The most worrying design technology for transmission and pipeline systems is Li Bo, the General Planning Institute of China National Petroleum Corporation, Beijing University of Petroleum (Beijing). Optimal design technology for gas pipeline system. Key technologies used in petroleum planning and design, 2000. By summarizing the three kinds of mathematical programming algorithms used in this field, the dynamic programming method: the constrained derivative method and the generalized reduced gradient method, the various calculations are listed :: Lian :: the scope of use, the process parameters for the natural gas pipeline system Optimal design: Fool's knee: x-tune: calculation method for optimization design parameters of gas pipeline process Composition) and the end state when delivered to the user, the layout of the pipeline system, the output of natural gas and the demand of the user, the optimal design of this pipeline system, the design goal is usually to solve the pipe diameter of each pipe section, the number of compressor stations, The tube length between the two compressor stations and the suction and discharge pressure of the compressor in each compressor station to minimize the total cost (including pipeline investment and operating costs). Constraints usually include equality constraints that reflect the length and flow balance of the pipe section, and inequality constraints that reflect restrictions on the source pressure, supply pressure, and compressor pressure ratio.
In the 1960s, foreign people began to engage in the theory and method research of gas pipeline optimization design. Because the optimization design of gas pipeline system is a very complicated constrained nonlinear optimization problem, and the dimension is very large, Therefore, its research work is to solve some design variables in the problem under the assumption that some variables are known, that is, to carry out optimization research on the local part of the design problem. In these studies, the work of B. Rothfarb, Flanigan and others has made significant progress. The mathematical programming methods used are dynamic programming method, constrained derivative method and generalized reduced gradient method.
The dynamic programming method is to optimize the pressure of each node in a pipe network m, and select the corresponding pipe network element pipeline and compressor from the equipment list through the obtained optimal pressure, so as to minimize the construction and operation costs of the pipe network. This method requires the number and location of gas stations and the length and diameter of each pipe section to be predetermined, and is not suitable for large-scale network systems with many network elements (including pipelines, gas stations, gas storage, etc.). The reason is that there is a dimensional disaster when solving with the dynamic programming method: if the one-dimensional state variable has m values, then for the dimensional problem, the state xt has a value, and each state value must be calculated and stored optimally The value function / (:. The calculation of practical problems that are slightly larger (even = 3) is often unrealistic, and there is currently no general method to overcome the dimensionality disaster in dynamic programming.
In a tree-like natural gas pipeline network system, if the number of nodes in the pipeline network is 1, then there are -1 pipeline sections. If there are 7 pipe diameters to choose from for each pipe section, there will be 71 pipe diameter combinations. The problem to be solved is to select an optimal combination from these pipe diameter combinations to minimize the investment and operating costs of the pipe network. Obviously, enumerations are unrealistic. The key to solving the problem is to find a method that can eliminate those uneconomical diameter combinations without enumeration. B. Rothfarb et al. Have developed a merger technique that can eliminate uneconomic pipe diameter combinations without enumeration, so that the number of possible pipe diameter combinations and the number of nodes is generally linear rather than increasing exponentially. This technology provides an effective means for optimizing the natural gas pipeline network using dynamic programming. Take the example to illustrate the basic principle of this method.
The schematic diagram of the local gas pipeline network) has been determined. According to Panhande's formula: K, the number and position of gas stations in the pipeline network are given in advance, and the design variables are divided into decision variables and state variables. The optimization mathematical model is abbreviated as: there are P decision variables and there are households The constrained derivative, the constrained derivative of the objective function to the fourth decision variable 4 is: the determinant value of the Jacobian matrix of the B constraint equation.
The ratio of 4 to 5 reflects the impact of each decision variable change on the objective function. ) Use Newton-Raphson method to solve the constraint equation.
In matrix form, it can be expressed as: (2) Use the steepest descent method to find the minimum value of the objective function.
The direction of descent is determined as :.
The value of 7 is determined according to experience, but two criteria should be observed: First, each decision variable should be determined; the maximum value of 7 / 7max. For example, if the diameter is used as the decision variable, 7 is usually taken in the range of ~ 25.4mm If the power consumed by the compressor is used as a decision variable, 77 is usually between 0 and 400kW; the second is 7 / the value of the decision variable should not be less than zero. The Lagrangian cubic interpolation and golden section method were used to determine the 77 value.
Constrained derivative method is a classic mathematical programming method. When the optimal design of the process parameters of gas pipelines is used, it is easy to obtain the optimal solution for non-constrained or non-constrained nonlinear programming problems. But this method can only optimize part of the design problem.
Edgar et al. First applied the generalized reduced gradient method to the optimal design of the natural gas transmission network. This technology can simultaneously determine the optimal values ​​of design variables such as the number of compressor stations, the length and diameter of the pipe section between the two compressor stations, and the operating conditions (intake pressure, exhaust pressure) of the compressor in the compressor station. Minimize the investment and operation costs of the pipe network.
A function of the power consumption of the body, Gen P: A regression coefficient related to power; N power consumption of the compressor delivery gas.
Two solution techniques are used for two different situations. One is to use the generalized reduced gradient method when the initial investment of the compressor station is zero, that is, 4 = 0; the other is that there is a fixed initial investment in the compressor station. That is, at 4 o'clock, it is necessary to use the generalized reduced gradient method and the branch and bound method to solve the problem. For those more complicated pipe network systems, such as various branches and rings, the method is also applicable, but the calculation time increases.
Because the construction cost is a one-time investment, the operating cost is a perennial investment, and the funds have time value. Obviously, the unit construction cost and the unit operating cost are not equal. To make the value of the two equal, the construction cost and the operating cost are the same The annual interest rate is converted to the cost of the equipment at the end of the year. The optimized objective function is the total annual conversion cost of the total pipeline network investment: / 3 ~ the annual operating cost of the pipeline network.
Since the operating cost of the pipeline network is mainly based on the power cost of the compressor station, and is relatively stable, the operating cost of the pipeline network can also be approximated by the power consumed by the compressor to deliver gas.
The design variable X is divided into base variables (non-independent variables) and non-base variables Xy (independent variables), and the relationship is determined by constraints. The base variable is represented by a non-base variable and the base variable is eliminated from the objective function to obtain a simplified objective function with the non-base variable as the independent variable, ie =, and then the negative gradient of this function is used to construct a feasible downward direction.
The gradient d // dx of the objective function with respect to non-base variables is called the reduced gradient. The central problem of this algorithm is to use the reduced gradient to construct the search direction.
The generalized reduced gradient method is used to transform this kind of programming problem with non-linear and linear constraints into an unconstrained non-linear programming problem, and then the conjugate gradient method is used to determine the downward direction of the search. Since the conjugate gradient method has quadratic termination (that is, the quadratic function, the algorithm terminates in finite steps), which improves the effectiveness and reliability of the solution. Inequality constraints are satisfied by coordinating step sizes in the iterative process of solving X and 4; the value of I is satisfied, and its iteration formula is: The generalized reduced gradient method has high efficiency in solving the constrained nonlinear natural gas pipeline network planning problem . This algorithm makes it possible to optimize all design variables simultaneously. However, it should be noted that the optimized optimal pipe diameter can only be given in a continuous form. To obtain discrete optimal pipe diameter values, other optimization methods such as branch and bound method and sub-gradient optimization method need to be supplemented.
The optimization design of the process parameters of the natural gas pipeline system is a very complicated nonlinear programming problem. To improve the efficiency and scale of the optimization design, various mathematical programming methods must be applied, especially some new high-efficiency nonlinear programming methods, and It needs to be combined with the actual engineering of the gas pipeline process design.
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