Constrained Derivatives in Natural Gas Pipeline System Optimization
- Orin Flanigan (Arkansas Louisiana Gas Co.)
- Document ID
- Society of Petroleum Engineers
- Journal of Petroleum Technology
- Publication Date
- May 1972
- Document Type
- Journal Paper
- 549 - 556
- 1972. Society of Petroleum Engineers
- 4.2 Pipelines, Flowlines and Risers, 4.1.6 Compressors, Engines and Turbines, 4.6 Natural Gas
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Here is a promising new method for optimizing both the facilities and the operations of integrated natural gas pipeline systems. The balancing-of-nodes calculation method is used as a system model and the method of constrained derivatives is used in conjunction with this model to optimize the system represented.
Historically, differential calculus has been a prime tool in optimization work. When certain conditions are present, differential calculus can be a simple and powerful tool. These conditions are: powerful tool. These conditions are: 1. The system can be described by a set of continuous analytical equations.
2. The system contains no constraints.
3. The system equations are differentiable. When these (and sometimes other) conditions are met, the classical approach is to take the first derivative of the system equations with respect to certain specified variables. These derivatives are set equal to zero and the resulting equations are solved for their various roots. The second derivatives are then used to evaluate whether the roots are maxima, minima, or points of inflection. A physical representation of a simple unconstrained system is a single hill. Differential calculus will lead us to the top of the hill.
Unfortunately, the range of actual problems to which this procedure can be applied is quite limited. Most problems have at least some constraints; and many systems of interest are severely constrained. Some examples of these constraints are: (1) the pressure must not exceed a certain value; (2) the pressure must not exceed a certain value; (2) the temperature must be between certain limits; (3) limits are imposed on the flow rates. These constraints to the system may be visualized as a series of fences on the hill that usually prevent an explorer from reaching the top of the hill. The optimum or maximum is the highest point on the hill that can be reached without crossing the fence. In this case, the classical differential calculus approach is useless.
In 1965, Wilde developed the mathematics for representing a constrained system in the terms of differential calculus. This work was later amplified by Wilde and Beightler. Ratios of Jacobians are used, and these ratios are called constrained derivatives. Although these constrained derivatives are fairly complex, they may be thought of as somewhat analogous to the simple first derivative in calculus. They are gradients that point the direction to the optimum. These constrained derivatives may be used along with optimization criteria (such as the Kuhn-Tucker conditions) to guide a trial-and-error solution to an optimization problem. For the special case of equality constraints, the Kuhn-Tucker conditions are not required because each constrained derivative becomes zero at the optimum.
Natural gas pipeline systems are examples of nonlinear, equality-constrained systems. The pipeline system flow equations represent the constraints. The system cost equation represents the objective function, which may be either linear or nonlinear. When the method of constrained derivatives is applied to this system, the result is a calculation method that will solve for a local optimum.
For optimization work in general, the concept of state and decision variables is very helpful.
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