# Microgrid Optimization

Consider a microgrid system where there are several energy sources and colocated energy storage devices that can either sink or source power with their corresponding sources. The net power at each source/storage is metered to the grid main bus using a boost converter. For an efficient design of the microgrid, the number of storage elements (N) and their capacities need to be optimized. Storage is expensive and designing a microgrid, with storage sized properly, is an open problem. Associated with computing the optimal N is the optimal values for the duty ratios at the converters that controls the power metered to the main bus from each source. A more complex situation is when we have M microgrids that have the ability to interconnect. This is a Variable-Size Design Space (VSDS) optimization problem that has a large number of permutations for exchanging power.

# Traffic Network Signal Coordination Planning

Traffic Network Signal Coordination Planning (TNSCP) is usually formulated as an optimization problem [17, 61, 53, 54], and it is another application that motivates the need for VSDS optimization algorithms. The objective of this problem could be to maximize the traffic flow in one corridor (road), have higher overall vehicle speeds, reduce the number of stops, or some other objective. Most current methods for solving the TNSCP formulate the problem such that the signals’ green times are the design variables [95], which is a formulation of fixed-size design space. This is a complex optimization problem, and extensive research has been conducted in using different optimization methods for this formulation, with different objective functions [78, 27, 102, 113, 107, 110, 80]. Usually, the green times values obtained from optimization are such that groups of signals share common values for the green times. Figure 1.5 is an illustration for this observation. The network in Fig. 1.5 has 20 intersections; some signals on the main streams at some intersections may have the same green time value. Hence it is possible to formulate the TNSCP optimization problem as an optimal grouping problem where it is desired to find the optimal grouping for all the optimized signals. Specific input values for the green times define the groups (subsets). The TNSCP optimization problem is then to determine which signals should belong to which group in order to optimize the objective function. The advantage of this formulation is that signals will be coordinated (using green times), to optimize the given objective, rather than treated as individual variables. In this traffic problem, it is the coordination between signals that is more effective than the green times of individual signals. This coordination information becomes out of context when using the standard formulation, while in the suggested formulation it is more direct. The number of variables (number of groups and number of signals in each group) is variable and needs a VSDS optimization tool.

Figure 1.5: The TNSCP optimization problem is formulated as an optimal grouping problem where each group (subset .S',.V/) has a uniform green time value for all optimized signals in the group.

# Optimal Grouping Problems

Optimal grouping problems have received a great deal of interest. Examples include the optimal bin packing problems [44], where a set of boxes of different sizes should be packed in containers of a given size, in order to minimize the number of containers. Reference [45] points out that standard FSDS global optimization algorithms are inefficient in handling this type of problem. A Grouping Genetic Algorithm (GGA) was developed to handle this type of problem, see, e.g., reference [43]. The GGA is group oriented, and hence is characterized by a VSDS (one solution may have two groups while another may have three groups). Tailored new definitions for genetic operations had to be introduced in the GGA theory in order to handle the resulting VSDS optimization problem.

# Systems Design Optimization

Several systems design optimization problems are essentially VSDS problems. Here the thermodynamic cycle design optimization example is briefed. Thermodynamic cycle optimization is challenging as it involves several optimization variables and objectives. The first challenge (based on the choice of heat source, sink and working fluid) is to decide on the cycle topology or architecture. For example, in an exhaust waste heat recovery cycle that is applied as the bottoming cycle on a Gas Turbine (GT), the simplest cycle topology that can be applied is a Rankine cycle that utilizes a heater, a cooler/condenser, a compressor/pump and an expander (which is the separate turbine in Fig. 1.6). However, depending on the working fluid and the source/sink conditions, this basic cycle topology is not necessarily the best for achieving the most efficient cycle. Certainly, optimization of a given basic cycle configuration can be carried out with allowed inclu-

Figure 1.6: Schematic diagram of the Rankine cycle.

sions of intermediate cycles and components at appropriate locations (in terms of end-point pressures and temperatures). However, this is only part of the overall optimization needed and is typically straightforward. The broader and more challenging type of optimization is to consider several different types of allowed inclusions (of intermediate cycles and components) in the basic cycle configuration towards optimizing one or more well defined objective functions (such as overall power plant efficiency, cost, etc.) As discussed, the number of optimization variables in this broader optimization prospective changes depending on the particular cycle configuration and number of inclusions under consideration. In other words, in optimizing a given cycle, one solution may be completely specified with fewer variables whereas a range of valid choices may be available with larger number of variables.

# Structural Topology Optimization

In structural topology optimization, the design objective could be to minimize the mass while satisfying a set of requirements in terms of the loads that a structure can handle. The design variables selection may vary depending on the problem formulation. Structural topology optimization problems, in general, are characterized by complex design spaces due to the large number of design variables. Standard global optimization methods become inefficient in such problems. For example, consider Fig. 1.7 where an example of three different designs for a structural elements is shown. If it is desired to optimize some objective, what would be optimization variables when using a FSDS algorithm? Is it possible to use a VSDS optimization algorithm?

Figure 1.7: Three different designs of a structural element.

# Pixel Classification Problems

An important problem in satellite imagery is the classification of pixels for partitioning different landcover regions [23]. Satellite images usually have a large number of classes with nonlinear and overlapping boundaries. This problem has been formulated in the literature as a global optimization problem [35, 81], and several attempts have been made to solve it using GAs [35]. Reference [23], in particular, presents a method that minimizes the number of misclassified points, by assuming two possible sizes for the design space, and implementing a GA technique. The population in this case has two classes of chromosomes (males and females in [23] and new definitions for genetic operations had to be defined to handle the double-size design space. Clearly this type of problem would benefit from a VSDS optimization algorithm.