Development and computational tests of an undecoupled optimum short-term hydro-thermal schedulling code using network flows

Summary Optimizing the thermal production of electricity in the short term in an integrated power system when a thermal unit commitment has been decided means coordinating hydro and thermal generation in order to obtain the minimum thermal generation costs over the time period under study. Fundamental constraints to be satisfied are the covering of each hourly load and satisfaction of spinning reserve requirements. A nonlinear network flow model with linear side constraints with no decomposition into hydro and thermal subproblems was used to solve the hydrothermal scheduling. Hydrogeneration is linearized with respect to network variables and a novel thermal generation network is introduced. Computational results are reported.     

Medium-term hydro-thermal coordination via hydro and thermal subsystem decomposition

Summary This paper addresses the medium-term hydro-thermal coordination problem in an electric energy system. That is, the problem of finding the energy production of every power plant (hydro or thermal) in every subperiod of a given planning period, so that the customer load is supplied at minimum cost. The planning horizon is typically one to two months and the first week of this planning period is modeled in detail. The solution method proposed decomposes the problem in two subproblems corresponding to the hydro and thermal subsystems. These two subproblems are coordinated using a coordinating function for every subperiod. The coordinating function of a given subperiod expresses total production cost in that subperiod as a function of the total hydro production in that subperiod. The decomposition proposed makes it possible to use specialized algorithms to solve the hydro and thermal subproblems. This results in a very efficient computational procedure. From an experimental point of view the coordinating mechanism is robust. A case study is provided. It considers 61 thermal plants, a hydro system including 8 cascaded hydro plants and a 48 subperiods planning period.     

A Decomposition Methodology Applied to the Multi-Area Optimal Power Flow Problem

This paper describes a decomposition methodology applied to the multi-area optimal power flow problem in the context of an electric energy system. The proposed procedure is simple and efficient, and presents some advantages with respect to other common decomposition techniques such as Lagrangian relaxation and augmented Lagrangian decomposition. The application to the multi-area optimal power flow problem allows the computation of an optimal coordinated but decentralized solution. The proposed method is appropriate for an Independent System Operator in charge of the electric energy system technical operation. Convergence properties of the proposed decomposition algorithm are described and related to the physical coupling between the areas. Theoretical and numerical results show that the proposed decentralized methodology has a lower computational cost than other decomposition techniques, and in large large-scale cases even lower than a centralized approach.     

Stochastic Lagrangian Relaxation Applied to Power Scheduling in a Hydro-Thermal System under Uncertainty

A dynamic (multi-stage) stochastic programming model for the weekly cost-optimal generation of electric power in a hydro-thermal generation system under uncertain demand (or load) is developed. The model involves a large number of mixed-integer (stochastic) decision variables and constraints linking time periods and operating power units. A stochastic Lagrangian relaxation scheme is designed by assigning (stochastic) multipliers to all constraints coupling power units. It is assumed that the stochastic load process is given (or approximated) by a finite number of realizations (scenarios) in scenario tree form. Solving the dual by a bundle subgradient method leads to a successive decomposition into stochastic single (thermal or hydro) unit subproblems. The stochastic thermal and hydro subproblems are solved by a stochastic dynamic programming technique and by a specific descent algorithm, respectively. A Lagrangian heuristics that provides approximate solutions for the first stage (primal) decisions starting from the optimal (stochastic) multipliers is developed. Numerical results are presented for realistic data from a German power utility and for numbers of scenarios ranging from 5 to 100 and a time horizon of 168 hours. The sizes of the corresponding optimization problems go up to 200000 binary and 350000 continuous variables, and more than 500000 constraints.     

An augmented Lagrangian method for optimization problems with structured geometric constraints

This paper is devoted to the theoretical and numerical investigation of an augmented Lagrangian method for the solution of optimization problems with geometric constraints. Specifically, we study situations where parts of the constraints are nonconvex and possibly complicated, but allow for a fast computation of projections onto this nonconvex set. Typical problem classes which satisfy this requirement are optimization problems with disjunctive constraints (like complementarity or cardinality constraints) as well as optimization problems over sets of matrices which have to satisfy additional rank constraints. The key idea behind our method is to keep these complicated constraints explicitly in the constraints and to penalize only the remaining constraints by an augmented Lagrangian function. The resulting subproblems are then solved with the aid of a problem-tailored nonmonotone projected gradient method. The corresponding convergence theory allows for an inexact solution of these subproblems. Nevertheless, the overall algorithm computes so-called Mordukhovich-stationary points of the original problem under a mild asymptotic regularity condition, which is generally weaker than most of the respective available problem-tailored constraint qualifications. Extensive numerical experiments addressing complementarity- and cardinality-constrained optimization problems as well as a semidefinite reformulation of MAXCUT problems visualize the power of our approach.

     

Chaotic artificial immune approach applied to economic dispatch of electric energy using thermal units

The economic dispatch problem (EDP) is an optimization problem useful in power systems operation. The objective of the EDP of electric power generation, whose characteristics are complex and highly non-linear, is to schedule the committed generating unit outputs so as to meet the required load demand at minimum operating cost while satisfying system constraints. Recently, as an alternative to the conventional mathematical approaches, modern heuristic optimization techniques have been given much attention by many researchers due to their ability to find an almost global optimal solution in EDPs. As special mechanism to avoid being trapped in local minimum, the ergodicity property of chaotic sequences has been used as optimization technique in EDPs. Based on the chaos theory, this paper discusses the design and validation of an optimization procedure based on a chaotic artificial immune network approach based on Zaslavsky’s map. The optimization approach based on chaotic artificial immune network is validated for a test system consisting of 13 thermal units whose incremental fuel cost function takes into account the valve-point loading effects. Simulation results and comparisons show that the chaotic artificial immune network approach is competitive in performance with other optimization approaches presented in literature and is also an attractive tool to be used on applications in the power systems field.     

Decomposition methods for a spatial model for long-term energy pricing problem

We consider an energy production network with zones of production and transfer links. Each zone representing an energy market (a country, part of a country or a set of countries) has to satisfy the local demand using its hydro and thermal units and possibly importing and exporting using links connecting the zones. Assuming that we have the appropriate tools to solve a single zonal problem (approximate dynamic programming, dual dynamic programming, etc.), the proposed algorithm allows us to coordinate the productions of all zones. We propose two reformulations of the dynamic model which lead to different decomposition strategies. Both algorithms are adaptations of known monotone operator splitting methods, namely the alternating direction method of multipliers and the proximal decomposition algorithm which have been proved to be useful to solve convex separable optimization problems. Both algorithms present similar performance in theory but our numerical experimentation on real-size dynamic models have shown that proximal decomposition is better suited to the coordination of the zonal subproblems, becoming a natural choice to solve the dynamic optimization of the European electricity market.     

Two-phase optimization of coal strategies at a power station

Development of optimal coal mining and stockpiling strategies for a new 960 MW thermal power station depends strongly on the operation of New Zealand's mixed hydro/thermal power system. To overcome the large dimensionality of the problem the optimization proceeds in two phases for each future scenario studied. Phase 1 finds optimal hydro operating strategies with thermal generation in merit order of marginal fuel costs. It uses stochastic dynamic programming. The hydro generating strategies are used as input to the phase 2 stochastic simulation of the coal mining and stockpiling operation at the new thermal station. Finally, a strategy is identified which is a priori best over all future scenarios of industrial development and oil prices studied.     

多周期多种设备公用工程系统优化模型的改进及应用

本文提出了多周期多种设备公用工程系统改进的混合整数双线性优化模型,它含有两种优化变量和系统运行过程的离散动态约束,期望系统总设备投资(含设备折旧)与全周期运行操作费用之和最小。针对改进优化模型求解上的困难,给出将改进优化模型分解成有限多个关于连续变量的线性规划。论述了改进优化模型与分解模型的等价性以及两种模型的主要数学性质,并在此基础上提出了求解策略。最后将改进优化模型应用于某石化企业的蒸汽动力系统最优设计与运行优化集成实例。     

带阀点效应水火联合调度问题的一种半定凸松弛求解法

水火联合调度问题是电力系统中一类复杂的优化问题。合理安排调度周期内的水火电出力,确定一个最优发电计划,可以带来巨大的经济效益。在实际系统中,汽轮机调汽阀开启时出现的拔丝现象会使机组耗量特性产生阀点效应。忽略阀点效应,在一定程度上降低求解的精度。本文考虑带阀点效应的水火联合调度问题。该问题非凸非光滑,且带有非线性约束,直接使用确定性全局优化方法求解是相当困难的。本文使用高效的半定规划求解此问题。首先用耗量特性函数的初始周期代替其余有限的周期,并对其进行二次拉格朗日插值拟合。再通过引进0-1变量,得到整个耗量特性函数的近似,进而把问题松弛为半定规划模型。最后,采用凸规划应用软件包CVX求解一个仿真算例,得到一个近似全局最优解。     

A correction to “Single machine scheduling with release times,deadlines and tardiness objectives” [EJOR 133 (2001) 447–453]

In this study we consider the elevator operation problem of single-car elevator systems with destination hall call registration. In this part we construct a branch-and-bound algorithm to solve the dynamic operation optimization problem formulated in the first part. To calculate lower bounds of the subproblems generated in the course of the branch-and-bound algorithm, we first relax some of the constraints of the subproblems and decompose the relaxed subproblems into three parts. Then, we apply the Lagrangian relaxation method to the decomposed subproblems.     

Analysis of stochastic problem decomposition algorithms in computational grids

Stochastic programming usually represents uncertainty discretely by means of a scenario tree. This representation leads to an exponential growth of the size of stochastic mathematical problems when better accuracy is needed. Trying to solve the problem as a whole, considering all scenarios together, yields to huge memory requirements that surpass the capabilities of current computers. Thus, decomposition algorithms are employed to divide the problem into several smaller subproblems and to coordinate their solution in order to obtain the global optimum. This paper analyzes several decomposition strategies based on the classical Benders decomposition algorithm, and applies them in the emerging computational grid environments. Most decomposition algorithms are not able to take full advantage of all the computing power available in a grid system because of unavoidable dependencies inherent to the algorithms. However, a special decomposition method presented in this paper aims at reducing dependency among subproblems, to the point where all the subproblems can be sent simultaneously to the grid. All algorithms have been tested in a grid system, measuring execution times required to solve standard optimization problems and a real-size hydrothermal coordination problem. Numerical results are shown to confirm that this new method outperforms the classical ones when used in grid computing environments.     

Matrix minor reformulation and SOCP-based spatial branch-and-cut method for the AC optimal power flow problem

Alternating current optimal power flow (AC OPF) is one of the most fundamental optimization problems in electrical power systems. It can be formulated as a semidefinite program (SDP) with rank constraints. Solving AC OPF, that is, obtaining near optimal primal solutions as well as high quality dual bounds for this non-convex program, presents a major computational challenge to today’s power industry for the real-time operation of large-scale power grids. In this paper, we propose a new technique for reformulation of the rank constraints using both principal and non-principal 2-by-2 minors of the involved Hermitian matrix variable and characterize all such minors into three types. We show the equivalence of these minor constraints to the physical constraints of voltage angle differences summing to zero over three- and four-cycles in the power network. We study second-order conic programming (SOCP) relaxations of this minor reformulation and propose strong cutting planes, convex envelopes, and bound tightening techniques to strengthen the resulting SOCP relaxations. We then propose an SOCP-based spatial branch-and-cut method to obtain the global optimum of AC OPF. Extensive computational experiments show that the proposed algorithm significantly outperforms the state-of-the-art SDP-based OPF solver and on a simple personal computer is able to obtain on average a \(0.71\%\) optimality gap in no more than 720 s for the most challenging power system instances in the literature.     

Solving nonconvex economic load dispatch problem using particle swarm optimization with time varying acceleration coefficients

This article presents the design and application of an efficient hybrid heuristic search method to solve the practical economic dispatch problem considering many nonlinear characteristics of power generators, and their operational constraints, such as transmission losses, valve‐point effects, multi‐fuel options, prohibited operating zones, ramp rate limits and spinning reserve. These practical operation constraints which can usually be found at the same time in realistic power system operations make the economic load dispatch (ELD) problem a nonsmooth optimization problem having complex and nonconvex features with heavy equality and inequality constraints. A particle swarm optimization with time varying acceleration coefficients is proposed to determine optimal ELD problem in this paper. The proposed methodology easily takes care of solving nonconvex ELD problems along with different constraints like transmission losses, dynamic operation constraints, and prohibited operating zones. The proposed approach has been implemented on the 3‐machines 6‐bus, IEEE 5‐machines 14‐bus, IEEE 6‐machines 30‐bus systems and 13 thermal units power system. The proposed technique is compared with solve the ELD problem with hybrid approach by using the valve‐point effect. The comparison results prove the capability of the proposed method give significant improvements in the generation cost for the ELD problem. © 2015 Wiley Periodicals, Inc. Complexity 21: 299–308, 2016     

Optimal discrete long-term operation of nuclear-hydrothermal power systems

We discuss in this paper an algorithm for solving the optimal long-term operating problem of a hydrothermal-nuclear power system by application of the minimum norm optimization technique. The algorithm proposed here has the ability to deal with large-scale power systems and with equality and/or inequality constraints on the variables. A discrete model for the xenon and iodine concentrations is used, as well as a discrete model for hydro reservoirs. The optimization is done on a monthly time basis. For simplicity of the problem formulation, the transmission line losses are considered as a part of the load.This work supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. A4146.     

Optimum load scheduling of nuclear-hydro-thermal power systems

This paper considers the problem of short-term optimal operation of nuclear-hydro-thermal electric power systems. The solution is obtained by use of a functional analytic optimization technique that employs the minimum norm formulation.A power system with an arbitrary number of generating stations is considered. The limited flexibility exhibited by the thermal nuclear reactors, when operating in a load-following mode, is accounted for by means of a model of the xenon concentration in their cores. The nonlinear effects induced by trapezoidal water reservoirs and the time delay of the water flow between upstream and downstream hydroplants is taken into consideration as well.A two-level iterative scheme of the feasible type is proposed for implementing the optimal solution.This work was supported in part by the National Research Council of Canada, Grant No. A-4146.     

Capacity and flow assignment of data networks by generalized Benders decomposition

A mixed-integer non-linear model is proposed to optimize jointly the assignment of capacities and flows (the CFA problem) in a communication network. Discrete capacities are considered and the cost function combines the installation cost with a measure of the Quality of Service (QoS) of the resulting network for a given traffic. Generalized Benders decomposition induces convex subproblems which are multicommodity flow problems on different topologies with fixed capacities. These are solved by an efficient proximal decomposition method. Numerical tests on small to medium-size networks show the ability of the decomposition approach to obtain global optimal solutions of the CFA problem.     

A constrained nonlinear 0–1 program for data allocation

This paper analyzes the problem of allocating copies of relations from a global database to the sites of a geographically distributed communication network. The objective of the allocation is to minimize the total cost due to transmissions generated by queries from the various sites, including queries that access multiple relations. This allocation problem is modeled as a constrained nonlinear 0–1 subproblems generated during subgradient optimization are solved as optimization. Some of the unconstrained quadratic 0–1 subproblems generated during subgradient optimization are solved as maximum flow problems, while the others require implicit enumeration, depending on the nature of the objective function coefficients of the subproblems. Our solution approach is tested extensively on data allocation problems with as many as 100 sites and 20 relations. On a set of randomly generated test problems our approach was close to two orders of magnitude faster than the general purpose integer programming code OSL.     

Optimization of Optimal Power Flow Problems

Optimal power flow problems arise in the context of the optimization and secure exploitation of electrical power in alternating current (AC) networks. This optimization problem evaluates the flow on each line and to ensure that they are under line thermal limits. To improve the reliability of the power supply, a secure network is necessary, i.e., it has to be able to cope with some contingencies. Nowadays high performance solution methods, that are based on nonlinear programming algorithms, search for an optimal state while considering certain contingencies. According to the number of contingencies the problem size increases linearly. As the base case can already be large-scaled, the optimization time computation increases quickly. Parallelization seems to be a good way to solve quickly this kind of problem. This paper considers the minimization of an objective function and at least two constraints at each node. This optimization problem is solved by IPOPT, an interior point method, coupled with ADOL-C, an algorithmic differentiation tool, and MA27, a linear solver. Several results on employed parallelizing strategies will be discussed. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Design of optimal utility of wireless rechargeable sensor networks via joint spatiotemporal scheduling

In wireless rechargeable sensor networks, how to optimize energy resources for maximizing the sensor data is a challenging problem. In this paper, mobile charging vehicle scheduling, sensor charging time splitting and rate control with battery capacity constraints are considered together to maximize network utility. However, they are considered independently in exist works even though these problems are interdependent. In order to improve network performance through collaborative optimization of three problems, a joint optimization problem is formulated firstly. Then, a multistage approach is developed to jointly optimize the three subproblems iteratively. Furthermore, an accelerated distributed algorithm is integrated to improve the convergence speed of rate control. The results of extended experiments demonstrate that proposed approach can obtain higher network utility and charging efficiency compared to other charging scheduling methods.