S.M. Shahidehpour

Short-term generation scheduling with transmission and environmental constraints using an augmented Lagrangian relaxation

This paper proposes a new approach based on augmented Lagrangian relaxation for short term generation scheduling problems with transmission and environmental constraints. In this method, the power system constraints, e.g. load demand, spinning reserve, transmission capacity and environmental constraints, are relaxed by using Lagrangian multipliers, and quadratic penalty terms associated with power system load demand balance are added to the Lagrangian objective function. Then, the decomposition and coordination technique is used, and nonseparable quadratic penalty terms are replaced by linearization around the solution obtained from the previous iteration. In order to improve the convergence property, the exactly convex quadratic terms of decision variables are added to the objective function as strongly convex, differentiable and separable auxiliary functions. The overall problem is decomposed into N subproblems, multipliers and penalty coefficients are updated in the dual problem and power system constraints are satisfied iteratively. The corresponding unit commitment subproblems are solved by dynamic programming, and the economic dispatch with transmission and environmental constraints is solved by an efficient network flow programming algorithm. The augmented Lagrangian relaxation method enhanced by the decomposition and coordination techniques avoids oscillations associated with piece-wise linear cost functions. Numerical results indicate that the proposed approach is fast and efficient in dealing with numerous power system constraints. >

Effects of ramp-rate limits on unit commitment and economic dispatch

The authors propose an algorithm to consider the ramp characteristics in starting up and shutting down the generating units as well as increasing and decreasing power generation. They consider the inclusion of ramping constraints in both unit commitment and economic dispatch. Since implementing ramp-rate constraints is a dynamic process, dynamic programming (DP) is a proper tool to treat this problem. To overcome the computational expense which is the main drawback of DP, this study initially employs artificial intelligence techniques to produce a unit commitment schedule which satisfies all system and unit operation constraints except unit ramp-rate limits. Then, a dynamic procedure is used to consider the ramp properties as units are started up and shut down. According to this adjustment, maximum generating capabilities of units will change the unit operation status instead of following a step function. Finally, a dynamic dispatch procedure is adopted to obtain a suitable power allocation which incorporates the unit generating capability information given by unit commitment and unit ramping constraints, as well as the economical considerations. Two examples are presented to demonstrate the efficiency of the method. >

Transaction analysis in deregulated power systems using game theory

The electric power industry is in transition to a deregulated marketplace for power transactions. In this environment, all power transactions are made based on price rather than cost. A regional power pool is noted as the most straightforward path to a deregulated electricity marketplace. However, many questions remain unanswered regarding the economics of power pool participation. In a deregulated energy marketplace, participants are interested in maximizing their own profits, regardless of the system-wide profits. It is perceived that competition will reduce the price of electricity for retail customers, however, the key issue for participants is related with the price definition to remain competitive. In this paper, the authors use game theory to simulate the decision making process for defining offered prices in a deregulated environment. The outcome of this study may be used by power pool coordinators to discourage unfair coalitions. A modified IEEE 30 bus power system is used as a deregulated power pool to illustrate the main features of the proposed method.

An intelligent dynamic programming for unit commitment application

A heuristic improvement of the truncated window dynamic programming has been studied for the unit commitment application. The proposed method employs a variable window size according to load demand increments, and corresponding experimental results indicate a substantial saving in the computation time without sacrificing the quality of the solution. An iterative process for the number of strategies saved in every stage is incorporated to fine tune the optimal solution. >

Linear reactive power optimization in a large power network using the decomposition approach

A mathematical framework is presented for the solution of the economic dispatch problem. The application of the Dantzig-Wolfe decomposition method for the solution of this problem is emphasized. The systems optimization problem is decomposed into several subproblems corresponding to specific areas in the power system. The upper bound technique along with the decomposition method are applied to a 16-bus system and a modified IEEE 30-bus system, and numerical results are presented for larger systems. The results indicate that the presented formulation of the reactive power optimization and the application of the decomposition procedure will facilitate the solution of the problem. The algorithm can be applied to a large-scale power network, where its solution represents a significant reduction in the number of iterations and the required computation time. >

Unit commitment with transmission security and voltage constraints

In this paper, optimal power flow (with transmission security and voltage constraints) is incorporated in the unit commitment formulation. Using Benders decomposition, the formulation is decomposed into a master problem and a subproblem. The master problem solves unit commitment with prevailing constraints, except transmission security and voltage constraints, by augmented Lagrangian relaxation. The subproblem minimizes violations of transmission security and voltage constraints for a commitment schedule given in the master problem. Since transmission constraints are decoupled from voltage constraints in the subproblem, the subproblem could further be decomposed into two smaller subproblems. The resulting transmission subproblem minimizes transmission flow violations for the worst contingency case by adjusting unit generation and phase shifter controls. The corresponding reactive subproblem minimizes voltage constraints by reactive power generation and tap changing controls. In the case of transmission flow or voltage infeasibility, Benders cuts are produced for unit commitment rescheduling. The iterative process between the master problem and the subproblems provides a minimum production cost solution for generation scheduling while satisfying the constraints. A comprehensive unit commitment software package is developed using Visual C++. A 36-unit system in the IEEE-118 bus network is used to demonstrate the efficiency of the method.

State estimation for electric power distribution systems in quasi real-time conditions

This paper proposes a new method for the state estimation of electric power distribution system conditions oriented towards working with minimum number of remote measurements available in the network. This approach will require information concerning the network reconfiguration, remote measurements of voltages, real and reactive power as well as feeder currents in distribution substations. All kinds of statistical information pertaining to distribution transformer loads are used together with real-time remote measurements. In the proposed method, the existing scheme configuration is examined by a configuration pre-screening process based on the addressed references in an oriented graph. The permissible intervals for remote measurement errors and the reliability of statistical information are discussed. A practical example is presented and the exploitation of results has proved its accuracy and efficiency. >

Ramp-rate limits in unit commitment and economic dispatch incorporating rotor fatigue effect

In this study, a rigorous mathematical method is proposed for dealing with the ramp-rate limits in unit commitment and the rotor fatigue effect in economic dispatch. An iterative procedure is employed to coordinate the unit commitment and the power dispatch for obtaining an economical solution within a reasonable time. The Lagrangian relaxation method is used to generate the unit commitment schedule with relaxed power balance constraints. A network model is adopted to represent the dynamic process of operating a unit over the entire study time span, as the required unit commitment schedule can be achieved by searching for the shortest path in the network. In order to find the global optimal solution for the economic dispatch problem within personal computer resources, a piecewise linear model is used for thermal units. Furthermore, linear programming is used in optimizing the benefits of ramping the units, with low operating cost against the cost of shortening the service life of the turbine rotor. In this regard, linear programming is used to dispatch the power generation among committed units by considering a ramping penalty for the fatigue effect in rotor shafts, while preserving the operational constraints of the system as well as the generating units. >

Integrated generation and transmission maintenance scheduling with network constraints

Most unit maintenance scheduling packages consider preventive maintenance schedule of generating units over a one or two year operational planning period in order to minimize the total operating cost while satisfying system energy requirements and maintenance constraints. In a global maintenance scheduling problem, the authors propose to consider transmission line maintenance scheduling and line capacity limits along with generation and line outages. The inclusion of transmission and network constraints in generating unit maintenance will increase the complexity of the problem, so they propose to decompose the global generator/transmission scheduling problem into a master problem and sub-problems using Benders decomposition. In the first stage, a master problem is solved to determine a solution for maintenance schedule decision variables. In the second stage, sub-problems are solved to minimize operating costs while satisfying the network constraints. Benders cuts based on the solution of the sub-problem are introduced to the master problem for improving the existing solution. The iterative procedure continues until an optimal or near optimal solution is found.

A fuzzy-based optimal reactive power control

A mathematical formulation of the optimal reactive power control problem using fuzzy set theory is presented. The objectives are to minimize real power losses and improve the voltage profile of a given system. Transmission losses are expressed in terms of voltage increments by relating the control variables to the voltage increments in a modified Jacobian matrix. This formulation does not require Jacobian matrix inversion, and hence it will save computation time and memory space. The objective function and the constraints are modeled by fuzzy sets. Linear membership functions of the fuzzy sets are defined and the fuzzy linear optimization problem is formulated. The solution space is defined as the intersection of the fuzzy sets describing the constraints and the objective functions. Each solution is characterized by a parameter that determines the degree of satisfaction with the solution. The optimal solution is the one with the maximum value for the satisfaction parameter. Results for test systems reveal the advantages of the approach. >