Literature review

As described in Chapter 2, much research has been devoted to solving the optimal dispatch problem. A subset of this research also seeks to design the microgrid in addition to solving its dispatch problem. This research can be separated into four categories.

1. Non-optimal design approaches that use design of experiments or other methods to compare different design choices without performing design optimization;

2. Optimal design methods that do not solve for the optimal dispatch but instead use rule-based or fixed-controller methods to decide the microgrid operation;

3. Formulations that solve the optimal design and optimal dispatch problems as a combined All-in-One (AiO) problem;

4. Formulations that decompose the optimal design and optimal dispatch problems into separate, coordinated subproblems.

Non-optimal design approaches

HOMER Energy solves the optimal design problem using a full-factorial design of experiments on number and size of components, using a rule-based dispatch strategy to evaluate the microgrid operation. The HOMER approach solves the dispatch strategy for

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minimum cost in a single time increment without explicitly planning energy storage for future time increments [Lambert et al. 2006]. The approach is not formally an “optimal”

design problem as the full-factorial DOE chooses component sizes at a limited number of fixed intervals.

Zoka et al. compared the performance of four specific microgrid designs by solving the optimal dispatch problem for each, but they do not provide for more general design optimization capability [Zoka et al. 2007].

Optimal design without optimal dispatch

Vallem and Mitra used the simulated annealing algorithm to solve a mixed-integer, nonlinear sizing and siting (topology) of a microgrid to minimize system cost while maintaining system reliability [Vallem et al. 2005]. Their formulation uses generic

“distributed power” of constant cost per unit power and does not directly consider an optimal dispatch problem. They do include a subproblem for power flow to minimize loss-of-load.

Pelet et al. used genetic algorithms to solve the nonlinear sizing problem incorporating detailed, physics-based models of the microgrid components [Pelet et al.

2005]. Their objective is to minimize capital and operational costs, with a tradeoff objective (Pareto objective) of minimizing CO2 emissions. They use a rule-based control strategy to solve for the energy use of each component rather than an optimal dispatch problem.

Research by del Real et al. studied both optimal dispatch [del Real et al. 2009a]

and the optimal sizing of a microgrid [del Real et al. 2009b], though not concurrently. In the optimal sizing case, they formulate a sizing problem with renewable energy (wind) and hydrogen storage. Instead of solving for an optimal dispatch problem they use

“estimated operating conditions” to ensure that the design can meet the necessary load under various wind resource scenarios. Their objective function is formulated as a convex sum of equipment capital costs with no operating or maintenance costs, and the

constraints are all linear functions. Therefore, they are able to solve a mixed-integer quadratic program.

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Lu et al. solved a nonlinear microgrid design problem by decomposing it using MDO techniques to solve each time increment individually while coupled to other time increments through battery storage linking variables [Lu et al. 2010]. However, their approach does not calculate a multi-component optimal dispatch; instead, they use non-dispatchable components (wind and solar) to determine when battery storage should charge or discharge. In addition, they solve the problem using one-day time increments, which does not consider the hourly dynamics of renewable energy supply and energy storage.

Lu et al. extended their research to find the optimal design (sizing) of a hybrid energy system including energy storage, solar power, and wind power, with the objective of minimizing the capital cost of the system while also minimizing loss of load (LoL) given uncertainty in the renewable energy inputs [Lu et al. 2011]. As mentioned in Chapter 2, they solve the dispatch of energy storage for each design at one-hour increments using stochastic Markov chains to minimize the probability of LoL.

AiO approaches

Stadler et al. posed the design and dispatch problem as an All-in-One (AiO) problem for the entire year by linearization and solution using linear programming [Stadler et al. 2009], thus increasing solution efficiency but with the limitation of

linearizing submodels, such as diesel generator efficiency as a function of load, or battery internal resistance as a function of state-of-charge.

Asano et al. solved a mixed-integer, nonlinear design (optimal sizing and

component selection) and dispatch problem by separating the year into six representative days, then solving each day as a fixed AiO problem [Asano et al. 2007]. However, their approach does not consider the boundaries of the solved days by linking the time before and after the representative days. As such, the approach provides an approximation of the microgrid performance, but may differ from how the microgrid will be controlled and operated in practice.

Geidl’s dissertation includes a section discussing optimal sizing and layout of microgrid components [Geidl et al. 2007b]. Though Geidl has also addressed the optimal

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dispatching problem, he does not consider how to solve the optimal dispatch and optimal design problems concurrently.

Decomposed approaches

A decomposed method of microgrid sizing followed by controller design is proposed by del Real et al. but not implemented. In addition, their proposed framework does single-pass optimization, where an optimal controller is designed for an optimal design. This type of single-pass optimization has been shown to often result in a suboptimal design [Fathy et al. 2001].

The method presented in this chapter is a decomposition-based approach. The optimal dispatch problem is nested within the optimal design problem, as described further in Section 3.2.1 and first explained in [Whitefoot et al. 2011]. This type of nested optimization can result in a globally-optimal solution if the problem is shown to be convex, though non-convex functions are often present. This nested strategy is commonly used in combined optimal design and optimal control problems [Fathy et al. 2001, Peters et al. 2009], but has not been used previously for optimal design and optimal dispatch problems.