We consider the optimization and control of gas-flow in networks of pipelines. The main objective is to maximize the throughput through the pipeline from the providers to the customers by choosing proper valve-and release-configurations as well as compressors in order to maintain the pressure in given bounds.This problem is considered in the context of a multistage mathematical formalization. Namely on the first stage we use the graph model in order to define the "basic regime" for the pressure and flow gas. On the second stage the detailed description of dynamic processes in gas pipeline units based on nonlinear partial hyperbolic equation which are linearized in some neighborhood of the "basic regime" defined on the first stage. In the presented paper the mathematical model and the corresponding optimization problem of gas transport networks are introduced on the basis of the constructive approach in the graph and 2-D system setting.
Graph modelAt the initial planning level the simplest graph model is proposed to express potentially critical flow/pressure values within the given margins of inflows, outflows and setting of active components such as storage capacity gasholders, compressor stations and others in order to satisfy/optimize the demand distributed over different nodes. An approach is based on the assumption that the network consists of several supply points where the gas is injected into the system, several demand points where the gas flows out of system and other intermediate nodes and storage where the gas is rerouted or stockpiled. Pipelines are represented by edges linking the nodes. Let S = (I, U) be a stationary network, where I = 1, 2, ..., n denotes the set of nodes, U denotes the set of edges connecting these nodes. It is convenient to divide the set of nodes I in two subsets IΔ, I= : IΔ ∪ I= = I, IΔ ∩ I= = ∅. The elementsi ∈ IΔ are called the multiplication-nodes (i.e one input and several output). The elements i ∈ I= are called the balance nodes. We assume that each balance node i ∈ I= has a finite number ri of input flows z1, z2, ..., zr i and a finite number si of output flowsz1,z2, ...,zs i such thatwhere ai denotes the intensity or available storage capacity of the node i (the sign of ai are locally characterize the gas offtake/supply i ∈ I= ). The set of edges U are divided in two subsets UΔ = {(i, j) : i ∈ IΔ or j ∈ IΔ}, U * = U \ UΔ(i.e no direct connection to inflow node). Also we will use the following notation: q i the gas flow in the node, i ∈ IΔ; d * i , d * i -the upper and lower network throughput gas capacity in the node i ∈ IΔ; qij : i → j -the gas flow from the node i to the node j; d * ij , d * ij-the upper and lower network throughput gas capacity on the edge i → j, (i, j) ∈ U * . In addition for each edge (i, j) ∈ U * introduce another transformation coefficient aij, (i, j) ∈ U such that the initial gas flow qij coming out of node i is transformed into the new gas flow aijqij coming into node j. Hence, the flow into the gas network is formed by input flows of qi, i ∈ Δ ...
