# ITERATIVE POWER MAXIMIZATION BY ONE-HALF COST DICHOTOMY FOR OPTIMIZING WIND FARM DEPLOYMENT

## DOI:

https://doi.org/10.20535/kpi-sn.2019.4.177315## Keywords:

Energy, Costs, Wind farm deployment, Successive optimization, Wind turbine, Dichotomization, Annual energy## Abstract

**Background.** In deploying a wind farm, it is important to hold a balance of energy generated by wind turbines and costs spent on buying and installing them. However, ranking the energy and costs is commonly uncertain. Besides, the existing methods of optimal wind farm deployment are pretty slow, whereas even optimal numbers of wind turbines of certain types along with their costs and energy produced by them may need to be frequently recalculated.

**Objective.** The goal is to develop a practically rapid method of maximizing the produced energy by minimizing the costs. The method should get rid of ranking the energy and costs. Besides, it should speed up the process of finding optimal numbers of wind turbines. The optimality here is to be interpreted in wide sense implying also fitting wind statistics, controlling the costs and production, and adjusting to energy markets.

**Methods.** Once an expected power for every wind turbine type is calculated, a power maximization problem is formulated in the form of an integer linear programming problem, where optimal numbers of wind turbines of certain types are to be found. This problem involves a span of acceptable annual energy formed by a maximum and a minimum of the annual desired energy. Additionally, costs are constrained. The minimal and maximal numbers of wind turbines of a definite type are constrained also. First, the power maximization problem is solved by an arbitrary large constraining costs. Then the constraining costs are decreased until the solution is nonempty. If the solution is empty, the costs are increased. Every next step of either the decrement or increment is twice smaller than the previous one. This process is continued until the change in the costs becomes sufficiently insignificant.

**Results.** The optimization process is rapidly executed requiring only a few iterations to achieve an optimal solution. In particular, solving optimization problems with five known wind turbine types takes up to one tenth of a second, so a bunch of such problems is solved within a second or so. In general, the optimization requires no less than 3 iterations. After the first iteration, the constraining costs drop too low and the problem has no solution. However, the empty solution at the third iteration is not excluded, and a nonempty solution can appear after a few empty solutions. Nevertheless, an apparent economical impact after applying the wind farm deployment optimization is expectedly strong. There is an example with saving almost 18.4 million euros, which is 28.2 % of the initial (non-optimized) costs. Such gains, however, are expectedly decreasing as the relative difference between a maximum and a minimum of the annual desired energy is shortened.

**Conclusions.**The presented approach is a method of successive optimization. It allows to avoid solving the two-criterion problem for simultaneous energy maximization and cost minimization for deploying wind farms. The computational core of this method is that the expected power output is maximized via solving an integer linear programming problem. The successive optimization process starting with an initial power maximization problem by arbitrary large costs is always convergent if the problem has a nonempty solution. The costs then are dichotomized in order to produce energy between definite maximum and minimum so that further changes in costs would be ineffective. The dichotomization allows to rapidly achieve the optimal solution, which includes the final resulting annual energy, costs spent on it, and the respective numbers of wind turbine types to be installed.

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