A Pinch Analysis approach for minimizing compression energy and capital investment in gas allocation network

Abstract

Transportation of fluid is a very important aspect of process industries, especially for oil and gas industries. Pipelines have been considered as the most effective and safest way of transporting fluids. Transportation of gas through a pipeline is an energy-intensive process; hence, energy optimization in gas transportation networks is an important issue to be considered while framing the environmental policies. In this paper, a novel graphical methodology based on pinch analysis approach for simultaneous minimization of capital investment and compression energy requirement in gas allocation network with the aid of thermodynamic relations is developed. The results of the proposed methodology are expressed as a Pareto optimal front. The ε-constraint method is used to generate Pareto optimal front for the two objectives. Identifying the relationship between capital investment and energy requirement gives the opportunity to the decision-maker for choosing the suitable optimal operating point based on the operating and economic conditions of the process. This result allows the planner to calculate the effects via increasing or decreasing energy requirement or capital investment. The applicability of the proposed methodology is demonstrated through two illustrative examples.

Graphic abstract

Introduction

The gas transportation process is energy-intensive and needs capital investment (Gabbar and Kishawy 2011). Gas transportation via pipelines is considered as the most consistent and economical mode of moving gas from key stock sources to delivery centers (Mostafaei et al. 2015). As the gas flows through the network, pressure loss occurs due to friction between the gas and the pipe inner wall (Ganat and Hrairi 2018). To overcome this pressure loss and keep the gas moving, compressor stations (CS’s) are needed in the gas allocation network (GAN) (Woldeyohannes et al. 2011). The compression procedures involve huge amounts of energy that are typically supplied by the compressor with shaft works (Amirabedin et al. 2010). Optimizing the allocation networks with the objectives to minimize energy and investment will result into the policies which will reduce the investment along with negative effect on the environment. The energy consumption of the compressor is responsible for a major fraction of operating costs of GAN (Rose et al. 2016), and the CS’s restore the lost pressure in the GAN (Gopalkrishnan and Bieglar 2013). With the growing concern over global warming and sustainable energy usage, every region needs to be its energy-efficient (Hu et al. 2011). The reduction of the energy used in pipeline operations will not only have an economic impact but also environmental impact. The more efficient use of CS’s is reduced greenhouse emissions to the atmosphere (Sorrell 2015). The principal concerns with both designing and operating a GAN are minimizing energy consumption and capital investment while maximizing throughput (Ogbonnaya et al. 2016). However, energy consumption and capital investment are also correlated (Wolf and Smeers 1996). Understanding the relationship between these two objectives will help the designer to reach an optimal solution.

Researchers have worked on optimization in GAN with different objectives such as optimal sizing of gas pipeline network (Uraikul et al. 2004), optimizing the configuration of the pipeline (Munoz et al 2003), minimizing electric power consumption (Warchol et al. 2018) and number of the operating compressor in gas pipeline networks (Zhang and Liu 2017). These works utilize mainly mathematical modeling methods to determine the optimum results for a GAN. The mathematical model can be a deterministic or stochastic model, which is selected based on the nature of the system (Mikolajkováet al. 2018). Luongo et al. (1989) analyzed that the operational cost of CS’s contributes around 25% to 50% in the total operating budget of a GAN. One of the initial remarkable work has been carried out for utility cost minimization of GAN proposed by Wu et al. (2000) which considers pressure drop, mass flow rate, and the number of units functioning as choice variables at each CS. Later, A nonlinear programming (NLP) model is presented by Abbaspour et al. (2005) to include the compressor speed as an operating variable while minimizing total utility consumption. Further, Kabiriana and Hemmati (2007) proposed an integrated NLP optimization model for optimizing existing GANs including the long-run design prospects to minimize net present value that includes functional and investment costs. Later, Andre et al. (2009) proposed a modeling framework to minimize investment costs for retrofitting of existing GANs. This modeling framework initially determines the optimal location of pipeline segments that need to be reinforced and then the optimal dimensions of pipes. Next, a numerical optimization model for minimizing the overall investment cost of a GTN is developed by Ruan et al. (2009) which considers various features (such as pressure pipe diameter, compression ratio, length and thickness) affecting investment cost.

CS’s are the most energy-intensive parts in GAN. Mora and Ulieru (2005) presented a mathematical algorithm to minimize energy requirements in CS’s while satisfying demands with requisite pressure and flows at different locations. Later, an ant colony optimization algorithm is proposed by Chebouba et al. (2009) for calculating optimum utility consumption by CS for fixed flow GAN. Further, to apprehend the impact of various variables such as flow rate, pressure and temperature for high-pressure gas pipeline one-dimensional, a non-isothermal gas model is proposed by Chaczykowski (2010). Borraz-Sánchez et al. (2011) proposed an optimization model for minimizing compression fuel cost in transport network for transporting natural gas which involves two choice constraints: mass flow rate at each point and gas pressure level. To minimize energy requirement, design with the shortest distance is not always the optimum choice; it depends on various other factors (Sanaye and Mahmoudimehra 2013). Kashani and Molaei (2014) consider multiple objectives such as fuel consumption, profit and line pack in the proposed optimization formulation where results are expressed as Pareto optimal fronts between the various objectives. A stochastic optimization model for the day-to-day scheduling of GAN is proposed by Behrooz and Boozarjomehry (2017) for calculating the least pressure barrier essential for distribution stations within the energy limit. Further, in the stochastic work of Su et al. (2018), a model is developed using a grid concept for determining the pipeline capacities at changed situations and also investigated the significance of the failure of the unit in the pipeline transport system. Liao et al. (2019) proposed the MILP model for the scheduling of a branched pipeline system in the continuous-time formulation which allows multiple batches to be processed by a node over a single slot leading to the requirement of fewer event points to represent a schedule to reduce computational time. It should be noted that these approaches are based on mathematical programming techniques.

Graphical approaches, especially pinch analysis (PA)-based approaches, have proved their abilities in point of giving more physical insight to the problems than mathematical programming-based approaches (Klemeš et al. 2019). It has been successfully implemented in most of the areas in process industries (Bandyopadhyay 2006). PA is introduced by Linhoff (1993) for maximizing energy recovery in chemical processes. Initially, the concept of PA is developed based on thermodynamics for designing energy-efficient heat exchanger networks. Further, it becomes a dominant tool for minimizing resource utilization and waste generation in process industries. It has been extensively used in mass exchanger networks (Chen and Wang 2012), resource allocation network (Chandrayan and Bandyopadhyay 2014), total site planning (Nawi et al. 2016), batch water network (Chaturvedi 2017), heat exchanger network (Lai et al. 2017), production planning (Sinha and Chaturvedi 2018), heat integrated water network (Foo et al. 2018), carbon emission reduction (Sinha and Chaturvedi 2019), heat transfer fluid minimization (Chaturvedi 2019), electricity trading with carbon emission pinch analysis (Tan et al. 2020). Recently, the concept of PA has been applied to many new domains, especially in non-energy and non-material applications like the economic selection of projects (Roychaudhuri et al. 2017), environmental risks management (Wang et al. 2017), health-care sector (Basu et al. 2017), solid waste management (Singh 2019), urban recreational open spaces (Subramanian et al. 2019), fire susceptibility (Kumar et al. 2020), resource conservation networks with epistemic uncertainties (Bandyopadhyay 2020). Further, the PA is successfully utilized for solving multi-objective problems. Initially, Shenoy et al. (1998) suggested a methodology for determining the trade-offs of utility requirement and capital cost w.r.t. the minimum driving force (ΔT_min). Hallale (2002) developed a graphical approach for minimizing freshwater considering two different areas: process modifications and water regeneration. Later, Fraser et al. (2005) proposed a method for minimizing cost and flow rates of external mass separating agent (MSA) in mass exchanger network synthesis with the aid of grand composite curves. In this work, it is also found that the low-cost MSA is not necessary the one with the lowest cost per unit mass of MSA. Later, a methodology utilizing the concept of PC is introduced by Shenoy and Bandyopadhyay (2007) for minimizing the operational cost of water supply along with minimizing water flow rates. In another important work, Krishna Priya and Bandyopadhyay (2013) have adopted this concept in energy sector planning to determine the optimal condition that will meet the energy demands and emission limitations at least capital investment. This paper extends the theme of PA to another important domain of process industries, i.e., for optimization of GAN to address the two objectives compression energy and capital investment.

From the above literature review, it can be analyzed that there are two important objectives compression energy and capital investment which are addressed by researchers. However, in the methodologies developed for GAN, the relation between capital investment and compression energy requirement is less focused. So there is still a need for a systematic approach that is capable of addressing the two important objectives (compression energy and capital investment) simultaneously and establishes the relation between total capital investment (TCI) and total compression energy requirement (TCER). It should be noted that an increase in capital investment may not always be energy efficient. Similarly, an increase in compression energy may not always reduce capital investment. Addressing the two objectives simultaneously allows calculating the effects of changing one objective on another. Secondly, as highlighted in previous paragraphs graphical approaches are developed for almost every domain of process industries as it gives enhanced visualization and better physical understanding. This paper presents a novel graphical approach for the simultaneous minimization of TCER and TCI while the installation of new stations in GAN. The solution is expressed as a Pareto optimal front between TCER and TCI. The Pareto optimal front is generated using the ε-constraint method (Haimes and Freedman 1975). The Pareto optimal front captures the trade-off between TCER and TCI that may be utilized to choose the suitable functioning point based on the operating and economic conditions of the process. In the next section, problem definition for the simultaneous minimization for a GAN is given, which is then followed by mathematical formulation, graphical analysis and illustrative examples.

Problem definition and mathematical formulation

A schematic representation of the problem definition is presented in Fig. 1. In this segment, a general problem of the GAN for targeting the two objectives that are TCI and TCER while installing new CS’s in GAN is given as follows:

• The set of X_s existing CS’s is given where each supply station X_i {i, 0 ≤ i ≤ X_s} is characterized by a given pressure (P_si) and supplying a flow (F_si) to demand.

• The set of Y_r new CS’s where each supply station Y_j {j, 0 ≤ j ≤ Y_r} available at specified pressure (P_rj) and maximum accessible flow (${\text{F}}_{rj}^{\max }$). For the installation of new CS’s, the investment required per unit flow is given as an investment coefficient (C_rj).

• The new CS's and existing CS’s are used to satisfy Z_d demands. Each demand Z_k (k, 0 < k ≤ Z_d) has a specific flow requirement (F_dk) at a minimum acceptable pressure (P_dk).

• The primary objective of this study is to develop a Pareto optimal front between TCER and TCI involved in installing new CS’s. The Pareto optimal front represents the trade-off between TCER and TCI.

Fig. 1

A schematic representing a generalized transportation problem

Note that supplying flow from a station having a higher pressure than the lower demand station pressure does not require any compression energy. Further, it may be noted that in this formulation the demand pressures are assumed to be higher than the highest supply CS’s. The pressure drop during the flow transfer is neglected. However, it may be included to an extent via raising the demand side pressure. The mathematical model comprises the following sets, variables and parameters.

Nomenclature:

Sets.

$I$ {i / i = Existing compression station}.

$J$ {j / j = New compression station}.

K { k / k = Demand compression station}.

Variables.

$E_{{{\text{total}}}}$ Total compression energy requirement (kJ/s),(TCER).

$C_{{{\text{total}}}}$ Total capital investment ($), (TCI).

$f_{ik/jk}$ Flow from existing supply compression station (i) /new compressor station (j) to demand station (k), (Sm³/s).

Parameters.

$\mu_{{\left( {i/j/k} \right)}}$ Energy index of supply (i/j) and demand (k) CS, (kPa).

$C_{rj}$ Investment cost of the jth CS ($).

$n$ Polytropic index.

$F_{{{\text{dj}}}}$ Flow requirement of demand, (Sm³/s).

$F_{{{\text{si}}}}$ Supplying flow from CS, (Sm³/s).

$F_{rj}^{\max }$ Maxmum supply flow from new CS, (Sm³/s).

$F$ Volumetric flow rate (m³/s).

$F_{0}^{* }$ Standard volumetric flow rate (Sm³/s).

P₀ Pressure under standard conditions (kPa).

λ Compression Energy Index (kJ/Sm³).

Constraints

The mathematical model for GAN comprises constraints related to material balances, demand supply and availability limits and thermodynamic constraints detailed as follows:

Material balances

The flow from existing CS’s ($X_{s}$) to demand $\left( {Z_{d} } \right)$ is symbolized (f_ik). Similarly, supply from external station ($Y_{r}$) to demand ($Z_{d}$) is symbolized by (f_jk). The flow balance for each existing CS's, new station and demand station may be addressed as follows (Eq. 1):

$$\mathop \sum \limits_{i = 1}^{{X_{s} }} f_{ik } + \mathop \sum \limits_{j = 1}^{{Y_{r} }} f_{jk} = \, F_{dk} \quad \forall k \in K$$

(1)

Availability constraints

These constraints restrict the total flow from the new CS to demand below the capacity of that new CS.

$$\mathop \sum \limits_{j = 1}^{{Y_{r} }} f_{jk} \le F_{rj}^{\max } \quad \forall j \in J$$

(2)

Thermodynamic constraints

The compression energy requirement (Bandyopadhyay et al. 2014) is directed by the initial ($P_{{{\text{in}}}}$) and final ($P_{f}$) states along with the volumetric flow (F). For isothermal compression, the energy requirement ($E_{{{\text{iso}}}}$) can be expressed as Eq. (3);

$$E_{{{\text{iso}}}} = P_{{{\text{in}}}} F \left( {\ln \left( {\frac{{P_{f} }}{{P_{{{\text{in}}}} }}} \right)} \right)$$

(3)

The energy requirement expressed in Eq. (4) can also be stated as follows:

$$E_{{{\text{iso}}}} = { }F_{0}^{*} \left[ {\left( {P_{0} {\text{ln}}\left( {\frac{{P_{k} }}{{P_{0} }}} \right)} \right) - { }\left( {P_{0} {\text{ln}}\left( {\frac{{P_{{\left( {i/j} \right)}} }}{{P_{0} }}} \right)} \right)} \right]$$

(4)

The compression energy ($E_{{{\text{poly}}}}$) for the polytropic process can be expressed in Eq. 5;

$$E_{{{\text{poly}}}} = \left( {\frac{n}{n - 1}} \right)P_{0}^{\frac{1}{n}} P_{{\left( {i/j} \right)}}^{{\frac{{\left( {n - 1} \right)}}{n}}} F_{0}^{*} \left[ {\left( {\frac{{P_{k} }}{{P_{{\left( {i/j} \right)}} }}} \right)^{n - 1/n} - 1} \right]$$

(5)

where $F_{0}^{{* }}$ represents the standard volumetric flow rate being compressed, $P_{0} { }$ is the pressure under standard conditions, and $P_{k}$ and $P_{{\left( {i/j} \right)}}$ are the demand and supply pressures, while ‘n’ is the polytropic index. Note that for the adiabatic process ‘n’ is replaced by heat capacity ratio (γ).

The Energy Index (EI) at a pressure level (existing station (i) / new station (j) / demand station (k)) of an isothermal process and the polytropic process can be calculated by using Eqs. (6) and (7).

$$\mu_{{\left( {i/j/k} \right)}} = P_{0} {\text{ln}}\left( {\frac{{P_{{\left( {i/j/k} \right)}} }}{{P_{0} }}} \right)$$

(6)

$$\mu_{{\left( {i/j/k} \right)}} = \left( {\frac{n}{n - 1}} \right)P_{0} \ln \left( {\frac{{P_{{\left( {i/j/k} \right)}} }}{{P_{o} }}} \right)^{{{\raise0.7ex\hbox{${\left( {n - 1} \right)}$} \!\mathord{\left/ {\vphantom {{\left( {n - 1} \right)} n}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$n$}}}}$$

(7)

where ${\upmu }_{{\left( {i/j/k} \right)}}$ signify the EI for stations. Hence, the compression energy requirement ($E_{ik} )$ can be calculated as Eq. (8);

$$E_{ik} = f_{ik} \left( {{\upmu }_{k} - {\upmu }_{i} } \right) \quad \forall i \in I, k \in K$$

(8)

TCER ($E_{{{\text{total}}}}$) may be expressed mathematically as follows (Eq. (9)):

$${\text{ TCER}} \left( {E_{{{\text{total}}}} } \right) = \mathop \sum \limits_{k = 1}^{{Z_{d} }} \mathop \sum \limits_{j = 1}^{{Y_{r} }} f_{jk} \left( {{\upmu }_{k} - {\upmu }_{j} } \right) + \mathop \sum \limits_{k = 1}^{{Z_{d} }} \mathop \sum \limits_{i = 1}^{{X_{s} }} f_{ik} \left( {{\upmu }_{k} - {\upmu }_{i} } \right)$$

(9)

where $\mu_{k}$ signifies the EI of the demand and (${\upmu }_{{\left( {i/j} \right)}}$) is the EI of existing and new supply station. The quantity $\left( {{\upmu }_{k} - {\upmu }_{{\left( {i/j} \right)}} } \right)$ can be defined as the compression energy index, ${\uplambda }_{{\left( {i/j} \right)k}}$ (CEI) and can be expressed as Eq. (10);

$${\text{CEI}}, \lambda_{{\left( {i/j} \right)k}} = \left( {\mu_{k} - \mu_{{\left( {i/j} \right)}} } \right) \quad \forall i \in I,j \in J, k \in K$$

(10)

TCI ($C_{{{\text{total}}}}$) can be expressed as follows (Eq. 11):

$${\text{TCI}} ( C_{{{\text{total}}}} ) = \mathop \sum \limits_{k = 1}^{{Z_{d} }} \mathop \sum \limits_{j = 1}^{{Y_{r} }} {\text{C}}_{rj} f_{jk}$$

(11)

where ${\text{C}}_{rj}$ is the investment coefficient for installing jth new compressor station.

Objective functions

The two objective functions TCER and TCI of the problem are expressed in Eqs. (12) and (13);

$${\text{Minimize}} {\text{ TCER}} \left( {E_{{{\text{total}}}} } \right){ }$$

(12)

$${\text{Minimize}} {\text{ TCI }}\left( {C_{{{\text{total}}}} } \right){ }$$

(13)

To address these conflicting objectives and to identify the most suitable optimal point based on the decision-maker choice for GAN, a Pareto multi-objective optimization method is utilized. In this method, the Pareto set is generated using $\varepsilon$-constraint method which converts multi-objective optimization to constraint single-objective optimization. The primary objective of this study is to develop a Pareto optimal front between TCER and TCI. Considering the constraint (Eqs. 1 and 2), the formulation can be solved for one of the objectives (Eqs. 12 or 13). Later other Pareto optimal points of the Pareto set are obtained by placing evenly distributed incremental scalar weights on the aggregate objective function. The Pareto optimal front represents the trade-off between TCER and TCI.

Graphical analysis

The overall problem formulated in the previous section is a linear programming problem (LPP). This LPP is solved using the concepts of PA. PA has emerged as an efficient tool in solving typical kind of linear programming problems having a particular mathematical configuration. Various algorithms have been developed and applied successfully to solve a linear programming problem, some particular mathematical structure (Bandyopadhyay 2020). Also, the easy implementation, accurate solutions, better visualization and increased physical understanding of the underlying issues make PA an appropriate tool for solving this problem. In this paper, the targets from PA are identified and captured in a Pareto optimal front. These targets help in framing environmental policies and allocating appropriate financial budgets to reduce TCER in GAN.

The development of this PA-based solution to this problem is as follows. Based on the number of demands, the two possible cases may arise.

Case 1: Flow transportation to a single demand.

Case 2: Flow transportation to multiple demands.

Case 1: Flow transportation to a single demand

The method of energy composite curve (ECC) has been proposed for this analysis. This technique is similar to the limiting composite curved (LCC) presented by Wang and Smith (1994) for targeting water requirements in the water allocation network based on the mass transfer model. The concept of LCC is widely adopted in different areas of system engineering such as power system planning and production planning. Minimum and maximum compression energy limits for satisfying demand can be calculated by supplying flow from higher-pressure CS and lower-pressure CS. Further, a value is selected ($E_{c}$) between these limits, and hence, based on this value demand compression energy index (DCEI) ($\mu_{d}$) can be calculated via dividing this value by total flow requirement. EI for every CS can be calculated by Eq. (6) for the isothermal process and Eq. (7) for the polytropic process. Next, CEI is calculated by using Eq. 10 which is the difference between EI of demand to the EI of existing station, i.e., $({\upmu }_{k} - {\upmu }_{{\left( {i/j} \right)}}$). Further, to construct the ECC, the CEI is plotted against cumulative compression energy as shown in Fig. 2. The initial pinch point is calculated via drawing an initial work line from the beginning and rotating it anti-clockwise until it touches the ECC, the point where it touches is the pinch point.

Fig. 2

Energy composite curve and work curve for GAN

Next, PC of each new station can be calculated using this pinch point (Eq. (14)). Pinch point divides the total zone of CEI into two sections. New CS’s having CEI below the pinch point can be installed, while new CS’s having CEI above the pinch point should not be installed. For several CS’s, the question arises which CS can be added first. Adding a station with the lowest cost is not always profitable. This order is determined by ranking the stations using the concept of PC, developed by Shenoy and Bandyopadhyay (2007). The PC is the function of pinch point, and hence pinch point also affects this ranking of CS’s.

$${\text{Prioritized Cost of}}\,j{\text{th CS}},{\text{PC}}_{j} = \frac{{C_{rj} }}{{\left( {{\text{CEI}}_{p} - {\text{CEI}}_{j} } \right)}}$$

(14)

In Eq. (14) $C_{rj}$ is the cost of the jth CS, while ${\text{CEI}}_{p}$ and ${\text{CEI}}_{j}$ are the pinch CEI and CEI of jth CS. PC of new supply CS’s can be arranged in increasing order to get a prioritized sequence of new supply stations. It may be noted that due to the introduction of new stations, pinch point may shift. If after shift the new prioritized sequence still matches the initial prioritized sequence, the procedure may be continued with the shifted pinch point. However, in case PC sequence changes due to the pinch shift, then the flow should be adjusted such that the work lines (curve) pass through both the old pinch point and the shifted pinch point. The trade-off between the TCI and TCER provides the Pareto optimal front which includes set of all optimal point.

Case 2: Flow transportation to multiple demands

A typical case of two demands is shown in Fig. 3; in this case, there are two demands D₁ and D₂ at different pressure levels P₁ and P₂ (P₂ > P₁). Next, demand D₁ can be shifted to pressure level P₂. The energy required for this shifting ($E_{s}$) can be calculated by Eq. (15), and this additional energy requirement can directly add to the TCER (Eq. (17)).

$${\text{E}}_{{\text{s}}} = {\text{ F}}_{{Z_{1} }} \left( {{\upmu }_{{k_{2} }} - {\upmu }_{{k_{1} }} } \right)$$

(15)

Fig. 3

Schematic representing general multiple demand problems

Similarly, if there are more than two demands, all demands can be shifted to the highest pressure using Eq. (16).

$${\text{E}}_{{\text{s}}} = { }\mathop \sum \limits_{k \ne H} {\text{F}}_{dk} \left( {{\upmu }_{k}^{{\text{H}}} - {\upmu }_{k} } \right)$$

(16)

where $\mu_{k}^{H}$ is the highest pressure demand EI. Then, the total energy ($E_{{{\text{total}}}}$) can be calculated by using Eq. (17).

$$E_{{{\text{total}}}} = E_{{{\text{Given}}}} + E_{s}$$

(17)

Hence, the problem converges to a single demand problem and Pareto optimal front can be generated in the same way as it is carried out in case of a single demand problem.

Targeting algorithm

Based on the above analysis, the following algorithm is proposed for obtaining the Pareto optimal front between TCER and TCI involved in installing new supply stations. Figure 4 shows the flowchart of the proposed algorithm.

Fig. 4

Flowchart showing proposed algorithm for obtaining Pareto optimal front

Generation of energy composite curve

Step 1 Minimum ($E^{\min }$) and maximum ($E^{\max }$) possible compression energy limits for satisfying the demand can be calculated using the concept that the higher the pressure, the lower the energy requirement as explained in Sect. 3.1. In the case of multiple demands, shift all the lower pressure demands to the highest pressure demand. The energy required for shifting (E_s) is calculated by Eq. (18) (refer to Sect. 3.2).

$$E_{s} = { }\mathop \sum \limits_{k \ne H} {\text{F}}_{dk} \left( {{\upmu }_{k}^{{\text{H}}} - {\upmu }_{k} } \right)$$

(18)

New limits $E^{{\min^{^{\prime}} }}$ and $E^{{\max^{^{\prime}} }}$ can be calculated as follows (Eq. (19)):

$$E^{{\min^{^{\prime}} }} /E^{{\max^{^{\prime}} }} { } = { }E^{\min } /E^{\max } + {\text{E}}_{{\text{s}}}$$

(19)

Step 2 Calculate the EI of existing stations, new stations and demands using Eq. (6) for the isothermal process and by Eq. (7) for the polytropic process (refer to Sect. 2).

Step 3 Select a value (E_c) in between $E^{{\min^{^{\prime}} }}$ to $E^{{\max^{^{\prime}} }}$ to be TCER. The value of E_c should be in between the minimum to maximum TCER. The trade-off between TCER and TCI is generated via placing evenly distributed incremental scalar weights, on the scale of this minimum to maximum TCER. DCEI can be calculated by Eq. (20).

$${\text{DCEI}} = \frac{{E_{c} }}{{\mathop \sum \nolimits_{k = 1}^{{Z_{d} }} F_{dk} }}$$

(20)

Step 4 CEI which is the difference of energy index of a demand station to supply station that can be calculated using Eq. (10) as explained in Sect. 2.

Step 5 Arrange CEI in increasing order and correspondingly flow $\left( {f_{m} } \right)$ in the next column taking a negative sign of existing flow and positive for the demand (Bandyopadhyay and Sahu 2010).

Step 6 In next column, net flow is calculated via cumulating the flow values (Eq. 21). Note that m denotes the sequence number of the row.

$${\text{Net flow }}F_{m} = \mathop \sum \limits_{l = 1}^{m} f_{m}$$

(21)

where l denotes the row.

Step 7 The equivalent energy to each interval is calculated by multiplying the flow debit of that interval and variance in CEI binding that interval. The net energy for each interval can be intended by adding the energy for the entire preceding rows (Eq. (22)).

$${\text{Net Energy }}E_{m} = \left\{ {\begin{array}{*{20}l} {0,} \hfill & {{\text{for}} m = 1} \hfill \\ {\left( {{\uplambda }_{m} - {\uplambda }_{m - 1} } \right)\left( {\mathop \sum \limits_{l = 1}^{m} F_{m} } \right),} \hfill & { {\text{for}} m > 1} \hfill \\ \end{array} } \right.$$

(22)

Step 8 Calculate the cumulative compression energy via cascading the net energy calculated in the previous step. Plot the graph of CEI versus cumulative compression energy to get ECC. Draw the initial with the highest slope that does not cross ECC starting from the origin.

Step 9 Pinch point is determined from the graph as the point where ECC touches the initial work line.

Note that Steps 5–8 are the algebraic procedure to calculate cumulative compression energy versus CEI. Plotting CEI against cumulative compression energy will result in ECC.

Generation of Pareto optimal front

PC is calculated by using Eq. (14). PC of new CS can be arranged in increasing order to get a prioritized sequence of new CS and accordingly capital investment is calculated as discussed in Sect. 3 also. Next, to get all the optimal points, the ε-constraint method is adopted in this paper. In the ε-constraint method, the optimization problem is solved for one objective while other objectives are included in the form of inequality constraints. Next, a set of optimal solutions are generated which minimizes the capital investment while fixing the upper bound of energy at a given value (E_c). The set of all optimal points will give a trade-off between the TCI and TCER.

Illustrative examples

The developed procedure is explained via two illustrative examples in this section. In the first example, there is only one demand which is to be satisfied by existing stations along with three new stations. The second example explains the suitability of the proposed procedure for multiple demands in GAN which is satisfied by existing stations along with two new stations.

Transportation network with three new stations and single demand

For this example, pressure and flow rate for existing CS’s ($X_{1}$,$X_{2}$, $X_{3}$, $X_{4}$, and $X_{5}$) and new CS’s ($Y_{1}$, $Y_{2} ,$ and $Y_{3}$) are given in Table 1. Demand pressure and the flow rate requirement are 200 kPa and 700 Sm³/s. The compression process is assumed to be isothermal.

Table 1 Source and demand data for illustrative Example 1

It is a single demand problem; hence, no shift is needed, and initially compression energy limits ($E^{\min }$ and $E^{\max }$) are calculated to be 5330 kJ/s to 9200 kJ/s (Step 1). As discussed in the algorithm problem is solved multiple times varying the energy cap (E_c) value. Here, calculation for a particular value E_c, i.e., 5600 kJ/s, is shown which is within the energy limits. Next, EI of existing stations, new stations and demands is to be calculated (Step 2). DCEI is calculated to be 8 kJ /Sm³ using Eq. (20).

CEI of CS’s can be calculated as per Step 4 (column 2, in Table 2). CEI is arranged in increasing order and correspondingly flow in the next column taking a negative sign of existing flow and positive for the demand as shown in column 3 in Table 2 (Step 5). The net flow for each interval is calculated by using Eq. (21). The compression energy corresponding to each interval is calculated by multiplying the difference between compression energy indexes to net flow (Step 6). In column 6, cumulative compression energy can be calculated by Eq. (22) (Step 7). Cascaded energy versus CEI is plotted to get ECC (Fig. 5). Next, to calculate pinch point, the initial work line with the highest slope is plotted starting from the origin such that it does not cross ECC (Step 8). The pinch point is calculated to be at CEI 24 kJ/Sm³ which is used for calculating PC (Step 9). The PC’s of $Y_{1}$, $Y_{2}$ and $Y_{3}$ are calculated to be 0.91 $.s/kJ, 0.71 $.s/kJ and 1.66 $.s/kJ using Eq. 14.

Table 2 Generation of ECC for Example 1

Fig. 5

Energy composite curve and work curve for Example 1

It is found that the $Y_{2}$ is at least PC. Flow from $Y_{2}$ is calculated to be 332.35 Sm³/s, but the maximum flow limit from the $Y_{2}$ is 180 Sm³/s. Hence, $Y_{2}$ can supply 180 Sm³/s. Next in the prioritized sequence is $Y_{1}$. The maximum flow from this CS is calculated as 146.81 Sm³/s but the limit is 140 Sm³/s. Therefore, 140 Sm³/s flow can be made through this CS. Lastly, CS $Y_{3}$ is selected and the maximum flow is calculated 25 Sm³/s. Note that the flow from the existing CS and new CS is 75 Sm³/s more than the demand flow which is equal to 700 Sm³/s. Hence, the flow of 75 Sm³/s has to be reduced from existing CS starting from the route having the highest CEI. The optimal value of the TCI is obtained $ 5750 for the TCER of 5600 kJ/s. The optimal network for Example 1 is shown in Fig. 6.

Fig. 6

Optimal network of GAN for Example 1

In similar way, TCI is calculated for the different values of E_c. The value E_c is varied by putting evenly distributed incremental scalar weights, on the scale of this minimum to maximum TCER. The plot of TCI vs TCER is the resultant Pareto optimal front shown in Fig. 7. From this Pareto optimal front, it is found that the TCI decreases with an increase in TCER. Moreover, it can be observed and further calculated from Fig. 7 that, in the range of TCER from 5330 to 5750 kJ/s, all three new CS’s need to be installed to fulfill the demand, along with some flow reduction from existing CS’s in order to adjust extra flow.

Fig. 7

Pareto optimal front for two objectives TCI and TCER (Example 1)

While increasing TCER from 5750 kJ/s to 6850 kJ/s, only new compressor stations Y₁ and Y₂ are utilized. Next, in the range of TCER from 6850 to 8290 kJ/s all three CS's need to be installed. Further, increasing TCER from 8290 kJ/s to 9200, new compressor stations Y₂ and Y₃ are needed. Additionally, it may be noted that from the point where TCER is 5330 kJ/s, increasing in TCI does not affect the TCER. In the same way, after the TCER value of 9200 kJ/s, further, addition in the TCER does not affect the TCI. The Pareto frontier provides the flexibility to the decision-maker to choose an optimized trade-off solution. Additionally, note that sensitivity analyses can be carried out via changing the input variables (such as the pressure of compressor stations and demand pressure) and then calculate the objective values and hence the Pareto optimal front.

Transportation network with four existing stations and two demands

For this example, data are given for four existing CS’s ($X_{1}$,$X_{2}$,$X_{3}$ and $X_{4}$) and two new CS’s ($Y_{1}$ and $Y_{2}$) for satisfying two demand stations ($Z_{1}$ and $Z_{2}$) in Table 3. The compression process is assumed to be isothermal.

Table 3 Source and demand data for Example 2

Initially, EI of existing stations, new stations and demands is calculated. Minimum ($E^{\min }$) and maximum ($E^{\max }$) compression energy limits for satisfying the demand for this problem are calculated using Step 2 which is equal to 41,769.3 kJ/s and 47,756.7 kJ/s. There are two demands in this example. Compression energy follows the additive property. So lower pressure demand is shifted to the highest pressure demand. The energy required for shifting (E_s) is calculated by Eq. (15) which is equal to 5340 kJ/s. The energy limits are modified accordingly using Eq. (17). The modified limits of energy are 47,109.3 kJ/s and 53,097 kJ/s.

Next, an energy cap (E_c) is selected in between $E^{{\min^{^{\prime}} }}$ and $E^{{\max^{^{\prime}} }}$ The calculation for a particular value E_c, i.e., 50,340 kJ/s, is illustrated. Now, the DCEI can be calculated via Eq. (20). CEI can be calculated as per Step 4.

The numerical values calculated by applying the proposed algorithm are shown in Table 4. Next plot the cascaded energy versus CEI to get ECC as shown in Fig. 8. The initial work line with the highest slope is drawn that does not cross ECC starting from the origin. The pinch point is calculated to be at 90 kJ /Sm³ which is further utilized for calculating PC. The PC’s of $Y_{1}$ and $Y_{2}$ are calculated to be 0.08 $.s/kJ and 0.12 $.s/kJ using Eq. 14.

Table 4 Generation of ECC for Example 2

Fig. 8

Energy composite curve and work curve for Example 2

It should be noted that on the introduction of a new station $Y_{1}$ which has the least PC pinch jumps to 60.58 kJ/Sm³. The PC for station $Y_{1}$ and Y₂ based on the new pinch point can be calculated as 0.66 ($.s/kJ) and 0.24 ($.s/kJ). Due to pinch point jump, PC sequences are changed; therefore, for the optimum solution, the flow from station $Y_{2}$ should be such that the points (11,393.85, 90) and (3891.75, 60.58) should hold the pinch. The inverse slope of this work line (a section of work curve) passing through these points is 255 Sm³/s. The work curve also passes through the point (2741.5, 56.07). Next, sectional line of work curve passes through the points (2741.5, 56.07) and (0, 32.59), and its inverse slope is 116.76 Sm³/s. Therefore, the final flows that need to supply can be summarized as follows: for the station $Y_{1}$ 138.24 Sm³/s and for $Y_{2}$ 116.76 Sm³/s that corresponds to investment of $ 1234.62.

For the given TCER, the optimal network is shown in Fig. 9 which shows supplying the flow from existing, new CS’s to demand CS’s. In similar way, TCI can be calculated for the different values of ${\text{E}}_{{\text{c}}} .$ To get all the optimal points, the value of E_c is varied between 47,109.3 kJ/s ($E^{{\min^{^{\prime}} }} )$ and 53,097 kJ/s $\left( {E^{{\max^{^{\prime}} }} } \right)$. The set of optimal points are identified, and hence, Pareto optimal front is plotted (Fig. 10). This Pareto optimal front provides the trade-off between TCER and TCI.

Fig. 9

Optimal allocation network for Example 2

Fig. 10

Pareto optimal front for objectives: TCI and TCER (Example 2)

It can be observed and further calculated from Fig. 10 that, at the point where TCER is 47,109.3 kJ/s station Y₂ is needed while increasing from TCER 47,109.3 to 53,097 kJ/s, CS's Y₁ and Y₂ are needed. Above TCER 53,097 kJ/s, CS Y₁ needs to be installed. Further, it may be noted that similar to Example 1 in this case also, after a certain point, an increase in the TCER does not affect the TCI and vice versa.

Conclusion

A novel graphical methodology based on PA to solve a bi-objective optimization problem in GAN is proposed in this paper. The two objectives of the problem are TCER and TCI. The proposed methodology identifies the relationship between these two objectives. The paper extends the idea of PA and prioritization to GAN in order to solve this dual-objective problem graphically. As the problem is bi-objective, the solution is expressed in terms of Pareto optimal front. To solve this multi-objective problem, ε-constraint method is used. In this paper, the upper bound of an objective is fixed and formulation is solved for other objective; in this way, a set of optimal solutions are generated. The most important feature of this solution is that the effect of variation in one objective on the other objective can be calculated. In this way, the resultant Pareto optimal front can be utilized to choose the suitable functioning point. Also, this Pareto optimal front can be used to determine the stations that need to be installed along with their flow capacities. Hence, the proposed methodology can be utilized while framing energy policies in GAN. Current research is directed toward incorporating other important issues such as compressor station characteristics, distance between CS’s and sizing of pipelines in the framework.