Optim Eng (2007) 8: 259–275 DOI 10.1007/s11081-007-9018-y
Deterministic global optimization approach to steady-state distribution gas pipeline networks Yue Wu · Kin Keung Lai · Yongjin Liu
Published online: 31 July 2007 © Springer Science+Business Media, LLC 2007
Abstract Natural gas is normally transported through a vast network of pipelines. A pipeline network is generally established either to transmit gas at high pressure from coastal supplies to regional demand points (transmission network) or to distribute gas to consumers at low pressure from the regional demand points (distribution network). In this study, the distribution network is considered. The distribution network differs from the transmission one in a number of ways. Pipes involved in a distribution network are often much smaller and the network is simpler, having no valves, compressors or nozzles. In this paper, we propose the problem of minimizing the cost of pipelines incurred by driving the gas in a distribute non-linear network under steady-state assumptions. In particular, the decision variables include the length of the pipes’ diameter, pressure drops at each node of the network, and mass flow rate at each pipeline leg. We establish a mathematical optimization model of this problem, and then present a global approach, which is based on the GOP primal-relaxed dual decomposition method presented by Visweswaran and Floudas (Global optimization in engineering design. Kluwer book series in nonconvex optimization and its applications. Kluwer, Netherlands, 1996), to the optimization model. Finally, results from application of the approach to data from gas company are presented.
Y. Wu School of Management, University of Southampton Highfield, Southampton, UK K.K. Lai () Department of Management Sciences, City University of Hong Kong, Hong Kong, Hong Kong e-mail:
[email protected] K.K. Lai College of Business Administration, Hunan University, Hunan, China Y. Liu Department of Applied Mathematics, Dalian University of Technology, Dalian, China
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Keywords Distribution network · Gas pipeline network · Nonlinear network optimization · Steady-state 1 Introduction The consumption of natural gas in Hong Kong has increased steadily in recent years. Its clean-combustion characteristics and its ease of distribution at low pressures justify its wide use. As the demand for natural gas increases, gas pipeline networks have evolved over the decades into a complex system. In Hong Kong, about 8940 km of pipelines are laid under and over ground. According to the company, the system claims to be within economic reach of an estimated 80% of Hong Kong homes. Therefore, new challenges are being imposed on decision makers. Efforts are needed to intensify the development of an appropriate network capable of reducing the costs while still satisfying the demand for natural gas. A mathematical model capable of simulating pressure and the mass flow rate of the pipeline network under different operating conditions would greatly facilitate the evaluation of new strategies and plans. It is estimated that the global optimization of operations can save at least 15% of the network cost. Hence, the problem of minimizing the network cost is of tremendous importance. As natural gas is generally produced from gas wells, it must be delivered through a pipeline network of buried and overground pipelines to end-users. These pipeline networks are established in order either to transmit gas at high pressure from coastal supplies to regional demand points (transmission network) or to distribute gas to consumers at low pressure from the regional demand points (distribution network). There is a wide range of literature on this subject which may be consulted by readers wishing to obtain more details (Engl 1996; Wong and Larson 1968; Lall and Percell 1990; Carter 1998; Wu et al. 2000). The distribution network differs from the transmission network in a number of ways. The distribution system consists of a network of relatively small-diameter pipes, operating at low and medium pressure. Also the distribution network is simpler, having no valves, compressors or nozzles. The main problem in the design process of gas distribution networks, once the topology has been defined by the technical team, is searching for the features that its pipelines and pumps should possess in order to meet the nodes’ flow and pressure requirements. Furthermore, we are interested in finding optimal pipeline features that ensure meeting the requirements, thus defining the cheapest network. There are two important approaches, numerical simulation and optimization, to gas networks. The main purpose of simulation is to determine the actual behavior of a gas network under given conditions. Simulation basically answers the question: what happens if we run our grid with given control variables and known boundary flows. Typical questions like finding a control regime which achieves several target values (goal seeking across the net) usually require a series of simulation runs by an expert users who are familiar with the network. Finding the best regime may even take a large number of runs. There are two disadvantages of numerical simulation. First, it cannot ensure that the cost achieved is minimal. Second, due to the fact that adjustment of pipe diameters merely depends on users’ experience, it is not surprising to find that different users always arrive at different decisions for the same problem.
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The optimization approach differs from numerical simulation. The aim of optimization is to find the best set of control variables within a single optimization run, possibly executed by less experienced users. The searching or experimenting process must be substituted with more sophisticated algorithms. Moreover, optimization generally works with simplified models, but it yields optimum results where limits or certain target values will be achieved automatically if they are defined as optimization problem constraints. To the best of our knowledge, there has been a great deal of work on the optimization approach to gas pipe distribution networks. Rothfarb et al. (1970) investigate three problems in the design and expansion of offshore natural gas pipeline networks. Suggestions were for optimizing pipeline diameters for a fixed tree structure by trying out a variety of diameter assignments. However, this optimization approach can only deal with small problems. As the number of possible diameter assignments grows, it becomes more difficult to solve the problems. Recently, Mawengkang and Murtagh (1986) use MINOS, a large scale optimization software, to solve the distribution problem. Because of the nonconvexity of the model, a local minimum is often obtained. More recently, Castillo and González (1998) study the optimal features in a distribution network and find the most economic solution using genetic algorithms. The purpose of this paper is to provide an in-depth study of the underlying mathematical structure of the distribution gas pipe networks. Then, based on this study, we present a mathematical model of the distribution gas pipe networks. It is noted that the primal model is a nonsmooth and nonconvex problem which is very difficult to solve. By introducing new variables, we can convert the original model to a nonconvex quadratically constrained problem and derive a global optimization approach which is based on GOP primal-relaxed dual decomposition method presented by Visweswaran and Floudas (1996) and then make remarks on the implementation. Finally, we present numerical results. The rest of this paper is organized as follows. In Sect. 2, the steady-state distribution gas pipe network is investigated, and a mathematical model of this problem is presented. We provide a global optimization approach to this problem in Sect. 3. In Sect. 4, some numerical results are given. We end this paper in Sect. 5 with our conclusions and directions for future work.
2 Model formulation In this section, we focus on the pipes cost minimization problem. We first present the modeling assumptions and the problem description in Sect. 2.1. In Sect. 2.2, we provide the mathematical model of the optimization problem. 2.1 Problem description The network cost concerned here is the construction cost which includes the investment cost, labor cost and material cost. Mostly, the cost components are proportional to the diameter size. The smaller the diameter is, the lower the cost. Therefore, the
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Fig. 1 Sudden squeezing or enlargement of pipes
cost of realizing the network is a linear function of the lengths with different diameters. However, the pipes’ diameter cannot be too small since there are energy head constraints at each node. The energy heads at each node must satisfy some upper and lower limits. Hence, the problem is to minimize a linear function of the lengths with different diameters subject to some constraints. In the flow of compressible fluids there can be significant changes in fluid density: this implies significant variation of pressure and temperature as well. Therefore, the analysis of such systems involves four equations: the equation of state, continuity, momentum and energy. This makes the analysis rather complicated. To simplify it, it is usually assumed that the flow is reversible and adiabatic, these conditions implying isentropic flow. Furthermore, it is often assumed that the fluid is a perfect gas with constant specific heat (average value). Moreover, it is assumed that the following conditions are satisfied in a gas pipe distribution network: 1. The pipe-diameter configuration must follow a descending order, i.e., sudden squeezing or enlargement of pipes (see Fig. 1) is not allowed. 2. There are no compressors, valves, and nozzles in the network. 3. The network is in a steady state, i.e., velocity, pressure and the cross-section of the stream may vary from point to point, but do not vary with time. 4. All pipes in a network are of the same type of material, such as all pipes are made of steel or ductile iron. 2.2 Mathematical model The objective function of the problem is a linear function of the lengths with different diameters. This problem involves the following constraints: (i) mass flow balance equation at each node; (ii) gas flow equation through each pipe; (iii) pressure limits in constraints at each node. The first two are also referred to the steady-state network flow equations. Note that while the mass flow balance equations are linear, the pipe flow equations are nonlinear: for details, refer to Furey (1993) and Wu (1998). When we take into account the fact that a change in the flow direction of the gas stream may take place in the network, the pipe flow equation takes the following form: p12 − p22 = cLd −5 q|q|α , where p1 and p2 are the upstream and downstream pressure of the pipe respectively, q is the mass flow rate through the pipe, d is the inside diameter of the pipe, L is
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Fig. 2 An example network with m = 8 nodes and n = 7 pipes
the pipe length, α is a constant, α ≈ 1, the pipe resistance c is a positive quantity depending on the pipe’s physical attributes, which is give by c = Kf, where K = (1.3305 × 105 )ZSg T . These parameters refer to the following: Z gas compressibility factor Sg gas specific gravity T average temperature (constant) f frictional factor The steady-state network flow equations can be stated in a very concise form by using incidence matrices. Let us consider a network with m nodes and n pipes. Each pipe is assigned a direction which may or may not coincide with gas flow through the pipe. Let A be the m × n matrix whose elements are given as follows ⎧ if j th pipe comes out from i th node, ⎨ 1, aij = −1, if j th pipe goes into from i th node, ⎩ 0, otherwise. The distribution gas pipe network example with m = 8 nodes and n = 7 pipes is shown in Fig. 2. The matrix A for this example is given by ⎞ ⎛ 1 0 0 0 0 0 0 1 1 0 0 0 ⎟ ⎜ −1 1 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 −1 0 ⎟ ⎜ 0 0 −1 0 1 1 ⎟ ⎜ 0 A=⎜ ⎟. 0 0 0 −1 0 0 ⎟ ⎜ 0 ⎟ ⎜ 0 −1 0 0 0 0 ⎟ ⎜ 0 ⎠ ⎝ 0 0 0 0 0 −1 0 0 0 0 0 0 0 −1 The matrix has some special characteristics. For instance, each row in matrix A, corresponds to a node, and each column corresponds to a pipe in the network. In addition, each column contains exactly two nonzero elements, one is 1 and the other −1,
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Fig. 3 Segment length used in pipe j with diameter size dl
which correspond to the pipe’s two end nodes. In fact, A is the network’s incidence matrix. Hence, the matrix A is a totally unimodular matrix. Let q = (q1 , . . . , qn )T be the mass flow rate through the pipes and p = (p1 , . . . , pm )T the pressure vector, where pi , i = 1, . . . , m, is the pressure at the i th node. Let s = (s1 , . . . , sm )T be the source vector, where si , i = 1, . . . , m, is the source at the i th node. Component si is positive if the node is a supply node, negative if it is a delivery node, and zero otherwise. Without loss of generality, it is assumed that the sum of the sources is equal to zero, i.e., m
si = 0.
i=1
Suppose the pipe’s diameters are chosen from among a discrete number set, Ω = {d1 , . . . , dk }, with d1 > d2 > · · · > dk . According to our assumptions pipe-diameter configuration must follow a descending order and sudden squeezing or enlargement of pipes is not allowed. We can divide a pipe into segments so that each segment corresponds to one diameter size: the decomposition is shown in Fig. 3 for pipe j . It is noted that there is no segment of diameter dl when Lj l = 0. Assume that the length of pipe Lj is given, then the network flow equations and the length equality constraints can now be stated as follows: Aq = s, ¯ q), AT p 2 = g(L, B L¯ = L,
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2 )T , L = (L , . . . , L )T , g(L, ¯ q) = (g1 (L¯ 1 , q1 ), . . . , where p 2 = (p12 , . . . , pm 1 n T T T T ¯ ¯ ¯ ¯ ¯ gn (Ln , qn )) , L = (L1 , . . . , Ln ) , Li = (Li1 , Li2 , . . . , Lik )T , Lik is the segment length for pipe i with k-th diameter size, and gi (L¯ i , qi ) = c( kl=1 Lil dl−5 )qi |qi |α , B is a m × (mk) pipe-length incident matrix, which is shown as follows ⎛ T ⎞ e 0 ··· 0 ⎜ 0 eT · · · 0 ⎟ , B =⎜ .. .. .. ⎟ ⎝ ... . . . ⎠
0
0
· · · eT
where e is a vector with k dimension whose elements are 1. Now given the bounds p L , p U of pressures at every node, the problem is to determine the segment length for pipe with different diameter sizes, pressure vector p and the flow vector q so that the total pipe costs are minimized. The objective function is a linear function of the pipe lengths with different diameters. The model is stated as follows: min
n
hT L¯ i
i=1
s.t. Aq = s, ¯ q), AT p 2 = g(L,
(1)
B L¯ = L, pL ≤ p ≤ pU ,
L¯ ≥ 0
where h = (h1 , . . . , hk )T is a vector with hi being the consolidated cost per unit length for the k-th diameter size. Remark 1 We emphasize that in model (1) the input parameters involve the length of each pipe(Li ), the cost per unit length for each diameter(hi ), source and user node(si ), the bounds of pressure at each node(piL , piU ), while the other variables are decision variables. Remark 2 It can be seen that the functions gi (L¯ i , qi ) = c( kl=1 Lil dl−5 )qi |qi |α are nonsmooth functions. In general, it can be very difficult to solve a problem with these characteristics. What we do in this work is to exploit the special structure of problem and approximately transform it into optimization problem with quadratic constraints that allow us to develop an optimization approach. 3 Global optimization approach After establishing problem (1), we approach the solution to this problem. In Sect. 3.1, we approximately transform problem (1) into an optimization model with quadratic constraints. In Sect. 3.2, we present an approach, which is based on GOP primalrelaxed dual decomposition method presented by Visweswaran and Floudas (1996), to the optimization model.
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3.1 Model transformation As pointed out in Sect. 2.1, problem (1) is a nonsmooth problem. It is also a nonconvex program where the nonconvexities are due to pressure drop constraints. It is very difficult to solve these problems. As a result, we approximately transform the problem into an easier model. Denote qi+ = max{0, qi }, qi− = max{0, −qi }, then we have qi+ qi− = 0, qi = + 2 = ((q + )2 , . . . , qi − qi− , and |qi | = qi+ + qi− . Also denote q+ = (q1+ , . . . , qn+ )T , q+ 1 2 = ((q − )2 , . . . , (q − )2 )T . Note that (qn+ )2 )T , q− = (q1− , . . . , qn− )T , and q− n 1 gi (L¯ i , qi ) ≈ c( kl=1 Lil dl−5 )qi |qi |, hence problem (1) is approximately equivalent to the following model. min
n
hT L¯ i
i=1
s.t. Aq+ − Aq− = s, ¯ q+ , q− ), ˜ L, AT p 2 = g( B L¯ = L,
(2)
T q− = 0, q+
pL ≤ p ≤ pU ,
¯ q+ , q− ≥ 0 L,
¯ q+ , q− ) = (g˜ 1 (L¯ 1 , q + , q − ), . . . , g˜ n (L¯ n , qn+ , qn− ))T , g˜ i (L¯ i , q + , q − ) = where g( ˜ L, i i k 1 1 c( l=1 Lil dl−5 )qi |qi | = c( kl=1 Lil dl−5 )((qi+ )2 − (qi− )2 ). Remark 3 After introducing new transformation variables, we find that problem (2) has m + 2n + kn variables and m + 2n + 1 constraints except for lower and upper bounds, while problem (1) has m + n + kn variables and m + 2n constraints except for lower and upper bounds. Problem (2) has only n more variables and one more constraint than problem (1). By introducing new variables in problem (2), we can transform it into an optimization model with quadratic constraints. By introducing u+ , u− , and adding constraints 2 = 0, u− − q 2 = 0, the problem can be converted into a form as shown below: u+ − q+ − min
n
hT L¯ i
i=1
s.t. Aq+ − Aq− = s, ¯ u+ , u− ), ˜ L, AT p 2 = g( B L¯ = L, T q− = 0, q+
(3)
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2 u+ − q+ = 0, 2 u− − q− = 0,
pL ≤ p ≤ pU ,
¯ q+ , q− , u+ , u− ≥ 0 L,
+ − − T ¯ u+ , u− ) = (g˜ 1 (L¯ 1 , u+ , u− ), . . . , g˜ n (L¯ n , u+ ¯ where g( ˜ L, n , un )) , g˜ i (Li , ui , ui ) = 1 1 k − c( l=1 Lil dl−5 ) × (u+ i − ui ), i = 1, . . . , n.
Remark 4 After introducing new transformation variables in problem (2), we find that problem (3) is a nonconvex optimization problem with quadratic constraints where the nonconvexities are due to the pressure drop constraints and the presence of the complementarity constraints q + q − = 0. 3.2 Algorithm Within this section, a novel global optimization approach for the solution of the gas pipe distribution network problem is presented. The approach is based on the GOP primal-relaxed dual decomposition method presented by Visweswaran and Floudas (1996). First, the basic concepts of the GOP algorithm are given, then the approach to this particular problem is discussed. 3.2.1 Basic concepts As problem (3) is a nonconvex program, this leads to the existence of multiple local minima. Here, we present a global GOP optimization approach to this problem. The GOP optimization method guarantees convergence to an -global minimum for a kind of specific structure program in which the objective function and constraints must satisfy some conditions. The approach generates a nondecreasing sequence of lower bounds and a nonincreasing sequence of upper bounds on the global solution. The sequence of upper bounds on the global solution is obtained by solving the primal problem from different starting points. Lower bounds are generated by solving relaxed dual problems. To develop the GOP approach to this problem, we introduce other new variables. Therefore, by introducing new variables q¯+ , q¯− and adding q+ − q¯+ = 0, q− − q¯− = 0 to constraints in problem (3), this problem can be further converted to the following problem with a linear objective function, linear and bilinear constraints: min
n
hT L¯ i
i=1
s.t. Aq+ − Aq− = s, ¯ u+ , u− ), AT p¯ = g( ˜ L, B L¯ = L, T q+ q− = 0,
(4)
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q+ − q¯+ = 0, q− − q¯− = 0, + + u+ i − qi q¯i = 0,
i = 1, . . . , n,
− − u− i − qi q¯i = 0,
i = 1, . . . , n,
p¯ L ≤ p¯ ≤ p¯ U ,
¯ q+ , q− , u+ , u− ≥ 0 L,
2 )T , p¯ L = ((p L )2 , . . . , (p L )2 )T , p¯ U = ((p U )2 , . . . , (p U )2 )T . where p¯ = (p12 , . . . , pm m m 1 1
Remark 5 Problem (4) has m + 2n + kn + 2n + 2n = m + (k + 6)n variables and m + 2n + 1 + 2n + 2n = m + 6n + 1 constraints. Problem (4) has some specific characters. For instance, its objective function is a linear function and its constraints are linear or bilinear. In what follows, we present an approach to this problem. First, we introduce some definitions used in the algorithm. T , q¯ T )T ∈ R N , y ≡ (uT , uT , q T , q¯ T )T ∈ R M , where N ≡ Denote x ≡ (L¯ T , p¯ T , q− + + − + − ¯ q− , q¯+ ≥ 0}, Y ≡ {y | y ≥ 0}. (k + 2)n + m, M ≡ 4n, and X ≡ {x | p¯ L ≤ p¯ ≤ p¯ U , L, Definition 1 (Primal problem) Fixing the y variables (say to y k ∈ Y ) in problem (4) results in a subproblem, which is called a primal problem. Remark 6 The primal problem of problem (4) is a linear programming, which is easier to solve. The solution of this problem also provides a set of multipliers λk . Definition 2 (Relaxed dual problem) The relaxed dual problem is shown as follows:
min μB | μB ≥ min L(x, y, λk ) , (5) y∈Y,μB
x∈X
where μB is a scalar, and the function L(x, y, λ) is the Lagrange function for problem (4). The multipliers λk are obtained by solving a primal problem. Definition 3 (Connected variables) Define the vector g k (y) as follows: g k (y) = ∇x L(x, y, λk )|x k
and gik (y) = ∇xi L(x, y, λk )|x k ,
where xi is the i th element of the vector x, i = 1, . . . , N . Then, it can be seen that for every fixed y, the Lagrange function at x k is given by L(x, y, λk ) = L(x k , y, λk ) +
N
gik (y)(xi − xik ).
i=1
The KKT optimality conditions for the primal problem require that gik (y k ) = ∇xi L(x, y k , λk )|x=x k = 0,
∀i = 1, . . . , N.
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Then, for y = y k , every variable xi for which gik (y) is a function of y is defined to be a connected variable. The set of all such connected variables is given by Ick , that is, Ick := {i : gik (y) is a function of y}. Remark 7 Because the objective function is linear and the constraints are linear or bilinear, L(x, y, λk ) is linear in x for fixed y, and L(x, y, λk ) is linear in y for fixed x. The Lagrange function is shown as follows: L(x, y, λk ) =
n
T hT L¯ i + λk1 (Aq+ − Aq− − s)
i=1 T ¯ u+ , u− ) + λk T (B L¯ − L) ˜ L, + λk2 (AT p¯ − g( 3 T + λk4 (q+ q− ) +
n
+ + λk5i (u+ i − qi q¯i )
i=1
+
n
− − k λk6i (u− i − qi q¯i ) + λ7 (q+ − q¯+ ) T
i=1 T
+ λk8 (q− − q¯− ). In this case, gik (y) can be easily obtained. 3.2.2 GOP algorithm First, we present some properties which are the basis of the GOP algorithm. The proof can be found in Floudas (2000). Property 1 The optimal solution of minx∈X L(x, y, λk ) depends only on those xi , for which gik (y) is a function of y. Remark 8 This property is important from the computational point of view. This property suggests that for such problems, the computational effort required to obtain a global solution is not determined by the number of variables in the problem, but rather by the number of connected variables, which can result in reductions by several orders of magnitude in the time needed to solve the problem. Property 2 For any given y k and for all i ∈ Ick , let xiL and xiU be the lower and upper bounds on the variable xi . Also, let Bj be a combination of lower/upper bounds of these connected variables x and let CB be the set of all such combinations. Then, ⎫ ⎧ Bj k ⎪ ⎪ ⎪ ⎪L(x , y, λ ), ⎬ ⎨ Bj k L k min L(x, y, λ ) = min with gi (y) ≥ 0 ∀xi = xi , . x ⎪ ⎪ ⎪ ⎪ B ⎩ g k (y) ≤ 0 ∀x j = x U ⎭ i
i
i
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Next, based on the above properties, the GOP algorithm will be stated in the following. The terminology used in the algorithm is presented as follows. Given a node j in the branch and bound tree, Pj is its parent node, and Ij is the iteration at which node j is created. Rj is the set of constraints defining the region corresponding to node j . At any point, N¯ denotes the total number of nodes in the tree, and C denotes the current node. In addition, F denotes the set of iterations with a feasible primal problem, while I denotes the set of iterations when the primal problem is infeasible. GOP Algorithm Step 0: Initialization (a) (b) (c) (d) (e)
Read in the data for the problem, including tolerance for convergence . Define initial upper and lower bounds (f U , f L ) on the global optimum. Generate initial bounds for the x variables x L and x U . Choose a starting point y 1 for the algorithm. Set K = 1, C = PC = 1, N¯ = 1.
Step 1: Selection of current region (a) If K = 1, Set RC = ∅ and goto Step 2. (b) If K ≥ 2, set RC = ∅, m = C. Then: (i) Add the Lagrange function and qualifying constraints for node m to RC . (ii) Set m = Pm . If m = 1, then goto Step 2. (iii) Repeat steps (i) and (ii). Step 2: Primal problem (a) Solve the primal problem to give P K (y K ). (i) If feasible, set F = F ∪ K and update f U = min[f U , P K (y K )]. (ii) If infeasible, solve a relaxed primal problem. Set I = I ∪ K. (b) Store y K and λK . Step 3: Determination of current partitions (a) Generate the current Lagrange function LK (x, y, λK ). (b) Determine the set of connected variables IcK and the corresponding partial derivatives giK (y)(i = 1, . . . , IcK ) of the current Lagrange function. (c) For each connected variable, determine tight lower and upper bounds xiL and xiU in the current region y ∈ RC . Otherwise, use the original bounds. (d) Evaluate lower and upper bounds on giK (y) in the region y ∈ RC . (i) If giK (y) ≥ 0 ∀y ∈ RC , set xiB = xiL in the current Lagrange function, and remove i from the set IiK . (ii) If giK (y) ≤ 0 ∀y ∈ RC , set xiB = xiU in the current Lagrange function, and remove i from the set IiK . Step 4: Relaxed dual problem (a) Select a combination of the bounds Bl of the connected variables, say Bl = B1 .
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(b) Find the solution (μ∗B , y ∗ ) to the following relaxed dual subproblem: min μB
y∈Y,μB
s.t. μB ≥ LK (x Bl , y, λK ), B
giK (y) ≥ 0 if xi l = xiL ,
(6)
B
giK (y) ≤ 0 if xi l = xiU , (y, μB ) ∈ RC . (i) If μ∗B < f U − , set j = N¯ + 1, P (j ) = C, N¯ = N¯ + 1, and store the solution j
in μB , yj .
(ii) If μ∗B > f U − , fathom the solution. (c) Select a new combination of bounds, say Bl = B2 , for the connected variables. (d) Repeat steps (b) and (c) until all the combinations of bounds for connected variables have been considered. Step 5: Selecting a new lower bound j p p Select the infimum of all μB , say μB . Set C = p, y K+1 = y p , f L = μB . Step 6: Check for convergence U L | < , stop; Otherwise, set K = K + 1 and return to step 1. If | f f−f U
4 Numerical results In this section, we text the numerical performance of the proposed algorithm. To achieve this, this paper considers two examples. The first one is relatively small and the second one is a larger and more complex network. In our numerical experiments, the following data are used through all the examples. Gas compressibility factor: Z = 0.95; Gas specific gravity: Sg = 0.6248; Average temperature: T = 459.67 + 60; Friction factor: f = 0.0085. 4.1 Example 1 The first example is a 6-node, 5-pipe network as shown in Fig. 4. There is only one source node (node 1) and one user node (node 6) with s1 = 600 and s6 = −600, respectively. For all other nodes si = 0, i = 2, 3, 4, 5. The range pressure of each node is [600, 800](Pa). For each pipe, length=10 (miles). We further assume that there are six available diameters and their costs are shown in Table 1.
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Fig. 4 Example 1 network
Table 1 Cost of different diameters in Example 1
Table 2 Example 1 results
K
Diameter (mile)
1
8
4500
2
7.5
3900
3
7
3700
4
6.5
2900
5
6
2100
6
5.5
1900
Pipe
Cost ($)/mile
Diameter
Upstream
Downstream
Flow rate
(mile)
pressure (Pa)
pressure (Pa)
(SCMH)
1–2
8.5
800
764
600
2–3
8.5
764
727
600
3–4
8.5
727
687
600
4–5
8.5
687
645
600
5–6
8.5
645
600
600
In our Matlab implementation of algorithm shown in Sect. 3, where convergence of our program is set with the convergence tolerance = 10−6 , we solved the example. The results are displayed in Table 2. As can be seen, all pipes consist of only segment 8.5 miles in diameter and the total pipe cost of the network is $225,000. 4.2 Example 2 The second example is a larger and more complex as shown in Fig. 5. The network consists of only one source node (No. 1), 13 user nodes (No. 12–14) and 14 pipes. We further assume that source node No. 1 is 30 (kPa) and minimum pressure at all nodes is 2.5 (kPa). In this case, we suppose there are six available diameters, cost per meter of these diameters is shown in Table 3, length of each pipe is displayed in Table 4. The source supply rate is 4200 SCMH and the demand rate for each user is given in Table 5. After applying the algorithm in Matlab, where convergence of our program is set with the convergence tolerance = 10−6 , we obtain the solution of the configuration displayed in Fig. 4. The direction of gas flow is shown in Fig. 5, and the solution including pressure, flow rate of each node, is displayed in Table 3. Furthermore, it is known that the total cost of the gas pipeline network shown in Fig. 4 is $2,996,718. From the results shown in Table 6, we can see that only two pipes (2–4 and 10–11) consist of two diameter sizes. For No. 2–4, the length of diameter 300 mm is 16 m and the length of diameter 200 mm is 4 m; for No. 10–11, the length of diameter
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Fig. 5 Example 2 network
Table 3 Cost of different diameters in Example 2
K
Diameter (mm)
Cost ($)/meter
1
400
4568
2
300
3482
3
250
3896
4
200
2901
5
150
2167
6
100
1905
Table 4 Length of each pipe in Example 2 Pipe
1–2
2–3
2–4
3–6
4–5
5–7
6–8
Length (m)
25
90
20
180
130
110
55
Pipe
7–14
8–9
9–10
10–11
11–12
14–13
13–9
Length (m)
120
70
20
70
60
90
120
Table 5 Demand rate of each user in Example 2 2
3
4
5
6
7
8
9
10
11
12
13
14
0
460
460
460
350
460
350
0
350
350
350
0
460
200 mm is 50 m and the length of diameter 150 mm is 20 m. Other pipes in the network consist of only one diameter size.
5 Conclusions The purpose of the paper is to propose a global approach to steady-state distribution gas pipeline networks. First, we present a study of the mathematical structure of pipe configuration in steady-state natural gas distribution networks and establish the pipe
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Fig. 6 The direction of gas flow
Table 6 Example 2 results Pipe
Diameter
Upstream
Downstream
Flow rate
(mm)
pressure (kPa)
pressure (kPa)
(SCMH)
1–2
300
30
29.46
4200
2–3
200
29.46
25.86
1945
2–4
300–16, 200–4
29.46
29.15
2255
3–6
200
25.86
20.66
1485
4–5
200
29.15
24.53
1795
5–7
200
24.53
21.93
1335 1135
6–8
150
20.66
15.68
7–14
200
21.93
15.38
875
8–9
150
15.68
11.37
785 1120
9–10
200
11.37
10.55
10–11
200–50, 150–20
10.55
7.04
850
11–12
150
7.04
2.5
500
14–13
150
15.38
13.81
415
13–9
150
13.81
11.37
415
cost minimization model. We have highlighted that this is difficult problem to solve because it is a nonsmooth and nonconvex problem. By introducing new variables and adding constraints, we can convert the primal problem to a quadratic model. The proposed approach is tested on two examples of gas pipeline network. The results are also given. Further research needs to be conducted liking into reliability of the steady-state gas network. Also, producing a transient model is another important future project.
References Carter RG (1998) Pipeline optimization: dynamic programming after 30 years. In: Proceedings of the 30th PSIG annual meeting, Denver, CO
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