Hydro Subproblems Research Articles

A two-stage planning model for power scheduling in a hydro-thermal system under uncertainty

A two-stage stochastic programming model for the short- or mid-term cost-optimal electric power production planning is developed. We consider the power generation in a hydro-thermal generation system under uncertainty in demand (or load) and prices for fuel and delivery contracts. 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 that couple power units. It is assumed that the stochastic load and price processes are given (or approximated) by a finite number of realizations (scenarios). Solving the dual by a bundle subgradient method leads to a successive decomposition into stochastic single 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 primal problem 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 400.000 binary and 650.000 continuous variables, and more than 1.300.000 constraints.

Multiperiod hydrothermal economic dispatch by an interior pointmethod

This paper presents an interior point algorithm to solve the multiperiod hydrothermal economic dispatch (HTED). The multiperiod HTED is a large scale nonlinear programming problem. Various optimization methods have been applied to the multiperiod HTED, but most neglect important network characteristics or require decomposition into thermal and hydro subproblems. The algorithm described here exploits the special bordered block diagonal structure and sparsity of the Newton system for the first order necessary conditions to result in a fast efficient algorithm that can account for all network aspects. Applying this new algorithm challenges a conventional method for the use of available hydro resources known as the peak shaving heuristic.

Multiperiod Hydrothermal Economic Dispatch by an Interior Point Method

This paper presents an interior point algorithm to solve the multiperiod hydrothermal economic dispatch (HTED). The multiperiod HTED is a large scale nonlinear programming problem. Various optimization methods have been applied to the multiperiod HTED, but most neglect important network characteristics or require decomposition into thermal and hydro subproblems. The algorithm described here exploits the special bordered block diagonal structure and sparsity of the Newton system for the first order necessary conditions to result in a fast efficient algorithm that can account for all network aspects. Applying this new algorithm challenges a conventional method for the use of available hydro resources known as the peak shaving heuristic.

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 200 000 binary and 350 000 continuous variables, and more than 500 000 constraints.

Scheduling hydrothermal power systems with cascaded and head-dependent reservoirs

An optimization-based algorithm is presented for scheduling hydrothermal power systems with cascaded and head-dependent reservoirs. Within the Lagrangian relaxation framework, the hydro river catchment subproblems are difficult to solve because of the continuous reservoir dynamics and constraints, discontinuous operating regions, discrete operating states and hydraulic coupling of cascaded reservoirs. The head-dependent water-power conversion adds another dimension of difficulty since the objective functions of hydro subproblems are no longer stage-wise additive with respect to water discharge. It is difficult to solve the subproblems by relaxing the reservoir limits or the hydraulic coupling among units as in previous work. A new algorithm with a novel relaxation structure is presented in this paper to solve hydro river catchment subproblems. Numerical testing based on the data sets of a practical system shows that this method is efficient and effective to deal with hydrothermal systems with cascaded and head-dependent reservoirs, and discrete hydro constraints. The results indicate that near-optimal schedules are obtained.

A bundle method for hydrothermal scheduling

Lagrangian relaxation has been widely used in hydrothermal scheduling. Complicating constraints are relaxed by multipliers which are usually updated by a subgradient method (SGM). The SGM suffers from slow convergence caused by the nondifferentiable characteristics of dual functions. This paper presents an algorithm that utilizes the bundle trust region method (BTRM) to update the multipliers within the Lagrangian relaxation framework. The BTRM is shown to converge faster than the SGM as well as other bundle type methods in optimizing nondifferentiable dual functions. The application of BTRM for solving hydro subproblems results in greatly improved convergence over the SGM. Comparing BTRM with another bundle type method in updating the high level multipliers shows that better solution can be obtained by BTRM.

Hydrothermal scheduling based Lagrangian relaxation approach to hydrothermal coordination

This paper presents an improved Lagrangian relaxation based hydrothermal coordination algorithm. By using Lagrangian multipliers to relax system-wide demand and reserve requirements, the problem is decomposed into thermal and hydro subproblems. The thermal subproblem is solved by using dynamic programming without discretizing generation levels. Instead of solving the hydro subproblems independently as done in the standard Lagrangian relaxation formulation, the hydrothermal scheduling is used to solve for the output levels of hydro units. The hydrothermal scheduling uses the commitment status of thermal units obtained from the thermal subproblem. Extensive constraints are considered such as power balance, spinning reserve, minimum up/down time, capacity limits, ramp rate and hydro constraints. Accurate transmission losses are incorporated and nonlinear cost functions are used. Numerical results based on a practical electric utility data show that this new approach consistently provides better schedules than the standard Lagrangian relaxation approach.

USE OF PUMPED-HYDRO AS PEAK-LOAD MANAGEMENT PLANT IN OPTIMAL SCHEDULING OF POWER SYSTEMS

ABSTRACT The system demand, during some intervals of a day, may exceed the available system generation (excluding the pumped-hydro generation). A pumped-hydro plant may then, be used as a peak-load management unit to safeguard the power system by minimizing the power blackout in order to avoid large deviation in system frequency. In this paper, an algorithm is presented to obtain the optimal schedule for hydro, thermal plants including the pumped-hydro unit by proper selection of initial feasible trajectory for the pumped-hydro plant. An additional constraint is introduced to ensure the power balance in each time interval during the perturbation of water storage trajectory. The proposed method decomposes the problem into hydro and thermal subproblems and solves them alternately. The hydro subproblem is solved using a search procedure namely, the local variation method and the thermal subproblem is solved using a judicious combination of participation factors/linear programming method. Optimal scheduling was conducted on a 9-bus system and a 66-bus Utility system. The results obtained for the above two systems demonstrate the effectiveness of the algorithm proposed.

Hydraulically coupled power‐plants commitment within short‐term operation planning in mixed hydro‐thermal power systems

AbstractA new method for short‐term operation planning and power exchange scheduling of interconnected hydro‐thermal power systems is proposed in the paper. The criterion function is the total variable profit obtained by sales of energy to customers within the system, exchange of energy with other systems, diminished by the cost of system operation. The method is based on Lagrangian relaxation, decomposition technique and successive solution of dual and primal problems. The dual and primal problems are decomposed and solved using appropriate optimization methods, while Lagrangian multipliers are corrected using subgradient method. Dual hydro subproblems are solved using linearization and network flow method. Special modification of the algorithm is made so as to enable commitment of hydro power plants, taking into account bilogical constraints on minimal flow of water downstream of the hydro power plant. The efficiency of the method is illustrated on a test problem.

An optimization-based algorithm for scheduling hydrothermal power systems with cascaded reservoirs and discrete hydro constraints

An optimization-based algorithm is presented for the short-term scheduling of hydrothermal power systems using the Lagrangian relaxation technique. This paper concentrates on the solution methodology for hydro subproblems with cascaded reservoirs and discrete hydro constraints. Continuous reservoir dynamics and constraints, discontinuous operating regions, discrete operating states and hydraulic coupling of cascaded reservoirs are considered in an integrated fashion. The key idea is to substitute out the reservoir dynamics and to relax the reservoir level constraints by using another set of multipliers, making a hydro subproblem unit-wise and stage-wise decomposable. The optimal generation level for each operating state at each hour can be obtained simply by minimizing a single variable function. Dynamic programming is then applied to optimize the operating states across the planning horizon with a small number of well structured transitions. A modified subgradient algorithm is used to update multipliers. After the dual problem converges, the feasible solution to the hydropower subsystem is obtained by using a network flow algorithm, with operating states obtained in the dual solutions, and possibly adjusted by heuristics. Numerical testing based on practical system data sets show that this method is efficient and effective for dealing with hydrothermal power systems with cascaded reservoirs and discrete hydroelectric constraints.

A flexible approach to short-term hydro-thermal coordination. II. Dual problem solution procedure

The dual problem formulation is the crucial point in all Lagrangian relaxation based hydro-thermal generation coordination approaches. Such an approach is presented in Part I of this work. In this paper, an original dual problem solution procedure is presented. The dual is divided into independent thermal and hydro subproblems. Thermal subproblems are solved using the dynamic programming method. Hydro subproblems are treated by employing the special approximation in the state-space within the multi-pass dynamic programming method. Particular attention is paid to hydraulically coupled reservoirs. The special structure of these hydro subproblems is used and the successive approximation is applied to make the dynamic programming method more effective. Simulations performed on a realistic power system are presented to illustrate the efficiency of the proposed approach and the sensitivity of the dual problem solution regarding values of Lagrangian multipliers.

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

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

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.

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.

Power generation scheduling for multi-area hydro-thermal systems with tie line constraints, cascaded reservoirs and uncertain data

The authors propose an approach to the short-term generation scheduling of hydro-thermal power systems (GSHT). The objective of GSHT is to minimize the total operation cost of thermal units over the scheduling time horizon. To solve the problem within a reasonable time, the problem is decomposed 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. A probabilistic method is employed in considering load forecasting errors and forced outages of generating units to satisfy system reliability requirements. For the hydro subsystem, network flow concepts are adopted to coordinate water use over the entire study time span and the reduced gradient method is used to overcome the linear characteristic of the network flow method to obtain the optimal solution. Three case studies for the proposed method are presented. >