed that the total emission level should be achieved for both aluminum products together. The upper emission limits are the same as those for the first case. Equation (5) is again reformulated to consider the upper limit on emissions for both products combined:

$$\sum\_{i,j} x\_{ij} F\_i \le \sum\_j E\_j \tag{9}$$

The models for the three cases determine the minimum cost of the electricity supply mix considering emission limits. The main results of the transportation model for the three cases are summarized in Table 7, where the fractions of the electricity sources used to produce each product and the total cost of electricity supply are shown. It should be noted that, for reasons of confidentiality, the values of cost are normalized, where a value of 1 represents the cost for current electricity mix.

Table 7 shows that the cost of the current electricity supply mix is lower by 26% compared to cases considering emission limits (Case 1 and 2). Currently, there is a relatively high share of fossil sources in the electricity mix, which is cheaper compared to nuclear and renewable sources. With the current electricity mix, however, emission levels cannot be achieved because of the higher emission factor of fossil energy (see also Table 1). For both the current case and Case 1, the fractions of electricity sources in the mix are the same for both products. Case 2 is a hypothetical case, where the fractions of sources in the electricity mix for slugs and evaporator panels are different. Renewable energy is not selected in either Case 1 and 2, owing to the higher emission factor and lower cost compared to nuclear energy.

The Pinch diagrams in the next sections show where an electricity consumption Pinch occurs for each product (Case 1), and when only one Pinch for all the products combined should be achieved (Case 2).


**Table 7.** Main results from optimization.
