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This paper considers the pipeline network design problem (PND) in ethanol transportation, with a view to providing robust and efficient computational tools to assist decision makers in evaluating the technical and economic feasibility of ethanol pipeline network designs. Such tools must be able to address major design decisions and technical characteristics, and estimate network construction and operation costs to any time horizon. The specific context in which the study was conducted was the ethanol industry in Sao Paulo. Five instances were constructed using pseudo-real data to test the methodologies developed.

People use networks daily (sometimes unwittingly) in a wide range of contexts. Telephony networks, logistics networks for freight transportation and distribution networks (water, energy) provide services that are essential to modern life and, therefore, must be properly designed to ensure service quality and meet user demands.

Brazil is projected to be both a major producer and supplier of biofuels, particularly biodiesel and ethanol. These fuels are gaining importance on the world stage in view of the environmental, technological and economic impacts caused by global dependence on fossil fuels, particularly oil [

However, Brazil’s transport logistics system still rests largely on its highway network, which—although economically competitive—is not safe for transporting fuel, especially in large volumes. A paradigm shift must take place and Operations Research can help in that process.

This study aims to provide robust and efficient computational tools to assist decision makers in evaluating the technical and economic feasibility of ethanol pipeline network designs. Such tools must be able to address major design decisions and technical characteristics, and estimate network construction and operation costs to any time horizon.

The paper is organized as follows: Section 2 presents the context that motivated the research; Section 3 describes the problem under study, characterizing it in technical and theoretical terms; Section 4 presents the optimization approaches taken to solving the problem; Section 5 reports the results and conclusions; and Section 6 offers some final remarks on the study.

Current economic growth rates have generated increasing demand for oil. Consequently, market laws exert strong pressure on oil prices, which fluctuate, significantly increasing the cost of living. In parallel with this, the environmental effects of global warming are an issue globally. Pollutant emissions from burning fossil fuels aggravate the phenomenon, drawing criticism and spurring the search for viable alternatives [

Ethanol will play an important role in this process: firstly, because large-scale production costs have been competitive with those of oil since 2004, and secondly, because it is a cleaner energy source [

Brazil is now the world’s second-largest producer [

This result is worrying given that production is not evenly distributed across Brazil. Though concentrated mostly in the southeast [

Designing such networks is a complex task due to the large number of technical and economic requirements to be considered simultaneously. Proper project management is imperative, and projects are commonly divided into conceptual and hierarchically dependent phases that characterize their life cycle [

• Planning: early stage of the potential enterprise, where technical, economic and social data are collected to assist feasibility evaluation.

• Conceptual Engineering: identifies project technical and economic feasibility, and sets the basic and detail engineering agenda.

• Basic Engineering: the basic design is developed and detailed to consolidate engineering aspects before any procurement and implementation expenditures are made.

• Implementation: step execution and control. Activities typical of this stage include construction and assembly, commissioning and conditioning.

• Operation and Maintenance; Decommissioning: operation of pipeline is terminated because it is obso lescent, supplanted by other systems or no longer of interest to the owner.

From an Operations Research viewpoint, meanwhile, Minoux [

Investment and operating costs tend to be of different orders of magnitude [

This section will give an overview of ethnol pipeline network design, indicating the key theoretical and technical characteristics. The problem is formally stated, presenting the technical decisions that are being addressed and the main assumptions. Next a cost analysis is made of pipeline network projects. Finally, the problem is formulated mathematically.

Flows are caused by the action of a shear stress on a fluid initially at rest. Temporal analysis of a flow allows it to be classified into one of two regimes, namely the transitional (early stage) regime, in which flow velocity, density and temperature parameters at any point vary with time, and the steady-state (perennial stage) regime in which these parameters do not vary with time.

Steady-state flows, in turn, can be classified into two types: laminar, in which fluid particles move along parallel streamlines (paths) without macroscopic interaction between the layers; and turbulent, in which fluid particles move along irregular trajectories with major macroscopic interaction. The dimensionless Reynolds number is used to indicate whether the flow is laminar or turbulent [

In ducts, flow may occur under the predominant, but not exclusive, action of two different mechanisms: gravity or a pressure gradient. The main characteristic of flow under gravity, known as open-channel flow, lies in the fact that the fluid flows without completely filling the duct. In the case of flows promoted by a pressure gradient. The ducts are completely filled by the fluid transfer.

Finally, it is possible to characterize two regions in flow along a pipeline: the entrance region, where the velocity field profile is not yet fully developed, and the fully-developed flow region, where the velocity field is the same at any cross-section. The fully-developed region may not exist, but generally, for a pipeline length and diameter, if, it can be assumed that the steady state has been reached.

Pipeline networks comprise ducts and other components such as valves, bends, fittings, pumps, turbines and tanks. All these elements contribute to system energy dissipation. There are two types of losses, one due to friction between duct walls and the outer layers of the fluid, and the other due to geometric changes in cross-section profiles along the pipeline route. The D’Arcy-Weisbach Equation enables losses to be calculated for incompressible, permanent and fully-developed flows:

where is the pressure drop along the pipeline whose transferred flow is the parameter pi, the gravity (m/s^{2}) and the friction factor.

Flow machines are mechanical devices capable of extracting energy from fluids (turbines) or adding energy to them (pumps) by dynamic interactions between the device and the fluid [

In 1738 Bernoulli formulated the basic equation governing mechanical energy conservation in a fluid flow system. Generalized a few years later, the energy balance for an isothermal, confined, steady and turbulent flow of a viscous fluid that receives work from, and performs work on, flow machines is given by:

where is the density of the transferred fluid. It represents the sum of pressure (1st term), kinetic (2nd term) and gravitational potential (3rd term) energies in the system, the total power loss, the work done by the system on its border, and the work performed by device B on the system.

Pumps to ensure flows must be selected appropriately not to overestimate or underestimate the equipment’s capacity, thus avoiding unnecessary expense. The system curve is a curve relating energy flow to the load supplied to the fluid, considering the network’s technical and operational features. Consider the system in

where and are appropriate parameters. The Equation (3) characterizes the system curve, relating the energy supplied and the flow. This is shown in

System and pump curves are used to choose equipment. Given a system curve and an operating flow, the best-suited pump B to provide the required energy is a pump whose characteristic curve intersects the system curve at the desired flow operating point (

Ethanol production is a seasonal activity, and the productive capacity of any region is subject to the conditions necessary for crop development. Such issues make production minimally random in terms of constancy and amount to a given time horizon. Also sugar is produced from sugar cane, and high-demand situations impact ethanol production [

There is no minimum pressure requirement at the end of each section, provided the flow reaches its destination. Hence, pressures will be considered zero at these points. Economic gains also motivate this simplification. Still, assuming that the flows are promoted pressure gradients alone (Section 3.1.1), this hypothesis requires that at least one pumping station be installed at the entrance to each section. This requirement is naturally well conceived, since geographical dimensions require the use of storage tanks, which themselves result in pressure discontinueties.

In an ideal operating regime, storage tanks would be unnecessary in the producer regions, because flow balance would be automatically guaranteed. In practice, in

addition to production inconstancy over time, maintenance system halts are usually necessary, requiring a minimum installed storage capacity in each region. Estimates of such capacities should consider both the randomness of the production and the system downtime schedule, leading again to a stochastic approach. Ideal balance is assumed and also that no storage capacity is required.

The essential nature of services provided by networks requires the system to be able to satisfy design demands even under failure. A redundant network ensures there are at least two distinct paths between any two network nodes. However, this solution requires construction of more pipelines than minimally required to connect all nodes in the network, which in geographically extensive networks, can represent overinvestment. The common solution in such cases is to implement a preventive maintenance policy. It is assumed therefore that the network is designed with no redundance.

On the other hand, for a given set of technical specifications (diameter, thickness and material output), ducts are certified to withstand a certain maximum operating pressure. That relationship is given by the following equation:

where is a safety factor [

where estimates the maximum value of the flow in the duct.

Route engineering entails certain difficulties. First, it is impossible to build pipelines in straight lines, since there may be impassable terrain between points of interest. Additionally, many slopes may be present in the altitude profile between two regions and this has considerable impact on pump dimensioning. Such issues are to be addressed at data level: the former, when considering distances among the regions, and the latter, using an “equivalent” altitude difference to reflect effects of slopes along routes.

Pump dimensioning is an important issue. This is done to avoid cavitation (Section 3.1.2) by estimating these pumps’ NPSH, which is a function of the height of an equivalent liquid column upstream of the pump. Estimating NPSH, however, would entail estimating the maximum capacity of the tanks upstream of the pump, which would make the mathematical model much more complex. Thus, dimensioning will be done by estimating the downstream energy.

Geographically distributed pipeline networks are subject to temperature variations. This means that fluid property values may change significantly, which may invalidate use of the equations presented in Section 3.1. Thus, flow is assumed to be isothermal. Moreover, it is assumed to occur in a turbulent regime, but in fully developed state, since the distance the flow travels in each network segment is much greater than the pipeline diameter, a necessary condition for such a hypothesis to be true (Section 3.1.1).

Finally, the flow along the network is not energyconserving (Section 3.1.2). Energy losses are usually offset using pumps in order to ensure system energy balance. Equation (2), used to guarantee this balance in the mathematical model, furnishes an estimate of the energy to be provided to the system at pumps downstream. Losses due to geometric changes will be considered using an estimator for this loss, since there is no way a priori to define the set of components installed. Finally, it is assumed that the flow does not perform work on its surroundings.

The problem of designing an ethanol pipeline network, given a set of production regions, a set of associated flows and available diameters, consists of selecting a subset of links between pairs of these regions to constitute the topology, an associated diameter for each of these connections and, finally, calculating the energy to be introduced into the system to ensure the flow to destination region, while minimizing total project cost.

Economically, there are many significant costs in a pipeline project [

Topology decisions impact on procurement and assembly costs. The same is true of diameters, since different diameters have different prices and are more or less costly to assemble. Pumps are dimensioned (Section 3.1.3) indirectly by calculating from path, length, diameter and flow. Thus, fixed and variable pumping costs are directly influenced by the decisions taken and will be considered in the model.

The first three costs are associated with the network construction phase and the last one with the operation phase (to a time horizon). This can be expressed mathematically as:

where is total network cost; total pipeline procurement cost; total pipeline installation cost: the total fixed pumping cost; and the variable pumping cost to operating time horizon. accounts for approximately 17% of total cost [

Acquisition cost can be estimated using the cost per unit mass of the material from which they are manufactured. For this, consider the cross-section of sector of length. Defining as the cost of procuring a unit length of pipe with internal diameter, one obtains:

That is, the unit procurement cost of length of a pipe with diameter is a function of the cost per unit mass of the material, the density, the thickness of the pipe and the diameter. Thus:

where is the set of sectors that constitute the topology. The Equation (8) represents a linear approximation to the total procurement cost.

It is unlikely that an analytical expression exists or can be derived (as in Section 3.4.1) to estimate pipeline installation costs, as installation is a practical task that depends on many factors. However, it may be possible, by empirical analysis of historical data (past projects), to determine approximate, probably non-linear, equations for such costs. However, one can assume that the cost of installing a pipeline sector is directly proportional to its length (the longer the sector length, the greater the amount of resources needed to install it) and depends on the chosen diameter of pipe (handling pipes with different diameters can be more or less costly).

Thus, the cost of installing is given by:

where is the cost of installing a unit length of pipewith diameter. More complex and realistic cost functions can be used to replace (9), although this introduces nonlinearities into the total project cost function, which would complicate the solution to the problem.

The function obtained takes discrete values that depend on the pressure range sector is operating in. But selecting a reference pressure for each pressure interval, one can apply a linear regression to obtain the linear pumping cost function shown in

where is the cost of providing a unit of pressure, is the fixed cost associated with operating sector, and its operating pressure. The regression must satisfy the condition (negative costs

for viable pressure values are meaningless). Considering all the sectors that constitute:

Regressions for higher degree polynomials can be performed, resulting, however, in the inclusion of nonlinearities in the project cost function.

The procedure for estimating the variable pumping cost is the same, except that the costs associated with the selected pumps represent the cost of operating them to time horizon. Thus, the operating cost of for can be estimated by:

where is the cost of providing a unit of pressure in sector, is the fixed cost of operating the sector, and the operating pressure. The resulting expression is not time-dependent, but is addressed indirectly in (12). The positivity issue also applies here, as do higher-order regressions.

Substituting (8), (9), (11) and (12) in (6), the overall cost of network may be estimated by equation:

Note that (13) was derived considering only one diameter value, but will be generalized later.

The set of producing regions and candidate links can be represented by a complete digraph, where vertices are regions and edges the links between them. Values for distances and inter-region height differences are associated with edges. Similarly, a graph can be associated with the designed network structure. The graph that satisfies the coverage condition for all regions with the fewest connections is a spanning tree (ST) [

The set of regions is represented by and diameters by. The reduced set of regions is defined by. The indexes _{ }and _{ }and index _{ }represent the regions and the diameters, respectively. An edge or link interconnecting any two vertices is called a sector of the network. For a sector, the flow takes place in the direction. Accordingly, is the initial (input) vertex and the final (output) vertex of the sector considered.

Define binary variable, with value 1 if the sector is selected and 0 otherwise. To quantify the flow associated with each sector, we define the nonnegative continuous variable. We define the continuous variable to quantify the energy supplied in the region of a sector.

Initially, it must be ensured that all regions will be served by the network. Therefore, except for the destination region, each must have at least one sector through which to dispatch its production. Since the desired topology is a ST, the condition that only one flow exit exist per region shall be forced. Mathematically:

That is, for every region, only one pair may be selected. It is important to note that the equation does not restrict the number of sectors ending at any region.

It is necessary to ensure the flow balance in the regions. For every region, the incoming flow from at, plus its own production, is sent to, where to avoid cycles. Mathematically:

There is flow in sector if it is selected, i.e.,. To ensure that condition, the following constraints can be used:

where:

The lower bound is defined as the lowest production of all regions. The upper bound is defined as the lesser value of the maximum flow supported by the pipe or the total output being drained along the network. That is, at no time can any flow traversing the sector be below or above, and flow is zero if the sector is not chosen.

No explicit constraint was introduced to ensure that the solution topology is an ST, or even to ensure flow to destination; nonetheless, restrictions (14), (15) and (16) together, ensure both conditions [

Finally, to ensure energy balance, the amount of energy that must be provided to compensate for system losses must be estimated. In energy terms, each sector of the network can be viewed as an independent system with its own characteristics (flow, diameter, length, altitude difference etc.). Applying the assumptions of Section 3.2 to (2), it follows for every sector that:

where is the estimated losses due to geometry changes for the sector. By hypothesis,

.

So represents the energy to be provided at and not a pressure gradient within. Finally, the variable domain constraints:

The Equation (13) will be taken as the objective function. Using the decision variables defined in Section 3.5.2, gives:

which was generalized to consider all diameters in. Note that when no contribution by any cost related to is added to the project total cost.

The mathematical programming formulation (MF) of the problem is as follows:

MF: Minimize (23) subject to (14)-(16), (19)-(22).

MF can be manipulated. The variables appear only in the energy balance constraint (19), domain constraints (22) and objective function (23). The constraint (19) is an equality and, therefore, may be incorporated into the objective function. The variables can, in turn, be deleted from the formulation, together with constraints (22). Substituting (19) in (23), gives the new objective function of MF, where and are appropriate parameters:

We consider (24) to develop the algorithm. By evaluating its parameters, three characterizations of a good solution to the problem can be inferred. First, consider the total length in a feasible solution:

where. Both portions of cost in (24) are, in some sense, directly proportional to. The smaller the total length of the solution network, the lower the cost associated with it. Second, consider (

Additionally, suppose that is one of the vertices with highest output among producing regions and that, where represents the total length of the path connecting vertices and, and represents the direct connection between and. The following expression is computed for in (24):

That is, the cost is proportional to the square of the flow in and the length of the sector. In addition, once flows increase in the direction of the destination, it

can be advantageous to connect to through the path that minimizes the distance between them (directly, for example) in order to reduce the cost of transportation (operation) between and, in exchange for building additional units of pipe length. The

Finally, the cost of is inversely proportional to the fifth power of the selected diameters and directly proportional to the flows transferred. Since flows increase in the direction of the destination, this increase can be offset by forcing the diameters at least not to decrease in the direction of flow. On the other hand, pipes of larger diameter are more expensive. Thus, it may be beneficial to increase the diameter of sectors with large flows, providing they are as short as possible.

Initially, the heuristic PND_{H} calculates the minimum spanning (MST) tree for the case being solved. This is because the MST is the best estimate in terms of total network length and may be used as initial search point for problems of this nature [

The MF is solved via an exact Outer Approximation algorithm proposed by Duran and Grossmann [

Computational experiments were conducted to test the efficiency and robustness of the model and algorithms. Because it is an original application to the PND problem, the literature on the subject contains no instances; these had first to be built. This section presents the cases constructed, the results obtained with the algorithms and the main conclusions from the analysis of these results.

To build instances, the sugar industry of São Paulo was taken as context. In particular,

It can be readily seen how decentralized production is and how essential it is to use an appropriately designed and economically competitive logistics network to collect and concentrate production.

Five instances were built, respectively with 4, 8, 12, 16 and 20 producing regions selected from among those listed in

The distances between regions were determined using the geographic coordinates of the main town. The height differences were calculated directly via the towns’ altitudes. The outputs to be dispatched are listed in

To compose the instances, a set of six possible values for the pair “pipe diameter and thickness” was considered. Four of these pairs were associated with the smallest instance and all of them with the largest one.

Classical empirical equations were used to calculate the friction factors. There are three parameters to be considered in this calculation: diameter, relative roughness [

In all cases, the loss estimator was considered similar

for every, i.e.,. Construction unit costs were determined using Equation (7). The unit cost of carbon steel was obtained in [

The experiments were performed on an Intel Core 2 Duo 2.66 GHz with 8 GB of RAM. The PND_{H} heuristic was coded in ANSI C++. The MF was programmed in the commercial software AIMMS [

_{H} and MF execution times, and a comparative analysis of results in terms of economic gains offered by the solutions. PND_{H} runtimes were null in all cases. MF produced the best solution for each case, which was obtained in low runtimes, as shown. By comparison, PND_{H} was able to find two optimal solutions (~0%). The others solutions entail small economic losses ranging from 2% to 6.7%.

We now analyze solutions in non-financial technical terms.

The PND_{H} solution to PND_SP_08_05 is shown in

1.72% saving in project cost (

PND_{H} built the 1040.0 km solution to PND_SP_16_06, as sketched in

As compared with the optimal design (

Finally, _{H} solution to PND_SP_20_06 (1361.0 km). Interestingly, Ribeirão Preto was connected to Campinas and not to a closer region (Araraquara or Limeira). This is because connecting it to Araraquara entails increasing diameters all the way to Campinas, thus also increasing the solution cost. Nevertheless, the Ribeirão Preto-Campinas connection does not form part of the optimal solution (

These findings emphasize the quality of the solutions built by PND_{H} (economic losses of less than 6.7% at null computational time cost). This result confirms the potential of the set of characterizations of a good solution. However, the most encouraging results were those generated by the exact algorithm. All except the largest case were solved in very low computational times. Improvement may possible be achieved by inputting information from the heuristic solution to the OA algorithm.

The tools developed to solve the PND addressed here met the expectations of robustness and efficiency and can be used as decision support tools for projects of this nature. Indeed, as the PND relates to a pipeline network whose purpose is to channel fluids to some central location,

and where the main characteristic of the required solution is a topology without cycles and with pressure discontinuities at vertexes, these tools can be used.

In this study, the problem of designing pipeline networks to transport fuel was addressed with a view to developing robust and efficient computational tools able to consider and assess the main design-related technical characteristics and decisions, as well as to estimate the construction cost and the operating cost (to any time horizon), in order to assist decision makers when evaluating the technical and economic feasibility of such projects.

Technical decisions are interdependent. For example: for a given flow rate, the larger the diameter of the duct, the lower the power dissipation. Thus pumps can be smaller and cheaper. However, the larger the diameter, the more expensive the pipe becomes. Conversely, if pipe diameter is diminished in order to reduce construction cost for the same flow rate, power dissipation is higher and thus higher-capacity (and more costly) pumps have to be installed. The larger the network, the harder it is to address these issues.

Accordingly, the combinatorial nature of the problem coupled with non-linear equations governing the flow energy balance, recommended the use of MINL mathematical programming models. To solve the problem, a heuristic algorithm that traces the main features of a technically feasible solution was developed to search for the best possible solutions in terms of cost. The mathematical program was coded in the AIMMS development environment, which provided the necessary optimization tools.

The particular context adopted was the ethanol industry in São Paulo State. The tools developed were tested on five cases built on the basis of output data from around the state. The computational results were extremely satisfactory, since all cases were solved efficiently by the heuristics, and also to global optimality by the exact algorithm, with low execution times, thus confirming the robustness of the approaches adopted.

The most radical paradigm that can be broken is the deterministic approach. The assumption that the supply of ethanol is uncertain to the operating time horizon would probably lead to addressing the time variable directly, in order to design a network that best suits many different possible supply scenarios, as well as to using stochastic mathematical programs.

Technically, there are many options. Allocation of more than one diameter per sector of pipeline in the network may be considered. Pressure loss will be calculated as the sum of the losses in each of the sub-sections with different diameters. The energy balance equation should consider the different kinetic energy in the sectors, since different cross-sections produce different flow velocities for the same flow rate.

Another variant might consider finite storage capacities in the regions, to be decided in view of the related cost. This capacity is influenced by the stochasticity of production and the NPSH_{R} pump parameter. If only the latter is considered, a stochastic approach to the problem can be avoided by estimating the minimum storage capacity of each region.

Finally, it would be useful to determine the number and location of pumps needed to power the system. Motivations to treat this decision are that it may not be possible to provide all the power necessary in each region by using one single station (technical limitation) and that, since pipes have maximum pressure tolerances, it may be necessary to distribute the energy supplied so that theses limits are observed.

The authors would like to thank the Agência Nacional do Petróleo, Gás Natural e Biocombustíveis (ANP) for financial support provided under grant PRH-ANP-21.