A multi-stage stochastic programming model for adaptive biomass processing operation under uncertainty

Abstract

Variations of physical and chemical characteristics of biomass reduce equipment utilization and increase operational costs of biomass processing. Biomass processing facilities use sensors to monitor the changes in biomass characteristics. Integrating sensory data into the operational decisions in biomass processing will increase its flexibility to the changing biomass conditions. In this paper, we propose a multi-stage stochastic programming model that minimizes the expected operational costs by identifying the inventory level and creating an operational decision policy for equipment speed settings. These policies take the sensory information data and the current biomass inventory level as inputs to dynamically adjust inventory levels and equipment settings according to the changes in the biomass’ characteristics. We ensure that a prescribed target reactor utilization is consistently achieved by penalizing the violation of the target reactor feeding rate. A case study is developed using real-world data collected at Idaho National Laboratory’s biomass processing facility. We show the value of multi-stage stochastic programming from an extensive computational experiment. Our sensitivity analysis indicates that updating the infeed rate of the system, the processing speed of equipment, and bale sequencing based on the moisture level of biomass improves the processing rate of the reactor and reduces operating costs.

Appendices

Appendix A detailed mathematical notation and formulation

1.1 A.1 Detailed mathematical notation

See Table 11.

Table 11 Detailed mathematical notation

1.2 A.2 Optimization problem to be solved in each stage t: detailed formulation

We provide a detailed formulation for the optimization problem to be solved in each stage t below. This formulation corresponds to the generic formulation (2) presented in the main body of the paper.

$$\begin{aligned} Q_t({\textbf{I}}_{t-1},{\tilde{m}}_t):= \min \ {}&\sum _{i \in \textbf{E}^m}{\textbf{c}}^{h \top } I_{it} + c^pp_t + {\mathcal {Q}}_{t+1}({\textbf{I}}_{t}) \end{aligned}$$

(A1a)

$$\begin{aligned} \text {s.t. } & X_{0} = \gamma _{0} d_{0}({\tilde{m}}_t), \end{aligned}$$

(A1b)

$$\begin{aligned} & { X_{it} = (1-\phi _i-\varphi _{i,\kappa _t})\sum _{j \in \varvec{\delta _i^-}}X_{jt}} , \ \forall i \in \textbf{E}^p, \end{aligned}$$

(A1c)

$$\begin{aligned} & {X_{its} \le \gamma _{i}d_{it}({\tilde{m}}_t)\frac{l_0}{v_{1,\kappa _t}} V_{it}}, \ \forall i \in \textbf{E}^r, \end{aligned}$$

(A1d)

$$\begin{aligned} & X_{it} = \sum _{j \in \varvec{\delta _i^-}}X_{jt}, \ \forall i \in \textbf{E}^r \setminus \{i_{b1}, (i_{b2})\}, \end{aligned}$$

(A1e)

$$\begin{aligned} & { X_{i_{b1},t} = (1-\theta _{\kappa _t}) X_{i_b,t}}, \end{aligned}$$

(A1f)

$$\begin{aligned} & { X_{(i_{b2}),t} = \theta _{\kappa _t} X_{i_b,t}}, \end{aligned}$$

(A1g)

$$\begin{aligned} & { X_{it} = \gamma _{i}d_{it}({\tilde{m}}_t)\frac{l_0}{v_{1,\kappa _t}V_{it}}}, \ \forall i \in \textbf{E}^m, \end{aligned}$$

(A1h)

$$\begin{aligned} & I_{i1} = I_{i0} + \sum _{j \in \varvec{\delta _i^-}}X_{j1}-\sum _{l \in \delta _i^-} X_{i1}, \ \forall i \in \textbf{E}^m, \end{aligned}$$

(A1i)

$$\begin{aligned} & I_{it} = I_{i(t-1)} + \sum _{j \in \varvec{\delta _i^-}}X_{jt} -\sum _{l \in \delta _i^-} X_{it}, \ \forall i \in \textbf{E}^m, \end{aligned}$$

(A1j)

$$\begin{aligned} & { (1-{\tilde{m}}_{t})X_{it} \le {\bar{x}}_{i,\kappa _t}}, \ \forall i \in \textbf{E}^p, \end{aligned}$$

(A1k)

$$\begin{aligned} & (1- {\tilde{m}}_{t}) X_{i_r,t} \le {\overline{r}}, \end{aligned}$$

(A1l)

$$\begin{aligned} & p_t \ge r - X_{i_r,t}, \end{aligned}$$

(A1m)

$$\begin{aligned} & { 0 \le V_{it} \le {\bar{v}}_{i,\kappa _t}}, \ \forall i \in \mathbf{N \setminus \{ 1 \}}, \end{aligned}$$

(A1n)

$$\begin{aligned} & { 0 \le I_{it} \le {\bar{\iota }}({\tilde{m}}_t)}, \ \forall i \in \textbf{E}^m, \end{aligned}$$

(A1o)

$$\begin{aligned} & X_{0t}, X_{it} \ge 0, \ \forall i \in \textbf{N}, \end{aligned}$$

(A1p)

$$\begin{aligned} & p_t \ge 0. \end{aligned}$$

(A1q)

We explain the constraints in formulation (A1) and describe how they link to the general formulation (2) in the main document as follows:

• The objective (A1a) is to minimize the expected total inventory holding cost and the expected penalty of not achieving the target reactor feeding rate r. The latter objective helps ensure consistently high reactor utilization, as the reactor is the most expensive equipment in the biorefinery.

• Constraints (A1b), (A1c) and (A1h) represent the flow calculations for processing and storage equipment. These constraints correspond to constraints (2c) in the succinct, general formulation (2) in the main document. For example, constraints (A1c) calculate the biomass flow with respect to the moisture and dry matter losses during the grinding and pelleting processes.

• Constraint (A1b) represents the amount of biomass entering the system in stage t. It is calculated by multiplying the volume of the biomass bale and wet biomass density. Biomass wet density depends on the moisture content, ${\tilde{m}}_t$.

• Processing and transportation equipment, and the reactor have limits on the amount of biomass flowing into or out of them. Constraints (A1d), (A1k) and (A1l) represent these upper limits. These constraints correspond to constraints (2d) in the succinct, general formulation (2).

• In the transportation equipment, the biomass flowing into the equipment equals to the biomass flowing from it. Constraints (A1e), (A1f), and (A1g) represent these flow balance equations, just like constraints (2b) in the general formulation (2) in the main document.

• Constraints (A1i) and (A1j) are the inventory balance constraints of the storage equipment $i \in \textbf{E}^m$. They are a part of constraints (2e) in the succinct formulation (2).

• Constraints (A1m) and (A1q) calculate the shortfall of achieving the target reactor feeding rate, which is penalized in the objective. These constraints correspond to constraints (2f) and (2f) in the succinct, general formulation (2) in the main text.

• Constraints (A1n) and (A1o) set bounds on the processing speed of equipment and amount of inventory stored, respectively. These constraints are formulated in vector form in the general formulation (2) via constraints (2h) and (2i).

Appendix B Data tables

See Tables 12, 13, 14, 15.

Table 12 Particle size distribution percentiles

Table 13 Secondary grinder bypass ratios

Table 14 Equipment infeed rate limits

Table 15 Moisture and dry matter changes

Appendix C Regression analysis for biomass density

See Table 16.

Table 16 Regression analysis statistics