Marian G. Marcovecchio

On the Computational Studies of Deterministic Global Optimization of Head Dependent Short-Term Hydro Scheduling

This paper addresses the global optimization of the short term scheduling for hydroelectric power generation. A tailored deterministic global optimization approach, denominated sHBB, is developed and its performance is analyzed. This approach is applied to the optimization of a mixed integer nonlinear programming (MINLP) model for cascades of hydro plants, each one with multiple turbines, and characterized by a detailed representation of the net head of water, and a nonlinear hydropower generation function. A simplified model is also considered where only the linear coefficients of the forebay and tailrace polynomial functions are retained. For comparison purposes, four case studies are addressed with the proposed global optimization strategy and with a commercial solver for global optimization. The results show that the proposed approach is more efficient than the commercial solver in terms of finding a better solution with a smaller optimality gap, using less CPU time. The proposed method can also find alternative and potentially more profitable power production schedules. Significant insights were also obtained regarding the effectiveness of the proposed relaxation strategies.

Deterministic optimization of the thermal Unit Commitment problem: A Branch and Cut search

Abstract This paper proposes a novel deterministic optimization approach for the Unit Commitment (UC) problem, involving thermal generating units. A mathematical programming model is first presented, which includes all the basic constraints and a set of binary variables for the on/off status of each generator at each time period, leading to a convex mixed-integer quadratic programming (MIQP) formulation. Then, an effective solution methodology based on valid integer cutting planes is proposed, and implemented through a Branch and Cut search for finding the global optimal solution. The application of the proposed approach is illustrated with several examples of different dimensions. Comparisons with other mathematical formulations are also presented.

Global Optimal Design of Mechanical Vapor Compression (MVC) Desalination Process

Abstract This paper deals with the optimal design of Mechanical Vapor Compression (MVC) desalination process. Precisely, a detailed mathematical model of the process and a deterministic global optimization algorithm are applied to determine the optimal design and operating conditions for the system. The resulting model involves the real-physical constraints for the evaporation process. Nonlinear equations in terms of chemical-physical properties and design equations (efficiencies, Non-Allowance Equilibrium, Boiling Point Elevation, heat transfer coefficients, momentum balances, among others) are used to model the process. The model has been solved by using a deterministic global optimization algorithm previously developed by the authors [7] and implemented in a General Algebraic Modeling System GAMS [1]. The generalized reduced gradient algorithm CONOPT 2.041 [2] is used as NLP local solver. The model was successfully solved for different seawater conditions (salinity and temperature) and fresh water production levels. The influence of the production requirements on the process efficiency as well as the algorithms performance is presented.

A deterministic optimization approach for the unit commitment problem

Reliable power production is critical to the profitability of electricity utilities. This concern, together with the need for less dependence on fossil fuels consumption and for CO2 mitigation, is leading to the prospective use of combined forms of conventional and alternative forms of energy generation as the most promising means to meet an increasing demand for electric power. Unit commitment (UC) arises in this context as a most critical decision process, involving a large number of interacting factors and underlying therefore a complex optimization problem. As such, the UC problem has been receiving a good deal of attention in the literature, with heuristic approaches being most dominant. As an alternative, a deterministic optimization approach is proposed in this paper and applied to the thermal UC problem. The model developed is a mixed integer quadratic programming problem (MIQP) having the objective of minimizing the fuel consumption (calculated by a quadratic function) and start up costs, with a strategy proposed for its solution that exploits the characteristics of the UC problem. This consists of valid integer cutting planes and a Branch and Bound (B&B) search, which are developed and combined resulting in a Branch and Cut (B&C) algorithm particular to the UC problem. The approach is described and implemented to solve a reference case study. Although the UC problem is NP-hard, the results show that the proposed technique is capable of providing the optimal solution for real-world sized instances.

Global optimal synthesis of integrated hybrid desalination plants

Abstract Seawater desalination is one of the main alternatives to overcome the problem of fresh water supply. Thermal and membrane processes are the conventional technologies widely used to obtain potable water from the sea. Hybrid desalination plants integrating reverse osmosis (RO) and multi stage flash (MSF) systems result interesting alternatives. The hybrid systems combine the advantages of the high separation efficiency of thermal processes with the low energy consumption of reverse osmosis. These systems reduce dramatically the final fresh water cost; meanwhile the recovery ratio is increased. In the present work, a mathematical model for MSF and RO hybrid systems is derived on energy, mass and momentum balances meanwhile the most important aspect of the processes are introduced. Precisely, a superstructure is proposed embedding alternative and feasible arrangements for hybrid plants that will be simultaneously optimized. The resulting mathematical model is solved in order to determinate the global optimal plant layout, optimal equipment sizes and operating conditions, minimizing the fresh water cost. The cost equations evaluate accurately the presence or absence of equipments, streams or systems. The model was successfully solved for different seawater conditions (salinity and temperature). The optimal solutions depend strongly on the salt concentration and temperature of the feed water. Due to space limitations only one case study is presented here.

Hybrid Desalination Systems: Alternative Designs of Thermal and Membrane Processes.

Abstract In this paper, desalination systems integrating thermal and membrane processes are investigated. Specifically, a hybrid desalination plant integrating reverse osmosis (RO) and multi stage flash (MSF) systems is mathematically modeled. The non linear programming problem is developed in order to optimize the configuration and operating conditions in order to satisfy fresh water demands at minimum costs. Both implementations, with one and two reverse osmosis stages are considered at the particular framework RO-MSF adopted. The system is modeled in such a way that the RO unit receives brine feed from the MSF. In addition, partial extractions of flashing brine stream from flash chambers of MSF can be performed in order to feed the RO system. In fact, the flow-patterns and flow-rates of the brine streams of MSF are optimization variables. Heat transfer areas of pre-heaters and the geometric design of stages are design variables to be optimized. The optimization procedure will also decide between one and two RO stages and the number of modules operating in parallel at each stage. Moreover, all the operative conditions will be optimized. Different case studies considering various seawater conditions were successfully solved without convergence difficulties. It can be concluded that the optimal arrangement of hybrid system depends strongly on the seawater conditions (salinity and temperature) and the fresh water demand as well.

Improve-and-Branch Algorithm for the Global Optimization of Nonconvex NLP Problems

A new algorithm to solve nonconvex NLP problems is presented. It is based on the solution of two problems. The reformulated problem RP is a suitable reformulation of the original problem and involves convex terms and concave univariate terms. The main problem MP is a nonconvex NLP that outer-approximates the feasible region and underestimate the objective function. MP involves convex terms and terms which are the products of concave univariate functions and new variables. Fixing the variables in the concave terms, a convex NLP that overestimates the feasible region and underestimates the objective function is obtained from the MP. Like most of the deterministic global optimization algorithms, bounds on all the variables in the nonconvex terms must be provided. MP forces the objective value to improve and minimizes the difference of upper and lower bound of all the variables either to zero or to a positive value. In the first case, a feasible solution of the original problem is reached and the objective function is improved. In general terms, the second case corresponds to an infeasible solution of the original problem due to the existence of gaps in some variables. A branching procedure is performed in order to either prove that there is no better solution or reduce the domain, eliminating the local solution of MP that was found. The MP solution indicates a key point to do the branching. A bound reduction technique is implemented to accelerate the convergence speed. Computational results demonstrate that the algorithm compares very favorably to other approaches when applied to test problems and process design problems. It is typically faster and it produces very accurate results.

Unit Commitment Scheduling Including Transmission Constraints: a MILP Formulation

Abstract The benefits of good scheduling of electrical units are widely known (De la Torre et al, 2008). In this paper, a new approach to account for the Security-Constrained Unit Commitment (SCUC) is presented. A model is developed as a deterministic optimization problem, giving rise to a MILP formulation. Demand, reserve, and unit constraints are taken from a previous paper (Marcovecchio et al, 2014). Transmission constraints including buses balance, lower and upper bound for line power flows, and bus voltage angle constraints are included in this paper. Identification of loops matrix is not necessary as it is the case in several formulations (Stagg et al, 1968). Scheduling was solved for a 6-bus 3-generator and 11 transmission line problem and a 31-bus 16-generator and 43-transmission line problem. Computational times are very low, being 0.189 and 118.160 CPU sec. respectively. Relationship between capacity usage and occupied time for all generators is analyzed. In a similar way, power flow in each line related to the power flow in output and input bus connected to each line is addressed. This information is depicted by simple graphs.

Security constrained unit commitment scheduling: A new MILP formulation for solving transmission constraints

Abstract This paper presents a new Mixed Integer Linear Programming model (MILP) to account for the Security-Constrained Unit Commitment Problem (SCUC). Transmission Constraints are introduced through bus balances, line power bound flows, and bus voltage angle differences. Line status is also considered. Binary variables regarding line status (active or inactive) are introduced for this purpose. These variables allow discrete decisions on the connectivity of buses, reducing the angle coupling between buses, with several advantages. Three examples are solved. The results indicate that this method can obtain feasible solutions with CPU times of 2.5 s (for a 6-bus system) and 500 s (for the IEEE 118-bus system), and they reached cost savings up to 4.9% of the total generating cost for one day of programming horizon, in comparison with classical models. Relations of the network are illustrated graphically, and an analysis of the results is presented through new evaluation indexes.