Abstract
The optimal design of pipeline networks has inspired process systems engineers and operations research practitioners since the earliest times of mathematical programming. The nonlinear equations governing pressure drops, energy consumption and capital investments have motivated nonlinear programming (NLP) approaches and solution techniques, as well as mixedinteger nonlinear programming (MINLP) formulations and decomposition strategies. In this overview paper, we present a systematic description of the mathematical models proposed in recent years for the optimal design of pipeline networks in the energy industry. We provide a general framework to address these problems based on both the topology of the network to build, and the physical properties of the fluids to transport. We illustrate the computational challenges through several examples from industry collaboration projects, published in recent papers from our research group.
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Abbreviations
 \(a \in {A}_{i}\) :

Subset of nodes adjacent to node i
 \(c \in {C}\) :

Components in the fluid stream
 \(d \in {D}\) :

Alternative pipeline diameters
 \(i \in {I},j \in {J}, k \in {K}, l \in {L}\) :

Nodes in the network
 \(s \in {S}\) :

Fluid states
 \(t \in {T}\) :

Time periods
 \(ec_{t}\) :

Unit energy cost during period t
 \(g\) :

Gravity constant
 \({\text{glr}}\) :

Gastoliquid ratio in multiphase flows
 \(k_{{i,j}}\) :

Constant of the HazenWilliam correlation for a pipeline connecting i and j
 \(l_{{i,j}}\) :

Length of the pipeline connecting i and j
 \({\text{lt}}\) :

Leadtime for pipeline construction
 \(r\) :

Discount rate for cashflows
 \({\text{sg,\, sl}}\) :

Specific gravity of the gas/liquid
 \(s_{{\text{L}}}\) :

Head loss per unit length for water pipeline design purposes
 \(P_{{\text{o}}} ,~T_{{\text{o}}}\) :

Pressure and temperature at standard conditions
 \({\text{tc}}_{{i,j,d}}^{{}}\) :

Transportation capacity of a pipeline with diameter d connecting i and j
 \({\text{vmax}}_{{{\text{LP/GP}}}}\) :

Maximum linear velocities admitted for liquid and gas phases
 \(z\) :

Gas compressibility factor
 \(\alpha ,\beta\) :

Parameters of economyofscale functions
 \(\gamma\) :

Constant of the Weymouth correlation for gas flows
 \(\delta _{{d}}\) :

Numerical value of the diameter d of a pipeline
 \(\Delta {\text{sp}}\user2{~}_{{i,j}}^{{~{\text{Max}}}}\) :

Maximum difference of square pressures allowed between nodes i and j
 \(\Delta t\) :

Length of a time period
 \(\Delta z_{{i,j}}\) :

Elevation difference between nodes i and j
 \(\varepsilon\) :

Roughness of the internal wall of the pipeline
 ζ:

SPE constant for multiphase pipeline sizing
 \(\eta\) :

Pump yield
 \(\theta\) :

Temperature of the fluid
 \(\nu\) :

Kinematic viscosity of the fluid
 \(\mu _{{\text{c}}}\) :

Relative contribution of component c to the calculation of the pressure drop
 \(\rho ,\rho _{{{\text{avg}}}}\) :

Density/average density of the multiphase flow
 \(\varphi _{{c,j}}\) :

Fraction of component c removed from the flow stream at node j
 \(\psi _{{s,s^{\prime},i}}\) :

Yield of s’ per unit of s processed in node i
 \(\omega\) :

Exponent of the HazenWilliams correlation
 \({\text{Capex}}\left( {d_{{i,j,t}} } \right)\) :

Capital expenditures on a pipeline with diameter d connecting i and j built in period t
 \(d_{{i,j,t}}\) :

Diameter of a pipeline connecting i and j built in period t
 \(dl_{i,j,t}\) :

\(= d_{{i,j,t}} ^{2}\)for liquid pipelines
 \(dg_{{i,j,t}}\) :

\(=d_{{i,j,t}} ^{{2.667}}\)for gas pipelines
 \(D_{{c,k,t}}\) :

Demand of component c at node k during period t
 \(D_{{s,i,t}}\) :

Demand of fluid in state s at node i during period t
 \(C_{{s,i,t}}\) :

Amount of fluid in state s processed at node i during period t
 \(f\) :

Friction factor
 \(F_{{c,i,j,t}}\) :

Amount of component c flowing from i to j during period t
 \(F_{{s,i,i^{\prime},t}}\) :

Amount of fluid in state s flowing from i to i’ during period t
 \(h_{{\text{L}}}\) :

Head loss due to friction
 \(I_{{s,i,t}}\) :

Inventory level of s at node i at the end of period t
 \({\text{NPC}}\) :

Total net present cost
 \({\text{Opex}}\left( {F_{{c,i,j,t}} } \right)\) :

Operating expenditures of a pipeline carrying F units of c from i to j during period t
 \(P_{i,t}\) :

Pressure at node i during period t
 \(P_{{j,t}}^{{{\text{sq}}}}\) :

\(=P_{{i,t}} ^{2}\) for gas and multiphase pipelines
 \({\text{PW}}_{L}\) :

Pump power required to overcome head loss due to friction
 \(Q_{{c,i,t}}\) :

Production of component c at node i during period t
 \(Q_{{s,i,t}}\) :

Production of fluid in state s at node i during period t
 \(R_{{s,i,t}}\) :

Amount of fluid converted into state s at node i during period t
 \({\text{Re}}\) :

Reynolds number
 \({\text{TC}}\left( {d,P_{{i,t}} ,P_{{j,t}} } \right)_{i,j,\tau }\) :

Transportation capacity of a pipeline with diameter d connecting i and j built in period τ, according to inlet and outlet pressures in period t
 \({\text{TC}}_{{i,j,t}}^{{{\text{oil/gas/water}}}}\) :

Transportation capacity of oil/gas/water through pipeline i–j built in period t
 \(U\) :

Mean linear velocity of the flow
 \(X_{{c,j,t}}\) :

Concentration of c in the flows leaving node j during period t
 \(X_{{{\text{LM}}}}\) :

LockhartMartinelli parameter for multiphase flows
 \(\Delta P~_{{i,j,t}}^{{{\text{sq}}}}\) :

Difference of square pressures at nodes i and j during period t
 \(\Delta Pg,~\Delta Pl\) :

Pressure drops for gas and liquid phases in multiphase flows
 \(u_{{i,j,t}}\) :

= 1 if the flow direction is from i to j during period t
 \(x_{{i,j,d,t}}^{{}}\) :

= 1 if a pipeline with diameter d is built between i and j in period t
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Acknowledgements
Financial support from National University of Litoral under Grant CAI+D 2020 50620190100163LI, and CONICET under Grant PIP 11220200103053CO is gratefully acknowledged. Financial support from the Center of Advanced Process Decisionmaking at Carnegie Mellon University is also gratefully acknowledged.
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Appendix
Appendix
1.1 Calculation of energy consumption in a pipeline segment
A method to compute pumping cost for multiproduct pipelines as a function of the flowrate is presented by Cafaro et al. (2015), under the assumption of batch flow and neglecting “transmix” volumes. The mean velocity (U) in a straight segment is given by the ratio between the pump rate and the pipeline section, while the Reynolds number is given by Re = 4F/πdν, with ν being the fluid kinematic viscosity. If refined products flow in turbulent regime into the pipeline segments (Re > 4 × 10^{5}) the relationship between the head loss due to friction (h_{L}) and the pump rate can be derived from the Darcy's law (Darcy 1857) given in Eq. (26). This equation introduces the dimensionless friction factor f. Indices of nodes and time periods are omitted for simplicity.
Moreover, the friction factor f can be calculated through the ColebrookWhite equation, as in Eq. (27).
Equation (27) involves an implicit function accounting for two contributions: the pipeline wall roughness (ε), and the flow turbulence. Finally, the power required to overcome the friction loss is given by Eq. (28), representing a nonlinear function rapidly increasing with the flowrate F.
The parameter ρ is the liquid density, g is the gravitational constant, l is the length of the pipeline and η is the pump efficiency. Equations (29) and (30) are introduced in the MINLP model presented in Sect. 3.1.1 to calculate the energy consumption in every pipeline segment over time period t.
1.2 Lockhart–Martinelli procedure to compute pressure drops in multiphase flows
The Lockhart–Martinelli (LM) procedure has been devised to predict the pressure drop for fully developed gas–liquid flows. The first step is to calculate the individual effects of liquid and gas phases, i.e., the pressure drops \(\Delta Pl\) and \(\Delta Pg\) that would be expected if the liquid and gas streams were flowing alone through the same pipeline. The second step is to obtain the LM parameter (\(X_{{{\text{LM}}}}\)), as in Eq. (29).
Finally, the overall pressure drop in the pipeline segment is estimated from the pressure drop of the gas phase, as in Eq. (30).
Equation (30) follows the Wilkes function (Wilkes 2005) to fit the data of the LM empirical results. The parameter n depends on the flow regime of each phase, and is equal to 4.12 if both liquid and gas phases flow in turbulent regime, as usually seen in industrial applications. In horizontal pipelines, the pressure drop of the liquid phase \(\Delta Pl\) can be obtained from the Darcy equation (see Eqs. (31) and (32)), while the gas pressure drop \(\Delta Pg\) is related to the pipeline diameter and the gas flow through the Weymouth correlation (see Eq. (17) in Sect. 3.2). Note that the flowrates of the two different phases need to be tracked through separate variables (\(F_{{{\text{LIQ}},i,j,t}}\) and \(F_{{{\text{GAS}},i,j,t}}\)), as explained in Sect. 2.1.
1.3 SPE guidelines for sizing multiphase pipelines
Similar to the liquid pipeline sizing problem, the SPE (Lake et al. 2006) suggests maximum velocities for multiphase flows to prevent erosion, corrosion, noise or water hammer effects. Reference maximum velocities are 18 m/s to inhibit noise and 15 m/s to prevent corrosion. A more accurate procedure to obtain maximum velocities is suggested as follows.

1.
Obtain the average multiphase density (\(\rho_{{{\text{avg}}}}\)) from the gas to liquid ratio (\({\text{glr}}\)) as:
$$\rho_{{{\text{avg}}}} = \frac{{12409{\text{ sl }}P_{{{\text{in}}}} + 2.7{\text{ glr sg }}P_{{{\text{in}}}} }}{{198.7{ }P_{{{\text{in}}}} + z{\text{ glr }}\theta }},$$(31)where z is the gas compressibility factor, r the gas/liquid ratio [ft^{3}/bbl], \(\theta\) the flow temperature [°R], \(P_{{{\text{in}}}}\) the inlet pressure in PSI, sl the specific gravity of the liquid phase relative to water, and sg the specific gravity of the gas, relative to air.

2.
Set the maximum velocity for the liquid phase as:
$$v_{{{\text{max}}}} = \zeta\,\rho_{{{\text{avg}}}}^{  0.5} ,$$(32)where ζ is an SPE specific constant with a value of 150 for solidsfree fluids and continuous service operation, and \(v_{{{\text{max}}}}\) is measured in ft/s.

3.
Impose an upper bound on the liquid flowrate, as shown in Eq. (33).
$$F_{{{\text{LIQ}},i,j,t}} \le \xi\, { }v_{{{\text{max}}}} d_{i,j,t}^{2}\,,$$(33)with \(\xi = 0.64516\left( {11.9 + z{\text{ glr }}\theta /16.7{ }P_{{{\text{in}}}} } \right)^{  1}\).
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Cafaro, D.C., Presser, D.J. & Grossmann, I.E. Recent contributions to the optimal design of pipeline networks in the energy industry using mathematical programming. TOP 30, 618–648 (2022). https://doi.org/10.1007/s11750022006353
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DOI: https://doi.org/10.1007/s11750022006353