Real world problems have to do with relations and constraints. By the simulation of gas networks the relations are nodal balance equations and pressure drop equations. The constraints (minimal and maximal values) are on pressure and flow. If there is equipment like reducers and compressors there are constraints on flow, pressure difference and power. If there are constraints, there must be an objective. There can be a preferred supply flows, and one can have the objective to operate a network with the least amount of compressor power. If the supplies have gas of different quality, there may be constraints on quality on certain nodes. Putting relations and constraints in one model is very powerful. Using this model over the years has shown great benefit to the planning department.
Simulation of a gas network involves by nature a set of equations and a set of constraints. In the early days one started with solving equations. The constraints were translated to equations, where a priory a choice was made which constraint was constraining. With modern tools (solvers) these a priory choices don't have to be made. This will be made clear in this paper. In the next section the primary relations of a network will be shown. Then the traditional approach to solve them. Some problems will be found which brings us to the section piecewise linear programming. Constraints and Objectives are involved for the different pieces of equipment, which will be described. Given the iterative nature some words will be spend on sequential linear programming and then some final words in the conclusion.
Given a network with all types of elements. From the pipelines, the relations are already known. An offtake is quite simple, the flow of this branch equals some constant value. For supplies, reducers and compressors it is more complicated. The equation that is connected with this device is dependent of the control mode. Table 1 shows the control modes. Having chosen for each device on forehand its control mode, there is an equation for each device. However it should be done with care. You should not have conflicting control modes. Here follows some examples of conflicting control modes: ² two devices are connected at one point and both control the pressure of that point. ² two devices are connected to two pressure controlled points and there is an open valve between that two points ² two devices are connected to two pressure controlled points and there is a compression ratio controlled compressor between that two points ² a compressor and a reducer are in series and are both flow controlled In the network the pressures are unknown and the flows are unknown. For each node there is an equation and for each branch there is an equation, so the number of unknowns is equal to the number of equations.