Optimal generation scheduling for constrained multi-area hydrothermal power systems with cascaded reservoirs

The optimization problem in this paper is targeted at large-scale hydrothermal power systems. The thermal part of the system is a multi-area power pool with tie-line constraints, and the hydro part is a set of cascaded hydrostations. The objective is to minimize the operation cost of the thermal subsystem. This is an integer nonlinear optimization process with a large number of variables and constraints. In order to obtain the optimal solution in a reasonable time, we decompose the problem into thermal and hydro subproblems. The coordinator between these subproblems is the system Lagrange multiplier. For the thermal subproblem, in a multi-area power pool, it is necessary to coordinate the area generations for reducing the operation cost without violating tie limits. For the hydro subsystem, network flow concepts are adopted to coordinate water usage over the entire study time span, and the reduced gradient method is used to overcome the linear characteristic of the network flow method in order to obtain the optimal solution. In this study, load forecasting errors and forced outages of generating units are incorporated in system reliability requirements. Three case studies for the proposed method are presented.     

A mathematical programming model and solution for scheduling production orders in Shanghai Baoshan Iron and Steel Complex

In this paper, we investigate the production order scheduling problem derived from the production of steel sheets in Shanghai Baoshan Iron and Steel Complex (Baosteel). A deterministic mixed integer programming (MIP) model for scheduling production orders on some critical and bottleneck operations in Baosteel is presented in which practical technological constraints have been considered. The objective is to determine the starting and ending times of production orders on corresponding operations under capacity constraints for minimizing the sum of weighted completion times of all orders. Due to large numbers of variables and constraints in the model, a decomposition solution methodology based on a synergistic combination of Lagrangian relaxation, linear programming and heuristics is developed. Unlike the commonly used method of relaxing capacity constraints, this methodology alternatively relaxes constraints coupling integer variables with continuous variables which are introduced to the objective function by Lagrangian multipliers. The Lagrangian relaxed problem can be decomposed into two sub-problems by separating continuous variables from integer ones. The sub-problem that relates to continuous variables is a linear programming problem which can be solved using standard software package OSL, while the other sub-problem is an integer programming problem which can be solved optimally by further decomposition. The subgradient optimization method is used to update Lagrangian multipliers. A production order scheduling simulation system for Baosteel is developed by embedding the above Lagrangian heuristics. Computational results for problems with up to 100 orders show that the proposed Lagrangian relaxation method is stable and can find good solutions within a reasonable time.     

计及多重阻塞的电动汽车移动储能特性建模

为实现城市交通电力耦合系统在城市道路、充电设施、输电线路阻塞环境下的优化运行,提出了计及多重阻塞的动态交通电力流联合优化方法。首先,基于时空网络模型,提出了计及电动汽车移动、静止、充电、排队模式的队列时空网络模型,构建了适用于电动汽车的车辆调度模型,进而形成动态交通分配模型,以减少交通出行损失。其次,通过优化发电机组、储能等的出力和备用计划,计及城市电网安全、备用约束,构建了安全约束动态经济调度模型,以降低碳排放及发电成本。随后,形成多目标动态优化模型,并将其转换为混合整数凸二次规划问题。最后,在耦合IEEE-30、Sioux Falls系统中验证了所提模型的有效性。     

pth Power Lagrangian Method for Integer Programming

When does there exist an optimal generating Lagrangian multiplier vector (that generates an optimal solution of an integer programming problem in a Lagrangian relaxation formulation), and in cases of nonexistence, can we produce the existence in some other equivalent representation space? Under what conditions does there exist an optimal primal-dual pair in integer programming? This paper considers both questions. A theoretical characterization of the perturbation function in integer programming yields a new insight on the existence of an optimal generating Lagrangian multiplier vector, the existence of an optimal primal-dual pair, and the duality gap. The proposed pth power Lagrangian method convexifies the perturbation function and guarantees the existence of an optimal generating Lagrangian multiplier vector. A condition for the existence of an optimal primal-dual pair is given for the Lagrangian relaxation method to be successful in identifying an optimal solution of the primal problem via the maximization of the Lagrangian dual. The existence of an optimal primal-dual pair is assured for cases with a single Lagrangian constraint, while adopting the pth power Lagrangian method. This paper then shows that an integer programming problem with multiple constraints can be always converted into an equivalent form with a single surrogate constraint. Therefore, success of a dual search is guaranteed for a general class of finite integer programming problems with a prominent feature of a one-dimensional dual search.     

Decomposition in large system optimization using the method of multipliers

Although the method of multipliers can resolve the dual gaps which will often appear between the true optimum point and the saddle point of the Lagrangian in large system optimization using the Lagrangian approach, it is impossible to decompose the generalized Lagrangian in a straightforward manner because of its quadratic character. A technique using the linear approximation of the perturbed generalized Lagrangian has recently been proposed by Stephanopoulos and Westerberg for the decomposition. In this paper, another attractive decomposition technique which transforms the nonseparable crossproduct terms into the minimum of sum of separable terms is proposed. The computational efforts required for large system optimization can be much reduced by adopting the latter technique in place of the former, as illustrated by application of these two techniques to an optimization problem of a chemical reactor system.The authors would like to acknowledge the valuable comments given by Professor D. G. Luenberger of Stanford University.     

A robust and informative method for solving large-scale power flow problems

Solving power flow problems is essential for the reliable and efficient operation of an electric power network. However, current software for solving these problems have questionable robustness due to the limitations of the solution methods used. These methods are typically based on the Newton–Raphson method combined with switching heuristics for handling generator reactive power limits and voltage regulation. Among the limitations are the requirement of a good initial solution estimate, the inability to handle near rank-deficient Jacobian matrices, and the convergence issues that may arise due to conflicts between the switching heuristics and the Newton–Raphson process. These limitations are addressed by reformulating the power flow problem and using robust optimization techniques. In particular, the problem is formulated as a constrained optimization problem in which the objective function incorporates prior knowledge about power flow solutions, and solved using an augmented Lagrangian algorithm. The prior information included in the objective adds convexity to the problem, guiding iterates towards physically meaningful solutions, and helps the algorithm handle near rank-deficient Jacobian matrices as well as poor initial solution estimates. To eliminate the negative effects of using switching heuristics, generator reactive power limits and voltage regulation are modeled with complementarity constraints, and these are handled using smooth approximations of the Fischer–Burmeister function. Furthermore, when no solution exists, the new method is able to provide sensitivity information that aids an operator on how best to alter the system. The proposed method has been extensively tested on real power flow networks of up to 58k buses.     

On computational complexity of the electric power flow optimization problem in market environment

Under consideration is the electric power flow optimization problem for an electric power system which typically arises in calculation of electrical power auctions in the “day-ahead” and balancing markets. It was established that the problem of finding a feasible flow in the balancing market is NP-hard in the strong sense even in case of one generator. The problem of finding an optimal flow in the day-ahead market is proved to be NP-hard even with one generator and without controlled cuts.     

A Lagrangian decomposition approach for the pump scheduling problem in water networks

Dynamic pricing has become a common form of electricity tariff, where the price of electricity varies in real time based on the realized electricity supply and demand. Hence, optimizing industrial operations to benefit from periods with low electricity prices is vital to maximizing the benefits of dynamic pricing. In the case of water networks, energy consumed by pumping is a substantial cost for water utilities, and optimizing pump schedules to accommodate for the changing price of energy while ensuring a continuous supply of water is essential. In this paper, a Mixed-Integer Non-linear Programming (MINLP) formulation of the optimal pump scheduling problem is presented. Due to the non-linearities, the typical size of water networks, and the discretization of the planning horizon, the problem is not solvable within reasonable time using standard optimization software. We present a Lagrangian decomposition approach that exploits the structure of the problem leading to smaller problems that are solved independently. The Lagrangian decomposition is coupled with a simulation-based, improved limited discrepancy search algorithm that is capable of finding high quality feasible solutions. The proposed approach finds solutions with guaranteed upper and lower bounds. These solutions are compared to those found by a mixed-integer linear programming approach, which uses a piecewise-linearization of the non-linear constraints to find a global optimal solution of the relaxation. Numerical testing is conducted on two real water networks and the results illustrate the significant costs savings due to optimizing pump schedules.     

Application of chaos-based chaotic invasive weed optimization techniques for environmental OPF problems in the power system

This paper presents efficient chaotic invasive weed optimization (CIWO) techniques based on chaos for solving optimal power flow (OPF) problems with non-smooth generator fuel cost functions (non-smooth OPF) with the minimum pollution level (environmental OPF) in electric power systems. OPF problem is used for developing corrective strategies and to perform least cost dispatches. However, cost based OPF problem solutions usually result in unattractive system gaze emission issue (environmental OPF). In the present paper, the OPF problem is formulated by considering the emission issue. The total emission can be expressed as a non-linear function of power generation, as a multi-objective optimization problem, where optimal control settings for simultaneous minimization of fuel cost and gaze emission issue are obtained. The IEEE 30-bus test power system is presented to illustrate the application of the environmental OPF problem using CIWO techniques. Our experimental results suggest that CIWO techniques hold immense promise to appear as efficient and powerful algorithm for optimization in the power systems.     

A Decomposition Method for a Unilateral Contact Problem with Tresca Friction Arising in Electro-elastostatics

This article is concerned with the numerical modeling of unilateral contact problems in an electro-elastic material with Tresca friction law and electrical conductivity condition. First, we prove the existence and uniqueness of the weak solution of the model. Rather than deriving a solution method for the full coupled problem, we present and study a successive iterative (decomposition) method. The idea is to solve successively a displacement subproblem and an electric potential subproblem in block Gauss-Seidel fashion. The displacement subproblem leads to a constraint non-differentiable (convex) minimization problem for which we propose an augmented Lagrangian algorithm. The electric potential unknown is computed explicitly using the Riesz's representation theorem. The convergence of the iterative decomposition method is proved. Some numerical experiments are carried out to illustrate the performances of the proposed algorithm.     

Interior point models for power system stability problems

In this paper, a detailed analysis of the use of optimization techniques in the study of voltage stability problems, leading to the incorporation of voltage stability criteria in traditional Optimal Power Flow (OPF) formulations is presented. Optimal power flow problems are highly nonlinear programming problems that are used to find the optimal control settings in electrical power systems. The relationship between the Lagrangian Multipliers of the OPF problem and the classification of the maximum loading point level of the system is given. Finally, the paper presents a sequential OPF technique to enhance voltage stability using reactive power and voltage rescheduling with no increase in real (active) generation cost.     

Robust placement and tuning of UPFC via a new multiobjective scheme‐based fuzzy theory

This article addresses a new modified honey bee mating optimization namely multiobjective honey bee mating optimization (MOIHBMO) based fuzzy multiobjective methodology for optimal locating and parameter setting of unified power flow controller (UPFC) in a power system for a long‐term period. One of the profits obtained by UPFC placement in a transmission network is the reduction in total generation cost due to its ability to change the power flow pattern in the network. Considering this potential, UPFC can be also used to remove or at least mitigate the congestion in transmission networks. The other issue in a power system is voltage violation which could even render the optimal power flow problem infeasible to be solved. Voltage violation could be also mitigated by proper application of UPFC in a transmission system. These objectives are considered simultaneously in a unified objective function for the proposed optimization algorithm. At first, these objectives are fuzzified and designed to be comparable against each other and then they are integrated and introduced to a MOIHBMO method to find the solution which maximizes the value of integrated objective function in a 3‐year planning horizon, considering the load growth. A power injection model is adopted for UPFC. Unlike, the most previous works in this field the parameters of UPFC are set for each load level to avoid inconvenient rejection of more optimal solutions. IEEE reliability test system is used as an illustrative example to show the effectiveness of the proposed method. © 2014 Wiley Periodicals, Inc. Complexity 21: 126–137, 2015     

Convergent Lagrangian and domain cut method for nonlinear knapsack problems

The nonlinear knapsack problem, which has been widely studied in the OR literature, is a bounded nonlinear integer programming problem that maximizes a separable nondecreasing function subject to separable nondecreasing constraints. In this paper we develop a convergent Lagrangian and domain cut method for solving this kind of problems. The proposed method exploits the special structure of the problem by Lagrangian decomposition and dual search. The domain cut is used to eliminate the duality gap and thus to guarantee the finding of an optimal exact solution to the primal problem. The algorithm is first motivated and developed for singly constrained nonlinear knapsack problems and is then extended to multiply constrained nonlinear knapsack problems. Computational results are presented for a variety of medium- or large-size nonlinear knapsack problems. Comparison results with other existing methods are also reported.     

Semi-infinite programming method for optimal power flow with transient stability and variable clearing time of faults

This paper presents a new nonlinear programming problem arising in the control of power systems, called optimal power flow with transient stability constraint and variable clearing time of faults and abbreviated as OTS-VT. The OTS-VT model is converted into a implicit generalized semi-infinite programming (GSIP) problem. According to the special box structure of the reformulated GSIP, a solution method based on bi-level optimization is proposed. The research in this paper has two contributions. Firstly, it generalizes the OTS study to general optimal power flow with transient stability problems. From the viewpoint of practical applications, the proposed research can improve the decision-making ability in power system operations. Secondly, the reformulation of OTS-VT also provides a new background and a type of GSIP in the research of mathematical problems. Numerical results for two chosen power systems show that the methodology presented in this paper is effective and promising.     

A Branch-and-Bound Algorithm to Solve a Multi-level Network Optimization Problem

Multi-level network optimization problems arise in many contexts such as telecommunication, transportation, and electric power systems. A model for multi-level network design is formulated as a mixed-integer program. The approach is innovative because it integrates in the same model aspects of discrete facility location, topological network design, and dimensioning. We propose a branch-and-bound algorithm based on Lagrangian relaxation to solve the model. Computational results for randomly generated problems are presented showing the quality of our approach. We also present and discuss a real world problem of designing a two-level local access urban telecommunication network and solving it with the proposed methodology.     

Lagrangian relaxation based algorithm for trigeneration planning with storages

Trigeneration is a booming power production technology where three energy commodities are simultaneously produced in a single integrated process. Electric power, heat (e.g. hot water) and cooling (e.g. chilled water) are three typical energy commodities in the trigeneration system. The production of three energy commodities follows a joint characteristic. This paper presents a Lagrangian relaxation (LR) based algorithm for trigeneration planning with storages based on deflected subgradient optimization method. The trigeneration planning problem is modeled as a linear programming (LP) problem. The linear cost function poses the convergence challenge to the LR algorithm and the joint characteristic of trigeneration plants makes the operating region of trigeneration system more complicated than that of power-only generation system and that of combined heat and power (CHP) system. We develop an effective method for the long-term planning problem based on the proper strategy to form Lagrangian subproblems and solve the Lagrangian dual (LD) problem based on deflected subgradient optimization method. We also develop a heuristic for restoring feasibility from the LD solution. Numerical results based on realistic production models show that the algorithm is efficient and near-optimal solutions are obtained.     

Success Guarantee of Dual Search in Integer Programming: p-th Power Lagrangian Method

Although the Lagrangian method is a powerful dual search approach in integer programming, it often fails to identify an optimal solution of the primal problem. The p-th power Lagrangian method developed in this paper offers a success guarantee for the dual search in generating an optimal solution of the primal integer programming problem in an equivalent setting via two key transformations. One other prominent feature of the p-th power Lagrangian method is that the dual search only involves a one-dimensional search within [0,1]. Some potential applications of the method as well as the issue of its implementation are discussed.     

The crane scheduling problem: models and solution approaches

In this paper, we study the crane scheduling problem for a vessel after the vessel is moored on a terminal and develop both exact and heuristic solution approaches for the problem. For small-sized instances, we develop a time-space network flow formulation with non-crossing constraints for the problem and apply an exact solution approach to obtain an optimal solution. For medium-sized instances, we develop a Lagrangian relaxation approach that allows us to obtain tight lower bounds and near-optimal solutions. For large-sized instances, we develop two heuristics and show that the error bounds of our heuristics are no more than 100%. Finally, we perform computational studies to show the effectiveness of our proposed solution approaches.     

On modelling the maximum workload allocation for parallel unrelated machines with setups

Consider a manufacturing process in which a group of machines (or people) perform a single operation on a number of different parts. The processing time depends on both the part and the machine. In addition, each machine requires significant setup time between processing different part types. The problem consists of obtaining a feasible allocation of parts to machines such that the makespan (i.e. greatest machine workload) is minimized. We present two equivalent 0–1 models. The first model arises by considering the assignment of individual parts to machines. It is amenable to Lagrangian decomposition techniques. The second model is more hierarchical in nature; it considers the two options of assigning an entire part type to a single machine, or of splitting the type across machines. The second model is more amenable than the first to branch-and-bound techniques. We report about our computational experience for finding lower bounds of the optimal solution by appending violated cuts and, ultimately, obtaining the optimal solution of real-life problems.     

Extended Fuzzy C-Means and Genetic Algorithms to Optimize Power Flow Management in Hybrid Electric Vehicles

The need for personal transportation must be harmonized by considering the impact of so huge number of vehicles on the environment. The adoption of hybrid electric vehicles can provide a sensible improvement from an environmental viewpoint, but at the same time makes more difficult the definition and implementation of the overall powertrain control mechanism. In fact, powertrain control problems are known to be very complex due to conflicting requirements, and this difficulty augments in case of hybrid electric vehicles. Most of the features of the future hybrid electric vehicles are enabled by a new energy flow management unit designed to split the instantaneous power demand between the internal combustion engine and the electric motor, ensuring both an efficient power supply and reduced emissions. Classic approaches that rely on static thresholds, optimized on a fixed drive cycle, cannot face the high dynamicity and unpredictability of real-life drive conditions. The need to actually control a real vehicle stimulates the research of innovative methodologies for the real-time identification of the operating points of each energy source. This paper is framed into this context: after a brief discussion about a non-conventional formalization of the energy flows problem based on a multiobjective function, a knowledge-based control system for splitting the vehicle's power demand between the engine and motor is presented. The proposed approach exploits a fuzzy clustering criterion that combined with a genetic algorithm, permits to achieve better results, both in terms of a reduced computational effort and an improved efficiency of the control system over various driving cycles. To validate the proposed approach, simulation tests and comparisons with other energy management strategies are discussed.