Qiaozhu Zhai

Unit commitment with identical units successive subproblem solving method based on Lagrangian relaxation

When the Lagrangian relaxation based methods are applied to solve power system unit commitment, the identical solutions to the subproblems associated with identical units may cause the dual solution to be far away from the optimal solution and serious solution oscillations. As a result, the quality of the feasible solution obtained may be very unsatisfactory. This issue has been long recognized as an inherent disadvantage of Lagrangian relaxation based methods. In this paper, the homogeneous solution issue is identified and analyzed through a simple example. Based on this analysis, a successtve subproblem solving method is developed. The new method combines the concepts of augmented Lagrangian relaxation and surrogate subgradient to produce a good search direction at the high level. The low-level subproblems including those corresponding to the identical units are solved successively so that the commitments of the identical units may not be homogeneous in the dual solution. Compared with the standard Lagrangian relaxation method, the new method can obtain better dual solutions and avoid the solution oscillations. Numerical testing shows the new method is efficient and the quality of the feasible solution is greatly improved.

The conditions for obtaining feasible solutions to security-constrained unit commitment problems

The core of solving security-constrained unit commitment (SCUC) problems within the Lagrangian relaxation framework is how to obtain feasible solutions. However, due to the existence of the transmission constraints, it is very difficult to determine if feasible solutions to SCUC problems can be obtained by adjusting generation levels with the commitment states obtained in the dual solution of Lagrangian relaxation. The analytical and computational necessary and sufficient conditions are presented in this paper to determine the feasible unit commitment states with grid security constraints. The analytical conditions are proved rigorously based on the feasibility theorem of the Benders decomposition. These conditions are very crucial for developing an efficient method for obtaining feasible solutions to SCUC problems. Numerical testing results show that these conditions are effective.

Fast Identification of Inactive Security Constraints in SCUC Problems

Security constrained unit commitment (SCUC) is one of the most important daily tasks that independent system operators (ISOs) or regional transmission organizations (RTOs) must accomplish in daily electric power market. Security constraints have long been regarded as difficult constraints for unit commitment problems. If the inactive security constraints can be identified and eliminated, the SCUC problem can be greatly simplified. In this paper, a necessary and sufficient condition for a security constraint to be inactive is established. It is proved that all inactive constraints can be identified by solving a series of small-scale mixed integer linear programming (MILP) problems. More importantly, an analytical sufficient condition is established and most of the inactive constraints can be quickly identified without solving MILP or linear programming (LP) problems. A very important feature of the conditions obtained is that they are only related to the load demands and parameters of the transmission network. Numerical testing is performed for three power grids and the results are impressive. Over 85% of the security constraints are identified as inactive and the crucial transmission lines affecting the total operating cost are among those associated with the remaining security constraints, providing useful information for transmission planning.

A new method for unit commitment with ramping constraints

Abstract Unit commitment with ramping constraints is a very difficult problem with significant economic impact. A new method is developed in this paper for scheduling units with ramping constraints within Lagrangian relaxation framework based on a novel formulation of the discrete states and the integrated applications of standard dynamic programming for determining the optimal discrete states across hours, and constructive dynamic programming for determining optimal generation levels. A section of consecutive running or idle hours is considered as a commitment state. A constructive dynamic programming (CDP) method is modified to determine the optimal generation levels of a commitment state without discretizing generation levels. The cost-to-go functions, required only for a few corner points with a few continuous state transitions at a particular hour, are constructed in the backward sweep. The optimal generation levels can be obtained in the forward sweep. The optimal commitment states across the scheduling horizon can then be obtained by standard dynamic programming. Numerical testing results show that this method is efficient and the optimal commitment and generation levels are obtained in a systematic way without discretizing or relaxing generation levels.

A Systematic Method for Constructing Feasible Solution to SCUC Problem With Analytical Feasibility Conditions

Obtaining high-quality feasible solution is the core and the major difficulty in solving security-constrained unit commitment (SCUC) problems. This paper presents a systematic method for constructing feasible solutions to SCUC problem based on a group of analytical feasibility conditions. The feasibility check is performed based on the analytical necessary conditions such that most of infeasible UC states can be identified without solving LP problem. If a UC state is infeasible, it is adjusted with the possibly minimal operating cost increase based on the cost information. This UC adjusting issue is formulated as a zero-one programming problem and a branch and bound (B&B) method is established based on these feasibility conditions. Numerical testing is performed for a 31-bus system, an IEEE 24-bus system, and an IEEE 118-bus system. The testing results suggest that over 95% of infeasible UC states are identified by the analytical necessary conditions. The near-optimal feasible schedules for SCUC problem can be obtained efficiently by the proposed method. The feasible schedules obtained are compared with those obtained from mixed integer programming-based method in the IEEE 118-bus system. It is shown that the new method can produce competitive results in terms of solution quality and computational efficiency.

An MILP Based Formulation for Short-Term Hydro Generation Scheduling With Analysis of the Linearization Effects on Solution Feasibility

Mixed integer linear programming (MILP) based formulations and solution methods for short-term hydro generation scheduling (HGS) have been widely adopted by researchers, hydropower producers, and system operators in recent years. This approach calls for the nonlinear forebay level, tailrace level, penstock loss, and hydropower production functions to be replaced with their piecewise linear approximations. However, the effects of the linearization of the nonlinear functions and related constraints on solution feasibility have not been fully discussed in the literature. In this paper, the issues concerning solution feasibility are discussed in detail and a method is presented to ensure that the solution obtained based on the approximated MILP formulation remains feasible for the original nonlinear formulation. Furthermore, it is found that the real number water delay can be handled in the formulation without destroying the linear structure of the water balance constraints. Numerical testing results show that the method presented in this paper is effective.

An integrated optimization model for generation and batch production load scheduling in energy intensive enterprise

Reducing electricity cost is important for energy intensive enterprises (EIEs) with self generation power plant, like iron and steel plants. Such an EIE integrates power generation, power consumption and energy storage into a unified whole, and become an enterprise microgrid. Recent methods considering only load scheduling or only power generation scheduling cannot get the minimum total electricity cost. In this paper, an integrated optimization model for generation and batch production load scheduling in EIE is researched to get the minimum electricity cost, without violating constraints of production process. Linearization techniques are utilized to duel with nonlinear factors of the model. Then, mixed integer linear programming (MILP) is used for model solution. The case study of an iron and steel plant shows that the model is effective under a time-of-use (TOU) tariff implemented by the utility.

Storage-Reserve Sizing With Qualified Reliability for Connected High Renewable Penetration Micro-Grid

The major challenge in high renewable penetration microgrid is the power mismatch between stochastic renewables and demand. Energy storage and reserve purchase are main techniques in reducing that. Storage sizing problem is widely investigated in literatures without reserve capacity co-optimizing. And due to innate no loss-of-load assumption, a large proportion of capacity is used to handle inessential energy deficiency with small probability. In this paper, storage-reserve sizing problem with qualified reliability is raised to integrate reserve sizing and loss-of-load probability (LOLP) index into existing storage sizing problem. Two-stage probabilistic model is established to minimize total cost with optimizing storage capacity during first-stage and reserve strategy during second-stage. Since the time-consuming Monte Carlo simulation and stage iteration are usually required in problem solving, Markovian steady-state sizing method is proposed to improve efficiency. Probability constraint is tested by mathematical quantile. And two-stage model is transformed to single-stage one attributed to analytical solution of second-stage. Meanwhile obtained relationship among storage capacity, reserve capacity, and LOLP index can help designers balance between capacity and reliability. Numerical test shows: needed capacity is significantly reduced with little sacrifice of reliability; storage-reserve combination is economical, since they are probabilistically complementary; proposed solution method is fast and accurate.

A New Method for Solving Routing and Wavelength Assignment Problems in Optical Networks

The standard Lagrangian relaxation (SLR) method is an efficient method for solving the routing and wavelength assignment (RWA) problems in optical networks. However, previous work did not deal with multiple connection requests with identical source and destination pairs, which are frequently encountered in practice and can cause serious issues when using SLR. More specifically, in solving the dual subproblems after the wavelength capacity constraints are relaxed, the shortest path algorithms such as Dijkstras typically assign the same route to such connection requests, which possibly leads to a poor RWA solution. In this paper, we introduce a new method, i.e., the successive subproblem solving (SSS) method and one of its implementations, within the Lagrangian relaxation framework. The essence of SSS is to introduce coupled penalty terms and use the surrogate subgradients for search direction at the high level. The homogenous subproblems at the low level are then solved sequentially to avoid the nondecomposable difficulty. Theoretical analysis is performed to provide convergence proof. Numerical results are presented to show that the new method is effective and efficient.

Security-Constrained Unit Commitment Based on a Realizable Energy Delivery Formulation

Security-constrained unit commitment (SCUC) is an important tool for independent system operators in the day-ahead electric power market. A serious issue arises that the energy realizability of the staircase generation schedules obtained in traditional SCUC cannot be guaranteed. This paper focuses on addressing this issue, and the basic idea is to formulate the power output of thermal units as piecewise-linear function. All individual unit constraints and systemwide constraints are then reformulated. The new SCUC formulation is solved within the Lagrangian relaxation (LR) framework, in which a double dynamic programming method is developed to solve individual unit subproblems. Numerical testing is performed for a 6-bus system and an IEEE 118-bus system on Microsoft Visual C# .NET platform. It is shown that the energy realizability of generation schedules obtained from the new formulation is guaranteed. Comparative case study is conducted between LR and mixed integer linear programming (MILP) in solving the new formulation. Numerical results show that the near-optimal solution can be obtained efficiently by the proposed LR-based method.