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| Fig. 1: An active solar tracking array. (Source: Wikimedia Commons) |
Mathematical optimization methods specify a design infrastructure to maximize a given cost or objective while satisfying problem constraints. Often these design optimization problems trade off objectives like maximizing the amount of power output of a given design while simultaneously minimizing cost. These methods have tremendous potential to shape the energy landscape, by allowing designers to specify intelligent designs for harvesting energy while meeting the designers objectives. Furthermore, mathematical optimization provides a means for controlling physical processes in order to optimize physical systems. In this context, the system designer aims to create a system that continuously solves an optimization problem in order to maximize its given objective over time. One concrete example of this is Model Predictive Control (MPC), where at each point in time an optimization problem is solved in order to determine the correct control inputs. [1] These techniques can be used in energy applications to increase efficiency, output, and decrease cost when implemented correctly.
Optimization methods have been successfully applied in both active and passive solar designs. Active solar designs can consist of panels that actively move their incidence angle in order track the sunlight and hence increase their power production output (see Fig. 1). In solar optimization applications designers often trade off competing objectives such as energy consumption, financials costs, and environmental performance. [2] In order to solve these optimization problems, a model of the process if required. For example, if the optimization problem is designing array placements, the process requires a model of solar radiation as a function of solar array placement. These models may be inferred from solar data, but low resolution data requires interpolation and extrapolation methods. [2] Artificial neural networks have proven successful at using recorded sparse data in order to build solar radiation maps for use in optimization. [2] In order to try to provide continuous power flow, solar systems often use energy storage devices because of the intermittent nature of obtaining energy from the sun. [2] These designs have been successfully optimized to provide continuous power while estimating the inherent design parameters using genetic algorithms (GA). [2] Genetic algorithms optimize a given cost or objective by continuously evolving candidate populations of solutions and selecting the most fit population in accordance with a fitness function.
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| Fig. 2: Decrease in incident solar radiation occurs when the solar panels do not track the sun and is limited by the Cosine of the angle of incidence. (Source: N. Spielberg) |
In the case of perfect solar tracking, as shown in Fig. 2, the maximum efficiency occurs when the solar tracker perfectly tracking the solar radiation. This occurs when the angle of incidence is at zero and hence the cosine function is maximized at 1. Imperfect tracking causes a decrease in efficiency as shown by the Cosine of the angle of incidence.
Similar to designing solar arrays using optimization methods, a common application of optimization methods in wind applications is designing wind farms. In these applications, the design to be optimized includes the spacing and placement of the wind turbines in order to maximize the output of the farm. [2] In addition a common application of design optimization is for design of the wind turbine rotors themselves. These designs can be optimized using GA or similar approaches to pick their radius of gyration, blade length, and cross sectional areas. [2] Similarly, models are needed to predict wind power but the magnitude of wind fluctuations often seen in off shore winds makes these models difficult to construct. [2]
In hydropower applications, power is harnessed from moving water, often water falling due to gravity. Sizing of a hydroplant is an optimization problem in which the designer attempts to optimize the power output of the plant for a given river type and maximize the cost effectiveness of the design. [2] Additionally optimally scheduling hydroplants can be formulated as a nonlinear optimization problem in order to increase plant efficiency. [2]
Another application in which optimization methods have successfully been used to decrease energy consumption or maximize efficiency is in building HVAC systems. These systems regulate the temperature of a building by turning on the heating element in the building or air conditioner. In turn these systems require a thermal model of the building in order to predict how the building's temperature model will evolve over time given inputs from the heater or air conditioning element in the building. These models can be learned from data and used for solving the temperature regulation optimization problem in real time. [3] This technique which uses MPC and solved an optimization problem in a receding horizon fashion has been shown to result in a 30-70% reduction in energy consumption compared to traditional two position control of HVAC systems. [3]
Optimization methods have the potential to increase the efficiency of the energy sector and provide a promising framework to do so for the future.
© Nathan Spielberg. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] D. Mayne and J.B. Rawlings, Model Predictive Control: Theory and Design (Nob Hill Publishing, 2009).
[2] R. Baños et al., "Optimization Methods Applied to Renewable and Sustainable Energy: A Review," Renew. Sust. Energ. Rev. 15, 1753 (2011).
[3] A. Aswani, J. Taneja, and C. Tomlin, "Reducing Transient and Steady State Electricity Consumption in HVAC Using Learning-Based Model-Predictive Control," Proc. IEEE 100, 240 (2012).
