Concluding Comments

The presence of complementarity constraints in decomposition-based design optimization models poses a numerical challenge which existing coordination approaches for decomposition-based design optimization could usually not handle. This chapter stated the research problem of multidisciplinary design optimization with complementarity constraints (MDO-CC), a generalized decomposition-based, complementarity model for engineering system design optimization that relates decomposition-based design optimization to mathematical program with complementarity constraints. A first approach to solving the MDO-CC problem has been presented along the direction of augmented Lagrangian decomposition (ALD). The correspondence of stationarity conditions between the AIO formulation and the ALD formulation has been established. Another approach for MDO-CC has been presented based on regularization and inexact penalty decomposition (RIPD) techniques. As an important contribution, it has been shown that existing theories can be adapted to map a limit point of stationary solutions of the parameterized RIPD formulation to a strongly-stationary solution of the AIO formulation. Following this result, a solution algorithm for the MDO-CC has been proposed with potential implementations of a nested loop framework and a single loop framework. Additionally, it has also been shown that superlinear convergence rate can be expected for the proposed RIPD algorithm following local convergence results of the standard master and subproblem solvers. The proposed methods were applied to two classes of numerical test problems and their results are encouraging. The algorithms provide numerical tools to solve the more realistic decomposition-based, complementarity models for renewable energy generation system design optimization, which will be investigated in the following chapters.

Chapter 4

Hybrid Power/Energy Generation System

Design through Multistage Design

Optimization with Complementarity

Constraints

Renewable resources such as solar and wind have received increasing attention as a means of energy generation be- cause of their environmental friendliness, omnipresence and free availability. As various social, economic and political concerns grow over conventional fossil fuels, renewable energy has been identified as one of the most promising so- lutions to the world’s energy problem in the future. The remainder of this dissertation investigates the optimal design of renewable energy generation systems meeting different aspects of mankind’s energy demand. The computational framework developed in the proceeding chapter will be utilized to address the problem.

4.1

Hybrid Power Generation System Design Optimization

One of the most promising applications of renewable energy generation is the installation of hybrid power generation systems (HPGS) to meet basic electricity needs at remote areas with no power infrastructures. Due to the inherent variability of renewable energy sources, reliability has been a critical issue in HPGS development, and hence re- quires deliberate consideration during the design stage. If the reliability can be assessed with reasonable accuracy, an optimally system configuration with minimal, while adequate, capacity can be employed, significantly reducing the system cost.

The HPGS design community relies on a variety of methods to estimate a system’s reliability. Among these methods, two main categories of approaches have been widely employed: deterministic simulation-based methods and probabilistic methods. Deterministic simulation-based methods[114, 115, 116] evaluate a system’s reliability through directly testing the system against historical meteorological series. Such approaches naturally accounts for the time-variability of renewable resources as well as power demand, which is critical in HPGS reliability assessment and design optimization; they are also straightforward to implement given that the data is available. On the other hand, deterministic simulation-based approaches usually require large weather data sets, that may not be readily available. To address this situation, synthetic data are occasionally applied. Probabilistic methods either follow a Monte Carlo simulation to perform reliability analysis or use various analytical models to solve for a reliability metric. Currently,

most of the available analytical reliability assessment models still have limited capacity in handling resource variation in time, especially for the scenarios where the probabilistic state of the energy storage needs to be tracked. As a result, deterministic simulation-based methods remain the major class of reliability analysis approaches applied in existing HPGS design optimization literature.

This dissertation studies optimization approaches to address HPGS design under either type of reliability assess- ment method: the current chapter presents a multistage optimization with complementarity constraints approach for HPGS design based on deterministic reliability analysis; and the next chapter presents an HPGS design optimization approach based on a probabilistic Markovian reliability assessment model.

The deterministic design optimization of HPGS’s involves discrete-time simulation of the system over a certain time period. This setting presents two challenges for numerical optimization. First, the dynamics of such systems is usually discrete by nature as it includes nonsmooth logical disjunctions, e.g., switching between different sets of equations based on working conditions, and other nonsmooth functions such as min and max operation. Due to this discrete nature, well established optimization techniques for smooth problems could not be applied to such systems. Second, the consideration of system performance at each time step introduces additional variables (referred to as time- dependent variables) into the optimization model, thus increasing the size of the problem. As the number of time step increases, solving the hybrid power generation system design optimization problem with an all-in-one (AIO) approach may become impractical, undesirable, or even impossible.

Traditionally, one way of capturing logical disjunctions is to introduce discrete (binary/integer) variables. How- ever, such a mixed integer optimization model usually incurs intensive computation cost for large problem as the worst case solution time grows exponentially with the number of discrete variables. Alternatively, nonsmooth solvers such as genetic algorithm can be integrated with the discrete-time simulation. While this type of approach is robust in general, it also suffers from the lack of guarantees for optimality. In this chapter, a different track of handling the discreteness is presented through the aid of complementarity constraints.

Complementarity constraints are useful in the optimization of discrete-time systems, In that they can be used to model certain types of logical disjunctions without the use of binary variables. Since traditional optimization approaches for logically disjunctive models may have some limitations, the complementarity constraint offers an alternative for some classes of disjunctive problems. Following established theories in mathematical programs with complementarity constraints (MPCC), nonlinear programming (NLP) solvers can be adapted to obtain fast solutions. In additions, certain level of local optimality can be ensured.

In order to handle the size issue of hybrid power generation system design optimization, various decomposition- based approaches[176] can be applied so that the AIO problem can be solved through iterative solution of smaller, interrelated subproblems and coordination among them. In this chapter, we utilize the repetition of simulation and

decision making of HPGS at individual time steps, and present a multistage decomposition framework which decom- poses the AIO HPGS design optimization problem into a set of consecutive stage optimization subproblems. Although the research area of multistage optimization is not frequently linked to that of MPCC, it has recently been related to multidisciplinary design optimization (MDO) which is connected to MPCC in this dissertation: Kim and Hidalgo[184] present a pseudo-hierarchical decomposition framework in which multistage optimization problems can be formulated as multilevel MDO problems; Kim et al.[185] extend this idea with a task parallel algorithm that enables optimal load balancing of the decomposition framework.

This chapter presents a mathematical model of HPGS design for cost minimization under the zero loss of power supply constraint. In addition, a multistage optimization with complementarity constraints approach for HPGS design is presented, which first reformulates the logical disjunction in the HPGS simulation into complementarity constraints, then solves the reformulated problem with a multistage decomposition framework. The proposed algorithm is tested with an HPGS design case study at Corsica Island in France and the numerical results are encouraging.

The chapter is organized as follows: In Section 4.2, the HPGS problem is stated, with component and system models described. In Section 4.3, the complementarity reformulation of the HPGS design problem is presented, fol- lowed by its multistage decomposition formulation. In addition, an augmented Lagrangian decomposition algorithm is presented based on the decomposed formulation. A numerical study of a hybrid power generation system design case is presented in Section 4.4, and conclusions are drawn in Section 4.5.