Maximum gas production under equipment related
restrictions
Dion Witteveen, s1812939
Master’s Thesis Econometrics, Operations Research and Actuarial Studies Specialization: Operations Research
Faculty of Economics and Business University of Groningen
Supervisors:
Prof. Dr. R.H. Teunter (RUG) M. de Wolff, MSc (ORTEC) Co-assessor:
Maximum gas production under equipment related
restrictions
Dion Witteveen, s1812939
Abstract
Contents
1 Introduction 1
2 Literature review 2
3 Problem description 3
4 Context and currently used method 5
5 Mathematical problem formulation 6
5.1 Overview of variables . . . 6
5.2 Ranges of variables . . . 7
5.3 Flow direction . . . 7
5.4 Objective function . . . 8
5.5 Network equations . . . 8
5.6 Fixed pressures and loads . . . 10
5.7 Equipment restrictions . . . 10
5.8 Complete mathematical problem formulation . . . 10
6 Approach 12 6.1 Piecewise linear approximations . . . 13
6.2 Iterative improvement procedure . . . 15
6.2.1 Breakpoint selection . . . 16
6.3 Modeling the control valve constraints . . . 17
6.4 Modeling the compressor constraints . . . 18
7 Results 19 7.1 Test instances . . . 19 7.2 Settings . . . 22 7.3 Comparing performance . . . 23 8 Extensions 26 9 Conclusions 27 Bibliography 29 Appendices 30
A Edge specific constants 30
1
Introduction
Natural gas is an important source of energy in the Netherlands and many other parts in the world. Gas production companies must meet the demand for natural gas, both now and in the future. Gas fields have only a limited capacity and the gas field pressure decreases when gas is produced. This is why gas companies make long term production plans to make optimal use of the available gas fields and plan investments wisely. One important step in these production plans is determining the maximum production capacity of a gas network. This maximum capacity depends not only on the gas field pressure, but also on the network of pipelines and equipment that connects the wells to the consumers.
This research considers the problem of finding the maximum production capacity of a gas net-work, taking a number of restrictions into account. The pressure at the points where gas flows in or out of the network is fixed. The reason is that gas is required to be at a given pressure at the delivery point (fixed outflow pressure) and the characteristics of the gas fields can be considered constant (fixed inflow pressure) within a certain time period. In addition, we have restrictions on the network equipment. These restrictions state a minimum or maximum on the flow rate or pressure in certain parts of the network. To meet these restrictions, the settings of the control valves and compressors can be altered. The goal is to find the optimal settings, such that the production (flow out of the network) is maximized and the restrictions are not violated. The flow of gas in a network can be modeled by a set of equations, some of them nonlinear. To solve the maximum production problem, these nonlinear equations must be solved together with the restrictions on the network equipment. This results in a nonlinear optimization problem, where the objective is to maximize the outflow of gas and the constraints consist of the gas flow equations and the restrictions on the network equipment.
This thesis proposes an algorithm to solve the nonlinear optimization problem. Using piecewise linear approximations, the problem is transformed into a mixed integer linear program (MILP), which is solved to optimality. Piecewise linear approximations are obtained by evaluating the nonlinear functions on a set of breakpoints and connecting the breakpoint evaluations. To en-sure an accurate approximation of the nonlinear problem, the algorithm includes an iterative procedure to select efficient breakpoints.
The direct context of this research is the gas production planning software GenREM, the Gen-eralized Reservoir Evaluation Model. The problem of finding the maximum capacity of a gas network is one of the steps that GenREM takes in planning the gas production. The algorithm introduced in this thesis can improve both the user convenience and the results. However, this thesis is mostly focused on the problem in general. A good method to solving this problem is useful in all gas production planning situations. Moreover, the findings of this research apply to many other nonlinear optimization problems.
2
Literature review
What all gas network optimization problems have in common, are the nonlinear equations. Geißler et al. (2012) investigate the use of piecewise linear functions in solving nonlinear opti-mization problems. They conclude that the method is very suitable for problems with functions of only a few variables, as is the case in our problem. They point out that the advantage is that linear solvers are well developed, fast, and robust. Piecewise linear approximations have been successfully used in other gas network optimization problems (F¨ugenschuh et al., 2013; Martin et al., 2006; M¨oller, 2004; Pfetsch et al., 2014).
F¨ugenschuh et al. (2013) and Pfetsch et al. (2014) solve the so called validation of nomination problem. Given a specified supply and demand, they need to find the settings such all constraints are met. Martin et al. (2006) and M¨oller (2004) use a piecewise linear function approach to minimize compressor fuel costs in a gas network, given supply and demand. This research considers a gas network optimization problem with a few differences compared to the problems in current literature. In our case, the objective is to find the maximum production, instead of minimizing compressor fuel or finding a feasible solution where demand is met. Another difference is that the aforementioned literature considers networks that distribute gas to consumers, while we are considering networks in the context of gas production that contain additional pieces of equipment, the wells and plants, each modeled with different equations.
Another difference is the method to formulate piecewise linear functions in a MILP context. A basic method to model this was given in Jeroslow and Lowe (1984), it requires as many additional binary variables as there are breakpoints. Since then, many contributions focused on different and more efficient formulations, as summarized in Lin et al. (2013) and Vielma et al. (2010). There is also an efficient method without using binary variables, as used in De Farias et al. (2001), De Farias et al. (2008), and Keha et al. (2006). It uses a special branching strategy in the branch and bound algorithm. However, the most efficient method in current literature requires a logarithmic number of binary variables. It is introduced by Vielma and Nemhauser (2011) and they show that it outperforms all other methods. The algorithm proposed in this thesis uses this state of the art method to formulate the piecewise linear functions.
3
Problem description
This problem description starts with explaining the structure of the gas networks. It continues with the method to model the flow of gas in a network. Finally, the actual problem is introduced. Consider a gas network consisting of nodes and edges, two examples are given in Figure 1. The edges represent different pieces of equipment: pipelines, control valves, compressors, plants, and wells. There are three types of nodes. The connection nodes represent a connection between pieces of equipment and no gas leaves the network at these nodes. The other nodes are source nodes and delivery nodes. The source nodes represent the gas fields, this is where gas flows into the network. The delivery nodes represent the connections to the consumer gas network, this is where gas flows out of the network.
In Figure 1, the dashed edges represent the wells, the solid edges represent the other types of equipment. The reason that this figure does not show a distinction between the various types of equipment, is to keep the figure clear and simple. The small round nodes represent connection nodes, the square nodes represent source nodes (i.e. gas fields) and the large round nodes with an ’X’ represent delivery nodes. The gas flows from the source nodes to the delivery nodes. Most elements of these networks are at approximately the same height (at the surface), with the exception of the gas fields and the wells. The gas fields are located several kilometers below the surface and the wells connect them to the surface. Therefore, in this figure, all solid edges represent a horizontal flow of gas and all dashed edges represent a vertical flow of gas.
We distinguish between two types of network structures, branched networks and looped networks. Branched networks converge to a single delivery point at the end. Looped networks contain several branches that come together in a loop, with one or more delivery points located in the loop. Examples of a branched and a looped network are given in Figure 1a and 1b, respectively.
x
(a) Branched network structure.
x x
(b) Looped network structure.
The variables that represent the behavior of gas inside the network are the flow rate (volume per time unit) at each edge and the pressure at each node. The reason that pressures are considered at the nodes, instead of te edges, is that pressure is not constant at an edge. For example, pressure decreases when gas flows through a pipeline and it increases when gas flows through a compressor. Therefore, we consider the pressure at the beginning and end of each piece of equipment, i.e. at the nodes. This method of modeling the flow rates and pressures is common in gas network literature (Osiadacz, 1987; Martin et al., 2006; Woldeyohannes and Majid, 2011). At the source and delivery nodes, the pressure is fixed at a given value. The fixed value at the source nodes corresponds with the current gas field pressure and the fixed pressure at the delivery nodes corresponds with the required pressure at the consumer gas network.
In addition to these two variables, we have the load, representing the amount of gas leaving or entering the network at each node. A positive load represents a flow out of the network (at the delivery nodes), a negative load represents a flow into the network (at the source nodes). Note that the load at the connection nodes is fixed at zero, since inflow equals outflow for these nodes. The goal is to maximize the flow of gas to the delivery nodes, subject to the equipment restric-tions. These restrictions apply to certain pieces of network equipment. For example, a plant or compressor is required to operate within a certain pressure range, resulting in a minimum and maximum pressure restriction on that piece of equipment. To meet these restrictions, the settings of the control valves and the compressors can be adjusted. By altering these settings, the gas flows and pressures can be changed such that they meet the equipment restrictions. We want to know the optimal settings of the control valves and compressors, keeping in mind that our goal is to maximize flow and meet all restrictions.
4
Context and currently used method
This section starts with a paragraph that briefly describes all the steps of the gas production planning software GenREM, explaining where our problem fits into this planning process. The next two paragraphs explain the method that is currently used to determine the production capacities.
GenREM is used to plan the gas production for a given period of time. The planning is done in fixed time steps, usually a step of one month. In each time step, GenREM considers the characteristics of the gas fields. The gas fields are divided into grids, the grid pressures are assumed to be constant during a time step. Based on the amount of gas that was produced in the previous period, the subsurface model determines the grid pressures in the current period. Using these grid pressures and all other constraints, the maximum capacity of the surface networks must be determined. The currently used method, explained in the next paragraph, determines a feasible capacity, but not always the maximum capacity. Determining this maximum capacity is the subject of this research. Given the maximum capacities, GenREM can make the optimal production plan. This plan states which amounts of gas are produced from each location, such that the total amount of gas meets the forecasted demand.
Currently, GenREM starts by solving only the network equations, considering the settings for the control valves and compressors fixed at their maximum values. The fixed inflow and outflow pressures are taken into account, but all restrictions on the network equipment are ignored at first. This results in solving a set of nonlinear equations, the solution gives us the gas flows and pressures in the network. When they are determined, the equipment restrictions are checked one by one, following a user defined priority list. For each restriction, the user has defined one or more so called action segments, the action segments are a set of control valves and compressors belonging to that restriction. A control valve or compressor can be an action segment to multiple restrictions.
When a violated restriction is encountered, the settings of the corresponding action segments (i.e. the corresponding control valves and compressors) are altered during a series of iterations. The effect of a change in settings can only be determined by solving the complete set of network equations again. Therefore, at each iteration, the network equations are solved to redetermine the flows and pressures in the network. The iterations stop when the restriction is satisfied or when the settings of the action segment have reached their upper or lower bound. If the latter is the case and another corresponding action segment is available, the process continues with that action segment. If the restriction is satisfied, all previous restrictions have to be checked again, since the change in the network could have caused a new violation.
5
Mathematical problem formulation
This section gives a mathematical formulation of the problem. The first subsection gives an overview of the variables used throughout this thesis. The following subsections consider the range of the variables, modeling the flow direction, the objective function, the network equations, the fixed load and pressure constraints, and the equipment restrictions. The final subsection combines the equations of the other subsections in one complete mathematical formulation of the problem.
5.1
Overview of variables
Let us consider a gas network with m edges and n nodes. The layout of the gas network is stored in an edge-node incidence matrix, this is an n × m matrix A with entries
aij=
1 if edge j enters node i , −1 if edge j leaves node i ,
0 if edge j is not connected to node i . We divide the set of edges and the set of nodes into the following subsets.
E: the complete set of edges
EP: the set of edges representing a pipeline,
EV: the set of edges representing a control valve,
EC: the set of edges representing a compressor.
EW: the set of edges representing a well,
ET: the set of edges representing a plant,
N : the complete set of nodes, NC: the set of connection nodes,
NS: the set of source nodes,
ND: the set of delivery nodes.
We consider the following variables, pi: pressure (bar) at node i,
Pi: Pi= p2i, squared pressure at node i,
Qj: flow rate (m3/s) at edge j,
Li: load (m3/s) at node i,
Vj: control factor at edge j ∈ EV, indicating the position of the valve,
Kj: compressor power (kW) at edge j ∈ EC.
αj, βj, γj, θj: constants at edge j,
Pif ix: fixed squared pressure at node i ∈ NS∪ ND,
Pmax
i , Pimin: max and min squared pressure at node i,
Qmax
j , Qminj : max and min flow at edge j,
Vjmax, Vjmin: max and min control factor at edge j ∈ EV,
Kmax
j , Kjmin: max and min compressor power at edge j ∈ EC,
Rmax
j : maximum compression ratio at edge j ∈ EC.
5.2
Ranges of variables
The actual pressure and squared pressure are nonnegative real numbers. The flow rate is a real number, the sign of the flow rate is used to model the flow direction, as is explained in the next subsection. The load is a real number. A positive load means that gas is leaving the network, a negative load means that gas is entering the network. The control valve factors have given lower and upper bounds between 0 and 1. The values of the compressor power have given positive lower and upper bounds. To summarize,
pi≥ 0 ∀ i ∈ N, Pi≥ 0 ∀ i ∈ N, Qj ∈ R ∀ j ∈ E, 0 < Vjmin≤ Vj ≤ Vjmax≤ 1 ∀ j ∈ EV, 0 ≤ Kjmin≤ Kj ≤ Kjmax ∀ j ∈ EC.
5.3
Flow direction
Each edge is specified in a single direction, from node i to node j. The sign of each flow rate is used to model the direction of the flow, a positive flow rate represents a flow in the specified direction and a negative flow rate represents a flow in the opposite direction. However, modeling the direction of the flow is only needed for some edges, for all other edges the gas is only allowed to flow in one direction. The reason is that equipments like compressors, wells, and plants are designed to handle a gas flow in a single direction (toward the delivery point(s) in our case). For branched networks, this means that the flow direction of all edges is known. For looped networks, this means that the flow direction of most edges is known. The only exceptions are the edges where both directions lead toward a delivery point and hence gas can move in two directions. If these edges exist, they are located inside the loop. Figure 2 gives an illustration, the large round nodes with an ’X’ are the delivery nodes and the arrows represent the direction of flow. There is only one edge where gas can move in both directions, it is located at the top of the loop in Figure 2b. It is represented by the dark dashed line with double arrows.
and flow rates are allowed to be negative, this way the direction will be known after solving the problem.
x
(a) Branched network.
x x
(b) Looped network.
Figure 2: Flow directions in a branched and looped network. The small round nodes represent connection nodes, the square nodes represent source nodes (i.e. gas fields) and the large round nodes with an ’X’ represent delivery nodes. The edges represent the equipments, the arrows point in the flow direction. The looped network shows the one edge where gas can flow in both directions, it is represented by the dark dashed edge with two arrows.
5.4
Objective function
The objective is to maximize the flow of gas to the delivery nodes, i.e. to maximize the sum of loads at the delivery nodes.
max X
i∈ND Li.
5.5
Network equations
The steady-state flow of gas in a network is described by a set of equations. These are the nodal equations, the pipeline pressure drop equations, and the equations modeling the other equipment types: control valves, compressors, wells, and plants. These equations contain edge specific constants, they represent characteristics of the gas and the equipment or are estimated using operational data. More information on the constants is given in Appendix A. In our model, the pressure variable pioften occurs in squared form, which can be replaced by the squared pressure
variable Pi. To simplify the problem formulation, all equations will be formulated in terms of
the squared pressure variable Pi. After solving the problem in terms of the squared pressures,
we can determine the actual pressures by taking the square root.
is equal to the load of that node (Osiadacz, 1987). X
j
aijQj= Li ∀ i ∈ N .
The pipeline pressure drop equations model the pressure drop that occurs when gas flows through a pipeline. The pressure drop is caused by friction as gas moves through the pipe. It depends on the flow rate and an edge specific constant αh (Osiadacz, 1987). The constant term represents
all relevant characteristics of the gas and the pipeline.
Pi− Pj = αh· |Qh| · Qh ∀ h = ij ∈ EP.
Note that we use |Qh| · Qhinstead of Q2h. This formulation is needed because the flow rate can be
negative in some cases. A negative flow rate means the flow direction is opposite to the specified edge direction, in this case gas flows from node j to node i. However, the pressure difference is still considered in the specified direction, i.e. Pi− Pj. This difference must then be negative as
well, hence this formulation.
The control valve equations model the pressure drop that occurs when gas flows through a control valve. They can be written in a similar way as the pipeline pressure drop equations, with the addition of the control factor Vh. The control factor is used to control the pressure drop and flow
rate by partially blocking the flow of gas. Decreasing the control factor means increasing the friction. The edge specific constant αh depends on the characteristics of the gas and the control
valve.
Pi− Pj= Vh−2· αh· |Qh| · Qh ∀ h = ij ∈ EV .
The compressor equations relate the pressure ratio with the flow rate and the compressor power. The pressure ratio is the ratio of the squared discharge pressure (at the exit, here Pj) and squared
suction pressure (at the entrance, here Pi). The edge specific constants αh and θh depend on
the characteristics of the gas and the compressor. Pj Pi = 1 + 1 Qh · Kh· αh· θh 2·θh ∀ h = ij ∈ EC.
The well equations model the pressure drop that occurs when gas flows from a gas field to the surface, through a well. They relate the squared tubing head pressure (at the end of the well, here Pj), the squared gas field pressure (at the start of the well, here Pi) and the flow rate. Note
that gas field pressures are fixed and known. The edge specific constants αh, βh and γh depend
on the characteristics of the gas and the well.
Pj= αh· Pi− βh· Qh− γh· Q2h ∀ h = ij ∈ EW.
The plant equations model the pressure drop that occurs when gas flows through a plant. Using operating data, the edge specific constants αh, βh and γh are estimated.
5.6
Fixed pressures and loads
The source and delivery nodes have a given fixed pressure, since the gas field pressures and the consumer gas network pressure are known. All connection nodes have a fixed load of 0, since no gas leaves or enters the network at these nodes.
Pi= P f ix
i ∀ i ∈ NS∪ ND,
Li= 0 ∀ i ∈ NC.
5.7
Equipment restrictions
The pieces of equipment in the network can be restricted to a given minimum or maximum flow rate or pressure. Consistent with our model, the flow rate restrictions apply to the edges and the pressure restrictions apply to the nodes. However, not every node or edge has a restriction. To avoid needing to define corresponding sets of nodes and edges, we set the default minimum and maximum values to 0 and ∞, respectively. Finally, there is a maximum on the compression ratio, this restriction applies to all compressors. The equipment restrictions can be written as
Pi≤ Pimax ∀ i ∈ N,
Pi≥ Pimin ∀ i ∈ N,
|Qj| ≤ Qmaxj ∀ j ∈ E,
|Qj| ≥ Qminj ∀ j ∈ E,
Pj ≤ Pi· Rmaxh ∀ h = ij ∈ EC.
5.8
Complete mathematical problem formulation
This subsection summarizes the equations and definitions in the previous two subsections to give one complete mathematical problem formulation.
6
Approach
We have an optimization problem with a number of nonlinear constraints. The proposed solution is to linearize these constraints, using piecewise linear approximations of the nonlinear network equations. Besides solving the gas network equations, the problem in this research considers additional restrictions on the network equipment. These are solved by adjusting the settings of the control valves and compressors.
The optimal solution is given by the control valve and compressor settings that maximize the flow out of the network. The optimal settings will depend on the quality of the linear approximation. The proposed algorithm is designed to improve the approximation quality during a number of iterations, including a final feasibility check. A general description of the algorithm is given in the next paragraphs, more detailed descriptions of specific elements are given in the coming subsections.
When using piecewise linear approximations, each nonlinear function is evaluated on a number of breakpoints. The smaller the distance between the breakpoints, the better the approximation. Selecting more breakpoints therefore gives a better approximation. However, this increases the problem size and hence increases the time needed to solve the problem. The algorithm uses a method to select efficient breakpoints, without increasing the number of breakpoints.
In each iteration, we use the calculated flow rates and pressures of the previous iteration to select the breakpoints in the current iteration. The new breakpoints are placed such that they are more concentrated around the values of the previous solution. This is achieved by gradually decreasing the distance between breakpoints as we move closer to the value of the previous solution. The idea is that the linear approximations will be more accurate around the points that we want to evaluate and sufficient everywhere else. A more detailed description of the iterative improvement procedure is given in Section 6.2.
To ensure the quality of the approximation, we include a final feasibility check in the last iteration. The check is performed after the linear solver has found the optimal solution to the approximated problem. We then use the control valve factors and compressor powers from the optimal solution and consider them fixed. That is, the decision variables are now simply parameters and the problem is no longer an optimization problem, it is now a set of nonlinear equations that needs to be solved. Solving the equations will give us the actual flow rates and pressures under these fixed control valve factors and compressor powers.
The set of nonlinear equations is solved by the Newton Rhapson procedure. It uses an absolute tolerance level of 1 · 10−8, this is accurate enough to perform the feasibility check. The check is performed to see if all equipment restrictions are still satisfied in the nonlinear model, given a certain tolerance level. A tolerance level used in practice is 0.2, we will use this level for our test instances as well. Moreover, this final step can be used to measure the error between the approximated model and the nonlinear model. Error terms will be defined in Section 7.2. The proposed algorithm will be tested on a number of test instances. The tests are used to find out the balance between solution quality, the number of iterations used, and the number of breakpoints at each iteration. Different test instances are then used to evaluate the effect of the iterative improvement procedure. Moreover, test instances are used to compare the proposed algorithm to the currently used method.
contains the entire production planning process. It is more convenient to concentrate on the method it uses to determine the maximum production capacity and rebuild that part in R. This section continues with a subsection on modeling the piecewise linear approximations, fol-lowed by a subsection on selecting efficient breakpoints in the iterative improvement procedure. The next subsections consider the formulation of the equipment constraints in the linear approx-imated model.
6.1
Piecewise linear approximations
Piecewise linear functions can be used to approximate a nonlinear function. The domain of the nonlinear function is divided into a number of breakpoints and the nonlinear function is evaluated at each breakpoint. The piecewise linear function is then constructed by connecting these function evaluations, an example is given in Figure 3.
(a) Four breakpoints. (b) Six breakpoints.
Figure 3: A nonlinear function (black solid line) and a piecewise linear approximation (blue dashed line) using four and six breakpoints.
To explain how to model the piecewise linear approximations in a linear programming context, we introduce the variable x and nonlinear function f (x). Let y be the approximation of f (x) and let the breakpoints be given by ˆxi with i = 1, .., n.
To approximate the nonlinear function at a value x∗, we want to make a convex combination of the function values of the neighboring breakpoints of x∗. To construct a convex combination, we
introduce the variables λi with i = 1, .., n. For example, consider a case where x∗ lies between
breakpoints 4 and 5. Then we want a convex combination where only λ4and λ5are positive and
the approximation of the function as x = n X i=1 λi· ˆxi, y = n X i=1 λi· f (ˆxi) , n X i=1 λi= 1 , λi≥ 0 ∀i , {λ1, λ2, ..., λn} ∈ SOS2
The first equation defines x as a convex combination of breakpoints. The second defines y as the same convex combination of function evaluations at the breakpoints. The third and fourth lines restrict the combinations to be convex. The last line states that the set of λ’s is required to be a special ordered set of type 2. In order to use this formulation in our algorithm, we need to model the SOS2 constraints in a MILP context. As argued in Section 2, we use the method of Vielma and Nemhauser (2011) to accomplish this.
Functions of two variables are also modeled using SOS2 constraints. Let us introduce a function f (x, y). We have m breakpoints for the x variable and n breakpoints for the y variable. Together, these series of breakpoints result in a grid of input values with n · m grid points. Each square in the grid is then divided in two triangles. One possible method is the so called union jack triangulation, as shown in Figure 4.
c(0, 4) x0 x1 x2 x* x3 x4 y0 y1 y * y2 y3 y4
Figure 4: Example of a union jack triangulation, including given points x∗ and y∗ and their neighboring grid points (solid black circles).
6.2
Iterative improvement procedure
For each nonlinear function that is approximated, we need a sequence of breakpoints of the input variable. There are two things we need to consider: the number of breakpoints and the selection of the breakpoints. As mentioned before, increasing the number of breakpoints increases the accuracy of the approximation and increases the size of the mixed integer linear program. In other words, selecting the number of breakpoints is a trade-off between accuracy and solution speed. However, an efficient selection of the breakpoints can increase the accuracy of the approximation without increasing the problem size.
The algorithm solves the problem multiple times, in each iteration it improves the accuracy of the approximation. The iterative improvement procedure uses breakpoint selection to improve the accuracy. In each iteration, new breakpoint sequences are constructed and the mixed integer linear program is solved. Note that we solve an approximated version of the original problem and we alter the breakpoint sequences in each iteration, so we solve a slightly different problem each time.
Constructing the breakpoint sequences is based on the solution of the previous iteration. In the first iteration, we equally distribute the breakpoints on the interval from lower to upper bound. After solving the problem in the first iteration, we use the obtained optimal solution to determine new breakpoint sequences. For each nonlinear function, we know the value that the input variable has in this solution. We can now construct sequences in which the breakpoints are more concentrated around these solution values, making the piecewise linear approximations more accurate around these solution values and less accurate for values far away from the previous solution.
the previous solution value. The gradual increase is achieved by using the idea of a logarithmic sequence, similar to the idea of a logarithmic scale. The degree of concentration can be altered by a density setting. To illustrate the idea, Figure 5 gives an example of a logarithmic sequence. Figure 6 gives an example of three breakpoint sequences, the first is a uniform sequence and the other two are based on the logarithmic sequence idea. In this example, the value of the input variable was 80 in the previous solution, the new breakpoint sequences are more concentrated around that value. The exact procedure to create the breakpoint sequences is described in Section 6.2.1.
Figure 5: Example of a logarithmic sequence.
0 20 40 60 80 100
d=5 d=1 Uniform
Figure 6: Example of three breakpoint sequences, corresponding to a function with input value 80 in the previous solution. The first line gives a sequence with uniform intervals, the second and third are a logarithmic sequence with density 1 and 5.
6.2.1 Breakpoint selection
First, we formally describe the uniform sequence and logarithmic sequence. Consider a sequence of n elements with A and B equal to the lower and upper bound, respectively. Let the sequence be written as {xi}, i = 1, .., n. For the uniform sequence, we construct a finite arithmetic
sequence, i.e. a sequence with a constant difference between each pair of consecutive elements. The elements are defined by
x1= A ,
xi= xi−1+
B − A
n − 1 , i = 2, .., n .
This way, all elements are equally distributed on the interval from lower bound to upper bound. Note that xn= x1+ (n − 1) ·B−An−1 = B.
be denoted by {ai}, i = 1, .., n and the logarithmic sequence by {bi}, i = 1, .., n. The elements are defined by a1= log(A) , ai= ai−1+ log(B) − log(A) n − 1 , i = 2, .., n , bi= exp(ai) , i = 1, .., n .
Rewriting these definitions results in the following definition of the logarithmic sequence. b1= A , bi= b1· B A n−11 , i = 2, .., n .
This way, the difference between elements increases toward the end of the sequence, see Figure 5 for an example. Note that bn = bi−1·
B A n−11 n−1 = B.
In order to construct a breakpoint sequence based on the idea of a logarithmic sequence, we must first deal with the following. When the distance between A and B increases and the number of breakpoints remains the same, the relative difference between the elements increases. To avoid that, let us introduce a variable indicating the density of the sequence, denoted by d. We now take a logarithmic sequence between 1 and d + 1, subtract 1, multiply by B−A
d and add A. This
way, the sequence density is controlled by the parameter d and does not depend on the distance between A and B.
Now consider a lower and upper bound on our input variable. In addition, we have the value of the input variable from the previous iteration, let us call that point C. The point C always lies between the lower and upper bound. We divide the breakpoint sequence into three parts. The first two parts are to the left and right of the point C, both of equal length. These parts are chosen as large as possible, i.e. until one of the parts ends at the lower or upper bound. The third part consists of the remaining part of the interval. We can now construct logarithmic sequences, one from C to the end of the first part and one from C to the end of the second part. In the third part, we construct a uniform sequence. The number of breakpoints are divided in such a way that the distance between breakpoints in the third part is never smaller than in the first or second part. In Figure 6, we have the case that C = 80. The first part of the sequence is from 80 to 60, the second from 80 to 100, in both parts a logarithmic sequence is constructed. The third part is from 0 to 60, in this part a uniform sequence is constructed.
6.3
Modeling the control valve constraints
Recall the control valve constraints,
Pi− Pj= Vh−2· αh· |Qh| · Qh,
Vhmin≤ Vh≤ Vhmax.
To linearize these constraints, we need a piecewise linear function ¯f (Vh, Qh) to approximate
Vh−2· αh· |Qh| · Qh. In order to avoid needing to use a piecewise linear function of two variables,
valve factor Vh to formulate a lower and upper bound on the pressure drop. Pi− Pj ≥ (Vhmax) −2 · αh· |Qh| · Qh, Pi− Pj ≤ Vhmin −2 · αh· |Qh| · Qh.
With this formulation we only need to model a piecewise linear function of one variable, since Qh
is the only variable on the right hand side. This significantly reduces the size of the problem. Let ¯
g(Qh) be the piecewise linear approximation of αh· |Qh| · Qh, the linearized constraints become
Pi− Pj≥ (Vhmax) −2 · ¯g(Qh) , Pi− Pj≤ Vhmin −2 · ¯g(Qh) .
Once the problem is solved in terms of Pi, Pj, and Qh, the values of the control factors Vh can
be derived.
6.4
Modeling the compressor constraints
Recall the compressor constraints, Pj Pi = 1 + 1 Qh · Kh· αh· θh 2·θh , Khmin≤ Kh≤ Khmax.
In order to approximate the first equation in a linear model, we write Pj= Pi· 1 + 1 Qh · Kh· αh· θh 2·θh .
For this equation, we need a piecewise linear function ¯f (Kh, Qh, Pi) to approximate the right
hand side. In order to avoid needing to use a piecewise linear function of three variables, we reformulate the constraints similar to the reformulation of the control valve constraints.
We use the lower and upper bound on the compressor power Kh to formulate a lower and upper
bound on Pj, as a function of Qh and Ph.
Pj≥ Pi· 1 + 1 Qh · Kmin h · αh· θh 2·θh , Pj≤ Pi· 1 + 1 Qh · Kmax h · αh· θh 2·θh .
With this formulation we only need to model a piecewise linear function of two variables, since Qh and Pi are the only variables on the right hand side. This significantly reduces the size of
the problem. Let ¯gl(Qh, Pi) and ¯gu(Qh, Pi) be the piecewise linear approximations to the right
hand sides of these two equations. The linearized constraints become Pj ≥ ¯gl(Qh, Pi) ,
Pj ≤ ¯gu(Qh, Pi) .
Once the problem is solved in terms of Pi, Pj, and Qh, the values of the compressor powers Kh
7
Results
This section presents the results of the algorithm and the comparison with the currently used method. Before considering the test instances, settings, and performance results, it is interesting to note some very important practical benefits in using the linear approximation algorithm. The benefits arise from the user input that is required in the currently used method. Recall that the currently used method solves the restrictions one by one. The order in which the restrictions are handled is given in a priority list defined by the user. Moreover, for each restriction the user needs to define one or more corresponding control valves and compressors, the so called action segments. Each restriction is solved by adapting only the corresponding action segment(s). This method does not necessarily give the optimal solution and the quality of the solution depends highly on the user defined priorities and action segments. In practice, much time can be spend in searching for a good choice in priority order and action segments. The linear approximation algorithm proposed in this thesis needs no additional input concerning the restriction handling. Instead, the linear solver finds the optimal solution to the approximated problem, thereby solving all restrictions using the complete set of control valves and compressors. This relieves the user from searching for good priority orders and actions segments. Moreover, the algorithm gives a solution that does not depend on these kind of user decisions.
7.1
Test instances
In order to test the proposed algorithm, we use a set of test instances. Using the layout charac-teristics of existing gas networks, we have created six fictional network layouts. There are three branched and three looped networks in various sizes, including realistic sized gas networks. Let us call the branched layouts B1, B2, and B3 and the looped layouts L1, L2, and L3, both in order of increasing network size. The layouts define the edges and nodes, the placement of the equip-ment types and the equipequip-ment restrictions. For each layout, multiple test instances are created by randomly generating the values of the edge specific constants. A detailed documentation of the test networks is given in Appendix B.
ExampleB1
*
*
*
*
X
ExampleL1
*
*
*
X X
7.2
Settings
This section considers the settings of the iterative improvement procedure: the number of break-points, the number of iterations and the density. In order to investigate different settings we use 30 test networks, five different networks for each layout. In addition, let us introduce a measure of accuracy of the obtained solution. This measure can be obtained by performing the final feasibility check and comparing the results with the optimal solution.
After we have obtained the actual flow rates and pressures from the feasibility check, we compare them to the values from the MILP solution. To measure the accuracy, three error terms are introduced, the pressure error p, flow rate error Q and total error. Let the flow rates and
pressures of the optimal solution be denoted as p∗i and Q∗j and those of the feasibility check as pc
i and Qcj, i = 1, .., n, j = 1, .., m. Note that the pressure is written as small letter p, we are
measuring the error of the actual pressure and not the squared pressure. The errors are given by p= max i (|p ∗ i − p c i|) , Q = max i (|Q ∗ i − Q c i|) , T = max (p, Q) .
Initial exploration of the algorithm resulted in the following conclusions. Using less than 20 breakpoints regularly resulted in an infeasible approximated problem, even though the actual problem is feasible. With less than 20 breakpoints, the difference between the approximated problem and the actual problem can become too large and result in an infeasible approximated problem. The same problem occurs with a high density setting of 5 or 10. In the first iteration, when the breakpoints are equally distributed, the MILP could be solved without problems. In the second iteration, where we concentrate the breakpoints around the previous solution, the MILP regularly turned out to be infeasible. This is a results of the high density setting, the breakpoints are now too concentrated around the previous solution, giving an approximated problem that differs too much from the actual problem.
The test instances are solved using 20, 30, and 40 breakpoints. A density setting of 1 is used and five iterations are performed for each instance. The results are presented in Figures 9 and 10, showing the average pressure error and flow rate error, respectively. Two good candidate settings are 30 and 40 breakpoints using two iterations. For both pressure error and flow rate error, these settings resulted in an error below 0.1, which is sufficient in most practical applications. There is however one downside to the setting with 40 breakpoints. For practical convenience during testing, the solving time of the algorithm was limited to 1200 seconds. When using 40 breakpoints, three out of the five instances corresponding to the largest network layout (B3) were not solved within this time limit. This was the only case where this problem occurred, all other test networks were solved within the time limit. The setting with 30 breakpoints and two iterations was able to solve all networks without problems. The total average solving time was 17.1 seconds, with an average solving time of 5.7 seconds for the smallest network layout (L1) and 30.2 second for the largest network layout (B3).
1 2 3 4 5 0.0 0.2 0.4 0.6 40 30 20 Iteration Pressure error
Figure 9: Pressure error at each iteration, for 20, 30, and 40 breakpoints.
1 2 3 4 5 0.0 0.2 0.4 0.6 40 30 20 Iteration Flo w r ate error
Figure 10: Flow rate error at each iteration, for 20, 30, and 40 breakpoints.
7.3
Comparing performance
In this section, two different comparisons are presented. The first compares the accuracy of the proposed algorithm to the accuracy of a piecewise linear approximation algorithm without the iterative improvement procedure. The second compares the proposed algorithm to the currently used method of solving the restrictions one by one, they will be compared by objective value (the found production capacity).
Accuracy with and without iterative improvement procedure
iterative improvement procedure and once without. The algorithm without iterative improve-ment simply divides 30 breakpoints evenly between the lower and upper bound of the input values. The algorithm with the iterative improvement procedure uses the settings as determined in Section 7.2
The accuracy is evaluated by the total error term, as given in Section 7.1. The average total errors for both methods are given in Figure 11. The averages are calculated for each network layout separately, the layouts are indicated by the number of nodes (on the x-axis). The 95% confidence intervals are included in the figure.
For these instances, using the iterative improvement procedure results in an average decrease in error of 71.9%. The figure does not show a clear relation between this effect and the network size. The figure does show that the error when using the iterative improvement procedure is more consistent, while the algorithm without iterative improvement procedure shows more fluctuation in the error for different test network layouts.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 Nr. of nodes Error 26(L1) 33(B1) 56(L2) 64(B2) 96(L3) 120(B3)
Figure 11: Mean error (solid lines) and 95% confidence interval (dashed lines) for method with breakpoint selection procedure (blue lower lines) and without (red upper lines).
Objective value of the proposed algorithm vs. the current method
For the second comparison, 20 test networks are used. Five test instances are created from each of the following layouts: B1, B2, L1, and L2. In this case it was more difficult to create test instances, because they needed to work on the currently used method as well. For this method, the restriction priority order and the action segments needed to be specified for each network. The instances are first solved using the currently used method of solving the restrictions one by one. The same instances are then solved by the proposed piecewise linear approximation algorithm, using the settings as determined in Section 7.2.
The methods are compared by the objective value, i.e. the found production capacity. Figure 12 gives the average percentage increase in objective value when using the proposed piecewise linear approximation algorithm, compared to the currently used method. The averages are calculated for each network layout separately, the layouts are indicated by the number of nodes (on the x-axis).
The proposed algorithm is able to find higher production capacities for all test instances. The average increase in production capacity is 5.3%. The size of the improvement will always depend on the order of restrictions and the choice of action segments, since this affects the currently used method. This brings us back to one of the benefits mentioned in the beginning of this chapter, the proposed algorithm does not require these types of user input and its results do not depend on them. 0 5 10 15 20 Increase (%) Nr. of nodes 26(L1) 33(B1) 56(L2) 64(B2)
8
Extensions
In this section, two extensions to the model are introduced to further highlight the practical potential of the algorithm. The extensions are very easy to realize when using the proposed algorithm. The two extensions can be combined as well.
The first extension is to use the model to produce a required amount of gas. This can be achieved by a constraint stating that the sum of flows going to the delivery nodes is smaller or equal to the required gas production. If the required production is possible, the maximization of the objective will cause the production to be exactly equal to the required amount of gas. If the required production is too high, the maximum possible production is given. Let the required production be denoted by U . The extension is realized by adding the following constraint to the model.
X
i∈ND
Li≤ U .
The second extension is to divide the gas production among wells according to a specified distri-bution. Consider a problem with k wells, where flow rates Q1, .., Qk are the corresponding well
productions. Let the distribution be given by rj, with j = 1, .., k andPkj=1rj = 1. It represents
the proportion of total production that each well produces. In addition, introduce a tolerance level b, representing the allowed percentage of deviation from this distribution. The extension is realized by adding the following constraints to the model.
Qj ≤ 100 + b 100 · rj· k X i=1 Qj, Qj ≥ 100 − b 100 · rj· k X i=1 Qj.
9
Conclusions
This thesis studied the problem of finding the maximum gas production capacity. The capacity is limited by the gas field pressure, the delivery point pressure, and the network of pipelines and equipments. There are restrictions on the network equipment, stating a minimum or maximum flow rate or pressure. The flow of gas in a network can be modeled by a set of equations, some of them nonlinear. This results in a nonlinear optimization problem, where the objective is to maximize the outflow of gas and the constraints consist of the gas flow equations and the restrictions on the network equipment. The solution gives the optimal settings of the control valves and compressors.
This thesis proposed a piecewise linear approximation algorithm, transforming the problem into a mixed integer linear program that is then solved to optimality. To ensure an accurate ap-proximation of the nonlinear problem, the algorithm includes an iterative procedure to select efficient breakpoints. The iterative procedure uses the solution found in the previous iteration and increases the density of breakpoints around that point. In the next iteration, the linear approximations will be very good around the points that we want to evaluate and sufficient everywhere else.
Test instances were used to determine the settings of the algorithm and iterative improvement procedure. Subsequently, the algorithm was compared to a piecewise linear approximation method without the iterative improvement procedure. Results show that the iterative improve-ment procedure reduces the approximation error with 71.9% on average. Finally, the algorithm was compared to the currently used method, which solves the restrictions one by one. The proposed algorithm found a 5.3% higher production capacity on average.
There is another practical benefit to the proposed algorithm as well. The currently used method requires a lot of additional user input and the results of the method strongly depend on this input. Fortunately, the proposed algorithm does not require this additional input and its results do not depend on them, giving it a nice practical benefit.
The algorithm gives the possibility to easily include some extensions of the problem. Instead of finding the maximum capacity, it is possible to find the settings to exactly produce a required amount of gas. A second extension is to specify the distribution of production among wells, for example to equally distribute the production among wells. These extensions can be combined as well. Moreover, they are realized by simply adding a few constraints to the MILP.
The innovation in the proposed algorithm lies in the iterative improvement procedure and the corresponding breakpoint selection method. The breakpoint selection method gradually de-creases the distance between breakpoints as we move closer to the solution value of the previous iteration. The gradual increase is achieved using the idea of a logarithmic sequence. A density parameter controls the speed of the decrease in breakpoint distances. High density parameters have an undesired effect, the resulting approximation is too different from the original nonlinear problem. A low density parameter, i.e. a very gradual decrease in distance between breakpoints, does have the desired effect. It increases the accuracy around the previous solution value while still having a sufficient accuracy everywhere else. This shows that the idea of a gradual increase is important in the iterative improvement procedure.
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Appendices
A
Edge specific constants
This section gives a description of the edge specific constants and their test range, using the variables listed below. Note that most of these variables are not mentioned in section 5, since they only occur as components of the edge specific constants.
l: length of pipeline (m), test range from 500 to 2000, D: pipeline diameter (m), test range from 0.3 to 0.6, G: specific gravity of gas, test range from 0.6 to 0.7, Z: compressibility factor of gas, test value 1.1, T : temperature (K), test range from 290 to 350, C: valve flow coefficient, test range from 400 to 1000, RP: pipeline efficiency, test range from 0.8 to 0.95,
RC: compressor efficiency, test range from 0.7 to 0.8,
κ: specific heat ratio, test range from 1.4 to 1.5, FD: Darcy flow coefficient, test range from 150 to 200,
FN: Non-Darcy flow coefficient, test range from 7 to 12,
B: Static pressure correction coefficient, test range from 1.4 to 1.65, W : Tubing friction coefficient, test range from 10 to 16.
For pipelines, the edge specific constant α is given by
α = 7.2814 · 10−10· l · G · Z · T R2
P· D5.333
. Using the range of its components, the test range is 0.001 ≤ α ≤ 0.38. For control valves, the edge specific constant α is given by
α = 3600
2· T · G
2802· C2 .
Using the range of its components, the test range is 0.03 ≤ α ≤ 0.25. For compressors, the edge specific constants α and θ are given by
α = 273.16 · RC 100 · Z · T , θ = κ
κ − 1.
For wells, the edge specific constants α, β, and γ are given by α = B−1,
β = FD· B−1,
γ = FN+ W
B .
Using the range of its components, the test ranges are 0.61 ≤ α ≤ 0.71, 91 ≤ β ≤ 143, and 10 ≤ γ ≤ 20.
For plants, the edge specific constants α, β, and γ are estimated using regression techniques on operating data. The test ranges are 50 ≤ α ≤ 200, 10 ≤ β ≤ 100, and 0 ≤ γ ≤ 1.
B
Test networks
We have three branched test network layouts (B1, B2, B3) and three looped test network layouts (L1, L2, and L3). In this section, the layouts are documented. For each layout, there is a table listing all edges and the nodes that they connect (from node and to node). Subsequently, the node types, equipment types and equipment restrictions are listed.
From each layout, a number of test networks are created by using different values for the edge specific constants and the restrictions. The values for the edge specific constants are randomly generated. The values are obtained from the intervals given in the previous chapter, they repre-sent realistic values of the constants. To generate random values, a truncated normal distribution is used with mean equal to the center of the interval and standard deviation equal to the center multiplied by 0.3. The bounds of the truncated normal distribution are given by the bounds of the interval. The values of the restrictions and fixed pressures are manually chosen, since randomly chosen values too often led to infeasible problem instances. Furthermore, the control valve factor bounds are Vmin
j = 0.001 and Vjmax = 1 and the compressor power bounds are
Kjmin= 0 and Kjmax = 500. The maximum compression ratio is set to 3. The data for all test networks, including solutions, can be found in an online database1.
Table 1 gives the layout of the nodes and edges of network B1 by listing all edges with corre-sponding nodes.
Edge Node Edge Node Edge Node Edge Node
from to from to from to from to
1 2 1 10 11 6 19 20 13 28 29 24 2 3 1 11 12 7 20 21 13 29 30 25 3 4 2 12 13 8 21 22 17 30 31 26 4 5 2 13 14 4 22 23 17 31 32 20 5 6 3 14 15 9 23 24 18 32 33 21 6 7 3 15 16 10 24 25 18 7 8 3 16 17 11 25 26 19 8 9 5 17 18 12 26 27 22 9 10 5 18 19 12 27 28 23
The following list gives the delivery and source node locations (by listing the node numbers) and all equipment types (by listing the edge numbers) of network B1.
- Delivery nodes: 1, - Source nodes: 14, 15, 16, 27-33, - Pipelines: 2, 8, 9, 19-25, - Control valves: 3, 4, 10, 11, 12, - Compressors: 16, 17, 18, - Plants: 1, 5, 6, 7, - Wells: 13, 14, 15, 26-32.
The following list gives all restrictions that apply to network B1. Their corresponding values are presented in the online database, following the same order as in the list.
- Minimum on the flow rate at edge 1, 5, 6, 7, - Maximum on the flow rate at edge 1, 5, 6, 7, - Minimum on the pressure at node 17, 18, 19, - Maximum on the pressure at node 2, 11, 12.
Table 2 gives the layout of the nodes and edges of network B2 by listing all edges with corre-sponding nodes.
Edge Node Edge Node Edge Node Edge Node
from to from to from to from to
1 3 1 17 18 7 33 34 15 49 50 24 2 2 1 18 19 8 34 35 16 50 51 25 3 4 2 19 20 8 35 36 16 51 52 26 4 5 1 20 21 9 36 37 17 52 53 27 5 6 3 21 22 9 37 38 17 53 54 27 6 7 3 22 23 10 38 39 18 54 55 28 7 8 3 23 24 10 39 40 18 55 56 28 8 9 4 24 25 11 40 41 19 56 57 29 9 10 4 25 26 11 41 42 19 57 58 29 10 11 4 26 27 12 42 43 20 58 59 30 11 12 5 27 28 12 43 44 20 59 60 30 12 13 5 28 29 13 44 45 21 60 61 31 13 14 5 29 30 13 45 46 22 61 62 31 14 15 6 30 31 14 46 47 23 62 63 32 15 16 6 31 32 14 47 48 23 63 64 32 16 17 7 32 33 15 48 49 24
Table 2: Edge and node specifications of layout network B2
The following list gives the delivery and source node locations (by listing the node numbers) and all equipment types (by listing the edge numbers) of network B2.
- Source nodes: 33-64, - Pipelines: 14-31, - Control valves: 2, 5, 7, 8, 10, 11, 13, - Compressors: 6, 9, 12, - Plants: 1, 3, 4, - Wells: 32-63.
The following list gives all restrictions that apply to network B2. Their corresponding values are presented in the online database, following the same order as in the list.
- Minimum on the flow rate at edge 1, 3, 4, - Maximum on the flow rate at edge 1, 3, 4, - Maximum on the pressure at node 3, 4, 5, - Minimum on the pressure at node 7, 10, 13.
Table 3 gives the layout of the nodes and edges of network B3 by listing all edges with corre-sponding nodes.
The following list gives the delivery and source node locations (by listing the node numbers) and all equipment types (by listing the edge numbers) of network B3.
- Delivery nodes: 1, - Source nodes: 83-120, - Pipelines: 1, 2, 3, 4, 6, 8, 9, 13, 23, 25-32, 34, 39, 40, 41, 44, 45, 46, 47, 48, 50-81, - Control valves: 7, 10, 11, 12, 20, 21, 22, 24, 33, 35, 36, 37, 38, 49, - Compressors: 17, 42, 43, - Plants: 5, 14, 15, 16, 18, 19, - Wells: 82-199.
The following list gives all restrictions that apply to network B3. Their corresponding values are presented in the online database, following the same order as in the list.
Edge Node Edge Node Edge Node Edge Node
from to from to from to from to
1 2 1 31 32 22 61 62 40 91 92 54 2 3 1 32 33 22 62 63 40 92 93 55 3 4 1 33 34 23 63 64 40 93 94 56 4 5 1 34 35 23 64 65 41 94 95 57 5 6 1 35 36 24 65 66 41 95 96 58 6 7 1 36 37 24 66 67 41 96 97 59 7 8 2 37 38 26 67 68 41 97 98 60 8 9 3 38 39 26 68 69 42 98 99 61 9 10 4 39 40 27 69 70 42 99 100 62 10 11 5 40 41 27 70 71 29 100 101 63 11 12 5 41 42 28 71 72 30 101 102 64 12 13 6 42 43 34 72 73 31 102 103 65 13 14 6 43 44 35 73 74 32 103 104 66 14 15 7 44 45 36 74 75 33 104 105 67 15 16 8 45 46 36 75 76 43 105 106 68 16 17 9 46 47 36 76 77 44 106 107 69 17 18 10 47 48 37 77 78 44 107 108 70 18 19 11 48 49 37 78 79 44 108 109 71 19 20 12 49 50 17 79 80 44 109 110 72 20 21 13 50 51 25 80 81 44 110 111 73 21 22 14 51 52 38 81 82 44 111 112 74 22 23 15 52 53 38 82 83 45 112 113 75 23 24 16 53 54 38 83 84 46 113 114 76 24 25 17 54 55 38 84 85 47 114 115 77 25 26 18 55 56 39 85 86 48 115 116 78 26 27 19 56 57 39 86 87 49 116 117 79 27 28 20 57 58 39 87 88 50 117 118 80 28 29 21 58 59 40 88 89 51 118 119 81 29 30 21 59 60 40 89 90 52 119 120 82 30 31 22 60 61 40 90 91 53
Table 3: Edge and node specifications of layout network B3
Table 4 gives the layout of the nodes and edges of network L1 by listing all edges with corre-sponding nodes.
Edge Node Edge Node Edge Node Edge Node
from to from to from to from to
1 1 2 8 8 1 15 15 12 22 22 16 2 1 3 9 9 6 16 16 12 23 23 17 3 4 2 10 10 7 17 17 13 24 24 18 4 5 3 11 11 8 18 18 13 25 25 19 5 4 5 12 12 9 19 19 14 26 26 20 6 6 4 13 13 10 20 20 14 7 7 5 14 14 11 21 21 15
Table 4: Edge and node specifications of layout network L1
- Delivery nodes: 2, 3, - Source nodes: 21-6, - Pipelines: 1-5, - Control valves: 6, 7, 8, 15-20, - Plants: 9, 10, 11, - Compressors: 12, 13, 14, - Wells: 21-26.
The following list gives all restrictions that apply to network L1. Their corresponding values are presented in the online database, following the same order as in the list.
- Maximum on the pressure at node 6, 10, 11, - Minimum on the flow rate at edge 9, 11, - Maximum on the flow rate at edge 9, 11.
Table 5 gives the layout of the nodes and edges of network L2 by listing all edges with corre-sponding nodes.
Edge Node Edge Node Edge Node Edge Node
from to from to from to from to
1 1 2 15 15 8 29 29 19 43 43 27 2 3 2 16 16 8 30 30 19 44 44 28 3 2 4 17 17 9 31 31 20 45 45 29 4 3 5 18 18 10 32 32 20 46 46 30 5 6 4 19 19 11 33 33 21 47 47 31 6 8 5 20 20 12 34 34 21 48 48 32 7 6 7 21 21 13 35 35 22 49 49 33 8 8 7 22 22 14 36 36 22 50 50 34 9 9 1 23 23 15 37 37 23 51 51 35 10 10 2 24 24 16 38 38 23 52 52 36 11 11 3 25 25 17 39 39 24 53 53 37 12 12 3 26 26 17 40 40 24 54 54 38 13 13 6 27 27 18 41 41 25 55 55 39 14 14 6 28 28 18 42 42 26 56 56 40
Table 5: Edge and node specifications of layout network L2
The following list gives the delivery and source node locations (by listing the node numbers) and all equipment types (by listing the edge numbers) of network L2.
- Delivery nodes: 4, 5, 7, - Source nodes: 41-56,
- Pipelines: 2, 4, 6, 7, 8, 11, 14, 15, 16, 18, 25, 26, 29, 30, 31, 32, 33, 34, 35, 36, 39, 40, - Control valves: 1, 3, 5, 12, 21, 27, 28, 37, 38,
- Plants: 9, 10, 13, 20, 24, - Wells: 41-56.
The following list gives all restrictions that apply to network L2. Their corresponding values are presented in the online database, following the same order as in the list.
- Maximum on the flow rate at edge 9, 10, 13, 20, - Minimum on the flow rate at edge 20, 9, 10, 13, - Maximum on the pressure at node 3, 15, - Minimum on the pressure at node 19.
Table 6 gives the layout of the nodes and edges of network L3 by listing all edges with corre-sponding nodes.
Edge Node Edge Node Edge Node Edge Node
from to from to from to from to
1 2 1 25 25 17 49 49 25 73 73 47 2 3 1 26 26 18 50 50 38 74 74 48 3 2 4 27 27 19 51 51 38 75 75 49 4 3 7 28 28 20 52 52 38 76 76 50 5 5 4 29 29 21 53 53 39 77 77 51 6 6 5 30 30 21 54 54 39 78 78 52 7 6 7 31 31 22 55 55 39 79 79 53 8 8 2 32 32 22 56 56 40 80 80 54 9 9 4 33 33 22 57 57 40 81 81 55 10 10 5 34 34 23 58 58 40 82 82 56 11 11 6 35 35 23 59 59 41 83 83 57 12 12 6 36 36 24 60 60 41 84 84 58 13 13 7 37 37 24 61 61 42 85 85 59 14 14 7 38 38 26 62 62 42 86 86 60 15 15 3 39 39 26 63 63 29 87 87 61 16 16 8 40 40 27 64 64 30 88 88 62 17 17 9 41 41 27 65 65 31 89 89 63 18 18 10 42 42 28 66 66 32 90 90 64 19 19 11 43 43 34 67 67 33 91 91 65 20 20 12 44 44 35 68 68 43 92 92 66 21 21 13 45 45 36 69 69 44 93 93 67 22 22 14 46 46 37 70 70 44 94 94 68 23 23 15 47 47 37 71 71 45 95 95 69 24 24 16 48 48 17 72 72 46 96 96 70
Table 6: Edge and node specifications of layout network L3
The following list gives the delivery and source node locations (by listing the node numbers) and all equipment types (by listing the edge numbers) of network L3.
- Pipelines: 1, 2, 3, 4, 5, 6, 7, 9, 10, 24, 26-33, 35, 40-70,
- Control valves: 8, 11, 12, 13, 21, 22, 23, 25, 34, 36, 37, 38, 39, 48, - Compressors: 18, 43, 44,
- Plants: 14, 15, 16, 17, 19, 20, - Wells: 71-96.
The following list gives all restrictions that apply to network L3. Their corresponding values are presented in the online database, following the same order as in the list.
