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

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 twocriterion 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.


Introduction
Wind power is the most progressive renewable energy source, which has been intensively growing since early 2000s. This growth nearly reminds exponential one, and it is strongly believed to continue through the next decade [1,2]. The central place in the wind power technology is taken by wind farms whose task is to "capture" the wind power and convert it into electric energy. The con-verted energy must be transferable with a low loss factor [2,3].
Deploying a new wind farm, situated in a new location, requires fulfilling a series of very important actions. First of all, wind statistics should be collected and analyzed. Independently of this, energetic needs are assessed letting know thus an annual energy, which is desired to be obtained on the location. An economic analysis of the functioning wind farm is carried out also including costs of buying and installing wind turbines. Such an analysis requires knowing an exact number of wind turbines and their costs [4,5].
It was shown in [6,7] that energy generated by wind turbines having diverse power curves is the best way to achieve the highest performance of the wind farm. The power curve diversity is intentional to fit as ineffectiveness of the single wind turbine against wind statistics, as well as to fit wind speed distribution uncertainty. For this, an algorithm of simultaneously maximizing the produced energy and minimizing the costs was suggested [6,7]. By the algorithm, the respective two-criterion problem is solved with an efficient solution (to a singlecriterion problem). The output of the algorithm is a set of wind turbine types, which should be bought and installed, and a respective set of numbers of wind turbines in every type [8].
However, such a solution has three difficulties. Firstly, the ranks of energy criterion and cost criterion are presumed to be equal. This is not always true, especially for terrains where the wind power technology is just at its beginning and minimization of costs is more important than maximization of energy [9]. Secondly, uncertainty of the parameters of wind statistics (i. e., a wind speed distribution) cannot be completely removed. Finding the guaranteed and expected values of the costs and the wind farm power outputs, as it is proposed in [7], makes nonetheless the optimization efficiency weaker [10]. And, finally, what partially relates to the second difficulty is that there is no fast way for recalculating the wind farm deployment parameters. Indeed, volatility of the wind speed distribution parameters along with volatility of wind turbine market pricing both necessitate an intensive series of recalculations before a trade-off is made [4,10].

Problem statement
Despite the algorithm stated in [6,7] allows optimizing wind farm energy and costs simultaneously, its practical implementation requires further development. Namely, optimal numbers of wind turbines of certain types along with their costs and energy produced by them may need to be frequently recalculated. Such recalculations may be needed even after the wind farm is already deployed and functions but either wind conditions or energy demands start changing, or both them differ from those previously embedded into the wind farm model before its deployment. Therefore, the goal is to develop a practically rapid method of maximiz-ing the produced energy by minimizing the costs. The method should get rid of ranking the energy and costs. In fact, it is going to be a contribution to wind farm deployment optimization by speeding up the process of finding optimal numbers of wind turbines. Nevertheless, 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.

Wind statistics
Apart from wind turbine types and their numbers, an average wind power capacity strongly depends on wind speed distribution of an area, whereon a wind farm is projected. It is commonly modeled as the Weibull distribution of wind speed s [2, 3, 6, 7]: with a positive shape parameter b and a positive scale parameter a . The shape parameter relates to a factual range of wind speeds, whereas the scale parameter corresponds to the mean wind speed. These parameters of wind speed distribution (1) cannot be assessed as point estimates. They are commonly assessed as interval estimates, although the intervals are relatively pretty narrow. Besides, there are no probabilistic measures over those intervals as eliciting them from the observed wind statistics leads to uncertainty among a continuum of probabilistic measures over an interval. Being preliminarily assessed prior to the wind farm deployment, the parameters of wind speed distribution (1) are then determined more accurate after the wind farm starts its functioning.

Initial parameters of wind farm deployment
Let K be a number of available wind turbine types. Denote the power curve of the k-th wind turbine type by ( , ) w k s ( 1, k K ), where ( , ) w k s by a fixed wind speed is a power in megawatts (MW) produced by turbine k at that speed. An expected power produced by the k-th wind turbine type is 0 ( ) ( , ) k p s w k s ds (2) with using wind speed distribution (1) or other.
Let max E be an annual desired energy given in MWh. Then the respective expected power output in MW is [1,7]  Let an amount of financial resources required for a wind farm deployment be max C . These are maximal costs which can be spent on it. They include costs of buying and installing wind turbines. Thus, denote the costs for buying and installing the k -th wind turbine type by ( )

Power maximization problem
In general, a wind farm is built of wind turbines of various types [3,6,7]. Thus, let k n be a number of wind turbines of type , In particular, this means that, by 0, k n a wind farm is projected without wind turbines of type .
k An expected power output of the entire wind farm is This power output costs Therefore, power (4) is to be maximized along with minimizing costs (5) by the following constraints.
Firstly, power (4) should not exceed maximum (3), nor should it be less than i. e.
Secondly, the financial resources are limited to max C : Thirdly, a maximal number of wind turbines max N exists, which is determined by the area, whereon a wind farm is projected. This factually is a geographic capacity of the area that is tightly connected to a reasonable and efficient geographic concentration of wind turbines. Thus, the total number of wind turbines is limited to max N : Another peculiar constraint, which supplements constraint (9), is that a maximal number of wind turbines of a definite type may be limited also. Besides, a minimal number of wind turbines of that type may exists, which must be necessarily bought being imposed by, e. g., a contract binding the wind farm projector and a supplier of wind turbines. So, by a formal inequality The described maximization problem, without yet considering minimization of costs (5), can be written as an integer linear programming problem. Let Then the integer linear programming problem is subject to: Note that constraint (20) is required to be attached to inequality (19) for ensuring that all the K variables in vector (12) be integers.

Wind farm deployment optimization
First, problem (17) subject to constraints (18) -(20) is solved by an arbitrary large max .
C Denote such a solution by ( 2 1 ) The respective costs, according to (5), are The idea behind getting rid of ranking the energy and costs is quite simple. Starting off (21), the costs are decreased until the solution of problem (17) subject to constraints (18) -(20) is nonempty. Every next step of the decrement is smaller than the previous one. If the energy maximization problem has no solution, the costs are increased. Every next step of the increment is smaller likewise than the previous one. This process is continued until the change in the costs (either decrement or increment) becomes sufficiently insignificant. Then the last nonempty solution of the energy maximization problem is saved and considered as the optimal one for the wind farm deployment. The iterative process can be organized with using a simple dichotomization approach [11]. Therefore, once solution (21) is found, problem (17) subject to constraints (18) -(20) is further solved as follows. Value 1 is initially set along with some integer Q . This integer defines a maximal number of energy maximization problems to be solved, including the initial problem with solution (21). After solution (21) is found, the new constraining costs into vector (16) are set: max 0 C C . The iterative process is continued while 1/2 Q . After the next energy maximization problem is solved, the change in the costs is twice reduced: (obs) and (obs) /2 .
If a solution of problem (17) and inequalities (9) and (10)  ) is shown in Fig. 1,  whereon the resulting annual energy and its costs for every empty solution are set at the preceding values. The cost bounds which, generally speaking, are 0 ( ) /2 Q C Q C , ( 3 2 ) allow to relatively estimate the accuracy of the cost minimization. Bounds (32) "trap" the final resulting costs, and they do not necessarily "trap" constraining costs max C at the final iteration. Compared to how this example was solved in [6], where the energy and costs were maximized and minimized simultaneously, the final resulting costs are decreased from 26.551 to 23.59072852 million euros (see Fig. 2). The difference is more than significant: by sustaining the required annual energy minimum, it is 2960271.48 euros. By the way, executing more than eight iterations in this example does not improve the obtained solution.
Another example with the twice greater annual desired energy, where the rest of parameters are the same, is shown in Fig. 3. It takes only six iterations to get the optimal annual energy and costs. It is noticeable that here the initial solution is  Fig. 1; by this version, the wind farm should have been built of 6 wind turbinesone turbine Gamesa, 4 turbines Nordex, and one turbine REpower (despite the annual energy in 17353.1736 MWh, this version is almost 3 million euros more expensive than that in Fig. 1) implying that the wind farm would be built of 14 wind turbines involving each of those five wind turbine types. Such a solution ensuring an annual energy in 34998.436412 MWh (which is 99.9955 % of the maximum) would cost 65.27493723 million euros, which is 18.40611058 million euros more expensive than the optimal costs in 46.86882665 million euros. So, this is an example of that the suggested optimization process can have a really great positive impact on the wind farm deployment. On the other hand, this exemplifies that the most profitable deployment does not always imply a great diversity of wind turbine types. The wind farm here is optimally deployed by two types of wind turbines, whereas solution (33) stands for involving five types. The fact that the optimized wind farm consists of two wind turbines more can seemingly be a trade-off. Other numerical experiments show (Fig. 4) that the polyline of changes in max C resembles those ones in Figs. 1 and 3. Every iterative optimization process is convergent, although there are always many empty solutions. The only condition of a correct start is to obtain a nonempty solution (21) of the initial problem. This can be sometimes an uneasy task due to minimal power output (6) may be required and correspondingly set at a higher rate.
It is also clear that the optimization process must have iterations with empty solutions. Such it- erations, whose number can be even greater than the number of iterations with nonempty solutions, are unavoidable. This and other peculiarities of the suggested iterative optimization are to be discussed below.

Discussion
To start the optimization, apart from expected power (2)  are set at 0. Special attention must be paid to calculation of expected powers 1 { } K k k , which strongly depend on the wind speed distribution parameters and representation accuracy of wind turbine power curves [12,13]. The expected power is indeed very sensitive to the shape and scale parameters, so their inaccurate determination may badly bias an optimal solution if even the power curves are all represented accurately.
Vectors (13) for constraints (19) should be determined carefully as well, but any inexactnesses in their elements are easily corrected by recalculations, which are rapidly executed by the suggested successive optimization process. If some parameters concerning a functioning wind farm change, the rapid recalculation by a set of adjusted parameters will give a solution * ( ) Q N prompting which type and how many wind turbines are to be turned off/on. Other needs for the recalculation may emerge from volatility of the prices for buying and installing wind turbines. The example of that installing prices (27) are tied to buying prices (26) is just a naive pattern. So, the rapid recalculation of the optimal solution by the suggested successive optimization excellently fits such situations. In particular, solving optimization problems like that with objective (31) takes up to one tenth of a second, so a bunch of such problems is solved within a second or so.
It is easy to see that the optimization requires no less than 3 iterations. After the first iteration, the constraining costs drop too low and the problem has no solution. Then, at the third iteration, the constraining costs are increased according to (25), where 1 4 . So, this "triangle" turned over with its middle vertex downward is a distinct feature of the optimization process. However, the empty solution at the third iteration is not excluded (see Fig. 4). This nonetheless is normal, because a nonempty solution (in which the resulting costs are lower) can appear after a few empty solutions (see it in Fig. 1, wherein the final nonempty solution appears after four iterations with empty solutions).
An apparent economical impact after applying the wind farm deployment optimization is expectedly strong. The example with saving almost 18.4 million euros (Fig. 3), which is 28.2 % of the initial (non-optimized) costs, is quite remarkable. Such gains, however, are expectedly decreasing as the relative difference between a maximum and a minimum of the annual desired energy is shortened.

Conclusions
An optimization approach, presented in this paper, allows to avoid solving the two-criterion problem for simultaneous energy maximization and cost minimization for deploying wind farms. As solving the two-criterion problem is partially uncertain due to not surely known priority relationship between energy demands and financial resources, the presented approach is a method of successive optimization. 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.
A promising way to improve the suggested optimization process is to consider dichotomization steps different from one-half. This must be tried along with tolerating non-fixed parameters of the wind speed distribution. Despite it is not obvious that an improvement will be significant, such a supplement to the suggested successive optimization will be a more complete contribution to the theory and practice of wind farm deployment.