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Operational decisions for multi-period industrial gas pipeline networks under uncertainty

Abstract

An industrial gas company (IGC) manages a large pipeline network consisting of several gas production plants and many customers. Natural gas is an input to the production process at each plant, and a different industrial gas (e.g., hydrogen or oxygen) is the plant output. Each day the company must propose 24 h in advance a “nomination value” for the amount of natural gas they plan to use at each plant over the next day. The nomination value goes to the natural gas provider who then offers a set price for the nominated amount. If the nomination amount underestimates actual usage, the IGC must purchase the extra gas at a (typically higher) spot market price. If the nomination value overestimates usage, the IGC sells extra gas at a (typically lower) spot market price. In this study, we develop a multi-period model to determine decisions for daily nomination values of natural gas for multiple plants as well as production and distribution decisions for a steady-state behaving industrial gas at distribution network, where the direction of the gas flow is not known at each pipe. The model considers demands from multiple customers to be uncertain and varying throughout the day. Daily planning requires a multi-period formulation that includes ramping constraints (physical constraints that limit how much a plant’s production can change over each period). The model also dictates that customer demands be met with high probability and that the physical constraints of the pipeline are met. The result is a challenging multi-period, chance constrained, nonlinear integer optimization problem. We propose an optimization-based heuristic approach to find a high-quality feasible solution for industry-sized problems. Also, we use a convex relaxation of the problem to obtain valid lower bound for the problem in order to obtain a measure of the near-optimality of the problem’s feasible solution found via the introduced heuristic approach. Numerical experiments show that the proposed heuristic is able to efficiently find near-optimal solutions for the problem instances in which using an exact solution approach is prohibitively time consuming.

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Acknowledgements

The first author was supported by the Pennsylvania Infrastructure Technology Alliance (PITA) Grant PIT-13-17.

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Correspondence to Pelin Cay.

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Appendices

Appendix 1: MSP-A model formulation

This problem is basically just the original problem but with a reduced number of periods (subproblems), such that it focuses only on periods in \(S_s\) (as in Eq. 3). There are two versions of this model, depending on the starting period. If we begin in period 0, then we include constraints (2j) and (2m). If the beginning period is not period 0, then we set this period’s initial production equal to the ending production level from the previous period (which exists because we solve these problems forward in time). Thus, we do not include constraints (2j) and (2m) in this case. In this problem, we solve the original formulation of the problem i.e. Model 2 with difference on considered period length (\(S_s^i\), \(\forall i \in \{1 \dots N\}\), for simplicity we refer \(S_s\) in the formulation). This set is updated for each subproblem. The model formulation for MSP-A is as follows

$$\begin{aligned}&\displaystyle \mathop {{\text{ minimize }}}\limits _{y,x,{\bar{p}},q,{\mathcal {H}},s,w}&\displaystyle \;\;\;\;\; \sum _{i \in {\mathcal {P}}} {\hat{p}}_{i}{\mathcal {H}}_{i} + q_{i} \end{aligned}$$
(4a)
$$\begin{aligned}&\text{ subject } \text{ to }&\displaystyle \;\;\;q_{i} \ge p^+_i\left( \sum _{t \in {\mathcal {S}}_s,k \in \{1 .. {\tilde{s}}\}}(\delta _{kit}s_{kit})- {\mathcal {H}}_{i}\right) \qquad&\qquad { \forall i \in {\mathcal {P}}} \end{aligned}$$
(4b)
$$\begin{aligned}&\displaystyle q_{i} \ge p^-_i\left( {\mathcal {H}}_{i} -\sum _{t \in {\mathcal {S}}_s, k \in \{1 .. {\tilde{s}} \}}(\delta _{kit}s_{kit})\right) \qquad&\qquad { \forall i \in {\mathcal {P}}} \end{aligned}$$
(4c)
$$\begin{aligned}&\displaystyle y_{it} = \sum _{k \in \{1 .. {\tilde{s}}\}}s_{kit} \qquad&\qquad {\forall i \in {\mathcal {P}}, \forall t \in {\mathcal {S}}_s} \end{aligned}$$
(4d)
$$\begin{aligned}&\displaystyle s_{ki}^Uw_{kit} \le s_{kit} \le s_{ki}^U \qquad&\qquad {\forall i \in {\mathcal {P}}, \forall t \in {\mathcal {S}}_s, k \in \{1\}} \end{aligned}$$
(4e)
$$\begin{aligned}&\displaystyle s_{ki}^Uw_{kit} \le s_{kit} \le s_{ki}^Uw_{k-1it} \qquad&\qquad {\forall i \in {\mathcal {P}}, \forall t \in {\mathcal {S}}_s, k \in \{2 .. {\tilde{s}}-1\}} \end{aligned}$$
(4f)
$$\begin{aligned}&\displaystyle 0 \le s_{kit} \le s_{kit}^Uw_{k-1it} \qquad&\qquad {\forall i \in {\mathcal {P}}, \forall t \in {\mathcal {S}}_s,k \in \{{\tilde{s}}\}} \end{aligned}$$
(4g)
$$\begin{aligned}&\displaystyle \;\;\;y_{it} + \sum _{(j,i) \in {\mathcal {E}}} x_{jit} = \sum _{(i,j) \in {\mathcal {E}}} x_{ijt} \qquad&\qquad {\forall i \in {\mathcal {P}}, \forall t \in {\mathcal {S}}_s} \end{aligned}$$
(4h)
$$\begin{aligned}&\displaystyle \sum _{(j,i) \in {\mathcal {E}}} x_{jit} = \sum _{(i,j) \in {\mathcal {E}}} x_{ijt} \qquad&\qquad {\forall i \in {\mathcal {K}},\forall t \in {\mathcal {S}}_s} \end{aligned}$$
(4i)
$$\begin{aligned}&\displaystyle \sum _{(j,i) \in {\mathcal {E}}} x_{ji1} - \sum _{(i,j) \in {\mathcal {E}}} x_{ij1} \ge d_{i} \qquad&\qquad {\forall i \in {\mathcal {C}}} \end{aligned}$$
(4j)
$$\begin{aligned}&\displaystyle \sum _{(j,i) \in {\mathcal {E}}} x_{jit} - \sum _{(i,j) \in {\mathcal {E}}} x_{ijt} \ge \mu _{it} +\Phi ^{-1}(\eta )\sigma _{it} \qquad&\qquad {\forall i \in {\mathcal {C}}, \forall t \in {\mathcal {S}}_s \setminus \{1\}} \end{aligned}$$
(4k)
$$\begin{aligned}&\displaystyle \gamma _{ij}x_{ijt}\sqrt{x_{ijt}^2 + \epsilon }={\bar{p}}_{it}-{\bar{p}}_{jt} \qquad&\qquad {\forall (i,j) \in {\mathcal {E}}, \forall t \in {\mathcal {S}}_s} \end{aligned}$$
(4l)
$$\begin{aligned}&\displaystyle r_i^L \le y_{i1} - {\hat{y}}_{i0} \le r_i^U \qquad&\qquad {\forall i \in {\mathcal {P}}} \end{aligned}$$
(4m)
$$\begin{aligned}&\displaystyle r_i^L \le y_{it} - y_{i(t-1)} \le r_i^U \qquad&\qquad {\forall i \in {\mathcal {P}}, \forall t \in {\mathcal {S}}_s \setminus \{1\}} \end{aligned}$$
(4n)
$$\begin{aligned}&\displaystyle y^L_i \le y_{it} \le y^U_i \qquad&\qquad {\forall i \in {\mathcal {P}}, \forall t \in {\mathcal {S}}_s} \end{aligned}$$
(4o)
$$\begin{aligned}&\displaystyle \pi _i^- \le {\bar{p}}_{it} \le \pi _i^+ \qquad&\qquad {\forall i \in {\mathcal {V}}, \forall t \in {\mathcal {S}}_s} \end{aligned}$$
(4p)
$$\begin{aligned}&\displaystyle x_{ijt} \le x^U_{ij} \qquad&\qquad {\forall (i,j) \in {\mathcal {E}}, \forall t \in {\mathcal {S}}_s} \end{aligned}$$
(4q)
$$\begin{aligned}&\displaystyle - x_{ijt} \le x^U_{ij} \qquad&\qquad {\forall (i,j) \in {\mathcal {E}}, \forall t \in {\mathcal {S}}_s} \end{aligned}$$
(4r)
$$\begin{aligned}&\displaystyle w_{kit} \in \{0,1\} \qquad&\qquad {\forall i \in {\mathcal {P}}, \forall t \in {\mathcal {S}}_s, k \in \{1 .. {\tilde{s}}-1\}} \end{aligned}$$
(4s)

If the subproblem does not include period 1, then we drop constraint (4j) and (4m) from problem definition and solve without these constraints. At these cases, we introduce a new constraint such that we set previous subproblems’ last period production levels as starting production levels for the next subproblem and apply ramping constraint for the first period of the considered subproblem. For the rest of the problem formulations, if the time index is zero, it means that information is used either from previous subproblem’s last period information or current production information. We introduce a parameter \({\hat{y}}_{i {\hat{T}}_e}\) for all plant \(i \in {\mathcal {P}}\) which is used when the considered problem does not include period 1 in their subproblem. This parameter equals to the production levels for final period found from the previous subproblem. Then, the (4m) becomes

$$\begin{aligned} r_i^L \le y_{iT_s} - {\hat{y}}_{i{\hat{T}}_e} \le r_i^U \;\; \qquad \forall i \in {\mathcal {P}} \end{aligned}$$
(3m*)

This constraint mimics what (4m) does with the difference of using the solution from the previous subproblem.

Appendix 2: MSP-B model formulation

This problem includes the same constraints as Model (4), but the objective function differs because of the addition of penalty objective on the initial period production level of the considered subproblem. We introduce a new decision variable \({\bar{y}}_{iT_s}\) for all \(i \in {\mathcal {P}}\) that denotes what the initial production level should be instead of considering initial production levels, which are equal to final production levels from previous subproblem (\({\hat{y}}_{i{\hat{T}}_e}\)). Considering them in the penalty objective function deals to minimize the violation that leads infeasilbe solution of Model 4. This problem includes a penalty objective based on initial production value with penalty parameter \(\rho\) in the objective function as follows:

$$\begin{aligned}&\displaystyle \mathop {{\text{ minimize }}}\limits _{y,x,{\bar{p}},q,{\mathcal {H}},s,w,{\bar{y}}_{iT_s}}&\displaystyle \;\;\;\;\; \sum _{i \in {\mathcal {P}}} {\hat{p}}_{i}{\mathcal {H}}_{i} + q_{i}+\rho \sum _{i \in {\mathcal {P}}} | {\bar{y}}_{iT_s} - {\hat{y}}_{i{\hat{T}}_e}| \end{aligned}$$
(5a)
$$\begin{aligned}&\text{ subject } \text{ to }&\displaystyle \hbox {(4b), (4c), (4d), (4e), (4f), (4g), (4h)} \end{aligned}$$
(5b)
$$\begin{aligned}&\displaystyle \hbox {(4i), (4k), (4l), (4n), (4o), (4p), (4q), (4r), (4s)} \end{aligned}$$
(5c)
$$\begin{aligned}&\displaystyle r_i^L \le y_{iT_s} - {\bar{y}}_{iT_s} \le r_i^U \qquad&\qquad {\forall i \in {\mathcal {P}}} \end{aligned}$$
(5d)
$$\begin{aligned}&\displaystyle {\bar{y}}_{iT_s} \ge 0 \qquad&\qquad {\forall i \in {\mathcal {P}}} \end{aligned}$$
(5e)

In Model (5), we can see that the constraint (5d) is different than (3m*) since the ramping at (5d) is based on new introduced decision variable instead of parameter \({\hat{y}}_{i{\hat{T}}_e}\). This constraint considers which value of previous subproblem’s production levels make this subproblem feasible by considering the ramping constraints. Lastly (5e) is for non-negative production level.

Appendix 3: MSP-C model formulation

This problem includes the same constraints as Model 4 with different objective function. As in Model (5), we introduce a new parameter \({\bar{y}}_{i{\bar{T}}_e}\) for all \(i \in {\mathcal {P}}\) that denotes what the final production level should be that is passed from Model 5 solution (\({\bar{y}}_{iT_s}\) from Model 5, initial production level for the subproblem defined for Model 5). The penalty objective with penalty parameter \(\lambda\) minimizes the violation by how much the total last production level for all plants in this subproblem can be as close as the passed value \({\hat{y}}_{iT_e}\) for all \(i \in {\mathcal {P}}\) from Model 5. This model formulation becomes as follows:

$$\begin{aligned}&\displaystyle \mathop {{\text{ minimize }}}\limits _{y,x,{\bar{p}},q,{\mathcal {H}},s,w}&\displaystyle \;\;\;\;\; \sum _{i \in {\mathcal {P}}} {\hat{p}}_{i}{\mathcal {H}}_{i} + q_{i}+\lambda \sum _{i \in {\mathcal {P}}} | y_{iT_e} - {\bar{y}}_{i{\bar{T}}_e}| \end{aligned}$$
(6a)
$$\begin{aligned}&\displaystyle \hbox {(4b), (4c), (4d), (4e), (4f), (4g), (4h)} \end{aligned}$$
(6b)
$$\begin{aligned}&\displaystyle \hbox {(4i), (4k), (4l), (3m*), (4n), (4o), (4p), (4q), (4r), (4s)} \end{aligned}$$
(6c)
$$\begin{aligned}&\displaystyle r_i^L \le y_{it} - y_{i(t-1)} \le r_i^U \qquad&\qquad {\forall i \in {\mathcal {P}}, \forall t \in {\mathcal {S}}_s\setminus \{T_s\}} \end{aligned}$$
(6d)

In Model 6, the objective function is similar as in Model 5, but the penalty is applied on the last period production level and the solution provided from Model 5 for the one subproblem ahead for the considered subproblem (this model is solved relatively one of the previous subproblems where infeasible solution is first found, you can refer examples in pages 14–15). The constrant (6d) applies ramping constraint for the periods within \(S_s\) except starting period. The starting period’s ramping constraint is represented in constraint (3m*).

Appendix 4: MSP-D model formulation

This problem includes the same constraints as Model 5 that includes additional information as a constraint such that the solution from Model 5 which is used as parameter \({\bar{y}}_{i{\bar{T}}_e}\) in Model 6 is also considered in this formulation as a new parameter \(\upsilon\) i.e, \(\upsilon _i = {\bar{y}}_{i{\bar{T}}_e}\) for each plant i in \({\mathcal {P}}\). The model formulation is as follows:

$$\begin{aligned}&\displaystyle \mathop {{\text{ minimize }}}\limits _{y,x,{\bar{p}},q,{\mathcal {H}},s,w,{\bar{y}}_{iT_s}}&\displaystyle \;\;\;\;\; \hbox {(5a)} \end{aligned}$$
(7a)
$$\begin{aligned}&\text{ subject } \text{ to }&\displaystyle \;\;\; \hbox {(5b), (5c), (5d), (5e)} \end{aligned}$$
(7b)
$$\begin{aligned}&\displaystyle y_{iT_e} = \upsilon _{i} \qquad&\qquad {\forall i \in {\mathcal {P}}} \end{aligned}$$
(7c)

In Model 7, the objective function and the constriants are same as Model 5 with the different of constraint 7c which enforces the solution found at Model 5 is a hard constraint to for the final production levels for each plant i in \({\mathcal {P}}\).

Appendix 5: MSP-E model formulation

This problem considers the same constraints as in Model 4 with additional constraint as follows:

$$\begin{aligned}&\displaystyle \mathop {{\text{ minimize }}}\limits _{y,x,{\bar{p}},q,{\mathcal {H}},s,w}&\displaystyle \;\;\;\;\; \hbox {(4a)} \end{aligned}$$
(8a)
$$\begin{aligned}&\text{ subject } \text{ to }&\displaystyle \hbox {(4b), (4c), (4d), (4e), (4f), (4g), (4h)} \end{aligned}$$
(8b)
$$\begin{aligned}&\displaystyle \hbox {(4i), (4k), (4l), (3m*), (4n), (4o), (4p), (4q), (4r), (4s)} \end{aligned}$$
(8c)
$$\begin{aligned}&\displaystyle \hbox {(7c)} \end{aligned}$$
(8d)

The difference is, while we are solving the subproblem with the original objective function and same subproblem as in Model 4, we consider the last production value set to \(\upsilon _i\) for all \(i \in {\mathcal {P}}\), to find a feasible solution for the next subproblem.

Appendix 6: FullLp formulation of model 1

The lower bound formulation for the Model 1 is obtained by two steps of relaxation of the pressure drop constraint (1g). The first step is changing this constraint’s type from equality to inequality as introduced in Borraz-Sánchez et al. (2016)’s study which becomes as follows:

$$\begin{aligned} \gamma _{ij}x^2_{ijt} \le ({\bar{p}}_{it} - {\bar{p}}_{jt} ) (1 -2z_{ijt}) \qquad {\forall (i,j) \in {\mathcal {E}}, \forall t \in {\mathcal {S}}} \end{aligned}$$
(1*)

Then, we continue by applying the steps from Cay et al. (2018) to obtain FullLp formulation of Model 1. The right hand side of (1*) is represented explicitly as follows:

$$\begin{aligned} \gamma _{ij}x^2_{ijt} \le {\bar{p}}_{it} - 2{\bar{p}}_{it}z_{ijt} - {\bar{p}}_{jt} + 2{\bar{p}}_{jt} z_{ijt} \qquad {\forall (i,j) \in {\mathcal {E}}, \forall t \in {\mathcal {S}}} \end{aligned}$$
(2*)

Then, as a part of the first step, the integer reformulation of constraint (2*) is obtained by applying two times McCormick (1976) relaxations on bilinear terms. For this integer reformulation, extra variables are introduced \({\hat{w}}_{ijt}\) and \({\hat{s}}_{ijt}\) where \((i,j) \in {\mathcal {E}}, t \in {\mathcal {S}}\), where

$$\begin{aligned} {\hat{w}}_{ijt}&= {\bar{p}}_{it}z_{ijt} \qquad&{\forall (i,j) \in {\mathcal {E}}, \forall t \in {\mathcal {S}}} \end{aligned}$$
(a)
$$\begin{aligned} {\hat{s}}_{ijt}&= {\bar{p}}_{jt} z_{ijt} \qquad&{\forall (i,j) \in {\mathcal {E}}, \forall t \in {\mathcal {S}}} \end{aligned}$$
(b)

We can rewrite relation at (a) as follows:

$$\begin{aligned} {\hat{w}}_{ijt}&\le \pi _{it}^+ z_{ijt}&{\forall (i,j) \in {\mathcal {E}}, \forall t \in {\mathcal {S}}} \end{aligned}$$
(a1)
$$\begin{aligned} {\hat{w}}_{ijt}&\ge \pi _{it}^- z_{ijt}&{\forall (i,j) \in {\mathcal {E}}, \forall t \in {\mathcal {S}}} \end{aligned}$$
(a2)
$$\begin{aligned} {\hat{w}}_{ijt}&\ge {\bar{p}}_{it} - \pi _{it}^+(1- z_{ijt})&{\forall (i,j) \in {\mathcal {E}}, \forall t \in {\mathcal {S}}} \end{aligned}$$
(a3)
$$\begin{aligned} {\hat{w}}_{ijt}&\le {\bar{p}}_{it} - \pi _{it}^-( 1- z_{ijt}) \qquad&{\forall (i,j) \in {\mathcal {E}}, \forall t \in {\mathcal {S}}} \end{aligned}$$
(a4)

The same relation for (b) is as follows:

$$\begin{aligned} {\hat{s}}_{ijt}&\le \pi _{jt}^+ z_{ijt}&{\forall (i,j) \in {\mathcal {E}}, \forall t \in {\mathcal {S}}} \end{aligned}$$
(b1)
$$\begin{aligned} {\hat{s}}_{ijt}&\ge \pi _{jt}^- z_{ijt}&{\forall (i,j) \in {\mathcal {E}}, \forall t \in {\mathcal {S}}} \end{aligned}$$
(b2)
$$\begin{aligned} {\hat{s}}_{ijt}&\ge {\bar{p}}_{jt} - \pi _{jt}^+(1- z_{ijt})&{\forall (i,j) \in {\mathcal {E}}, \forall t \in {\mathcal {S}}} \end{aligned}$$
(b3)
$$\begin{aligned} {\hat{s}}_{ijt}&\le {\bar{p}}_{jt} - \pi _{jt}^-( 1- z_{ijt}) \qquad&{\forall (i,j) \in {\mathcal {E}}, \forall t \in {\mathcal {S}}} \end{aligned}$$
(b4)

Then, constraint (2*) is equivalent to

$$\begin{aligned} \gamma _{ij}x^2_{ijt} \le {\bar{p}}_{it} - 2{\hat{w}}_{ijt} - {\bar{p}}_{jt} + 2{\hat{s}}_{ijt} \qquad {\forall (i,j) \in {\mathcal {E}}, \forall t \in {\mathcal {S}}} \end{aligned}$$
(3*)

including the constraints (a1)–(a4) and (b1)–(b4). Then at the end of the first step, we obtain the mixed-integer second order cone constraint optimization (MISOCO) relaxation with integer reformulation of Model 1.

At the second step, we focus on relaxing the squared flow component of the constraint (3*). Since the objective of this formulation is to obtain a lower bound, outer linearization is applied on the squared flow term. To apply the outer linearization on \(x_{ijt}^2\) for all \((i,j) \in {\mathcal {E}}\) and \(t \in {\mathcal {S}}\), B number of uniformly distributed points \(a_{ij}^k\) for all \((i,j) \in {\mathcal {E}}\) and \(k \in \{1 ... B\}\) within the bounds of \(x_{ijt}\) are used and a new decision variable \({\hat{q}}_{ijt}\) where \((i,j) \in {\mathcal {E}}\) for each \(x_{ijt}^2\) is defined. Then, the relation between \({\hat{q}}_{ijt}\) and \(x_{ijt}^2\) is represented as follows:

$$\begin{aligned} {\hat{q}}_{ijt} \ge (a_{ij}^k)^2 + 2a_{ij}^k(x_{ijt} - a_{ij}^k) \qquad \qquad {\forall (i,j) \in {\mathcal {E}}, \forall t \in {\mathcal {S}}, k \in 1 .. B} \end{aligned}$$

As applied in Cay et al. (2018), we also include an upper bound on \({\hat{q}}_{ijt}\) for all \((i,j) \in {\mathcal {E}}\) and \(t \in {\mathcal {S}}\) by adding overestimator of that term derived from McCormick (1976) relaxation which is

$$\begin{aligned} {\hat{q}}_{ijt} \le (x_{ij}^L + x_{ij}^U)x_{ijt}-x_{ij}^Lx_{ij}^U \qquad \qquad {\forall (i,j) \in {\mathcal {E}}, \forall t \in {\mathcal {S}}} \end{aligned}$$

where \(x_{ij}^L\) is lower bound on amount of gas that can be in a pipe \((i,j) \in {\mathcal {E}}\) which is zero for this study. Then, FullLp model which is the relaxation of Model 1 is formulated as follows:

$$\begin{aligned}&\displaystyle \mathop {{\text{ minimize }}}\limits _{y,x,{\bar{p}},q,z,{\mathcal {H}},s,w,{\hat{q}},{\hat{w}},{\hat{s}}}&\displaystyle \;\;\;\;\; \hbox {(2a)} \end{aligned}$$
(9a)
$$\begin{aligned}&\text{ subject } \text{ to }&\displaystyle \hbox {(2b)}-\hbox {(2k)}\end{aligned}$$
(9b)
$$\begin{aligned}&\displaystyle \gamma _{ij}{\hat{q}}_{ijt} \le {\bar{p}}_{it} - 2{\hat{w}}_{ijt} - {\bar{p}}_{jt} + 2{\hat{s}}_{ijt} \qquad&\qquad {\forall (i,j) \in {\mathcal {E}}, \forall t \in {\mathcal {S}}} \end{aligned}$$
(9c)
$$\begin{aligned}&\displaystyle {\hat{q}}_{ijt} \ge (a_{ij}^k)^2 + 2a_{ij}^k(x_{ijt} - a_{ij}^k) \qquad&\qquad {\forall (i,j) \in {\mathcal {E}}, k \in 1 .. N, \forall t \in {\mathcal {S}}} \end{aligned}$$
(9d)
$$\begin{aligned}&\displaystyle {\hat{q}}_{ijt} \le (x_{ij}^L + x_{ij}^U)x_{ijt}-x_{ij}^Lx_{ij}^U \qquad&\qquad {\forall (i,j) \in {\mathcal {E}}, \forall t \in {\mathcal {S}}} \end{aligned}$$
(9e)
$$\begin{aligned}&\displaystyle \hbox {(a1)} -\hbox {(a4)}\end{aligned}$$
(9f)
$$\begin{aligned}&\displaystyle \hbox {(b1)} -\hbox {(b4)}\end{aligned}$$
(9g)
$$\begin{aligned}&\displaystyle \hbox {(2m)} -\hbox {(2s)} \end{aligned}$$
(9h)

Model 9 is a MILP reformulation, which is a relaxation of Model 1, with additional variables and constraints. Constraint (9d) represents the outer linearization for each flow \(x_{ijt}\) in pipe \((i,j) \in {\mathcal {E}}\) at period \(t \in {\mathcal {S}}\) and constraint (9e) is an upper bound on \({\hat{q}}_{ijt}\) by adding McCormick (1976) relaxation for all \((i,j) \in {\mathcal {E}}\) and \(t \in {\mathcal {S}}\).

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Cay, P., Mancilla, C., Storer, R.H. et al. Operational decisions for multi-period industrial gas pipeline networks under uncertainty. Optim Eng 20, 647–682 (2019). https://doi.org/10.1007/s11081-019-09430-9

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  • DOI: https://doi.org/10.1007/s11081-019-09430-9

Keywords

  • Gas network optimization
  • Uncertainty
  • Multi-period
  • Heuristic
  • Decision making