Optimization of a Steel Plant with Multiple Blast Furnaces Under Biomass Injection

Abstract

The allocation of resources between several blast furnaces in an integrated steelmaking plant is studied with the aim of finding the lowest specific operation cost for steel production. In order to reduce the use of fossil fuels, biomass was considered as an auxiliary reductant in the furnace after partial pyrolysis in an external unit, as a complement to heavy fuel oil. The optimization considers raw material, energy, and emission costs and a possible credit for sold power and heat. To decrease computational requirements and to guarantee that the global optimum is found, a piecewise linearized model of the blast furnace was used in combination with linear models of the sinter-, coke-, and power plants, hot stoves, and basic oxygen furnace. The optimization was carried out under different constraints on the availability of some raw materials as well as for different efficiencies of the hot stoves of the blast furnaces. The results indicate that a non-uniform distribution of the production between the furnaces can be advantageous, and some surprising findings concerning the optimal resource allocation under constrained operation are reported.

Appendix

The terms in the BF model (Eq. [1]) describing the effect of oxygen and energy in biomass[8] are non-linear with respect to the unknowns in optimizing F in Eq. [2], since η, Z _O,bio, and H _bio depend non-linearly on the unknown pyrolysis temperature (cf. Figure 2). This problem can be tackled by developing piecewise linearizations of the variables with a suitable discretization, as indicated schematically for the yield (η) in Figure A1. Next, the product terms in X ₇ and X ₈ must be linearized to allow for a MILP formulation. For the product of two variables, x and z, a discretization is first applied yielding a block structure (cf. Figure A2). For fixed discretization steps (Δx and Δz), the product at the center of a block is given by

$$ f_{ij} = \left( {x_{i} + \frac{\Updelta x}{2}} \right)\left( {z_{j} + \frac{\Updelta z}{2}} \right) $$

(A1)

where the lower index i is used to denote discretization points along the x variable and j similarly for the z variable. The slope in the x direction may be approximated as

$$ p_{ij} = \frac{{x_{i + 1} \left( {z_{j} + \frac{\Updelta z}{2}} \right) - x_{i} \left( {z_{j} + \frac{\Updelta z}{2}} \right)}}{\Updelta x} $$

(A2)

which after simplification, gives

$$ p_{ij} = z_{j} + \frac{\Updelta z}{2} $$

(A3)

By analogy, the slope in the z direction is

$$ q_{ij} = x_{i} + \frac{\Updelta x}{2} $$

(A4)

The intercept of the linearized model now is given by

$$ k_{ij} = \left( {x_{i} + \frac{\Updelta x}{2}} \right)\left( {z_{j} + \frac{\Updelta z}{2}} \right) - p_{ij} \left( {x_{i} + \frac{\Updelta x}{2}} \right) - q_{ij} \left( {z_{j} + \frac{\Updelta z}{2}} \right) $$

(A5)

which simplifies to

$$ k_{ij} = - \left( {x_{i} + \frac{\Updelta x}{2}} \right)\left( {z_{j} + \frac{\Updelta z}{2}} \right) $$

(A6)

Using these variables, the function of the product of x and z can now be expressed linearly as

$$ f = k + v_{x} + v_{z} $$

(A7)

In Eq. [A7], the variables k, v _x , and v _z are given by the conditions

$$ k \le k_{ij} + M(2 - y_{i} - \tilde{y}_{j} );\;k \ge k_{ij} - M(2 - y_{i} - \tilde{y}_{j} ) $$

(A8a,b)

$$ v_{x} \le p_{ij} x + M(1 - y_{i} );\;v_{x} \ge p_{ij} x - M(1 - y_{i} ) $$

(A9a,b)

$$ v_{z} \le q_{ij} z + M(1 - \tilde{y}_{j} );\;v_{z} \ge q_{ij} x - M(1 - \tilde{y}_{j} ) $$

(A10a,b)

where (“big”) M is a large number that serves to disconnect inactive values in detecting the block of interest for the product xz. The binary variables $y_{i} ,\tilde{y}_{j} \in \left\{ {0,1} \right\}$, in turn, have to satisfy the conditions

$$ \sum\limits_{i} {y_{i} = 1;\;} \sum\limits_{j} {\tilde{y}_{j} = 1;\quad \forall i,j} $$

(A11a,b)

and are connected to the values of x and z by

$$ x \le x_{i + 1} + M\left( {1 - y_{i} } \right);\;x \ge x_{i} - M\left( {1 - y_{i} } \right)\quad \forall i $$

(A12a,b)

$$ z \le z_{j + 1} + M\left( {1 - \tilde{y}_{j} } \right);\;z \ge z_{j} - M\left( {1 - \tilde{y}_{j} } \right)\quad \forall j $$

(A13a,b)