Efficient Wind Turbine Micrositing in Large-Scale ...

Proceedings of the ASME 2016 10th International Conference on Energy Sustainability ES2016 June 26-30, 2016, Charlotte, North Carolina

ES2016-59594

EFFICIENT WIND TURBINE MICROSITING IN LARGE-SCALE WIND FARMS David Guirguis Mechanical & Industrial Engineering University of Toronto Toronto, ON, Canada [email protected]

David A. Romero Mechanical & Industrial Engineering University of Toronto Toronto, ON, Canada [email protected]

Cristina H. Amon Mechanical & Industrial Engineering University of Toronto Toronto, ON, Canada [email protected]

ABSTRACT

KEYWORDS Wind Energy; Wind Farm Layout Optimization; Interior-Point Method; Wind Farms; Mathematical Programing.

As wind energy is established as a sustainable alternative source of electricity, very large-scale wind farms with hundreds of turbines are becoming increasingly common. For the optimal design of wind farm layouts, the number of decision variables is at least twice the number of turbines (e.g., the Cartesian coordinates of each turbine). As the number of turbines increases, the computational cost incurred by the optimization solver to converge to a satisfactory solution increases as well. This issue represents a serious limitation in the computer-aided design of large wind farms. Moreover, the wind farm domains are typically highly constrained including land-availability and proximity constraints. These non-linear constraints increase the complexity of the optimization problem and decrease the likelihood of obtaining even a feasible solution. Several approaches have been proposed for micrositing of wind turbines, including random searches, mixed-integer programs, and metaheuristics. Each of these methods has its own trade-off between the quality of optimized layouts and the computational cost of obtaining the solution. In this paper, we demonstrate the capability of non-linear mathematical programming for optimizing very large-scale wind farms by leveraging explicit, analytical derivatives for the objective and constraint functions, thus overcoming the aforementioned limitations while also providing convergence and local optimality guarantees. For that purpose, two large farms with hundreds of turbines and significant land-use constraints are solved on a standard personal computer.

1. INTRODUCTION The crisis of climate change has received serious global attention, where the impacts and consequences of global warming on weather, sea levels, health, ecosystems, water and food security threaten our lives on the planet. In December 2015, leaders of 195 countries adopted a historic agreement for mitigation of climate change at the United Nations climate change conference in Paris [1]. The agreement aims to limit the rise in global temperature to below 2 ℃, and promises were made to decrease greenhouse gas emissions [2]. The economic sector that emits the majority of carbon dioxide is the energy production sector; it contributes ~35% of the annual global gas emissions [3]. Therefore, constructing very large-scale wind farms, particularly onshore wind farms, as one of the lowest-cost renewable power generating solutions [4], should be a priority. It has been demonstrated that the optimization of wind farm layouts is required for maximum output power, lower cost, increased life span, better environmental engagement and aesthetics [5–7]. Since the pioneering work of Mosetti et al. in optimal micrositing of wind turbines with genetic algorithms [8], most published articles have followed their use of metaheuristics

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and stochastic approaches to approximate the solution to the problem. For wind farm layout optimization (WFLO) problems formulated in continuous-variable domains, the number of decision variables is at least double the number of wind turbines. In case of very large-scale wind farms, hundreds of turbines are located inside a predefined domain while complying with setbacks and other land-use constraints, maintaining a minimum inter-turbine distance. Under limited computational resources, these nonlinear constraints, in addition to the high-dimensional decision vector, would limit the advantage of metaheuristics and population-based optimization methods in exploring the decision space. For that reason, randomized search approaches have been proposed [9,10]. Although using random-assisted searches may decrease the computational cost, they lack optimality guarantees and the quality of the solutions is primarily a function of the number of function evaluations required for the algorithm. The prohibitive computational cost limits the quality of optimized layouts. The main objective of micrositing is to maximize the output power. This is achieved by minimizing the effects of wake turbulence that is produced by upstream turbines and reduces the effective wind speed at downstream turbines. To this end, models of wake behavior that quantify energy losses produced by turbine wakes are paramount [7]. To exploit the mathematical nature of the wake models, optimization approaches with mixedinteger linear programs have been proposed [11–15], albeit based on discrete problem formulations that restrict turbine locations to a set of pre-defined locations or a simplified model [15]. In order to push the limits of micrositing, the optimization approach must require a lower computational cost without scarifying the quality of the optimized solution, must be capable of tackling wake models that better approximate the real problem, and must provide optimality guarantees. Towards this goal, in this paper, we demonstrate the capability of nonlinear mathematical programing in solving very large-scale WFLO problems. In contrast to some previous work, our formulations use the turbine’s manufacturer power curve, use the sum of squares approach to better model multiple wakes interactions [16], and include significant land-availability constraints, while enforcing lower computational budgets than previous work with randomized searches [9,10]. Additionally, the potential benefits of a multi-start local search approach is further demonstrated in the context of multi-disciplinary optimization. Namely, we solve for the optimal electrical infrastructure of large scale wind farms as a post-processing step for each of the local optima obtained during the multi-start gradient-based optimization of the annual energy production. The rest of the article is organized as follows. Section 2 introduces the objective function and the mathematical model of the land-use constraints. Section 3 presents the optimization approach and exact gradients of the objective and constraints’ functions. The numerical experiments and the discussion of

obtained results are included in Section 4. Finally, the article ends with concluding remarks and the planned future work. 2. PROBLEM FORMULATION

2.1. Annual Energy Production The main objective of wind turbines micrositing is to maximize the real output electric power, however, most of proposed mathematical approaches consider maximizing the theoretical output power with the constant value of Betz limit (i.e., maximum theoretical limit that equals 16/27) for the power coefficient [11–14,17,18]. The power coefficient is a measure for the wind turbine’s performance, which is the ratio of power in the wind and turbine’s extracted power [19]. The power coefficient is not constant, but a function of wind speed. Thus, in order to capture the actual behavior of the wind turbine, it is required to consider the turbine’s manufacturer power curve. For the Vestas V1001.8MW turbine, the manufacturer-supplied electric power curve, power coefficient and theoretical wind power curve, as functions of wind speed, are illustrated in Fig. 1. Theoretically, the turbine’s output power [19] can be calculated by the following equation: 𝑃𝑜𝑢𝑡 =

1 𝜌 𝐴 𝑈3 𝐶𝑝 𝜂 2

(1)

where 𝜌 is the air density, A is the swept area, 𝑈 wind speed at the hub height, 𝐶𝑝 is the power coefficient and 𝜂 is the efficiency of the turbine. The real electric power curve of pitch-controlled turbines can be divided into three sections with respect to the wind speed as follows: ℎ(𝑈) 𝑃𝑜𝑢𝑡 = { 𝑃𝑟𝑎𝑡𝑒𝑑 0

, 𝑓𝑜𝑟 𝑈𝑐𝑢𝑡−𝑖𝑛 ≤ 𝑈 < 𝑈𝑟𝑎𝑡𝑒𝑑 , 𝑓𝑜𝑟 𝑈𝑟𝑎𝑡𝑒𝑑 ≤ 𝑈 ≤ 𝑈𝑐𝑢𝑡−𝑜𝑢𝑡 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

(2)

Figure 1: Vestas V100-1.8MW turbine's manufacturer electric power curve, power coefficient and theoretical wind power curve as functions of wind speed.

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The power in the section of the curve that lies between the cut-in speed 𝑈𝑐𝑢𝑡−𝑖𝑛 and the rated speed 𝑈𝑟𝑎𝑡𝑒𝑑 , is a function of wind speed ℎ(𝑈). The power is nearly constant for speeds higher than the rated speed and less than the cut-out speed 𝑈𝑐𝑢𝑡−𝑜𝑢𝑡 , while the turbine shuts down to prevent damage for speeds exceeding 𝑈𝑐𝑢𝑡−𝑜𝑢𝑡 . In order to establish a differentiable mathematical function for the annual energy production, the turbine data from the manufacturer’s reference document [20], is fitted by a polynomial function. Fig. 2 shows fitted polynomials with different orders for the Vestas V100-1.8MW [20]. In order to avoid oscillations in the fitted model (see the zoomed section in Fig. 2), the few data points which are close to the rated power can be adjusted to the same power value of 𝑃𝑟𝑎𝑡𝑒𝑑 . Moreover, the fitted curve must be connected to the horizontal line of 𝑃𝑟𝑎𝑡𝑒𝑑 . The corrected 4th order polynomial model is shown in Fig. 3.

Hence, equation (2) can be rewritten as: ℎ(𝑈) , 𝑓𝑜𝑟 𝑈𝑐𝑢𝑡−𝑖𝑛 ≤ 𝑈 < 𝑈∗ = { 𝑃𝑟𝑎𝑡𝑒𝑑 , 𝑓𝑜𝑟 𝑈∗ ≤ 𝑈 ≤ 𝑈𝑐𝑢𝑡−𝑜𝑢𝑡 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

(3)

ℎ(𝑈) = 𝛽1 𝑈4 + 𝛽2 𝑈3 +𝛽3 𝑈2 + 𝛽4 𝑈 + 𝛽5

(4)

𝑃𝑜𝑢𝑡

Where 𝑈 ∗ is the speed coordinate of the point that connects the polynomial curve ℎ(𝑈) with the horizontal line of 𝑃𝑟𝑎𝑡𝑒𝑑 , and 𝛽1, 𝛽2, 𝛽3 and 𝛽4 are the polynomial coefficients of the fitted model. Note that this representation of the turbine’s power curve is continuous and differentiable, but also more accurate than the ideal power curves used in some previous works. Finally, for a wind farm with 𝑁 turbines and a wind regime of S wind directions with a probability 𝑝, the annual energy production is expressed as: 𝐴𝐸𝑃 = 8766 ∑𝑆𝑑=1{∑𝑁 𝑘=1 𝑃𝑜𝑢𝑡,𝑘𝑑 } 𝑝𝑑

(5)

2.2. Wake Modelling Wind speeds behind wind turbines are reduced by wake turbulence produced by the rotating turbine blades. This deficit in wind speed results in a reduction in captured energy by downstream turbines. Excluding cases of high-complex terrains, the commonly used engineering wake models have acceptable accuracy by the industry [7]. Moreover, in far wakes, the velocity deficit is accurately described by a Gaussian profile rather than that with a “top-hat” shape [21,22], which leads to a continuous function for the wind velocity inside the wake region. In this paper, we use the Jensen’s model [23] for its simplicity and wide usage. However, for very large-scale wind farms, it is recommended to use models that take into account the two way interactions between the turbines and atmosphere, such as that by Frandsen et al. [24], or the deep-array wake model [25]. A future study may focus on the effects of using these alternative wake models on optimized solutions’ quality and computational cost. According to the Jensen’s model, the wind velocity at a downstream distance 𝑋 from a turbine with radius 𝑟𝑜 is:

Figure 2: Fitted polynomials to the data points of the manufacturer's power curve.

𝑣𝑘 = 𝑢 [1 −

2 2 𝑟𝑜 𝑓(𝜃) ( ) ] 3 𝑟𝑜 + 𝛼 𝑋

(6)

where 𝑢 is the undisturbed wind velocity, 𝛼 is the entrainment constant, and 𝜃 is the angular deviation of the location where the velocity is being calculated, from the wind direction. In Eq. (6), 𝑓(𝜃) is a Gaussian modulation that was proposed by Jensen [23] to smooth out the predicted velocity profile inside the wake, and which was validated by site measurements. In addition to the realistic description for the far wake, the modulation term 𝑓(𝜃) makes the wind velocity profile continuous at the boundaries of the wake region [18].

Figure 3: Adopted 4th order polynomial power curve in comparison with the manufacturer power curve.

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The entrainment constant at turbine’s hub height 𝑧 and for surface roughness 𝑧𝑜 is defined mathematically as: 1 𝛼= (7) 𝑧 2 (ln

𝑧𝑜

Forbidden zones

)

a

Feasible area

The Gaussian modulation of the velocity deficit is: 1 + cos(9𝜃) , 2 𝑓(𝜃) = { 0 ,

Bounding rectangle

π 𝜃≤ 9 π 𝜃> 9

(8)

i A

b

The affected wind velocity by multiple wake of 𝑛 turbines is calculated by Mosetti et al. [8] as: Figure 4: Modelling of land-use constraints.

𝑛

𝑣𝑘 2 𝑣𝑘𝑑 (𝑥, 𝑦) = 𝑢 1 − √∑ (1 − ) 𝑢 𝑘=1 [ ]

(9) 3. OPTIMIZATION APPROACH

It worth mentioning that, accounting the partial-wake in the model is beneficial for “top-hat” models mainly because of the high overestimation of the wake deficient near the boundaries of the wake region. On the other hand, the inaccuracies in the velocity estimation near the wake boundaries are much less in case of models with Gaussian velocity profile as concluded by Bastankhah and Porté-Agel in [22].

In the continuous-variable formulation of the WFLO problem, the turbines’ locations are represented by real-coded variables for their Cartesian coordinates (𝑥,𝑦). For problems with multi-directional wind regimes, during the simulation, the coordinates are rotated to align the layout with each wind direction [28]. The total number of turbines and turbine’s features are fixed, since our focus in this work is to solve the WFLO more efficiently. If successful, the proposed approach could be used to solve (in parallel) multiple WFLO problems for different number of turbines or turbine types. In other words, the proposed approach could serve as a building block, as an enabler for more comprehensive wind farm design efforts, should the need arise. In addition to the land-availability constraints, proximity constraints are defined explicitly, by means of which a minimum distance is enforced between each pair of turbines. For a pre-defined number of turbines 𝑁, the optimization problem is defined as:

2.3. Land-availability Constraints Modelling Most of real wind farms are associated with irregular boundaries [9,26]. Specific zones inside the wind farm’s bounding box may be excluded from turbine placement due to terrain conditions (e.g., ground unsuitable for construction), presence of dwellings, roads, natural features such as rivers and lakes. Other areas with community and environmental considerations as noise, wildlife and bird migration can also be modeled as forbidden zones. For stochastic and metaheuristic methods, these land-availability constraints can be approached implicitly as in [9,27], while in case of non-linear mathematical programming, a differentiable mathematical model is required. As illustrated in Fig. 4, for a forbidden zone “𝐴”, the landavailability constraint can be modeled as:

Objective: max 𝐴𝐸𝑃(𝑥, 𝑦) Subject to: 𝑔𝑚 (𝑥, 𝑦) ≤ 0, 𝑚 = 1, … , 𝑀 𝐶𝑙 (𝑥, 𝑦) ≤ 0, 𝑙𝑏 ≤ 𝑥𝑖 ≤ 𝑢𝑏, 𝑙𝑏 ≤ 𝑦𝑖 ≤ 𝑢𝑏,

𝐶𝑖 (𝑥𝑖 ,𝑦𝑖 ) = √(𝑥𝑖 − 𝑥𝑎 )2 + (𝑦𝑖 − 𝑦𝑎 )2 + √(𝑥𝑖 − 𝑥𝑏 )2 + (𝑦𝑖 − 𝑦𝑏 )2 − 𝐿𝑎𝑏 , (𝑥𝑖 𝑦𝑖 ) ∈ 𝐴 { 0, (𝑥𝑖 𝑦𝑖 ) ∉ 𝐴

𝑙 = 1, … , 𝐿 𝑖 = 1, … , 𝑁 𝑖 = 1, … , 𝑁

(11) (12) (13) (14) (15)

where the objective function is the annual energy production, as discussed in Section 2.1, 𝑀 and 𝐿 are the total numbers of proximity and land-availability constraints, lb and ub are the lower and upper bounding constraints of the decision variables, which represent the bounding box of the wind farm domain. The proximity distance is set to five times the turbine diameter, as is commonly reported in the wind farm literature. Thus, the proximity constraint for each pair of turbines 𝑖 and 𝑗 is defined as:

(10)

where (𝑥𝑖 ,𝑦𝑖 ) is the Cartesian coordinates of the turbine, (𝑥𝑎 , 𝑦𝑎 ) and (𝑥𝑏 , 𝑦𝑏 ) are the Cartesian coordinates of points 𝑎 and 𝑏 respectively, and 𝐿𝑎𝑏 is the length of line segment ̅̅̅ 𝑎𝑏. Thus, for any turbine located inside an infeasible area, the constraint violation is a positive number and decreases gradually until reaching the shared edge with the feasible area.

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𝑔𝑚 (𝑥, 𝑦) = 5𝐷 − √(𝑥𝑗 − 𝑥𝑖 )2 + (𝑦𝑗 − 𝑦𝑖 )2

(16)

𝐸 = 𝑠𝑖𝑛 (9 𝑡𝑎𝑛−1

The interior-point method (IPM) with barrier functions is used to solve the nonlinear constrained problem. At each iteration, the algorithm tries to satisfy the KKT optimality conditions. The detailed algorithm can be found in [29,30]. The gradients are coded and added to the program, whereas for limited memory, the Broyden–Fletcher–Goldfarb–Shanno (BFGS) quasi-Newton approach [30] is used to generate the Hessians at each iteration. The partial derivatives of the objective function are calculated by differentiating Eq. (5) with respect to Cartesian coordinates 𝑥𝑖 and 𝑦𝑖 , and are given by: 𝜕𝐴𝐸𝑃 𝜕𝑥𝑖

𝜕𝐶1 𝜕𝑥𝑖 𝜕𝐶1 𝜕𝑦𝑖

=

𝜕𝑣𝑘𝑑 𝜕𝑥𝑖

𝑓𝑜𝑟 𝑈𝑐𝑢𝑡−𝑖𝑛 ≤ 𝑣𝑘𝑑 < 𝑈 0 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

{

=

𝑦𝑖 −𝑦𝑎

=

𝜕𝑥𝑖

∂xj 𝜕𝑦𝑖

)} 𝑝𝑑 ,

𝑓𝑜𝑟 𝑈𝑐𝑢𝑡−𝑖𝑛 ≤ 𝑣𝑘𝑑 < 𝑈∗ 0 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

{

𝑣

𝜕𝑥𝑖

𝑘 ∑𝑛 𝑘=1( −1)

= −(

𝑢

𝜕𝑣𝑘 𝜕𝑥𝑖 2

𝑣𝑘 √∑𝑛 𝑘=1(1− )

)

(19)

𝑣

𝜕𝑦𝑖

𝑘 ∑𝑛 𝑘=1( −1)

= −(

𝑢

𝜕𝑣𝑘 𝜕𝑦𝑖 2

𝑣𝑘 √∑𝑛 𝑘=1(1− )

)

(20)

𝑢

The partial derivatives of wind speed for a downstream turbine 𝑖, which is affected by an upstream turbine 𝑗 if located in its wake region, are calculated as: 𝜕𝑣𝑘

={

𝜕𝑥𝑖

𝜕𝑣𝑘 𝜕𝑦𝑖

={

−

2𝑢 (𝐴−𝐵)(𝑥𝑗 −𝑥𝑖 ) 3

𝑖𝑓 𝜃 ≤

|𝑥𝑗 −𝑥𝑖 |

0 −

3 𝑢 𝐸 (𝑦𝑗 −𝑦𝑖 )

𝑖𝑓 0 < 𝜃 ≤

|𝑦𝑗 −𝑦𝑖 |

0 𝜕𝑣𝑘 𝜕𝑦𝑗

=−

𝜋 9

(21)

𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝜋 9

(22)

𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

𝜕𝑣𝑘 𝜕𝑣𝑘

,

𝜕𝑦𝑖 𝜕𝑥𝑗

𝜕𝑣𝑘

=−

(23)

𝜕𝑥𝑖

where 𝐴=

𝐵=(

𝛼 𝑟𝑜2 (𝑟𝑜+ 𝛼|𝑥𝑗 −𝑥𝑖 | )

𝑟𝑜

𝑟𝑜+ 𝛼|𝑥𝑗 −𝑥𝑖 |

2

3

(1 + cos (9 tan−1

) 𝑠𝑖𝑛 (9 𝑡𝑎𝑛−1

|𝑦𝑗 −𝑦𝑖 | |𝑥𝑗 −𝑥𝑖 |

))

(24)

|𝑦𝑗 −𝑦𝑖 |

4.5 |𝑦𝑗 −𝑦𝑖 |

|𝑥𝑗 −𝑥𝑖 |

(𝑥𝑗 −𝑥𝑖 ) +(𝑦𝑗 −𝑦𝑖 )

)(

2

2

𝑟𝑜

𝑟𝑜+ 𝛼|𝑥𝑗 −𝑥𝑖 |

2

) (26)

=

+ +

𝑥𝑖 −𝑥𝑏 √(𝑥𝑖 −𝑥𝑏 )2 +(𝑦𝑖 −𝑦𝑏 )2 𝑦𝑖 −𝑦𝑏 √(𝑥𝑖 −𝑥𝑏 )2 +(𝑦𝑖 −𝑦𝑏 )2 𝑥𝑖 −𝑥𝑗 2

2

√(𝑥𝑗 −𝑥𝑖 ) +(𝑦𝑗 −𝑦𝑖 ) 𝑦𝑖 −𝑦𝑗 2

√(𝑥𝑗 −𝑥𝑖 ) +(𝑦𝑗 −𝑦𝑖 )

(27) (28) (29) (30)

2

= −

∂𝑔𝑚 ∂𝑔𝑚 ∂xi

,

∂yj

= −

∂𝑔𝑚 ∂yi

(31)

The main objective of this study is to demonstrate the capability of nonlinear mathematical programming to push the computational limits of very large-scale wind turbines’ micrositing, without sacrificing the quality of optimized layouts, and without compromising their accuracy due to the use of oversimplified physical models. For that purpose, test cases of highly constrained wind farms with hundreds of turbines range from 100 to 400 are solved using the discussed approach in previous sections. The land-use constraints used in the test cases represent the shapes of the wind farm boundaries of the Danish Anholt offshore wind farm and the British West of Duddon Sands wind farm, with feasible areas of 51.2% and 59.6%, respectively (see Fig. 5). The boundaries of the test cases are adopted from [31]. The power curve and dimensional specifications of the modern turbine Vestas V100-1.8MW [20] are used in all numerical experiments. A real wind data that exhibits variations in the average velocity for different wind directions is adopted from [32], albeit with a different preferential direction to ease the analysis of the resulting optimal solutions. We use 24 wind directions to facilitate the comparison with other approaches in the literature; however, more directions are recommended for more accurate power calculations [33]. The used wind rose in our test cases is shown in Fig. 6. A highly constrained instance is shown in Fig. 7. In this example, 100 turbines are located inside a domain of 4.5x4.5 km. The severity of the proximity constraints is significant due to the high power density. It looks similar to a packaging problem where a possible solution could be a grid of 10x10 turbines with a spacing of 0.5 km (the minimum inter-turbine distance allowed by the constraints). The solver is started with an infeasible random solution, however, during the optimization iterations some turbines are moved long distances, in cases exceeding 25%

𝑢

𝜕𝑣𝑘𝑑

)(

4. NUMERICAL EXPERIMENTS

(18)

Where 𝜕𝑣𝑘𝑑

2

The derivatives of the proximity constraint with respect to coordinates of turbine 𝑗 are:

(17)

∂𝑔𝑚 𝜕𝑣𝑘𝑑

(𝑥𝑗 −𝑥𝑖 ) +(𝑦𝑗 −𝑦𝑖 )

√(𝑥𝑖 −𝑥𝑎 )2 +(𝑦𝑖 −𝑦𝑎 )2 𝜕𝑔𝑚

=

3 2 8766 ∑𝑆𝑑=1 {∑𝑁 𝑘=1 ((4𝛽1 𝑣𝑘𝑑 + 3𝛽2 𝑣𝑘𝑑 +2𝛽3 𝑣𝑘𝑑 + 𝛽4 )

2

𝑥𝑖 −𝑥𝑎

𝜕𝑦𝑖

)} 𝑝𝑑 ,

|𝑥𝑗 −𝑥𝑖 |

)(

√(𝑥𝑖 −𝑥𝑎 )2 +(𝑦𝑖 −𝑦𝑎 )2

𝜕𝑔𝑚

∗

𝜕𝑦𝑖

|𝑥𝑗 −𝑥𝑖 |

By differentiating Eqs. (10) and (16) with respect to the turbine coordinates 𝑥𝑖 and 𝑦𝑖 , the partial derivatives are:

=

3 2 8766 ∑𝑆𝑑=1 {∑𝑁 𝑘=1 ((4𝛽1 𝑣𝑘𝑑 + 3𝛽2 𝑣𝑘𝑑 +2𝛽3 𝑣𝑘𝑑 + 𝛽4 )

𝜕𝐴𝐸𝑃

|𝑦𝑗 −𝑦𝑖 |

) (25)

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(a)

of the wind farm’s length, and end up distributed in a semiuniform staggered layout. After 2,893 function evaluations, which were required for convergence, the annual energy production is improved by 9.1% and the maximum constraint violation of 452.2 m (at the starting infeasible solution) is diminished to zero. Five runs for each land-use constrained instance were performed in parallel under the MATLAB environment on a standard desktop computer with a quad-core 3.6GHz processor, and 1600MHz memory speed. The obtained results are shown in Table 1. Although all runs are started by random infeasible solutions, they are converged to feasible solutions with average improvements’ range of 3.14% to 5.09%. The computational costs range from 22 minutes for 100 turbines to 6.86 hours for 400 turbines. However, because different approaches reported in the literature have used different programming languages and may have used different code optimization strategies, the number of function evaluations is a more suitable comparison criterion to evaluate the computational efficiency of the proposed approach. The mean numbers of function evaluations range from hundreds to less than a couple of thousands, which are 1 to 2 orders of magnitude less than those that were reported in previous work [9,10,34]. In Fig. 8, the AEP values of the optimized layouts are sorted and plotted, along with the corresponding computational cost. As observed from Fig. 8, even though the AEP of the optimized turbine layouts does not change significantly between runs (standard deviation does not exceed 0.039), the computational cost, measured in terms for the number of function evaluations, ranges from a few hundreds to a few thousands, even for the same number of turbines. Thus, the sensitivity of the optimization algorithm to the initial solution affects the computational cost more than the quality of the optimized solutions.

(b)

Figure 5: Land-use constraints of the numerical experiments.

Figure 6: The wind rose shows average wind speeds in each direction.

(a)

(b)

Figure 7: Optimizing 100 turbines with high severity of proximity constraints: (a) history of turbines movements, “x” indicates the initial positions of the turbines, “o” indicates their final position. (b) Convergence behavior during 2,893 function evaluations.

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Table 1: Results of land-use constrained instances Number of Turbines

Annual Energy Production (TWh)

Layout Shape

Worst

Median

Best

100 150

0.5488 0.8058

0.5517 0.8105

0.5546 0.8128

Standard Deviation 0.0022 0.0028

200

1.0525

1.0553

1.0562

300

1.6005

1.6082

1.6111

350

1.7737

1.8405

400

2.0424

2.0837

Average Improvement %

Mean Computational Cost

3.14 4.31

Function Evaluations 996 1491

Elapsed Time (hours) 0.3587 1.2191

0.0017

5.09

1516

2.2946

0.0043

5.04

1781

5.7194

1.8515

0.0389

4.78

771

3.9349

2.0954

0.0235

5.04

961

6.8576

Function Eval’s

AEP

3500

2

3000 2500

1.5 2000 1

1500 1000

Function Evaluations

Annual Energy Production (TWh)

2.5

0.5 500 0

0 100 Turbines

150 Turbines

200 Turbines

300 Turbines

350 Turbines

400 Turbines

Figure 8: Sorted AEP values and associated computational cost of each optimization run, for different numbers of turbines.

Table 2: All optimized 400-turbines-layouts, with their associated AEP values and cabling costs. Note that each group of turbines connected to the same substation is color-coded.

Layout

AEP Cabling Cost

2.0837 TWh 202.85 M €

2.0521 TWh 211.70 M €

2.0954 TWh 204.07 M €

7

2.0424 TWh 210.60 M €

2.0878 TWh 202.78 M €

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In contrast to population-based optimization methods such as GA’s where the population size should be increased as the number of variables increases for extensive exploration of the decision space, the number of function evaluations required for convergence seems to be independent of number of turbines. We hypothesis that this is due to the information that the gradient of the objective function provides, allowing for a more efficient trajectory towards the nearest local optima than random-assisted or metaheuristic search methods. It is important to note that, if the optimization solver starts from a poor layout, the number of function evaluations required for convergence increases. Though it is generally not possible to determine a priori what a poor initial solution looks like in general optimization problems, domain-specific knowledge can be used to provide some guidance about what a good starting layout looks like. For example, in [18], we demonstrated the superiority of IPM utilizing exact gradients over GA and commonly used hybrid GA, under different wind regimes with moderate numbers of turbines. Therein, we showed that a uniform staggered layout provides a good starting point, frequently leading to a local optimum that is as good as or better than other local optimum, consuming less computational cost. The superiority over GA is expected to hold to the very largescale wind farms because of the decline in performance at high dimensionality. In the results shown in Fig. 8 and Table 1, we have noted that the standard deviation of AEP ranges from 0.0022 to 0.039 for all instances. This indicates that the optimized solutions are close in terms of their objective function value; however, the actual turbine layouts that lead to these AEP values should differ because they depend on the initial solutions, as a consequence of the inherent multimodality of the problem. This property can be utilized by a multi-start, gradient-based, local search approach to provide the wind farm developer with a set of high quality layouts, with close AEP values. From this array of solutions to the wind farm layout optimization problem, the designer can undertake other quantitative or qualitative assessments which cannot be easily considered during the optimization process, e.g., visual impact, shadow flicker, more accurate CFD-based assessments, or detailed infrastructure design which may be approached as a separate problem. In such cases, the multimodality of the problem and the availability of an array of locally optimal solutions from a multi-start approach, has the potential to inform the designer of designs tradeoffs. To illustrate this, we calculated the optimal cabling costs of the obtained turbine layouts for the cases with 400 turbines. To this end, the turbines were clustered into four groups using kmeans++ algorithm [35], where each group of turbines is supposed to be connected to an electrical substation. A cable network is established for each group by using Prim’s algorithm [36], where the objective is to find the minimum spanning tree. The cost parameters are taken from [37]. Although this implementation is unrealistic because the shortest path is not necessarily the most inexpensive way to connect the turbines in real wind farm design studies [37], it illustrates how the electrical

costs vary for different wind farm layouts that have similar AEP values. All optimized layouts with their scores of AEP and cabling costs are listed in Table 2. As shown, a tradeoff is observed between the layouts 3 and 5. It can also be seen that layouts 1 and 5 are different but they have close objective values, which supports the above discussion. CONCLUSIONS In this paper, very large-scale wind farms with hundreds of turbines and significant land-use constraints are optimized using an interior-point method, leveraging exact analytical derivatives of the objective and constraints’ functions. A power model, representing the turbine’s power curve, is developed by fitting a polynomial function to the manufacturer’s power curve. In addition, the modulation term proposed by Jensen to smooth the velocity profile inside the wake is used to ensure that the wake model is continuous across the wake boundary. Thus, the use of exact derivatives and nonlinear mathematical programming to solve the WFLO problem is enabled without introducing discontinuities, alternative simplified wake interaction models or using a theoretical power curve. The computational costs, expressed in terms of number of function evaluations that are required to converge to an optimal solution using the proposed approach, are 1 to 2 orders of magnitude less than those that have been reported in the literature for metaheuristics and stochastic optimization methods. It is demonstrated that using nonlinear mathematical programming based on exact derivatives of the objective and constraint functions in WFLO problems is feasible, requires less computational costs, and can lead to more efficient wind turbine layouts. ACKNOWLEDGMENTS This research work was financially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), through the Collaborative Research and Development (CRD) and the Discovery Grant Programs. REFERENCES [1]

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