Heuristic methods for wind energy conversion system positioning

Electric Power Systems Research 70 (2004) 179–185 Heuristic methods for wind energy conversion system positioning U. Aytun Ozturk a , Bryan A. Norman b,∗ a College of Business Administration, Hawai’i Pacific University, 1166 Fort Street Mall, Suite 305, Honolulu, HI 96813, USA b Department of Industrial Engineering, University of Pittsburgh, 1048 Benedum Hall, Pittsburgh, PA 15261, USA Received 8 May 2003; received in revised form 13 November 2003; accepted 9 December 2003 Abstract This paper considers the problem of determining the locations of wind generators in a wind farm consisting of many generators. The objective is to find a generator placement that maximizes profit, which is the product of the cost efficiency of the generators and the total power output from the wind farm. Generator placement is significant because if generator A is located close to generator B and is located downwind of generator B then the power output of generator A is reduced by an amount that varies with the distance between the generators. The problem can be formulated using mathematical programming but to solve the problem one cannot employ traditional optimization methods. Therefore, a greedy improvement heuristic methodology is developed and described in detail. The effectiveness of the proposed heuristic is demonstrated on a suite of test problems. These results indicate that the proposed method represents an effective solution strategy for this problem. © 2003 Elsevier B.V. All rights reserved. Keywords: Wind farm; Heuristic optimization; Power generation 1. Introduction Renewable energy technologies, such as photovoltaic, solar thermal, geothermal, biomass, tide, and wind are developing rapidly. Some of these technologies are now beginning to compete with existing energy production methods. Wind energy conversion, in particular, is becoming cheaper than traditional methods in some areas of the world. Wind electricity costs have declined from 35 to 5–7 cents per kWh in the last 22 years in the US [5]. One of the characteristics of Wind energy conversion is that the amount of electricity produced by one generator is less than what a conventional thermal generator would produce. Therefore, to generate more electricity, more than one generator is needed. This has led to the development of wind farms where many generators are located at a single site. Positioning of multiple generators in a wind farm, however, is affected by spacing constraints due to wake decay effects. That is, if generators are located too close to one another in the direction of the prevailing winds, the total amount of power generated is reduced because the upwind generator alters the wind pattern ∗ Corresponding author. Tel.: +1-412-624-9831; fax: +1-412-624-9831. E-mail addresses: [email protected] (U. Aytun Ozturk), [email protected] (B.A. Norman). 0378-7796/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2003.12.006 for the downwind generator. By mathematically modeling the generator placement problem, optimization methods can be used to mitigate the problem of wake decay effects and improve the efficiency of wind farms. In this paper, we present one such optimization methodology. The remainder of this paper is organized as follows. Section 2 contains a review of the relevant literature including a discussion of previous work concerning wind farm generator positioning and related problems from location theory. Section 3 discusses wind and power generation models. Section 4 presents the proposed solution methodology. Section 5 presents computational results from a suite of test problems. Finally, Section 6 presents conclusions and topics for future research. 2. Literature review There is only one paper to date that applies mathematical modeling to the problem of positioning generators for a wind farm. However, investigations have been done on other location problems that have some similar characteristics. We now consider relevant literature from the location literature followed by a discussion of the one paper that has been written that attempted to solve the generator positioning problem. 180 U. Aytun Ozturk, B.A. Norman / Electric Power Systems Research 70 (2004) 179–185 The wind farm generator positioning problem has some similarity to undesirable facility location problems. An undesirable facility is one where the facility adversely affects its environment and/or its customers [1]. There was virtually no work on this type of problem until the 1980s. One of the objectives in undesirable facility location problems is to locate multiple undesirable facilities but make them as far apart from each other as possible. This simplest version of the multiple undesirable facility location problem maximizes the distance between the undesirable facilities. However, even this simple version is recognized as NP-complete [2]. In our problem, the generators affect each other and they cannot be located too close together. This is the difference between an undesirable facility problem and the wind farm generator positioning problem—in our problem, we try to minimize the distances between generators, while maintaining feasibility, instead of maximizing the distance between them. The problem that we are investigating contains several possible sources of uncertainty. Uncertainty enters into this work because the wind changes intensity and direction with time, which in turn changes the amount of power produced by a wind farm. Numerous other location papers consider uncertainty but this typically involves uncertainty in customer demand, transportation costs, capacity, etc. [4]. In our problem, uncertainty is caused by the wind and this topic will be revisited in the next section. The only other research previously conducted that solves the wind farm generator positioning problem reduces the problem to a discrete one and uses a genetic algorithm to solve it [3]. The objective function used in [3] is given in (1):
    costtot 1 minimize w1 + w2 (1) Ptot Ptot This objective function has two parts. The first part seeks to maximize the total power (Ptot ) obtained from the wind farm and the second part, which includes the total cost (costtot ), minimizes the cost per unit power generated. However, choosing the right values for the weighting parameters w1 and w2 can be difficult and choosing an inappropriate value might cause the algorithm to converge to undesirable solutions. The wind farm generator cost function used in [3] is given in (2), where Nt represents the total number of generators placed in the wind farm. costtot = cost per generator × Nt ( 23 + 1 −0.00174Nt2 ) 3e (2) While this work represents a reasonable approach to the problem, we think that our proposed methodology, which considers a continuous representation for the location of the generators and uses an alternative objective function, is preferable. 3. Problem description There are two fundamental questions that arise in the wind farm generator positioning problem. The first is how many generators to place on the site. The second is where to locate the generators to minimize the power lost due to wake decay effects. The answers to these two questions are clearly interrelated and therefore an effective solution approach must consider both simultaneously. In this section, we discuss three main topics: the modeling of the wind direction and intensity, the choice of selecting a discrete versus a continuous problem representation, and the mathematical formulation of the relaxed version of the generator positioning problem. 3.1. Wind direction and intensity For actual wind farm sites, the wind direction and velocity change over time. However, the prevailing wind directions and wind speeds can be estimated by looking at historical wind data. Therefore, we chose to explicitly consider multiple wind directions. Assuming only one prevailing wind direction simplifies the problem, but is not realistic for most sites. There are different models that may be used to approximate the wind directions. We decided to use a discrete model with eight directions because eight directions represents a reasonable compromise between accuracy and computational effort required for evaluation. This is done in our algorithm using a binary array of length eight with each direction having a one if there is wind high enough to produce power coming from that direction and having a zero otherwise. It was also necessary to model the fact that the wind does not blow with the same frequency and intensity from each direction. For example, a given location might have wind blowing from three of the possible directions throughout the year but the frequency and intensity of the wind may be different for each direction. We modeled this by using a weighting factor for each direction and requiring that the sum of the weights be equal to 1.0. 3.2. Comparison of continuous and discrete approaches in location of generators It is possible to utilize either a discrete or a continuous representation for the placement area. In either case the solution space is very large. Consider a problem with an area of 40D × 40D where D represents the wind turbine rotor diameter. Assume that we utilize a discrete approach and divide this area into 100 squares each having an area of 16D2 . Then the solution space for the generator positioning problem would have a size of 2100 since in each square we can decide to either locate or not locate a generator. One possible combination can be seen in Fig. 1. Even this relatively small problem has a huge search space and is a difficult problem to optimize. We have chosen to use a continuous representation, which means that we have an infinite search space. We have selected a continuous representation because putting a grid structure on the problem will set a predetermined area for each generator (the grid U. Aytun Ozturk, B.A. Norman / Electric Power Systems Research 70 (2004) 179–185 181 X X X X X X X X Fig. 2. Prevailing and crosswind directions. X X 3.4. Power output and interference effects Fig. 1. An example of wind farm turbine layout using a discrete representation. square it occupies) and reduce the number of generators that can be located on the site. The discrete approach introduces another complication because one needs to decide the grid size. 3.3. Objective function and constraints The objective function used in this paper is different from the one that was previously used in [3]. We decided to use a different objective function (see Eq. (3)) for two main reasons:
  costtot maximize profit = k − × Ptot (3) Ptot First, the former is difficult to work with because it is not simple to determine the relative weights for w1 and w2 . Second, we think that a more realistic objective is to maximize profit, which is equal to the product of the profit per kWh obtained and the total amount of electricity generated. In (3), k represents the estimated selling price for a kilowatt-hour of electricity in a given market. Note that k may vary depending on the market. The ratio of costtot /Ptot represents the ratio of the total generator cost over the total power produced and equals the cost per kWh for the wind farm. Thus, the term in brackets represents the expected profit per kWh generated by the generators in the wind farm. The second term represents the total expected power output of the wind farm. The expected total profit equals the product of these two terms. For illustrative purposes, in our test problems we assumed the same numerical value for both the maximum power generated by one generator and the cost for one generator. Therefore, if there was no loss due to wake decay the ratio of costtot /Ptot would equal 1.0. In addition, if there are too many generators located close together, so that the power output relative to their cost has a ratio of more than k, then the first term can assume a negative value. In our test problems we used a k value of 1.0 to obtain a solution that would be close to ideal. Wake decay effects are encountered when there is another generator located in the neighborhood of a generator in a prevailing (energy producing) wind direction or in a crosswind direction. The regions considered as prevailing and crosswind are shown in Fig. 2. The interference neighborhood is defined as 0–12D for the prevailing wind direction and 0–4D for the crosswind directions where distance is defined using the Euclidean metric. The distance between generators in any prevailing wind direction must be more than 8D because of potential turbine damage from wake decay effects, which results if distances of less than 8D are utilized. If the distance between two generators is between 8 and 12D in a prevailing wind direction then the power of the downwind generator is reduced. If the distance between two generators is more than 12D the generators do not affect each other. These values change to 2 and 4D, respectively, for the crosswind directions [5]. The magnitude of the power reduction due to wake effects is determined differently depending on the type of interference. If the interference is in the prevailing wind direction, then we have assumed the reduction is calculated according to a quadratic function (5), where d represents the distance between interacting generators. However, if the interference is in the crosswind direction, the reduction is calculated according to a linear function (4). The reduction functions employed here are approximate and the prevailing direction reduction is more significant and hence quadratic. These functions are obtained using interpolation, the actual reductions will be different depending on the vegetation, topography and other factors in and around the wind farm. Different reduction formulas can be employed without a change in the rest of the formulation and the optimization approach described later can still be applied. crosswind linear reduction = 0.5 − 1 4 (d − 2) (4) prevailing wind quadratic reduction = 1.7289 − 0.2836d + 0.0116d 2 (5) 4. Methodology The positioning problem is mathematically complex due to the constraints on locating the generators and the 182 U. Aytun Ozturk, B.A. Norman / Electric Power Systems Research 70 (2004) 179–185 resulting loss in power that occurs if generators are placed close together in a prevailing or crosswind direction. We begin by considering an optimal solution methodology for a relaxed version of the problem and then discuss our heuristic approach for the problem as it is described in Section 4.2. 4.1. Non-linear programming approach A simplified version of this problem would have one prevailing wind direction and an objective of maximizing the number of generators placed on the site while not allowing any reduction in power for any of the generators. In this case, the problem reduces to fitting as many generators as possible into a rectangular area. Here the generators cannot be located in each other’s interference zone, which is similar to an ellipse for this case. This problem can be modeled as a non-linear programming problem and the placement constraints are shown in (6)–(8). (Xi − Xj )2 (Yi − Yj )2 + ≥1 (12D)2 (4D)2 ∀i = j (6) 0 ≤ Xi ≤ 40D ∀i ∈ N (7) 0 ≤ Yi ≤ 40D ∀i ∈ N (8) Note that only the constraints are shown for this mathematical programming formulation because we are only interested in determining if a feasible solution containing n generators exists. In this formulation (Xi , Yi ) represent the coordinates of generator i, N represents the set of generator locations, n = |N|, and the goal is merely to find a feasible solution that places all n generators on the site. Constraint (6) ensures that generator i is not located in the interference zone (in this case it is an ellipse) of generator j and constraints (7) and (8) ensure that the generators are located within the wind farm boundaries. In the formulation it is assumed that n is known and a non-linear solver such as LINGO can be used to find solutions to the problem. In order to find the maximum number of generators that can be located at the site, one can iteratively search for the maximum value of n solving the problem below at each iteration for a given n value. When more than one prevailing wind direction has to be considered, the problem becomes more complicated but can be modeled in a similar manner. In the case where the prevailing winds come from all eight directions, a circle can approximate the interference zone. However, for the more complicated problem with power reductions induced by wake decay effects this model cannot be readily applied. This is true because in the more general version of the problem, multiple wind directions with varying intensities are considered. There are also potential problems with scaling this model to solve problems where the optimal solution requires locating many generators. 4.2. Greedy heuristic approach Due to the complications with using mathematical programming to solve the problem, a heuristic approach was considered. The heuristic search algorithm begins with an initial solution and utilizes three types of operations to modify this solution. The three operations are (1) add a new generator to the solution, (2) remove a generator from the solution and (3) move a generator to a different location. These three operations are now described in more detail. In the add operation, num add locations are randomly generated and each is investigated one at a time to determine the change in the objective function, F, that would occur if a generator was located there. In the remove operation, the net change in the objective function is determined for removing each of the existing generators individually. In the move operation, each generator is moved up to 4D in each of the eight wind directions using variable increments and the corresponding effect on the objective is noted. At each iteration, the algorithm looks at all three operations and chooses the one that improves the objective the most. This enables the algorithm to add generators, remove generators, and move generators around in successive steps. If no improving move is possible, the algorithm maintains the same current solution set S and in the next iteration it only investigates the add operation since both the remove and move operations will still result in non-improving moves. If there is no improvement in the objective function for no imp limit consecutive iterations, then the algorithm terminates. Initial experiments with the greedy (takes the maximum improvement in each iteration) improvement algorithm indicated that it gave good solutions but that it often converged prematurely to locally optimal solutions. Therefore we added a basic diversification step to help the algorithm escape from these local optima. Once no imp limit consecutive non-improving moves have been performed, the search is stopped and perturb frac of the generators have their location perturbed by a random amount and the search is resumed. This gives the algorithm the flexibility to move to another solution and continue the improvement process. If there is no improvement in num perturb consecutive perturbation iterations then the algorithm stops (Figs. 3–5). 4.2.1. Seeding with a random solution versus a constructed solution Because the search algorithm is greedy in nature, the final solution is affected by the choice of the initial solution. We investigated three different methods for determining an initial solution. The first is to start from a solution with L randomly located generators. The second is to begin with a grid-like solution where generators have been packed into the site. A simple two-dimensional circle packing can be used to determine the maximum number of generators that could be located. The packing was constrained by the minimum separation distances in the prevailing and crosswind directions. Once the number of generators was determined, U. Aytun Ozturk, B.A. Norman / Electric Power Systems Research 70 (2004) 179–185 183 then the generators were evenly spaced throughout the site. Fig. 6 shows an example packing for a problem where the maximum number of generators was 16. There is white space around the generators to indicate that the additional space in both the X and Y directions was evenly distributed among the generators. The third initialization method is a constructive heuristic. The algorithm starts with zero generators and adjusts the number and locations of the generators using the add, remove, and move operations described previously. Preliminary testing indicated that the constructive method was better than random initialization, but that seeding the solution with a grid-like covering produced the best results. Therefore, in the remaining discussion and for the test problems, a grid seeded initial solution was used. Fig. 3. Greedy algorithm. Fig. 4. Function move search. Algorithm Greedy with Perturbation Initialize num_perturb to zero Initialize Global_F to zero While num_perturb <25 Call Algorithm Greedy Set TF to (F – Global_F) If TF 0 Increment num_perturb Else Set num_perturb to zero Perturb the set S Fig. 5. Greedy algorithm with perturbation. 4.2.2. Parameter setting The algorithm depends on several different parameters/settings including those listed in Fig. 7. Therefore, we conducted preliminary testing to determine the best values for the parameters and settings. Based on these results, the final algorithm parameters were set. The algorithm was relatively insensitive to the values of no imp, num perturb, and perturb fraction. Preliminary testing indicated that values of 25, 25, and 15%, respectively, produced good results. Another important factor in the algorithm is the step size value. Step sizes of 1.0D, 0.5D, and 0.1D were tested and it was observed that while the results were not drastically different, in general the smaller the value of step size the better the algorithm performs. There is a trade-off, however, since small values of step size increase the running time of the algorithm. Based on these tests, a value of 0.1D was used for the test problems. 5. Results To evaluate our solution methodology we constructed six test problems with different siting area sizes as listed in Table 1. All of the problems had a square siting area and the Table 1 Test problem sizes Problem Problem Problem Problem Problem Problem 1 2 3 4 5 6 Fig. 6. Example of packing. Fig. 7. Algorithm parameters. 31 32 38 40 41 46 × × × × × × 31 32 38 40 41 46 184 U. Aytun Ozturk, B.A. Norman / Electric Power Systems Research 70 (2004) 179–185 Table 2 Binary direction arrays Case 1 Case 2 1 1 0 1 0 1 1 0 0 1 0 1 1 0 1 0 Table 3 Direction intensity arrays Case 1 Case 2 0.30 0.10 0.00 0.30 0.00 0.20 0.10 0.00 0.00 0.25 0.00 0.25 0.40 0.00 0.10 0.00 dimensions are in terms of wind generator rotor diameter, D. Note that two of these problems have sizes that permit a perfect maximal packing since their dimensions are multiples of eight in both directions. For each problem size, we tested the two different wind direction and intensity scenarios shown in Tables 2 and 3. To determine the quality of the solutions found by the proposed heuristic methods we investigated upper bounds on the objective function. One upper bound is to determine the maximum number of generators that can be placed on the site and find the corresponding objective function value assuming there is no power reduction due to wake decay effects. Unfortunately, due to the irregular shapes that can arise for the interference zones for the generators, it is dif- ficult to determine the maximum number of generators that can fit in an area. It is possible to ignore the geometry of the interference zones and simply look at the total interference area to establish a bound on the maximum number of generators. However, this bound is not very tight and results in values that are considerably larger, approximately three to four times larger, than the best-known solutions for the test problems. Moreover, this bound produces values that are as much as 10 times more than the optimal solution for small problem instances with known optimal solutions. Due to the large gap between this upper bound and the optimal solution, we do not present the upper bound values in our analysis. We do compare the results of the algorithm with a feasible solution that has the maximum number of generators in it. Recall that this solution is also used as the initial seed solution for the improvement heuristic. Thus, the difference between these two values shows the amount of improvement that results from applying the improvement heuristic. For each combination of size and scenario, we have run the algorithm for 10 different random number seeds. We ran multiple replications because both the add and perturbation steps in the algorithm are stochastic. The results are shown in Table 4. Notice that in 10 of the problems the heuristic improved the solution. There was no improvement in the 32 × 32 and 40 × 40 problems for the second scenario because the solution constructed by the maximum packing provided an optimal spacing between the generators and therefore an optimal objective. The results indicate that across the 12 problems, the improvement heuristic on average improved the seed solutions by 4.3%. Because the algorithm only requires 1–3 min of computation time per replication and we are analyzing a design problem that is not real-time constrained, we propose running the algorithm 10 times and taking the best solution. Considering this strategy, we also analyzed the performance of the best of the 10 replications relative to the seed solution and found that the best replication improved the seed solutions by 8.5% on average. We conducted a t-test to see if the solutions found by the Table 4 Test problem results Problem size Mean across 10 replications S.D. across 10 replications Maximum across 10 replications Maximum packing Improvement for mean replication (%) Improvement for best replication (%) Case 31 32 38 40 41 46 1 × × × × × × 31 32 38 40 41 46 16734.9 20639.5 43604.3 52865.5 64164.4 83473.1 706.8 1706.0 2289.4 448.3 1319.1 2410.4 17860.8 23967.3 45929.2 53898.7 65079.7 86903.7 15175.1 18702.2 40545.3 52596.7 57885.7 79949.7 10.3 10.4 7.5 0.5 10.8 4.4 17.7 28.2 13.3 2.5 12.4 8.7 Case 31 32 38 40 41 46 2 × × × × × × 31 32 38 40 41 46 16740.6 30604.2 46586.5 70434.7 76185.2 89737.4 886.0 0.0 503.4 0.0 1714.0 1050.1 17874.8 30604.2 47150.2 70434.7 77795.7 91354.7 16502.1 30604.2 45445.2 70434.7 74384.7 89085.7 1.4 0.0 2.5 0.0 2.4 0.7 8.3 0.0 3.8 0.0 4.6 2.5 U. Aytun Ozturk, B.A. Norman / Electric Power Systems Research 70 (2004) 179–185 improvement heuristic were significantly better than those found using the maximum packing heuristic. The test statistic value had a P-value <0.001 indicating that there was a significant difference between the two methods. These results clearly indicate the merits of using the improvement heuristic. 185 areas, etc., into the search space. This would make the search harder but the results would be more useful for practical purposes. Another improvement could be to consider stochastic wind intensity and direction patterns. This would necessitate changing the evaluation function for a given layout but need not fundamentally change the search dynamics. We could also explore additional seeding heuristics and solution methodologies. 6. Conclusions and future research In this paper, we have described the need for developing methods for positioning generators for wind farms. We have introduced a heuristic methodology that has the potential to improve the efficiency of a wind farm. The effectiveness of this methodology was demonstrated on a suite of test problems. As a conclusion, combining this tool with the expertise of a wind analyst can enhance the wind farm installation process. There are many ways that this current research can be extended. One is to change the convex search space by introducing non-convexity due to lakes, residential usage References [1] P. Avella, et al., Some personal views on the current state and the future of locational analysis, Eur. J. Oper. Res. 104 (1998) 269–287. [2] E. Erkut, S. Neuman, A survey of analytical models for locating undesirable facilities, Eur. J. Oper. Res. 40 (1989) 275–291. [3] G. Mosetti, C. Poloni, B. Diviacco, Optimization of wind turbine positioning in large wind farms by means of a genetic algorithm, J. Wind Eng. 51 (1994) 105–116. [4] S.H. Owen, M.S. 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