Optimization-based scheduling of hydrothermal power systems withpumped-storage units

This paper presents an optimization-based method for scheduling hydrothermal systems based on the Lagrangian relaxation technique. After system-wide constraints are relaxed by Lagrange multipliers, the problem is converted into the scheduling of individual units. This paper concentrates on the solution methodology for pumped-storage units. There are, many constraints limiting the operation of a pumped-storage unit, such as pond level dynamics and constraints, and discontinuous generation and pumping regions. The most challenging issue in solving pumped-storage subproblems within the Lagrangian relaxation framework is the integrated consideration of these constraints. The basic idea of the method is to relax the pond level dynamics and constraints by using another set of multipliers. The subproblem is then converted into the optimization of generation or pumping; levels for each operating state at individual hours, and the optimization of operating states across hours. The optimal generation or pumping level for a particular operating state at each hour can be obtained by optimizing a single variable function without discretizing pond levels. Dynamic programming is then used to optimize operating states across hours with only a few number of states and transitions. A subgradient algorithm is used to update the pond level Lagrangian multipliers. This method provides an efficient way to solve a class of subproblems involving continuous dynamics and constraints, discontinuous operating regions, and discrete operating states     

Lagrangian relaxation hybrid with evolutionary algorithm for short-term generation scheduling

Short-term generation scheduling is an important function in daily operational planning of power systems. It is defined as optimal scheduling of power generators over a scheduling period while respecting various generator constraints and system constraints. Objective of the problem includes costs associated with energy production, start-up cost and shut-down cost along with profits. The resulting problem is a large scale nonlinear mixed-integer optimization problem for which there is no exact solution technique available. The solution to the problem can be obtained only by complete enumeration, often at the cost of a prohibitively computation time requirement for realistic power systems. This paper presents a hybrid algorithm which combines Lagrangian Relaxation (LR) together with Evolutionary Algorithm (EA) to solve the problem in cooperative and competitive energy environments. Simulation studies were carried out on different systems containing various numbers of units. The outcomes from different algorithms are compared with that from the proposed hybrid algorithm and the advantages of the proposed algorithm are briefly discussed.     

Group search optimization for economic fuel scheduling

The economical use of fuel available for the generation of power has become a major concern of electric utilities. This paper presents an approach for economic fuel scheduling problem by using group search optimization. This is a minimization technique that includes the standard load constraints as well as the fuel constraints. The generation schedule is compared to that which would result if fuel constraints were ignored. The comparison shows that fuel consumed can be adequately controlled by adjusting the power output of various generating units so that the power system operates within its fuel limitations and within contractual constraints. It has been found that small additional amount of fuel may be required to serve the same power demand but the additional cost of this fuel may well compensate for the penalty that might otherwise be imposed for not maintaining the fuel contract. Numerical results for two test systems have been presented and the test results obtained from group search optimization are compared with those obtained from particle swarm optimization and evolutionary programming.     

Hydro-thermal unit commitment problem using simulated annealing embedded evolutionary programming approach

This paper presents a new approach to solve the hydro-thermal unit commitment problem using Simulated Annealing embedded Evolutionary Programming approach. The objective of this paper is to find the generation scheduling such that the total operating cost can be minimized, when subjected to a variety of constraints. A utility power system with 11 generating units in India demonstrates the effectiveness of the proposed approach; extensive studies have also been performed for different IEEE test systems consist of 25, 44 and 65 units. Numerical results are shown comparing the cost solutions and computation time obtained by conventional 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     

Optimal Scheduling of Spinning Reserve with Transmission Constraints

An effective method is proposed to schedule spinning reserve optimally. The method considers the transmission constraint in the whole scheduling process. To get the feasible solution faster, transmission line limits are first relaxed using the Lagrangian Relaxation technique. In the economic dispatch, after unit generation and spinning reserve are allocated among the committed units to satisfy the system andunit constraints, the schedule is then modified by a linear programming algorithm to avoid line overloads. The schedule is then updated by a probabilistic reserve assessment to meet a given risk index. The optimal value of the risk index is selected via a cost/benefit analysis based on the tradeoff between the total Unit Commitment (UC) schedule cost and the expected cost of energy not served. Finally, a unit decommitment technique is incorporated to solve the problem of reserve over-commitment in the Lagrangian Relaxation–based UC. The results of reserve scheduling with the transmission constraint are shown by the simulation runs performed on the IEEE reliability test system.     

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.     

A solution to the ramp rate and prohibited operating zone constrained unit commitment by GHS-JGT evolutionary algorithm

Unit commitment (UC) problem on a large scale with the ramp rate and prohibited zone constraints is a very complicated nonlinear optimization problem with huge number of constraints. This paper presents a new hybrid approach of ’Gaussian Harmony Search’ (GHS) and ’Jumping Gene Transposition’ (JGT) algorithm (GHS-JGT) for UC problem. In this proposed hybrid GHS-JGT for UC problem, scheduling variables are handled in binary form and other constants directly through optimum conditions in decimal form. The efficiency of this method is tested on ten units, forty units and hundred units test system. Simulation results obtained by GHS-JGT algorithm for each case show a better generation cost in less time interval, in comparison to the other existing results.     

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.     

Differential evolution technique-based short-term economic generation scheduling of hydrothermal systems

This paper presents differential evolution (DE)-based optimization technique for solving short-term economic generation scheduling of hydrothermal systems. A multi-reservoir cascaded hydrothermal system with non-linear relationship between water discharge rate, power generation and net head is considered here. The water transport delay between the connected reservoirs is also taken into account. Several equality and non-equality constraints on thermal units as well as hydro units and the effect of valve-point loading are also included in the problem formulation. The effectiveness of the proposed method is demonstrated on two test systems comprising of hydro and thermal units. Convergence characteristic of the proposed technique has been found to be quite satisfactory. The results obtained by the proposed technique are compared with other evolutionary methods. It is seen that the proposed technique is capable of producing encouraging solutions.     

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.     

Short-term generation scheduling under a SO2 emissions allowances market

This paper presents a model to carry out the short-term scheduling of a thermal generation park ruled by a SO₂ emissions allowances market. Units are allowed to operate purchasing emissions allowances within the market or paying penalties due to the emissions excess. Each unit is lineally separated into three fictitious units which represent different situations of operation for it within the market: to have surplus allowances, to have to buy allowances and to have to pay a penalty. In order to solve the problem, Genetic Algorithm combined with Lagrangian Relaxation are used. The transmission network is modeled through DC Load Flow equations. The model is applied to a test system of 24 bus and 26 units. Results show that the model is a simple and useful method to handle operational alternatives within an allowances market.     

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.     

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     

Security-constrained optimal generation scheduling for GENCOs

This paper presents an approach for maximizing a GENCO's profit in a constrained power market. The proposed approach considers the Interior Point Method (IPM) and Benders decomposition for solving the security-constrained optimal generation scheduling (SC-GS) problem. The master problem represents the economic dispatch problem for a GENCO which intends to optimize its profit. The formulation of the master problem does not bear any transmission network constraints. The subproblem will be used by the same GENCO to check the viability of its proposed bidding strategy in the presence of transmission network constraints. In this case if the subproblem does not yield a certain level of financial return for the GENCO or if the subproblem results in an infeasible solution of the GENCO's proposed bidding strategy, the GENCO will modify its proposed solution according to the Benders cuts that stem out of the subproblem. The study shows a more flexible scheduling paradigm for a GENCO in a competitive arena. The proposed approach proves practical for modeling the impact of transmission congestion on a GENCO's expected profit in a competitive environment.     

Economic efficiency of coordinated multilateral trades in electricity markets

This paper presents economic efficiency evaluation of electricity markets operating on the basis of a coordinated multilateral trading concept. The evaluation accounts for the overall costs of power generation, network losses, and system and unit constraints. We assume a non-collusive oligopolistic competition. An iterative Cournot model is used to characterize the competitive behavior of suppliers. A supplier maximizes the profit of each of his generating units while taking rivals' generation as given. Time span is over multiple hours. This leads to a mixed integer non-linear programming problem. We use the augmented Lagrangian approach to solve iteratively for globally optimal schedules. An IEEE 24-bus, 8-supplier, and 17-customer test system is used for illustration. The results show that such a market at times of light demands exhibits little market power, and at times of large demands exhibits a great deal of market power. This contrasts with the PCMI and HHI concentration measures, which give fixed measurement values of market power. The results of two-year (730 round) market simulations show a range of deadweight efficiency loss between 0.9 and 6% compared to that of PoolCo which results in a range between 0.5 and 10% for the same test case.     

A solution method using neural networks for the generator commitment problem

This paper studies the feasibility of applying the Hopfield-type neural network to unit commitment problems in a large power system. The unit commitment problem is to determine an optimal schedule of what thermal generation units must be started or shut off to meet the anticipated demand; it can be formulated as a complicated mixed integer programming problem with a number of equality and inequality constraints. In our approach, the neural network gives the on/off states of thermal units at each period and then the output power of each unit is adjusted to meet the total demand. Another feature of our approach is that an ad hoc neural network is installed to satisfy inequality constraints which take into account standby reserve constraints and minimum up/down time constraints. The proposed neural network approach has been applied to solve a generator scheduling problem involving 30 units and 24 time periods; results obtained were close to those obtained using the Lagrange relaxation method.     

A new approach to security-constrained generation scheduling of large-scale power systems with a piecewise linear ramping model

This paper presents a new and efficient approach to determine security-constrained generation scheduling (SCGS) in large-scale power systems, taking into account dispatch, network, and security constraints in pre and post-contingency states. A novel ramp rate limit is also modeled as a piecewise linear function in the SCGS problem to reflect more practical characteristics of the generating units. Benders decomposition is applied to this constrained solution process to obtain an optimal SCGS problem based on mixed-integer nonlinear programming (MINLP). The formulation can be embedded in two stages. First, a MIP is formulated in the master problem to solve a unit commitment (UC) problem. This stage determines appropriate on/off states of the units. The second stage, the subproblem, is formulated as a NLP to solve a security-constrained economic dispatch (SCED) problem. This stage is used to determine the feasibility of the master problem solution. It provides information to formulate the benders cuts that connect both problems. The proposed approach is tested in the IEEE 118-bus system to show its effectiveness. The simulation results are more realistic and feasible, whilst assuring an acceptable computation time.     

精细化日发电计划模型与方法

电网调度精细化管理水平的提高客观上要求日发电计划的精细化管理。针对传统日发电计划功能单一且难以实现准确的安全校核与网损管理的问题,提出了精细化日发电计划模型及优化算法。该模型以电网状态估计的网络拓扑和参数构造约束条件,决策目标兼顾电网经济性与安全裕度价值,具有应对电网运行不确定性的能力。采用交直流混合迭代优化算法,提出了交流潮流分析与有功优化的一体化决策方法,工程实用性强,有助于实现网损管理和安全校核工作的精细化管理。最后采用IEEE 30节点算例验证了算法的合理性。     

A Practical Model for Generator Maintenance Scheduling with Transmission Constraints

In this paper an optimal maintenance scheduling of generating units in a power system has been developed with transmission network representation. Here a DC load flow has been embedded in the maintenance model to include network constraints resulting in a more practical maintenance schedule. The model developed here uses the minimization of system cost (production cost plus the unserved energy cost) as the objective criterion, whereas the reliability objective function used is the minimization of unserved energy. The optimization is achieved by integer linear programming. The incorporation of transmission network adds significant complexity to maintenance scheduling. The proposed model enables almost all practical maintenance scheduling constraints to be handled easily. The optimization has been carried out to minimize the cost function considering different cases (i.e., with and without incorporation of the transmission network). The effectiveness of the proposed method has been demonstrated by obtaining numerical results on sample and real scale test systems. A comparison of the cost objective function clearly indicates that the maintenance schedule obtained from the simple generation model alone is more expensive than the one with transmission, and that there is a considerable degree of suboptimality in the former case.