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/s2) 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 [
the circumferential tension in the pipeline, 
the diameter and 
the thickness. The operating pressure is a function of the amount of energy supplied to the system through pumping stations. To prevent limits being exceeded, several stations are placed along sections of the network. For simplicity, however, it is assumed that only one station will be installed in each region. As a result, the maximum operating pressure will occur exactly at the entrance to each section. In a duct with constant flow and speed, zero final pressure and no flow machines, it follows from (1), (2) and (4) that:
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.
of pipeline with diameter
. For each of the intervals constructed by partitioning the feasible flow space, defined by 
and
, a pump 
with procurement cost 
is selected (Section 3.1.3). This yields the step cost function represented in 
and
.
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).
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:
:
. 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 
(
is the destination vertex.
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):
and the length of the sector. In addition, once flows increase in the direction of the destination, it
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 
.
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.
. The smallest available diameter capable of carrying the flow calculated for each sector is selected. The cost of the initial solution is stored. The algorithm attempts to decrease the distance traveled by large flows and increase the diameter of some sectors, one at a time—in both cases, a predefined number of times. At each trial (path reduction and diameter increase), the new solution is stored if it improves cost.
, i.e.,
. Construction unit costs were determined using Equation (7). The unit cost of carbon steel was obtained in [
costs) were not obtained, the methodologies proposed in 3.4.3 and 3.4.4 were not used. Thus, the coefficients of the second portion of (23) were sorted so as at least to respect the percentage given in Section 3.4. 
