Scheduling hydro power systems with restricted operating zones anddischarge ramping constraints

An optimization-based algorithm is presented for scheduling hydro power systems with restricted operating zones and discharge ramping constraints. Hydro watershed scheduling problems are difficult to solve because many constraints, continuous and discrete, including hydraulic coupling of cascaded reservoirs have to be considered. Restricted or forbidden operating zones as well as minimum generation limits of hydro units result in discontinuous preferred operating regions, and hinder direct applications of efficient continuous optimization methods such as network flow algorithms. Discharge ramping constraints due to navigational, environmental and recreational requirements in a hydro system add another dimension of difficulty since they couple generation or water discharge across time horizon. Integrated consideration of the above constraints is very challenging. The key idea of this paper is to use additional sets of multipliers to relax discontinuous operating region and discharge ramping constraints on individual hydro units so that a two-level optimization structure is formed. The low level consists of a continuous discharge scheduling subproblem determining the generation levels of all units in the entire watershed, and a number of pure integer scheduling subproblems determining the hydro operating states, one for each unit. The discharge subproblem is solved by a network flow algorithm, and the integer scheduling problems are solved by dynamic programming with a small number of states and well-structured transitions     

Scheduling of hydrothermal power systems

The authors present a method for scheduling hydrothermal power systems based on the Lagrangian relaxation technique. By using Lagrange multipliers to relax system-wide demand and reserve requirements, the problem is decomposed and converted into a two-level optimization problem. Given the sets of Lagrange multipliers, a hydro unit subproblem is solved by a merit order allocation method, and a thermal unit subproblem is solved by using dynamic programming without discretizing generation levels. A subgradient algorithm is used to update the Lagrange multipliers. Numerical results based on Northeast Utilities data show that this algorithm is efficient, and near-optimal solutions are obtained. Compared with previous work where thermal units were scheduled by using the Lagrangian relaxation technique and hydro units by heuristics, the new coordinated hydro and thermal scheduling generates lower total costs and requires less computation time     

具有相同机组水火电调度问题的新算法

对Lagrangian松弛法求解水火电调度问题时由机同机级引起解震荡现象进行了研究。通过一个例子分析了震荡产生的根本原因。对此，在松弛函数中引入了惩罚项并采用了伪次梯度法来修正乘子。新算法在求解低层子问题时并不同时求解，使震荡现象在很大程度上得以克服，同时可大幅度地降低偶解对约束的违反程度。通过简单的例子和对一个包含两组机同机组的短期发电调度问题的计算表明，对偶解的约束违反程度明显地降低，解震荡明显地减弱且最后可行解的质量有显著的改善。     

A new method for unit commitment with ramping constraints

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.     

相同机组调度与竞标问题研究

本文在对利用拉格朗日松驰法解决大型电力市场综合资源的调度与竞标时所碰到的相同机组问题进行了讨论,认为改进电力市场竞标模式并不是解决相同机组调度与竞标问题的根本方法。     

Hydrothermal scheduling by augmented Lagrangian: consideration oftransmission constraints and pumped-storage units

This paper presents an augmented Lagrangian (AL) approach to scheduling a generation mix of thermal and hydro resources. AL presents a remedy to duality gap encountered with the ordinary Lagrangian for nonconvex problems. It shapes the Lagrangian function as a hyperparaboloid associating penalty in the direction of the coupling constraints. This work accounts further for the transmission constraints. We use a hydrothermal resource model with pumped-storage units. An IEEE 24-bus test system is used for AL performance illustration. Computational models are all coded in C. The results of the test case show that the AL approach can provide better scheduling results as it can detect optimal on/off schedules of units over a planning horizon at a minimal cost with no constraint violation. It requires no iteration with economic dispatch algorithms. The approach proves accurate and practical for systems with generation diversity and limited transmission capacity     

Short-term resource scheduling with ramp constraints [powergeneration scheduling]

This paper describes a Lagrangian relaxation-based method to solve the short-term resource scheduling (STRS) problem with ramp constraints. Instead of discretizing the generation levels, the ramp rate constraints are relaxed with the system demand constraints using Lagrange multipliers. Three kinds of ramp constraints, startup, operating and shutdown ramp constraints are considered. The proposed method has been applied to solve the hydro-thermal generation scheduling problem at PG&E. An example alone with numerical results is also presented     

Coordination of Midterm Outage Scheduling With Short-Term Security-Constrained Unit Commitment

The proposed model solves the coordinated generation and transmission maintenance scheduling with security-constrained unit commitment (SCUC) over the scheduling horizon of weeks to months. The model applies the Lagrangian relaxation technique to decompose the optimization problem into subproblems for generation maintenance scheduling, transmission maintenance scheduling, and short-term SCUC. The decomposition and cooperation strategy is applied to the first two subproblems for the scheduling of generation and transmission maintenance. The SCUC solution is based on the mixed integer programming (MIP) technique. The optimal hourly results for maintenance scheduling, generation unit commitment, and transmission flows are obtained using a chronological load curve. Effective strategies are applied for accelerating the convergence of the hourly solution. The numerical examples demonstrate the effectiveness of the proposed model.     

Unit commitment using Lagrangian relaxation and particle swarm optimization

Unit commitment (UC) is a NP-hard nonlinear mixed-integer optimization problem. This paper proposes ELRPSO, an algorithm to solve the UC problem using Lagrangian relaxation (LR) and particle swarm optimization (PSO). ELRPSO employs a state-of-the-art powerful PSO variant called comprehensive learning PSO to find a feasible near-optimal UC schedule. Each particle represents Lagrangian multipliers. The PSO uses a low level LR procedure, a reserve repairing heuristic, a unit decommitment heuristic, and an economic dispatch heuristic to obtain a feasible UC schedule for each particle. The reserve repairing heuristic addresses the spinning reserve and minimum up/down time constraints simultaneously. Moreover, the reserve repairing and unit decommitment heuristics consider committing/decommitting a unit for a consecutive period of hours at a time in order to reduce the total startup cost. Each particle is initialized using the Lagrangian multipliers obtained from a LR that iteratively updates the multipliers through an adaptive subgradient heuristic, because the multipliers obtained from the LR tend to be close to the optimal multipliers and have a high potential to lead to a feasible near-optimal UC schedule. Numerical results on test thermal power systems of 10, 20, 40, 60, 80, and 100 units demonstrate that ELRPSO is able to find a low-cost UC schedule in a short time and is robust in performance.     

系统分解协调算法的抽水蓄能电站优化调度

建立了包含抽水蓄能电站的电网统一调度优化模型,即以调度周期内火电燃料成本为最小目标函数,满足系统及各机组约束条件。利用系统分解协调思想,开发了一个结合拉格朗日松弛方法和粒子群优化算法的混合算法,将原优化问题分解为两层优化问题。上层拉格朗日算子优化利用次梯度算法求解,下层各子问题利用粒子群优化算法求解,经过迭代寻优得到最优对偶解后,利用一个启发式算法求得满足系统约束及各机组运行约束的原问题的可行解。最后通过算例验证了模型的合理性及算法的有效性。     

Power system operations scheduling using separable programming

The problem of electric power operations scheduling comprises maintenance scheduling, intermediate- and short-term hydro-thermal scheduling, unit commitment and production scheduling. Recognizing the inter-relationship among these subproblems, the effect on one another is analyzed. Computationally efficient formulations and algorithms for maintenance scheduling and intermediate- and short-term hydro-thermal scheduling are presented.A variation of the separable programming technique is presented. This results in a linear programming formulation of the nonlinear scheduling problem. This technique is applied to a hypothetical system containing nuclear, fossil, hydro-electric and pumped-storage units. The large problem of hydro-thermal scheduling due to the inclusion of a nuclear unit is decomposed into two stages. In the first stage, the relatively stable nuclear generation is optimized with respect to the generation from large fossil-steam units. Hourly generation levels for all units in the system are then determined in the second stage.     

Ramp rate constrained unit commitment by improved priority list and augmented Lagrange Hopfield network

This paper proposes an improved priority list (IPL) and augmented Hopfield Lagrange neural network (ALH) for solving ramp rate constrained unit commitment (RUC) problem. The proposed IPL-ALH minimizes the total production cost subject to the power balance, 15 min spinning reserve response time constraint, generation ramp limit constraints, and minimum up and down time constraints. The IPL is a priority list enhanced by a heuristic search algorithm based on the average production cost of units, and the ALH is a continuous Hopfield network whose energy function is based on augmented Lagrangian relaxation. The IPL is used to solve unit scheduling problem satisfying spinning reserve, minimum up and down time constraints, and the ALH is used to solve ramp rate constrained economic dispatch (RED) problem by minimizing the operation cost subject to the power balance and new generator operating frame limits. For hours with insufficient power due to ramp rate or 15 min spinning reserve response time constraints, repairing strategy based on heuristic search is used to satisfy the constraints. The proposed IPL-ALH is tested on the 26-unit IEEE reliability test system, 38-unit and 45-unit practical systems and compared to combined artificial neural network with heuristics and dynamic programming (ANN-DP), improved adaptive Lagrangian relaxation (ILR), constraint logic programming (CLP), fuzzy optimization (FO), matrix real coded genetic algorithm (MRCGA), absolutely stochastic simulated annealing (ASSA), and hybrid parallel repair genetic algorithm (HPRGA). The test results indicate that the IPL-ALH obtain less total costs and faster computational times than some other methods.     

Daily generation scheduling optimization with transmissionconstraints: a new class of algorithms

The authors describe a novel class of algorithm dealing with the daily generation scheduling (DGS) problem. These algorithms have been designed by adding artificial constraints to the original optimization problem; handling these artificial constraints by using a dual approach; using an augmented Lagrangian technique rather than a standard Lagrangian relaxation technique; and applying the auxiliary problem principle which can cope with the nonseparable terms introduced by the augmented Lagrangian. To deal with the DGS optimization problem these algorithms are shown to be more effective than classical ones. They are well suited to solve this DGS problem taking into account transmission constraints     

Coevolutionary Algorithm Based on Lagrangian Method for Hydrothermal Generation Scheduling

The coevolutionary algorithm (CEA) based on the Lagrangian method is proposed for hydrothermal generation scheduling. The main purpose of hydrothermal generation scheduling is to minimize the overall operation cost and the constraints satisfied by scheduling the power outputs of all hydro and thermal units under study periods, given electrical load and limited water resource. In the proposed method, a genetic algorithm is successfully incorporated into the Lagrangian method. The genetic algorithm searches out the optimum using multiple-path techniques and possesses the ability to deal with continuous and discrete variables. Regardless of the objective function characteristic the genetic algorithm does not have to modify the design rules and possesses the ability to go over local solutions toward the global optimal solution. The genetic algorithm can improve the disadvantages of the traditional Lagrangian method, which updates Lagrange multipliers according to the degree of system constraint violation by the gradient algorithm, and further searches out the global optimal solution. The developed algorithm is illustrated and tested on a practical Taiwan power system. Numerical results show that the proposed CEA based on the Lagrangian method is a very effective method for searching out the global optimal solution.     

A noise annealing neural network for hydroelectric generationscheduling with pumped-storage units

A new approach based on neural network is proposed for the hydroelectric generation scheduling with pumped-storage units at Taiwan power system. The purpose of hydroelectric generation scheduling is to determine the optimal amounts of generated powers for the hydro units in the system. To achieve an economical dispatching schedule for the hydro units including two large pumped-storage plants, a neural network is employed to reach a schedule in which total fuel cost of the thermal units over the study period is minimized. The neural network model presented can solve nonlinear constrained optimization problems with continuous decision variables. Incorporating the noise annealing concepts, the model is able to produce such a solution which is the global optimum of the original problem with probability close to 1. The proposed approach is applied to hydroelectric generation scheduling of Taiwan power system. It is concluded from the results that the proposed approach is very effective in reaching proper hydro generation schedules     

Optimal generation scheduling with ramping costs

In this paper, a decomposition method is proposed which relates the unit ramping process to the cost of fatigue effect in the generation scheduling of thermal systems. The objective of this optimization problem is to minimize the system operation cost, which includes the fuel cost for generating the required electrical energy and starting up decommitted units, as well as the rotor depreciation during ramping processes, such as starting up, shutting down, loading, and unloading. According to the unit fatigue index curves provided by generator manufacturers, fixed unit ramping-rate limits, which have been used by previous studies, do not reflect the physical changes of generator rotors during the ramping processes due to the fatigue effect. By introducing ramping costs, the unit on/offstates can be determined more economically by the proposed method. The Lagrangian relaxation method is proposed for unit commitment and economic dispatch, in which the original problem is decomposed into several subproblems corresponding to the optimization process of individual units. The network model is employed to represent the dynamic process of searching for the optimal commitment and generation schedules of a unit over the entire study time span. The experimental results for a practical system demonstrate the effectiveness of the proposed approach in optimizing the power system generation schedule     

Generation scheduling with thermal stress constraints

This paper describes a scheduling method for representing the thermal stress of turbine shafts as ramp rate constraints in the thermal commitment and dispatch of generating units. The paper uses Lagrangian relaxation for optimal generation scheduling. In applying the unit commitment, thermal stress over the elastic limit is used for calculating the ramping cost. The thermal stress contribution to generation cost requires the calculation of a set that includes thermal stress at the end of each time step; this requirement presents a complicated problem which cannot be solved by an ordinary optimization method such as dynamic programming. The paper uses an improved simulated annealing method to determine the optimal trajectory of each generating unit. Furthermore, the paper uses linear programming for economic dispatch in which thermal stress limits are incorporated in place of fixed ramp rate limits. The paper illustrates the economics of frequently ramping up/down of low cost generating units versus the cost of replacement of their turbine rotors with a shorter life span. The experimental results for a practical system demonstrate the effectiveness of the proposed method in optimizing the power system generation scheduling.     

发电经济调度可行解判据及其求解方法

发电经济调度问题是一个经典混合整数规划问题,然而用拉格朗日松弛法得到的对偶解对原问题通常是不可行的,要获得可行解必须先得到一种可行的机组组合.本文分析了现有可行化条件中存在的问题,提出了一个易于检验的可行化条件,并证明它是充分必要的.随后,介绍了获得可行解的方法.     

A Method for Solving the Fuel Constrained Unit Commitment Problem

The unit commitment problem involves finding the hourly commitment schedule for the thermal units of an electrical system, and their associated generation, over a period of up to a week. For some utilities, contractual or other factors limit the amount of fuel available to certain of the units or plants. This paper describes a new method which solves the unit commitment problem in the presence of fuel constraints. The method uses a Lagrangian decomposition and successive approximation technique for solving the unit commitment problem where the generation, reserve and fuel constraints are adjoined onto the cost function using Lagrange multipliers. All important operating constraints have been incorporated including minimum up and down times, standby operation, ramping limits, time-dependent start-up cost, spinning and supplemental reserve. The method is being applied to a production-grade program suitable for Energy Management Systems applications.     

具有爬升约束机组组合的充分必要条件

在Lagrangian松弛框架下，很难确定机组组合问题的一个可行解是否可通过调整对偶机组组合而获得。对于具有爬升约束的机组组合调度问题来说，由于机组出力在连续的2个开机区间的耦合性，求解可行解就更困难。在Lagrangian松弛框架下，开发1个机组组合新方法的核心是如何获得1个可行的机组组合。文中采用Benders分解可行性条件严格证明了在给定时段，机组组合可行的充分必要条件：即在该时段一个相应于系统负载平衡约束和旋转各用约束的不等式组成立。该条件不需要求解经济分配问题，就可以判定机组组合的可行性。有了此条件，可在发电功率经济分配前知道机组组合是否可行，若不可行，则可通过调整机组组合状态而获得可行的组合。该条件对于构造一个求解机组组合问题的系统方法是重要且有效的。数值测试表明该条件是判定机组组合可行性的有效方法。