Optimal configuration of polygeneration plants under process failure, supply chain uncertainties, and emissions policies

2.1. Problem statement

Let $M=\left\{1,2,\cdots ,m\right\}$ be the set of all production materials (e.g., raw materials, final outputs, and by-products) in the system and $P=\left\{1,2,\cdots ,p\right\}$ be the set of environmental by-products, i.e., emissions, such that $P\subset M$. Also, let $N=\left\{1,2,\cdots ,n\right\}$ be the set of all process units in the system. The specific problem tackled in this study can be formally stated as follows:

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  A multi-functional energy production system utilizes inputs, i.e., biomass feedstocks, to produce multiple outputs, e.g., power and heat.

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  The technological performance of the process units of the system is defined by a fixed set of proportions of input and output streams.

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  The demand and price of material streams are subject to uncertainty.

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  Production disturbances are in the form of process failures (e.g., damaged equipment), which lead to abnormal operations and reduced operational capacity.

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  Cap-and-trade or emissions tax policies are in effect and applied to the environmental emissions (elements in $P$) of the system.

The objective is to determine the energy production system that maximizes profit (i.e., revenue less costs) while considering uncertainties in supply chain-related parameters and economic pressures of emissions policies throughout the duration of abnormal operations.

2.2. Notations

From here on out, Greek letters denote decision variables, while the conventional Alphabetical letters denote model parameters. Uncertain variables are characterized by the accent tilde (~). The transpose of matrices is characterized by the superscript $⊺$. The diagonalization of a matrix is denoted by the operator $diag\left(\ast \right)$. Furthermore, the indices used in this study are as follows: $i\in M$ and $j\in N$, representing production materials and process units, respectively. The notations are briefly presented in the following subsections.

2.3. Model assumptions

The following assumptions are followed in this modeling work:

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  $\mathit{A}$ is scale-invariant. Thus, reductions (or increases) in the operational capacity of process units do not affect the proportions of the input–output streams in $\mathit{A}$. The values of $\mathit{A}$ are predetermined during the design stage.

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  The entire feasible operating range of process units is bounded by a lower limit (i.e., the minimum part-load capacity) and an upper limit (i.e., the rated capacity with safety factor).

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  The price and demand of material streams are uncertain.

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  The firm does not produce outputs above the demand limit. Similarly, the firm does not generate environmental emissions above the emissions cap.

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  The capital cost of each process unit is annualized and described by a piecewise linear function of capacity.

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  Two emissions policies are considered, namely emissions tax and cap-and-trade.

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  The price of an emissions allowance in the cap-and-trade program is constant.

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  Emission allowances are sold in competitive markets. There is a readily available demand for emissions allowances whenever the firm decides to sell its on-hand allowances.

2.4. Model development

This study considers the economic implications of inoperability, capital costs arising from process configurations, and emissions policies. The profit obtained from the configuration of the energy production plant is calculated as follows:

The total capital cost arising from decisions on vectors $\mathit{\kappa }$ and $\mathit{\varphi }$ is denoted by the term $W_{\mathit{AF}}\left({\mathit{V}}^{⊺}\mathit{\kappa }+{\mathit{F}}^{⊺}\mathit{\varphi }\right)$. The term ${\mathit{V}}^{⊺}\mathit{\kappa }$ represents the variable capital cost of scaling the production, while ${\mathit{F}}^{⊺}\mathit{\varphi }$ represents the fixed capital cost of operating the process units. The cost arising from the emissions policies is represented by the term $-{\mathit{E}}^{⊺}\mathit{G}\left(1-\psi \right)-{\mathit{P}}_{\mathit{AL}}^{⊺}\mathrm{\Delta }\psi$. Specifically, the cost arising from emissions tax is presented by ${\mathit{E}}^{⊺}\mathit{G}\left(1-\psi \right)$ while the costs arising from cap-and-trade are represented by ${\mathit{P}}_{\mathit{AL}}^{⊺}\mathrm{\Delta }\psi$. The model is linear when $\psi$ is fixed because in the term ${\mathit{P}}_{\mathit{AL}}^{⊺}\mathrm{\Delta }\psi$, the multiplication of the variable $\mathrm{\Delta }$ to another decision variable is avoided. Fixing the value of $\psi$ indicate that only one of the two emissions policies can take effect. However, when a hybrid emissions policy is in effect where an energy firm is given the flexibility to choose which policy to subscribe to, like the Canadian policy discussed in Wood (2018), ψ becomes a decision variable that makes the model quadratic (nonlinear). The quadratic term ${\mathit{P}}_{\mathit{AL}}^{⊺}\mathrm{\Delta }\psi$ in Equation (1) can be easily linearized by letting $\mathrm{\Omega }=\mathrm{\Delta }\psi$ and adding the following constraints: $L\psi \le \mathrm{\Omega }\le U\psi$, $\mathrm{\Omega }\le \mathrm{\Delta }-L\left(1-\psi \right)$, $\mathrm{\Omega }\ge \mathrm{\Delta }-U\left(1-\psi \right)$, where $L=-\mathcal{M}$, $U=\mathcal{M}$, and $\mathcal{M}$ is a known large number (i.e., $\mathcal{M}\to \infty$). It should be noted that $L\le \mathrm{\Delta }\le U$ because $\mathrm{\Delta }\in {\mathcal{R}}^p$. The details of the decision variable $\mathrm{\Delta }$ are presented in the subsequent discussions. Although the linearization of the nonlinear term in the objective function is apparent and straightforward, as previously demonstrated, it is not reflected in the succeeding models to keep the models’ form concise and as simple as possible.

On the other hand, revenue is represented by the term $W_{\mathit{OH}}{\stackrel{\sim }{\mathit{P}}}_{\mathit{PR}}^{⊺}\mathit{A}\mathit{\kappa }$ , where $\mathit{A}\mathit{\kappa }$ represents the total input-output streams (production) of the plant. In this study, production is limited by the annual demand and emissions limits as follows:

The production limit vector $\stackrel{\sim }{\mathit{D}}$ contains the annualized limits on the demand of outputs (final products) and emissions (environmental by-products). From the term $\mathit{A}\mathit{\kappa }$, the total emissions can be isolated as follows:

In the cap-and-trade policy, energy firms are restricted by regulators in their generation of emissions. However, they have the option to purchase emissions allowances $\mathrm{\Delta }$ to increase their emissions cap from the initial cap $\mathit{H}$ that they previously acquired. They may also sell the allowance that they have on hand for profit. Following these policy conditions, the emissions limit is represented as follows:

From Equation (5), the emissions limit can be extracted from vector $\stackrel{\sim }{\mathit{D}}$ as follows:

The emissions limit ${\stackrel{\sim }{D}}_i$ is meaningful when the cap-and-trade policy is in effect. When the emissions tax policy is in effect, ${\stackrel{\sim }{D}}_i$ is a variable with a sufficiently large value, which prevents it from interfering with production decisions under the emissions tax policy.

On the other hand, by performing some substitutions on Equations $(3)-(6)$, the scalar form of Equation $\left(3\right)$ that is solely on the emissions production limit can be obtained as follows:

Under this cap-and-trade policy, the development of negative emissions technologies is incentivized by providing firms with free allowances equivalent to their negative emissions. Such allowances can then be sold for profit. To regulate the amount of emissions allowances circulating in the carbon market, the government may interfere by reducing the amount of emissions allowances present in the system in the next planning horizon. In this way, this proposed policy can function in a stable fashion – at least in principle.

In this study, the production of an energy plant is subjected to process failures or inoperability in the form of damaged process units. These damages may be caused by climate change-induced extreme weather conditions. Such types of inoperability may be sustained throughout the entire planning horizon, i.e., one year, especially with the mobility restrictions or supply chain disruptions that may still linger after a disaster; or especially when the supplier of rare/specialized replacement parts for repair has been bankrupted because of the disaster. Inoperability is modeled as the reduction of the part-load operating level of process units, which is represented as follows:

The part-load operating level $\mathit{\kappa }$ scales the operations of the energy production plant. It is defined in an interval with a minimum part-load operating level (lower limit) denoted by ${\mathit{K}}_{\mathit{LB}}$. Below the lower limit, the part-load operation is assumed to be unstable. On the other hand, the maximum part-load operating level (upper limit) is denoted by $1-\mathit{Q}+{\mathit{K}}_{\mathit{SF}}$. The spare operating capacity ${\mathit{K}}_{\mathit{SF}}$ comes from the safety factors incorporated in the design of process units, which are predetermined at the design stage. The fractional inoperability level $\mathit{Q}$ originates from disturbances that affect the process units. It indicates whether a process unit is fully operable, i.e., $Q_j=0$, partially operable, i.e., $Q_j\in \left(0,1\right)$, or fully inoperable, i.e., $Q_j=1$. On the other hand, the binary unit selection vector $\mathit{\varphi }$ indicates whether a process unit operates, i.e., ${\varphi }_j=1$, or not, i.e., ${\varphi }_j=0$. It switches off a process unit whenever it cannot operate within the limits of its part-load operating level.

The deterministic model follows:

Model.$\left(1\right)$

subject to:

The uncertainty of the supply chain-related parameters, i.e., price and demand of the material streams, are modeled here using the target-oriented robust optimization (TORO) framework (Ng & Sy, 2014). The uncertain material price vector and production limit vector are represented as follows:

The vectors ${\mathit{P}}_{\mathit{PR}}^{\text{'}}$ and ${\mathit{D}}^{\text{'}}$ represent the most optimistic values of the material prices and production limits, respectively. On the other hand, the vectors ${\mathit{P}}_{\mathit{PR}}$ and $\mathit{D}$ represent the perturbations of the nominal values such that:

The robustness index $\theta \in \left[0,1\right]$ determines the perturbation of the uncertain values. It can be observed that the largest perturbations are obtained when $\theta =1$. The largest perturbations take the values ${\mathit{P}}_{\mathit{PR}}={\mathit{P}}_{\mathit{PR}}^{\text{'}\text{'}}$ and $\mathit{D}={\mathit{D}}^{\text{'}\text{'}}$. Thus, under the most favorable case, the values of ${\stackrel{\sim }{\mathit{P}}}_{\mathit{PR}}$ and $\stackrel{\sim }{\mathit{D}}$ would be at the maximum. It is also apparent that larger perturbations correspond to a higher value of $\theta$. Some practical insights can be obtained from these observations. First, an uncertainty-averse attitude would prefer a higher value of $\theta$. Second, a risk-seeking attitude would prefer a lower value of $\theta$. Lastly, this model provides a way to address uncertainties in process configuration by representing uncertainty as an interval.

The TORO framework hinges on robust optimization theory and satisficing theory, which promotes target-oriented decision making (Ng & Sy, 2014). It is desired to configure energy production plants that are under risk so that they can remain feasible for as wide a range of uncertainties as possible. To this end, target-oriented decision making is reflected by transforming the original objective function in Model $\left(1\right)$ into a constraint that corresponds to a target profit $F$. The modification of the model is presented as follows:

Model.$\left(2\right)$

subject to:

Equation $\left(12\right)$ represents the objective function mentioned earlier which describes robustness. Equations $\left(13\right)$ represents the transformation of the objective function of Model $\left(1\right)$ into a constraint that corresponds to a target profit $F$. Unlike conventional optimization procedures wherein the objectives are either maximized or minimized, a target-oriented approach is adopted here. In particular, the value $\pi ,$ which is maximized in Model (1), is converted into a fixed value, and, thus, represented as $F$, which corresponds to the target profit. Setting targets instead of aiming for the theoretically best possible outcome (i.e., maximum $\pi$) is a standard procedure employed in practice. This is commonly known as the “satisficing behavior” in economics (Ng & Sy, 2014). It means that decision-makers, in practice, tend to settle for good enough solutions (determined by the target) instead of seeking to achieve the theoretically best solution. On the other hand, Equations $\left(14\right)$ and $\left(15\right)$ are modifications of Equations $\left(3\right)$ and $\left(6\right)$, respectively, for incorporating the conditions on the uncertain parameters. The condition for their transformation is based on the definitions in Equations (9)-(11).

The terms $\forall {\stackrel{\sim }{\mathit{P}}}_{\mathit{PR}}\in {\mathrm{Z}}_{\theta }$ and $\forall \stackrel{\sim }{\mathit{D}}\in {\mathrm{Z}}_{\theta }$ require the evaluation of infinitely many constraints to solve Model $\left(2\right)$. It is because of the uncertain constraints containing ${\stackrel{\sim }{\mathit{P}}}_{\mathit{PR}}$ and $\stackrel{\sim }{\mathit{D}}$ that would lead to the creation of individual constraints for each of their infinite possible realizations in ${\mathrm{Z}}_{\theta }$. There is, therefore, a need to convert Model $\left(2\right)$ into an equivalent model that is amenable to solving using traditional mathematical programming techniques. To this end, the property of duality is employed.

To demonstrate the application of duality, Equation $\left(13\right)$ is modified such that the perturbation of ${\stackrel{\sim }{\mathit{P}}}_{\mathit{PR}}$ is maximized through the term ${\mathit{P}}_{\mathit{PR}}^{\text{'}}-\max {\mathit{P}}_{\mathit{PR}}$. The resulting expression is as follows:

The term $\max {\mathit{P}}_{\mathit{PR}}$ could then be represented as an isolated optimization (maximization) problem by following the conditions set in Equation $\left(11\right)$. The maximization problem is as follows:

Model.$\left(3\right)$

subject to:

Equation (17) represents the maximization term in Equation (16) which is treated as an objective function of a separate optimization problem. Equation (18) is the constraint derived from Equation (11). The duality property of linear programming is applied to this model to obtain its dual counterpart as follows:

Model.$\left(4\right)$

subject to:

The specific procedures of applying the duality property are omitted here for brevity. Instead, the readers are referred to the work of Bertsimas and Sim (2003) for a more detailed discussion on the property of weak and strong duality, which proves the equivalence of Model $\left(3\right)$ and Model $\left(4\right)$. The translation of the demand constraints in Equation $\left(14\right)$ and Equation $\left(15\right)$ follows the same procedure. After the application of the duality property on the uncertain constraints and the substitution of the dual models into Model $\left(2\right)$, the resulting robust optimization model is obtained as follows:

Model.$\left(5\right)$

subject to:

where ${\mathit{z}}_1$ and ${\mathit{z}}_2$ are dual variables.

Equations $\left(21\right)-\left(23\right)$ are obtained by substituting the dual models that correspond to the maximization of the perturbations of the uncertain variables ${\stackrel{\sim }{\mathit{P}}}_{\mathit{PR}}$ and $\stackrel{\sim }{\mathit{D}}$ . Specifically, Equation (21) is the result of substituting the objective function in Equation (19) for the term $\max {\mathit{P}}_{\mathit{PR}}={\mathit{P}}_{\mathit{PR}}^{\text{'}\text{'}}\theta {\mathit{z}}_1$ in Equation (16). On the other hand, Equation (22) and Equation (23) are the results of repeating the procedures applied to Equation (13), which was demonstrated through Equations (16)-(20), to Equation (14), and Equation (15), respectively. Lastly, Equation (24) represents the dual variables associated with Equations (21), (22), (23), which are obtained from the transformations performed on Equations (13), (14), (15).

$Z_{{\mathrm{\theta }}^{\prime }}\subseteq Z_{\mathrm{\theta }}$ whenever $\theta \ge {\theta }^{\text{'}}$. Thus, if a policy is feasible for an uncertainty set defined by $\theta$, then it will be feasible for all perturbations that would fall within this range. It can also be observed that given a fixed value of $\theta$ and $\psi$, Model $\left(5\right)$ is linear concerning the decision variables $\mathit{\kappa }$, $\mathit{\varphi }$, and $\mathrm{\Delta }$. Model $\left(5\right)$ could thus be conveniently solved for the maximum value of $\theta$ by performing a line search on $\theta \in \left[0,1\right]$. We could utilize well-known search algorithms like the bisection or golden search method to solve this model.

The bisection search method takes an interval $\left[a,b\right]$ in which the root of a function is believed to lie, such that $R(a)$ and $R(b)$ have opposite signs. The method then proceeds to find the midpoint $c$ of $\left[a,b\right]$ and then checks whether the root lies in $\left[a,c\right]$, or $\left[c,b\right]$. In contrast, the golden search method selects two points on the interval, $a<c<d<b$, and compares $R(c)$ and $R(d)$ to determine whether the root lies in $\left[a,c\right]$, or $\left[d,b\right]$. Ng and Sy (2014) introduced an efficient bisection search algorithm that is readily adaptable for solving TORO models. In this study, the bisection search algorithm promoted by Ng and Sy (2014) is applied to solve Model (5). The bisection search algorithm is implemented in this study using the following pseudo-code:

Model.$\left(6\right)$

Set tolerance level $\omega$ into a sufficiently small number;

Initialize ${\theta }^{+}=1$ and ${\theta }^{-}=0$;

WHILE ${\theta }^{+}-{\theta }^{-}>\omega$ DO:

${\theta }^{\ast }=\frac{{\theta }^{+}+{\theta }^{-}}{2}$;

Solve Model 5 using $\theta ={\theta }^{\ast }$;

IF a feasible solution exists THEN.

${\theta }^{-}={\theta }^{\ast }$

ELSE

${\theta }^{+}={\theta }^{\ast }$

END

END

3.1. System description

The case study considers a polygeneration system for producing syngas, power, heat, bio-oil, and biochar using biomass as the primary feedstock. The biomass considered in this study is forest residue. The scenario presented here is hypothetical. However, the assumptions used are based on realistic and plausible parameters; thus, this case should be seen as an illustrative example that demonstrates the advantages of the proposed model for this type of problem. The plant consists of the following process units:

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  Co-generation (Cogen) module – an integrated unit comprised of a gas turbine (GT) generator to produce electricity from synthetic gas (syngas), coupled with a heat recovery steam generator (HRSG) to generate heat from the residual heat in the GT exhaust.

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  Boiler – an auxiliary gas-fired boiler considered to potentially produce the additional heat demand on top of the heat generated from the Cogen module.

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  Gasification unit – a gasifier with a reactor temperature greater than 500 °C to produce bio-oil and syngas, with biochar co-production from biomass.

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  Pyrolysis unit – a pyrolizer for producing the same products as the gasifier at varying production yields using biomass as input.

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  Torrefaction – forest residue was torrefied at 250 °C in this process unit for 60 min with a calorific value of 19.56 MJ/kg (Ubando et al., 2019). The torrefaction upgrades the biomass to biochar with bio-oil co-production from the condensable gases.

The calorific value of syngas produced from the thermochemical processing of forest residues is 5.90 $\mathrm{M}\mathrm{J}\bullet {\mathrm{m}}^{-3}$ (Kostúr & Kačur, 2015). The operational setting of the process units considered here is sourced from Ubando et al. (2020). The input and output streams of the process units, their capital costs (fixed and variable), and part-load operating levels are presented in Table 1 . The prices of materials streams, along with their maximum perturbation values, are shown in Table 2 . The flowsheet of the polygeneration plant described here is presented in Fig. 1 .

The values of the parameters pertaining to emissions policy for the baseline scenario of this study are as follows: ${\mathit{P}}_{\mathit{AL}}=$ 8.24 $\mathrm{U}\mathrm{S}\mathrm{D}\bullet {\mathrm{t}}^{-1}$ (Zeng et al., 2017), $\mathit{E}=$ 8 $\mathrm{U}\mathrm{S}\mathrm{D}\bullet {\mathrm{t}}^{-1}$ (Jenkins, 2014), $\mathit{H}=$ 0 $\mathrm{t}$ (assumed). For the benchmark case, $\psi =1$ is assumed. The following further case-specific assumptions are followed in this study:

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  All economic parameters used here are expressed in 2021 U.S. Dollars (USD).

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  The emissions considered in this case study are only CO₂ emissions.

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  It is assumed that the maximum perturbation values are ${\mathit{D}}^{\text{'}\text{'}}=0.5{\mathit{D}}^{\text{'}}$ and ${\mathit{P}}_{\mathit{PR}}^{\text{'}\text{'}}=0.5{\mathit{P}}_{\mathit{PR}}^{\text{'}}$.

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  The polygeneration plant is assumed to operate for $W_{\mathit{OH}}=$ 8000 $\mathrm{h}\bullet {\mathrm{y}}^{-1}$ with a target profit of $F=1.0\times {10}^7\mathrm{U}\mathrm{S}\mathrm{D}$.

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  It is assumed that the process units have no salvage value at the end of their service lives (40 years), and their respective book values follow straight-line depreciation, with an annualizing factor of $W_{\mathit{AF}}=$ 0.06 ${\mathrm{y}}^{-1}$.

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  The fixed capital cost of process units is estimated using the average fixed-to-variable (FtV) cost ratio of the main and auxiliary process units in Sy et al. (2016). This assumes that the polygeneration plant investigated here follows the cost structure (Aboody et al., 2018) of the polygeneration plant in Sy et al. (2016), which has an FtV cost ratio of 0.4 for main process units (e.g., Cogen) and 0.26 for auxiliary process units (e.g., boiler).

• •

  In the torrefaction process, the generated non-condensable gases were assumed negligible in the analysis because they are relatively minimal compared with the biochar and bio-oil produced (Stelte, 2012) by the unit.

3.2. Analyses

3.3. Device and software

The mathematical program of Model $\left(5\right)$ and the search algorithm in Model (6) is coded using LINGO programming language and ran in the Global Solver of LINGO 19.0 software (Educational License) using a Dell (G5 5500) Laptop whose specifications are presented in Table 3 . The LINGO code used is presented in Appendix A. Furthermore, the Monte Carlo simulation is performed in Microsoft Excel using the Data Table feature to propagate the $1\times {10}^6$ simulation instances. The file containing Monte Carlo simulation results is provided in the supplementary file.

4.1. Influence of target setting and process inoperability on process configuration

The optimal configuration of the baseline scenario (CS1) is presented in Fig. 2 . The gasification unit is inactivated in this design. Syngas (1287.14 $\times {10}^3\bullet {\mathrm{m}}^3\bullet {\mathrm{h}}^{-1}$) is outsourced along with biomass (188.97 $\mathrm{t}\bullet {\mathrm{h}}^{-1}$) as primary feedstock. While the polygeneration plant produces all four outputs, it is a primary power-producing plant with a capacity of 394.69 $\mathrm{M}\mathrm{W}$. In this regard, the co-generation unit is operated the most for power production. The secondary product of the plant is heat (308.06 $\mathrm{M}\mathrm{W}$). Some of the heat is utilized as intermediate inputs for the pyrolysis and torrefaction units for the thermochemical processing of biomass. The two thermochemical process units produce bio-oil (24.27 $\mathrm{t}\bullet {\mathrm{h}}^{-1}$). They also sequester CO₂ to produce biochar (26.74 $\mathrm{t}\bullet {\mathrm{h}}^{-1}$) resulting in CO₂ neutrality. The baseline design is considered the benchmark in the analysis of the succeeding scenarios (CS2-CS4).

Fig. 3 presents the optimal configuration of the second scenario (CS2). This configuration shows a 5.67 % and 6.92 % increase in the operating capacity of the co-generation and pyrolysis units, respectively. It is apparent that the increase in operating capacities of these process units is in response to the ambitious target profit that characterizes this scenario, $F=3.0\times {10}^7\mathrm{U}\mathrm{S}\mathrm{D}$. Consequently, biomass (201.38 $\mathrm{t}\bullet {\mathrm{h}}^{-1}$) and syngas (1359.55 $\times {10}^3\bullet {\mathrm{m}}^3\bullet {\mathrm{h}}^{-1}$) consumption increased by 6.58 % and 5.63 %, respectively. On the other hand, power (417.09 $\mathrm{M}\mathrm{W}$), heat (325.02 $\mathrm{M}\mathrm{W}$), biochar (28.24 $\mathrm{t}\bullet {\mathrm{h}}^{-1}$), and bio-oil (25.91 $\mathrm{t}\bullet {\mathrm{h}}^{-1}$) production increased by 5.67 %, 5.50 %, 5.62 %, and 6.76 %, respectively. The increase in production capacity also resulted in the emissions of CO₂ amounting to 15.16 $\mathrm{t}$.

Fig. 4 presents the optimal configuration of the third scenario (CS3). In this configuration, the co-generation, gasification, and pyrolysis units are inactivated. The boiler unit operates at the same level as the baseline scenario, while the operating capacity of the torrefaction unit is reduced by 95 %. The syngas (5.96 $\times {10}^3\bullet {\mathrm{m}}^3\bullet {\mathrm{h}}^{-1}$) and biomass (0.42 $\mathrm{t}\bullet {\mathrm{h}}^{-1}$) consumption is reduced by 99.54 % and 99.78 %, respectively. Power production is completely ceased due to the inactivation of the co-generation unit while the heat (7.81 $\mathrm{M}\mathrm{W}$), bio-oil (0.03 $\mathrm{t}\bullet {\mathrm{h}}^{-1}$), and biochar (0.23 $\mathrm{t}\bullet {\mathrm{h}}^{-1}$) production are reduced by 97.47 %, 99.89 %, and 99.13 %, respectively. It is apparent that the operational capacity of the process units is reduced in response to the low target profit that characterizes the scenario, $F=2.0\times {10}^5\mathrm{U}\mathrm{S}\mathrm{D}$. The configuration is CO₂ neutral, and the primary product of the polygeneration plant is heat, along with bio-oil and biochar.

The optimal configuration of the last scenario (CS4) is presented in Fig. 5 . Note that this scenario has the same target profit as CS3, i.e., $F=2.0\times {10}^5\mathrm{U}\mathrm{S}\mathrm{D}$, but is under the condition of process inoperability of the boiler unit equivalent to $Q_{\mathit{BOILER}}=$ 0.60. Unlike CS3, in this configuration, only the gasification unit is inactivated. However, compared to the baseline case, the operational capacity of the co-generation, boiler, and pyrolysis units are reduced by 5.95 %, 54.17 %, and 7.85 %, respectively. The operating capacity of the torrefaction unit is the same as the baseline scenario. The CO₂ emissions of this configuration amount to 47.37 $\mathrm{t}$. The economic and environmental performance of the four configurations discussed in this section is summarized in Table 4 .

4.2. Viability of the robustness metric in the practical setting

The comparison of the calculated values of the robustness index ${\theta }^{\ast }$ for each of the four previously presented configurations against the obtained robustness from the Monte Carlo simulations performed on each of them is presented in Fig. 6 . Clearly, the calculated robustness is validated by the simulations. On the other hand, the robustness of the optimal configuration for the baseline case under various target profits $F$ is presented in Fig. 7 . The plot clearly shows that robustness decreases as the target profit increases.

4.3. Influence of emissions policies on the robustness of process configuration

The results of SA1 are summarized in Fig. 8 . The finding in Fig. 8(b) suggests that the optimal configuration that employs strategies to drawdown CO₂ emissions is motivated by the price of emissions allowance $P_{\mathit{AL}}>8.08\mathrm{U}\mathrm{S}\mathrm{D}\bullet {\mathrm{t}}^{-1}$. It is also observed in Fig. 8(a) that robustness of the optimal configurations decreases as the price of emissions allowance increases. Robustness also decreases as the initial carbon emissions cap $H$ decreases. The economic pressures introduced by these parameters on the polygeneration firm are the apparent cause of this influence on the robustness of optimal configurations. It is also important to note that the model may choose to configure the capacities of activated process units or activate process units that have CO₂ sequestration capabilities, i.e., pyrolysis, gasification, or torrefaction, in response to the disincentives of CO₂ emissions introduced by the cap-and-trade policy. The activation of the CO₂ sequestrating process unit may lead to neutral or negative CO₂ emissions. Hence, the red shade patterns observable in Fig. 8(c) are attributable to the activation of these process units. On the other hand, Fig. 9 shows that a hybrid emissions policy that provides the polygeneration firm with the flexibility to subscribe either to a cap-and-trade or emissions tax has a relatively similar impact on the robustness of the resulting optimal process configuration compared to pure cap-and-trade.

5.1. Risk aversion and the role of target setting in process configuration

The robustness of the optimal process configurations decreases as the target profit increases. Although setting conservative target profits increases the robustness of the resulting configuration, it limits the amount of profit that the polygeneration plant can obtain. The robustness of the resulting configuration also suffers from extremely ambitious targets. Thus, a risk-averse attitude would lean towards conservative targets, while a risk-seeking attitude would lean towards ambitious targets. A decision-maker would benefit economically, i.e., in terms of achieving desired profits, and psychologically, i.e., confidence towards the probability of achieving target profits, from a decision within their risk appetite – the level of risk that the decision-maker is willing to absorb. In the context of post-pandemic energy supply chains, which are permeated by manifold uncertainties, a moderately ambitious target profit setting, i.e., $F=1.0\times {10}^7\mathrm{U}\mathrm{S}\mathrm{D}$ with ${\theta }^{\ast }=0.41$ (CS1) would be desirable. The value of ${\theta }^{\ast }$ can be considered the probability of achieving the target profit given the implementation of the resulting process configuration. The observed trade-off between target profit setting and robustness is an obvious idea in practice. The utility of the proposed model of this study is that it obtains optimal process configurations that adhere to this obvious and practical idea. It should be noted that the process design implications of robustness and target profit setting are not readily available in practice.

5.2. The cost efficiency of process configuration under boiler inoperability

The analysis performed in CS4 demonstrated the influence of process inoperability on the optimal configuration of processes in the polygeneration plant. Fig. 10 compares CS3 (zero inoperability) and CS4 (the boiler is inoperable). Both scenarios are capable of achieving the target profit $F=2.0\times {10}^5\mathrm{U}\mathrm{S}\mathrm{D}$; however, the inoperability of the boiler unit incurred major drawbacks to the optimal process configuration in CS4. In CS3, the optimal process configuration is a primarily heat-producing polygeneration plant. The inoperability of the boiler unit introduced a challenge to heat production. Instead of relying solely on the boiler for heat production, in CS4, the co-generation unit is activated to produce heat and power. To sequester the CO₂ emissions consequent to the activation of the co-generation unit, the pyrolysis unit is also activated. The activation increased the production of auxiliary products such as bio-oil and biochar. However, the activation of capital-intensive units in response to the inoperability of the boiler unit drastically reduced the cost-efficiency of the resulting optimal configuration, which is evidenced by the dip in the value of the profit margin ratio of the plant (see Fig. 10). It has also reduced the robustness of the resulting configuration and made it less “clean” in terms of CO₂ emissions. Nevertheless, the target profit can be achieved through this design at a relatively high probability of achievement despite the challenges introduced by the inoperability of the boiler unit. Table 5 presents the performance of the polygeneration plant in scenario CS3 if not configured regardless of the inoperability of the boiler unit ($Q_{\mathit{BOILER}}=$ 0.60). In comparison to CS4, the polygeneration plant would have lost 180,663.08 USD in profit if its design was not configured to achieve targets amidst the inoperability of the boiler unit. The incorporation of inoperability analysis in process configuration is among the advantages of the proposed model. It introduces a viable approach to designing the polygeneration plant that adapts to the risks associated with process failures without compromising target profit.

Table 5 presents the performance of the polygeneration plant in scenario CS3 if not configured regardless of the inoperability of the boiler unit ($Q_{\mathit{BOILER}}=$ 0.60). In comparison to CS4, the polygeneration plant would have lost 180,663.08 USD in profit if its design was not configured to achieve targets amidst the inoperability of the boiler unit. The incorporation of inoperability analysis in process configuration is among the advantages of the proposed model. It introduces a viable approach to automatically generate designs for the polygeneration plant that can adapt to the operational challenges introduced by severe process failures without compromising target profit.

5.3. Influence of economic pressures of emissions policies on process configuration

The findings obtained from SA1 demonstrated the straightforward effect of emissions policies on polygeneration plants. The robustness of polygeneration plants suffers from strict emissions policies, i.e., policies with a high carbon price. While this observation is apparent in practice, its implication on process configuration is not easily determined. For instance, the findings suggest that investments in technologies that draw down CO₂ emissions are motivated by sufficiently high carbon prices. Although this finding is evident in practice, the selection of specific technologies and the decision on its operational capacity in such a way that meets economic targets are not readily apparent. The advantage of the proposed model lies in the integration of this analysis in response to emissions policies. On the other hand, the tests employed in SA2 showed that different emissions policies, i.e., pure or hybrid policies, have an approximately similar impact on the robustness of the resulting optimal process configurations. This is largely due to the “carbon price” set for both cap-and-trade and emissions tax policies being almost equivalent. However, it is noticeable that the rate of decrease in robustness with respect to the initial emissions cap is lesser (see Fig. 9) in the hybrid policy than in the pure cap-and-trade policy (see Fig. 8(a)). This is a consequence of the flexibility of the polygeneration firm under the hybrid policy, which allows it to subscribe to emissions tax, which incurs a lesser cost per ton of CO₂ when the initial emissions cap decreases to sufficiently low levels. The effect of a hybrid policy on the robustness of the resulting configuration may be more pronounced under conditions where there is a disparity between the price of emissions allowance and the carbon tax rate.

5.4. Viability of the proposed model

The previous discussions highlighted the practicality of the performance of the designs produced by the proposed model, which means that it follows what is expected of a polygeneration plant when it becomes operational in the real world. The sophisticated aspect of the model is its ability to execute an optimal process configuration in response to the factors considered in the modeling work while producing a practical performance. The computed performance of the model was assessed using Monte Carlo simulation. The findings from the analysis suggest that the robustness metric used in the study is a viable metric to assess the performance of the obtained polygeneration plant designs against uncertainties in supply chain-related factors (i.e., price and demand).