In a power grid system, utility is a measure of the satisfaction of users’ electricity consumption;cost is a monetary value of electricity generated by the supplier. The utility and cost functions represent the satis...In a power grid system, utility is a measure of the satisfaction of users’ electricity consumption;cost is a monetary value of electricity generated by the supplier. The utility and cost functions represent the satisfaction of different users and the supplier. Quadratic utility, logarithmic utility,and quadratic cost functions are widely used in social welfare maximization models of real-time pricing. These functions are not universal;they have to be discussed in detail for individual models. To overcome this problem, a piece-wise linear utility function and a piece-wise linear cost function with general properties are proposed in this paper. By smoothing the piece-wise linear utility and cost functions, a social welfare maximization model can be transformed into a differentiable convex optimization problem. A dual optimization method is used to solve the smoothed model. Through mathematical deduction and numerical simulations, the rationality of the model and the validity of the algorithm are verified as long as the elastic and cost coefficients take appropriate values. Thus, different user types and the supplier can be determined by selecting different elastic and cost coefficients.展开更多
A practical transportation problem for finding the “departure” time at “all source nodes” in order to arrive at “some destination nodes” at specified time for both FIFO (i.e., First In First Out) and Non-FIFO “...A practical transportation problem for finding the “departure” time at “all source nodes” in order to arrive at “some destination nodes” at specified time for both FIFO (i.e., First In First Out) and Non-FIFO “Dynamic ” Networks is considered in this study. Although shortest path (SP) for dynamic networks have been studied/documented by various researchers, contributions from this present work consists of a sparse matrix storage scheme for efficiently storing large scale sparse network’s connectivity, a concept of Time Delay Factor (TDF) combining with a “general piece- wise linear function” to describe the link cost as a function of time for Non-FIFO links’ costs, and Backward Dijkstra SP Algorithm with simple heuristic rules for rejecting unwanted solutions during the backward search algorithm. Both small-scale (academic) networks as well as large- scale (real-life) networks are investigated in this work to explain and validate the proposed dynamic algorithms. Numerical results obtained from this research work have indicated that the newly proposed dynamic algorithm is reliable, and efficient. Based on the numerical results, the calculated departure time at the source node(s), for a given/specified arrival time at the destination node(s), can be non-unique, for some Non-FIFO networks’ connectivity.展开更多
基金Supported by the Natural Science Foundation of China(11171221)
文摘In a power grid system, utility is a measure of the satisfaction of users’ electricity consumption;cost is a monetary value of electricity generated by the supplier. The utility and cost functions represent the satisfaction of different users and the supplier. Quadratic utility, logarithmic utility,and quadratic cost functions are widely used in social welfare maximization models of real-time pricing. These functions are not universal;they have to be discussed in detail for individual models. To overcome this problem, a piece-wise linear utility function and a piece-wise linear cost function with general properties are proposed in this paper. By smoothing the piece-wise linear utility and cost functions, a social welfare maximization model can be transformed into a differentiable convex optimization problem. A dual optimization method is used to solve the smoothed model. Through mathematical deduction and numerical simulations, the rationality of the model and the validity of the algorithm are verified as long as the elastic and cost coefficients take appropriate values. Thus, different user types and the supplier can be determined by selecting different elastic and cost coefficients.
文摘A practical transportation problem for finding the “departure” time at “all source nodes” in order to arrive at “some destination nodes” at specified time for both FIFO (i.e., First In First Out) and Non-FIFO “Dynamic ” Networks is considered in this study. Although shortest path (SP) for dynamic networks have been studied/documented by various researchers, contributions from this present work consists of a sparse matrix storage scheme for efficiently storing large scale sparse network’s connectivity, a concept of Time Delay Factor (TDF) combining with a “general piece- wise linear function” to describe the link cost as a function of time for Non-FIFO links’ costs, and Backward Dijkstra SP Algorithm with simple heuristic rules for rejecting unwanted solutions during the backward search algorithm. Both small-scale (academic) networks as well as large- scale (real-life) networks are investigated in this work to explain and validate the proposed dynamic algorithms. Numerical results obtained from this research work have indicated that the newly proposed dynamic algorithm is reliable, and efficient. Based on the numerical results, the calculated departure time at the source node(s), for a given/specified arrival time at the destination node(s), can be non-unique, for some Non-FIFO networks’ connectivity.
