Large-scale analysis and optimization of non-linear networks is among the most difficult of system analytic problems. “Non-linear” means an input to a network does not result in output from the network that is proportional to the input. Traditionally, linear programming and nonlinear programming methods have been used to solve network optimization problems. Due to their mathematical limitations, however, each of these methods typically is applied to network problems that first are simplified and/or reduced in scale compared to the problem that exists in the actual network. Often, the network problems that are to be analyzed are framed with objectives more limited in number and/or complexity than the multiple objectives that typically exist for network design and operation. In today's complex, high-speed, high-growth and deregulated world, the number of planning and operational goals for networks generally (networks, for example, for electric power generation, transmission, distribution, and use; telecommunications; data generation, transmission, and use; transportation; and shipping, among innumerable other types of networks) has increased considerably. The traditional network optimization approaches are increasingly unequal to the challenges. In general, there are some approaches to solving complex network problems.
A common approach to network analysis and optimization simplifies the optimization problem with a number of assumptions, and hence reduces complexity. One approach uses a piece-wise linear simplification. However, in the context of complex network optimization, most (if not all) objectives are nonlinear problems. A linear method cannot produce accurate solutions because the loss objective cannot be piece-wise linearized accurately. A power system, for example, needs to be optimized to meet a number of conflicting simultaneous objectives, but traditional methods cannot handle multiple objectives. This limitation necessitates application of a multitude of programs to solve a number of similar problems whose solutions really should work in close conjunction with one another. Simplified approaches lead to fast but inflexible, inaccurate, and limited solutions.
Another approach typically recognizes that non-linear networks often are best addressed for analytical and optimization purposes with non-linear mathematics, involving non-linear equations. Special methods are needed to solve nonlinear equations. Current non-linear approaches, however, typically map the original large network problem into another domain where the problem size is considerably smaller than the original, then uses non-linear programming methods capable only of running a problem of that reduced scale. These non-linear programming methods typically use penalty functions to enforce “inequality” constraints. Specifically, system “constraints” that are expressed as “limits” and that fluctuate dynamically within a permissible range in a functioning network system. In effect, these programming methods attempt to draw conclusions that oversimplify the complex relationships within the problem. The penalty factors increment at each iteration of the solution, with the final values being close to infinity, resulting in distorted “problem space” and hence ill-conditioning of the mathematical matrix model. In another such example, in power transmission optimization, the “Dual Augmented Lagrangian Approach for Optimal Power Flow” method does not maintain the sparsity structure of the incidence matrix. This results in large run-times. Such methods also lack a proper technique to determine infeasible cases. These oversimplifying methods tend to produce approximate solutions to complex optimization problems, where the significance of the approximation inaccuracy has significant incremental and cumulative dysfunction/cost implications.
Electric power systems are examples of large highly nonlinear networks. They must be planned and managed to meet system operating constraints under both major and minor changes in network generation, transmission, and distribution system resources, and to meet changing network consumer demand. When power system operators and designers make adjustments to the network power system, they must address multiple concerns. For example, they require a solution that is relatively inexpensive to implement, will not hinder or threaten the performance of the existing network, and is robust. This is an ongoing, expensive task, where poorly understood, ill-planned adjustments result in millions of dollars in revenue loss, diminished power quality, increased congestion and system losses, increased equipment damage, and increased liability.
The U.S. electric system includes over 6,000 generating units, more than 800,000 kilometers of bulk transmission lines, approximately 12,000 major transmission substations, and innumerable lower-voltage distribution transformers. This enormous network is controlled regionally by more than one hundred (100) separate control centers that coordinate responsibilities jointly for the impacts upon real-time network operations. A control center coordinates the operation of bulk power system components and is responsible for operating the power system within a geographic region called a control area. One or more utilities make up a control area. A control center is connected to other control centers with transmission interconnection (tie) lines. Through communications (metering and telemetry), the control center is constantly informed of generating plant output and the system condition of transmission lines, substations (including load demand at each substation), and ties to neighboring systems. A control center uses this information in attempting to optimize power flow across the grid during operations, while simultaneously ensuring reliability and interchange scheduling with other control centers. However, with dramatic increases in demand, even the most efficient transmission (and distribution) grid networks are running out of capacity, thus increasing requirements for improved optimization during network operations and in planning for system upgrades and new system facilities.
As electric power deregulation and market system development efforts continue, and as the number of power transactions increases, new power demand growth and supply requirements are pushing transmission and distribution systems to their limits. These T&D networks cannot be improved to meet new requirements at the rate or with the effectiveness needed to keep pace with new pressures on them. This has resulted in more volatile generation dispatch patterns, increased congestion, reduced operating margins, and significant challenges to system reliability. Increasing requirements of industry and commerce dependent on high-quality power supply are expected to accelerate this trend. The need is increasingly critical to optimize the functions and use of electric power systems, through increased accuracy and greater productivity in system planning and operation.