Optimization models for integrated biorefinery operations

Abstract

Variations of physical and chemical characteristics of biomass lead to an uneven flow of biomass in a biorefinery, which reduces equipment utilization and increases operational costs. Uncertainty of biomass supply and high processing costs increase the risk of investing in the US’s cellulosic biofuel industry. We propose a stochastic programming model to streamline processes within a biorefinery. A chance constraint models system’s reliability requirement that the reactor is operating at a high utilization rate given uncertain biomass moisture content, particle size distribution, and equipment failure. The model identifies operating conditions of equipment and inventory level to maintain a continuous flow of biomass to the reactor. The sample average approximation method approximates the chance constraint and a bisection search-based heuristic solves this approximation. A case study is developed using real-life data collected at Idaho National Laboratory’s biomass processing facility. An extensive computational analysis indicates that sequencing of biomass bales based on moisture level, increasing storage capacity, and managing particle size distribution, increases utilization of the reactor and reduces operational costs.

Appendices

Detailed notations

Table 16 Detailed mathematical notation

Detailed formulation

Introducing a surrogate linear objective function. The true objective of this problem is to minimize the total operational cost per dry tons of biomass processed during the planning horizon, which is computed using unit “dollar per dry ton” ($/dt). In this case the energy consumption and operation cost function, ${\hat{f}}(\cdot )$ is as follows:

$$\begin{aligned} {\hat{f}}(X_{ts}, I_{ts}, \omega _s) = \frac{ \displaystyle \sum _{i \in {\mathbf{N}}} T(e_i + c_i)}{\displaystyle \sum _{t\in {\mathcal {T}}} (1-m_{i_rts}) X_{i_rts} }. \end{aligned}$$

As a result the true objective that we evaluate is given by:

$$\begin{aligned} \min \ \sum _{i \in {\mathbf{E}}^m}h_i I_{i0} + \frac{1}{S} \frac{\displaystyle \sum _{i\in {\mathbf{N}}} T(e_i + c_i) }{\displaystyle \left[\sum _{s=1}^{S} \sum _{t\in {\mathcal {T}}} (1-m_{i_rts}) X_{i_rts} \right]}. \end{aligned}$$

However, this results in a non-convex objective function, which makes the problem hard to solve. Notice that, in order to minimize the total cost per dry ton, we need to maximize the amount of biomass processed, while using minimum initial inventory. By this observation, we consider the following surrogate linear objective function, which makes the problem much easier to solve.

$$\begin{aligned} \min \ \sum _{i \in E^m} h_i I_{i0} - \left( \frac{1}{S} \left[\sum _{s=1}^{S} \sum _{t\in {\mathcal {T}}} (1-m_{i_rts})X_{i_rts} \right]\right) . \end{aligned}$$

Note that two terms (i.e., holding cost and average amount of biomass processed) of this surrogate objective function have different units. In order to resolve this we used a weighted sum of the two terms, where the weight of the initial holding cost is much smaller.

Use of this surrogate objective function leads to the following chance-constrained stochastic linear program:

$$\begin{aligned} ({\hat{P}}) {:}{=} \min \,&\sum _{i \in E^m} h_i I_{i0} - \left( \frac{1}{S} \left[\sum _{s=1}^{S} \sum _{t\in {\mathcal {T}}} (1-m_{i_rts})X_{i_rts} \right]\right)&\end{aligned}$$

(21)

$$\begin{aligned} \text {s.t. }&0 \le V_{it} \le {\bar{v}}_i(\kappa _t), \,&\forall i \in {\mathbf{N}}, t \in {\mathcal {T}}, \end{aligned}$$

(22)

$$\begin{aligned}&0 \le I_{i0} \le {\bar{\iota }}_i(\kappa _0),\&\forall i \in {\mathbf{E}}^m, \end{aligned}$$

(23)

$$\begin{aligned}&X_{0ts} = \gamma _{0} d_{0ts}V_{1t}, \,&\forall t \in {\mathcal {T}}, s \in {\varvec{{\mathcal {S}}}} \end{aligned}$$

(24)

$$\begin{aligned}&X_{its} = (1-\phi _i-\varphi _{it}(\kappa _t))\sum _{j \in {\varvec{\delta _i^-}}}X_{jts} ,\&\forall i \in {\mathbf{E}}^g, t \in {\mathcal {T}}, s \in {\varvec{{\mathcal {S}}}} \end{aligned}$$

(25)

$$\begin{aligned}&X_{its} \le \gamma _{i}d_{its} V_{it}, \,&\forall i \in {\mathbf{E}}^r, t\in {\mathcal {T}}, s \in {\varvec{{\mathcal {S}}}} \end{aligned}$$

(26)

$$\begin{aligned}&X_{its} = \sum _{j \in {\varvec{\delta _i^-}}}X_{jts}, \,&\forall i \in {\mathbf{E}}^r \setminus \{i_{b1}, (i_{b2})\}, t\in {\mathcal {T}}, s \in {\varvec{{\mathcal {S}}}} \end{aligned}$$

(27)

$$\begin{aligned}&X_{i_{b1},ts} = (1 - \theta _{ts}) X_{i_bts}, \,&\forall t\in {\mathcal {T}}, s \in {\varvec{{\mathcal {S}}}} \end{aligned}$$

(28)

$$\begin{aligned}&X_{(i_{b2}),ts} = \theta _{ts} X_{i_bts}, \,&\forall t\in {\mathcal {T}}, s \in {\varvec{{\mathcal {S}}}} \end{aligned}$$

(29)

$$\begin{aligned}&X_{its} = \gamma _{i}d_{its}V_{it}, \,&\forall i \in {\mathbf{E}}^m, t \in {\mathcal {T}}, s \in {\varvec{{\mathcal {S}}}} \end{aligned}$$

(30)

$$\begin{aligned}&I_{i1s} = I_{i0} + \sum _{j \in {\varvec{\delta _i^-}}}X_{j1s} - X_{i1s}, \,&\forall i \in {\mathbf{E}}^m, s \in {\varvec{{\mathcal {S}}}} \end{aligned}$$

(31)

$$\begin{aligned}&I_{its} = I_{i(t-1)s} + \sum _{j \in {\varvec{\delta _i^-}}}X_{jts} - X_{its}, \,&\forall i \in \mathbf{E}^m, t \in {\mathcal {T}}\setminus \{1\}, s \in {\varvec{{\mathcal {S}}}}\nonumber \\ \end{aligned}$$

(32)

$$\begin{aligned}&I_{its} \ge {\underline{\iota }}_i d_{its}, \,&\forall i \in \mathbf{E}^m , t \in {\mathcal {T}}, s \in {\varvec{{\mathcal {S}}}}\nonumber \\ \end{aligned}$$

(33)

$$\begin{aligned}&I_{its} \le {\bar{\iota }}_i d_{its}, \,&\forall i \in {\mathbf{E}}^m, t \in {\mathcal {T}}, s \in {\varvec{{\mathcal {S}}}}\nonumber \\ \end{aligned}$$

(34)

$$\begin{aligned}&X_{its} = (1-\varphi _{it}(\kappa _t)) \left( (1-\frac{{\hat{t}}_i}{t}) \sum _{j \in {\varvec{\delta _i^-}}}X_{jts} + I_{i(t-1)s} \right) , \,&\forall i \in {\mathbf{E}}^{mill}, t \in {\mathcal {T}}, s \in {\varvec{{\mathcal {S}}}}\nonumber \\ \end{aligned}$$

(35)

$$\begin{aligned}&I_{its} = I_{i(t-1)s} + \sum _{j \in {\varvec{\delta _i^-}}}X_{jts} - X_{its}, \,&\forall i \in \mathbf{E}^{mill}, t \in {\mathcal {T}}, s \in {\varvec{{\mathcal {S}}}}\nonumber \\ \end{aligned}$$

(36)

$$\begin{aligned}&I_{its} \le {\bar{\iota }}_i d_{its}, \,&\forall i \in \mathbf{E}^{mill}, t \in {\mathcal {T}}, s \in {\varvec{{\mathcal {S}}}}\nonumber \\ \end{aligned}$$

(37)

$$\begin{aligned}&(1-m_{its})X_{its} \le {\bar{x}}_i(\kappa _t), \,&\forall i \in {\mathbf{E}}^p, t \in {\mathcal {T}}, s \in {\varvec{{\mathcal {S}}}}\nonumber \\ \end{aligned}$$

(38)

$$\begin{aligned}&(1- m_{i_rts}) X_{i_rts} \le {\overline{r}}, \,&\forall t \in {\mathcal {T}}, s \in {\varvec{{\mathcal {S}}}} \end{aligned}$$

(39)

$$\begin{aligned}&X_{0ts}, X_{its} \ge 0, \,&\forall i \in {\mathbf{N}}, t \in {\mathcal {T}}, s \in {\varvec{{\mathcal {S}}}} \end{aligned}$$

(40)

$$\begin{aligned}&I_{i0}\ge 0, \,&\forall i \in {\mathbf{E}}^m \cup {\mathbf{E}}^{mill}, \end{aligned}$$

(41)

$$\begin{aligned}&I_{its} \ge 0, \,&\forall i \in {\mathbf{E}}^m \cup {\mathbf{E}}^{mill}, s \in {\varvec{{\mathcal {S}}}} \end{aligned}$$

(42)

$$\begin{aligned}&\displaystyle \frac{1}{S}\sum _{s=1}^S {\mathbb {1}}\left[ \frac{1}{T} \sum _{t \in {\mathcal {T}}} (1- m_{i_rts}) X_{i_rts} \ge r \right] \ge 1-{\hat{\epsilon }}. \end{aligned}$$

(43)

The bypass ratio in the separation unit is calculated as follows:

$$\begin{aligned} \theta _{ts} {:}{=} {\left\{ \begin{array}{ll} \max \{0.5 - 0.4(\rho _{1ts}^{50}-6.35)/(\rho _{1ts}^{50} - \rho _{1ts}^{10}), 0\} &{}\text {for}\, \rho _{1ts}^{50} \ge 6.35\\ \min \{0.5 + 0.4(6.35 - \rho _{1ts}^{50})/(\rho _{1ts}^{90} - \rho _{1ts}^{50}), 1\} &{}\text {for}\, \rho _{1ts}^{50} < 6.35, \end{array}\right. }\nonumber \\ \end{aligned}$$

(44)

where, the value 6.35 (in mm) corresponds to the screen size used in the separation unit.

We explain the constraints in formulation $({\hat{P}})$ as follows:

• Constraints (22) and (23) correspond to the constraints (3) and constraints (7) in the succinct formulation (${\hat{P}}$) in the main document. They represent the bounds on the equipment processing speed and initial inventory, respectively.

• Constraints (24), (25), (30), and (35) represent the flow calculations for processing and storage equipment. These constraints correspond to constraints (10) in the succinct formulation (${\hat{P}}$) in the main document. For example, constraints (25) calculate the biomass flow with respect to the moisture and dry matter losses during the grinding process.

• Constraints (26) represent the upper limit on the amount of biomass flow from each transportation equipment $i \in {\mathbf{E}}^r$. These constraints correspond to constraints (11) in the succinct formulation (${\hat{P}}$) in the main document.

• In the transportation equipment, the biomass flowing into the equipment equals to the biomass flowing from it. Constraints (27), (28), and (29) represent these flow balance equations, just like constraints (9) in the succinct formulation (${\hat{P}}$) in the main document.

• Constraints (31) and (32) are the inventory balance constraints of the storage equipment $i \in {\mathbf{E}}^m$. They are also a part of constraints (10) in the succinct formulation (${\hat{P}}$) in the main document.

• Constraints (33) and (34) set the upper and lower thresholds for the inventory level. The inventory upper bound comes from the storage equipment’s capacity. The lower threshold is required for consistent flow out of the storage equipment ${\mathbf{E}}^m$. Although the volume of a storage equipment is fixed, variations in the biomass density impact the amount of biomass allowed to be stored. These constraints are the detailed version of constraints (12) in the succinct formulation (${\hat{P}}$) in the main document.

• Pelleting process takes ${\hat{t}}_i$ units of time (which is less than the chosen time period t) in the pelleting mill i. The pelleting mill has an in-process storage capacity, which keeps the biomass during the pelleting process. Constraints (36) and (37) represent the inventory balance and inventory capacity in pelleting mill ${\mathbf{E}}^{mill}$, respectively. In the succinct formulation in the main documents, we included these constraints within constraints (10) and (12), respectively.

• Processing equipment and the reactor have limits on the amount of biomass flowing into them. Constraints (38) and (39) represent these upper limits. These constraints are part of constraints (10) in the succinct formulation (${\hat{P}}$) in the main document.

Data tables

Table 17 Energy consumption and fixed costs

Table 18 Particle size distribution percentiles

Table 19 Moisture and dry matter changes

Table 20 Equipment infeed rate limits

A bisection search based heuristic algorithm for solving the SAA problem (${\hat{P}}$)

Regression analysis for biomass density

Table 21 Regression analysis statistics