Hydrothermal Coordination Research Articles

Interior point methods application in optimum operational scheduling of electric power systems

The topic of interior point (IP) methods has been one of the most active research areas since it was rediscovered by Karmarkar in 1984. A variety of IP algorithms have been applied to many areas of power systems optimisation. This study presents a survey of literature on the application of IP methods to planning and operation of power systems. A review of research papers that have been published since 1994 in three major power system optimisation problems is presented. The problems are economic dispatch, unit commitment and hydrothermal coordination.

Optimization methods applied for solving the short-term hydrothermal coordination problem

Short-term hydrothermal coordination (STHTC) is a very complicated optimization problem. It is a dynamic large-scale non-linear problem and requires solving unit commitment and economic power load dispatch problems. From this perspective, many successful and powerful optimization methods and algorithms have been employed to solve this problem. These optimization methodologies and techniques are widely diverse and have been the subject of ongoing enhancements over the years. This paper presents a survey of literature on the various optimization methods applied to solve the STHTC problem. A review and a methodology-based classification of most of the publications on the topic are presented.

Price of Water

Price of WaterThere are watercourses on the globe which as yet do not deliver up their energy to the needs of the people. How much energy their waters bear, is it worth to take away this energy? Those and alike questions must be (and they are) answered before start to build hydro power station. Similar problems must be solved to control hydro power plants in most gainful way which is known as hydrothermal coordination. The notion of price of water can be met lately in technical literature as one of numerical indices of these issues. The gross price of water and net price of water are considered in this paper. Gross price of 1 t water is the price of electric energy obtained by conversion of potential energy of 1 t of water, lifted to a height of power station water head. Net price of water is the difference between gross price and total expenses determined by hydro power station building and its exploitation costs in a year related to 1 m3of water. If net price of water is positive, it is worth building power station. The greater net price is, the more urgent is the building. Net price of water grows with water head but it continues only to some height of the dam because further increase of head sharply increases capital outlay and other exploitation expenses. To maximize net price of water, optimization of net price function can be done. Net price of water diminishes when some amount of water is diverted for other needs. When amount of diverted water is out of discussion, no controversy can emerge. However when by diverted water some goods with some monetary worth can be obtained, the task must be solved how much water can be diverted so that the water of watercourse be used to the maximum benefit. The environmental issues must be taken into account as well.

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.

Short-term hydrothermal coordination considering an AC network modeling

In short-term hydrothermal coordination, the transmission network is typically modeled through a DC power flow. However, this modeling can lead to inoperable solutions when verifying with AC power flow. This work presents a methodology that includes an AC power flow model to overcome the above problem. The approach takes into account issues like congestion management and quality control of service which are often present in large and weakly meshed networks—the typical pattern of South American electrical power systems. Generalized Benders decomposition, together with more traditional and well-known optimization techniques are used for the solution of this problem. The Master Problem stage defines the generation level by regarding the inter-temporal constraints, whereas the sub-problem stage determines both the active and reactive economical dispatch for each step of the load curve. It meets the electrical constraints through a modified AC optimal power flow. The methodology has been proven over a 9-bus hydrothermal system and on actual system to verify the robustness of convergence.

Dynamic optimal power flow approach to account for consumer response in short term hydrothermal coordination studies

This paper deals with the representation of price-responsive loads in the short term coordination of predominantly hydroelectric hydrothermal systems. The proposed approach allows the representation of cross-relationships between demand and price during different dispatch intervals of a given time horizon. Also, consumer energy constraints which behave as dynamic restrictions can be taken into account. By that means, a load shifting program in which the consumers' overall energy consumption remains constant throughout the time horizon can be simulated. The short-term hydrothermal coordination is solved as a single optimisation problem employing the primal–dual interior point method. Realistic subsystems, based on the Brazil interconnected power system, are used to illustrate the effect of price-responsive loads on reservoir storages.

Hydrothermal coordination using modified mixed integer hybrid differential evolution

This work presents a novel modified Mixed Integer Hybrid Differential Evolution (MIHDE) algorithm, for solving the unit commitment problem of a hydrothermal power system. Hydrothermal scheduling involves the optimisation of a non-linear objective function with a set of system and hydraulic constraints. Discrete and dynamic constraints such as unit start-up/shutdown and minimum-up/minimum-downtime limits are also included in the hydro unit commitment. The problem of hydrothermal coordination is a mixed integer non-linear optimisation problem, which is well-suited for the use of MIHDE. In this paper, the MIHDE algorithm is modified in order to handle the reservoir end-volume and load balance constraints in the hydrothermal scheduling. The performance of the proposed approach is validated by illustration with standard test systems. The results of the proposed approach are compared with those techniques reported in literature. From the numerical results, it is found that the modified MIHDE-based approach is able to provide better solution at a relatively lesser computational effort.

Short-term hydrothermal optimisation with congestion and quality of service constraints

In short-term hydrothermal coordination (STHC), the transmission network is typically modelled through a DC power flow. However, this modelling can lead to inoperable solutions when verifying with AC power flow. A methodology that includes an AC power flow model to overcome the problem applied to STHC is presented. The approach takes into account issues such as congestion management and control of quality of service, which are often present in large and weakly meshed networks – the typical pattern of Latin American electrical power systems. Generalised Benders' decomposition together with more traditional and well-known optimisation techniques, is used for this problem. The master problem stage defines the generation levels by considering the inter-temporal constraints, whereas the sub-problem stage determines both the active and reactive economical dispatches for each step of the load curve. It meets the electrical constraints (nodal balance, transmission limits and voltage levels) through a modified AC optimal power flow. The methodology was proved over a nine-busbar hydrothermal system and the solution found was validated with a quasi-exhaustive enumeration procedure to prove the optimality of the solution. Also proved over large system was the feasibility to realistic systems.

The Short-Term Hydrothermal Coordination via Genetic Algorithms

This article presents a solution for the short-term hydrothermal generation scheduling problem (HGSP) by using a genetic algorithm. The hydrothermal coordination is solved using an equivalent cost function for the available thermal units, while the unit commitment and the economic dispatch are solved by considering the more capable individuals obtained from the hydrothermal coordination stage. Quadratic cost functions for the thermal units and a second grade equation depending on the volume and the flow for the active power calculation of the hydraulic generation are considered. The appropriateness of the proposition is proved in two power systems: The first system is composed of 4 hydraulic power stations and 10 thermal units, while the second system is composed of 16 hydraulic power stations and 30 thermal ones.

Price-Based Unit Commitment: A Case of Lagrangian Relaxation Versus Mixed Integer Programming

This paper formulates the price-based unit commitment (PBUC) problem based on the mixed integer programming (MIP) method. The proposed PBUC solution is for a generating company (GENCO) with thermal, combined-cycle, cascaded-hydro, and pumped-storage units. The PBUC solution by utilizing MIP is compared with that of Lagrangian relaxation (LR) method. Test results on the modified IEEE 118-bus system show the efficiency of our MIP formulation and advantages of the MIP method for solving PBUC. It is also shown that MIP could be applied to solve hydro-subproblems including cascaded-hydro and pumped-storage units in the LR-based framework of hydro-thermal coordination. Numerical experiments on large systems show that the MIP-based computation time and memory requirement would represent the major obstacles for applying MIP to large UC problems. It is noted that the solution of large UC problems could be accomplished by improving the MIP formulation, the utilization of specific structure of UC problems, and the use of parallel processing.

STOCHASTIC MARKET EQUILIBRIUM MODEL FOR GENERATION PLANNING

It is widely accepted that medium-term generation planning can be advantageously modeled through market equilibrium representation. There exist several methods to define and solve this kind of equilibrium in a deterministic way. Medium-term planning is strongly affected by uncertainty in market and system conditions; thus, extensions of these models are commonly required. The main variables that should be considered as subject to uncertainty include hydro conditions, demand, generating units' failures, and fuel prices. This paper presents a model to represent a medium-term operation of the electrical market that introduces this uncertainty in the formulation in a natural and practical way. Utilities are explicitly considered to be maximizing their expected profits, and biddings are represented by means of a conjectural variation. Market equilibrium conditions are introduced by means of the optimality conditions of a problem, which has a structure that strongly resembles classical optimization of hydrothermal coordination. A tree-based representation to include stochastic variables and a model based on it are introduced. This approach to market representation provides three main advantages: Robust decisions are obtained; technical constraints are included in the problem in a natural way, additionally obtaining dual information; and large-size problems representing real systems in detail can be addressed.

An efficient method for nonlinearly constrained networks

Many nonlinear network flow problems (in addition to the balance constraints in the nodes and capacity constraints on the arc flows) have nonlinear side constraints, which specify a flow relationship between several of the arcs in the network flow model. The short-term hydrothermal coordination of electric power generation is an example of this type. In this work we solve this kind of problem using an approach in which the efficiency of the well-known techniques for network flow can be preserved. It lies in relaxing the side constraints in an augmented Lagrangian function, and minimizing a sequence of these functions subject only to the network constraints for different estimates of the Lagrange multipliers of the side constraints. This method gives rise to an algorithm, which combines first- and superlinear-order multiplier methods to estimate these multipliers. When the number of free variables is very high we can obtain a superlinear-order estimate by means of the limited memory BFGS method fitted to our problem. An extensive computational comparison with other methods has been performed. The numerical results reported indicate that the algorithm described may be employed advantageously to solve large-scale network flow problems with nonlinear side constraints.

On centralized power pool auction: a novel multipliers stabilization procedure

This paper addresses the Short-Term Hydro-Thermal Coordination (STHTC) problem. It is a large-scale, combinatorial and nonlinear optimization problem. It is usually solved using a Lagrangian Relaxation (LR) approach. LR procedure is based on the solution of the dual problem of the original one. The dual problem variables are the Lagrange multipliers. These multipliers have an economic meaning: electric energy hourly prices. This paper focuses in an efficient solution of the dual problem of the STHTC problem. A novel multiplier stabilization technique, which significantly improves the quality of the solution, is presented. The provided method could be the optimization tool used by the Independent System Operator of a centralized Power Pool. The solution procedure diminishes the conflict of interest in determining energy prices. A realistic large-scale case study illustrates the behavior of the presented approach.

A Genetic Algorithm Solution Approach to the Hydrothermal Coordination Problem

In this paper, a genetic algorithm solution to the hydrothermal coordination problem is presented. The generation scheduling of the hydro production system is formulated as a mixed-integer, nonlinear optimization problem and solved with an enhanced genetic algorithm featuring a set of problem-specific genetic operators. The thermal subproblem is solved by means of a priority list method, incorporating the majority of thermal unit constraints. The results of the application of the proposed solution approach to the operation scheduling of the Greek Power System, comprising 13 hydroplants and 28 thermal units, demonstrate the effectiveness of the proposed algorithm.

A nonlinear optimization package for long-term hydrothermal coordination

Long-term hydrothermal coordination is one of the main problems to be solved by an electric utility. Its solution provides the optimal allocation of hydraulic, thermal and nuclear resources at the different intervals of the planning horizon. The purpose of the paper is two-fold. Firstly, it presents a new package for solving the hydrothermal coordination problem. The model implemented accurately describes both the hydraulic and thermal subsystems. The resulting large-scale nonlinear and nonconvex problem is solved through either Minos or Snopt, two state-of-the-art nonlinear optimization packages. The second objective of the paper is to deliver to the optimization community the package. Since it implements a difficult, real, and large-scale problem, it can be considered a good test for new nonlinear optimization solvers. Computational results of the package in the solution of a set of real instances of up to 32,000 nonlinear variables, 13,000 linear and 2500 nonlinear constraints are reported.

Medium-term hydrothermal coordination by semidefinite programming

Hydrothermal coordination (HTC) is a problem that has been solved using direct and decomposition solution methods. The latter has shown shorter solution times than the former. A direct solution method for the HTC problem that is based in semidefinite programming (SDP) is presented in this paper. SDP is a convex programming method with polynomial solution time. The variables of the problem are arranged in a vector, which is used to construct a positive-definite matrix; the optimal solution is then found in the cone defined by the set of positive-definite matrices. An HTC problem can be formulated as a convex optimization problem without explicitly stating the integer value requirements for the thermal-plants discrete variables. Thus, it is possible to replace the nonconvex integer-value constraints by convex quadratic constraints, and then use SDP. Due to its polynomial complexity, it is not necessary to use decomposition or other tools for discrete optimization, such as enumeration schemes or other exponential-time procedures. No initial relaxation is necessary when applying a SDP algorithm; the solution shows only minor mismatches in the integer variables, which are easily corrected by a heuristic method. Different size test cases are presented. The solution quality is assessed by comparing with that produced by a Lagrangian relaxation method.

Short-term hydrothermal generation scheduling model using a genetic algorithm

A new model to deal with the short-term generation scheduling problem for hydrothermal systems is proposed. Using genetic algorithms (GAs), the model handles simultaneously the subproblems of short-term hydrothermal coordination, unit commitment, and economic load dispatch. Considering a scheduling horizon period of a week, hourly generation schedules are obtained for each of both hydro and thermal units. Future cost curves of hydro generation, obtained from long and mid-term models, have been used to optimize the amount of hydro energy to be used during the week. In the genetic algorithm (GA) implementation, a new technique to represent candidate solutions is introduced, and a set of expert operators has been incorporated to improve the behavior of the algorithm. Results for a real system are presented and discussed.

Wheeling cost influence in hydrothermal dispatch and series compensation allocation

This paper presents a methodology for optimizing both the transmission tariff and the thermal generation costs, considering the series compensation (SC) allocation problem in a multiperiod dispatch. The inclusion of the tariff cost in the objective function of the hydrothermal coordination problem can change the dispatch of remote hydro plants, especially if the SC is considered. The proposed methodology is important because the transmission cost tariff can affect the network users in a competitive environment. The OPF problem is based on the nonlinear primal-dual interior point technique associated with the Benders decomposition approach. The algorithm is tested using an equivalent of the southern system of Brazil and compared with existing techniques.

An implementation of Newton-like methods on nonlinearly constrained networks

The minimization of a nonlinear function with linear and nonlinear constraints and simple bounds can be performed by minimizing an augmented Lagrangian function, including only the nonlinear constraints. This procedure is particularly interesting when the linear constraints are flow conservation equations, as there exist efficient techniques for solving nonlinear network problems. It is then necessary to estimate their multipliers, and variable reduction techniques can be used to carry out the successive minimizations. This work analyzes the possibility of estimating the multipliers of the nonlinear constraints using Newton-like methods. Also, an algorithm is designed to solve nonlinear network problems with nonlinear inequality side constraints, which combines first and superlinear-order multiplier methods. The computational performance of this method is compared with that of MINOS 5.5. Scope and purpose Real problems with the structure of a nonlinear network flow problem with nonlinear side constraints exist and have a high dimensionality (e.g. the short-term hydrothermal coordination of electric power generation). They often need to be solved and it is important to find the procedure that will solve them with the highest efficiency. In previous works this author has analyzed the viability of estimating the multipliers of the nonlinear constraints from the Kuhn–Tucker system in order to improve the inexpensive Hestenes–Powell update. However, the conclusion was that there is no significant advantage for multipliers estimated through the Kuhn–Tucker system. The purpose of this paper is mainly to use the convergence properties of the Newton-like methods in order to develop a new estimator of the nonlinear constraint multipliers that improves the efficiency and the robustness of this augmented Lagrangian method.

Medium-Term Hydro-Thermal Coordination by Semidefimite Programming

Hydrothermal coordination (HTC) is a problem that has been solved using direct and decomposition solution methods. The latter has shown shorter solution times than the former. A direct solution method for the HTC problem that is based in semidefinite programming (SDP) is presented. SDP is a convex programming method with polynomial solution time. The variables of the problem are arranged in a vector, which is used to construct a positive-definite matrix; the optimal solution is then found in the cone defined by the set of positive-definite matrices. An HTC problem can be formulated as a convex optimization problem without explicitly stating the integer value requirements for the thermal-plants discrete variables. Thus, it is possible to replace the nonconvex integer-value constraints by convex quadratic constraints and then use SDP. Due to its polynomial complexity, it is not necessary to use decomposition or other tools for discrete optimization, such as enumeration schemes or other exponential-time procedures. No initial relaxation is necessary when applying a SDP algorithm; the solution shows only minor mismatches in the integer variables, which are easily corrected by a heuristic method. Different size test cases are presented. The solution quality is assessed by comparing with that produced by a Lagrangian relaxation method.