Tidal Turbine Array Optimization Based on the Discrete Particle Swarm Algorithm

In consideration of the resource wasted by unreasonable layout scheme of tidal current turbines, which would influence the ratio of cost and power output, particle swarm optimization algorithm is introduced and improved in the paper. In order to solve the problem of optimal array of tidal turbines, the discrete particle swarm optimization (DPSO) algorithm has been performed by re-defining the updating strategies of particles’ velocity and position. This paper analyzes the optimization problem of micrositing of tidal current turbines by adjusting each turbine’s position, where the maximum value of total electric power is obtained at the maximum speed in the flood tide and ebb tide. Firstly, the best installed turbine number is generated by maximizing the output energy in the given tidal farm by the Farm/Flux and empirical method. Secondly, considering the wake effect, the reasonable distance between turbines, and the tidal velocities influencing factors in the tidal farm, Jensen wake model and elliptic distribution model are selected for the turbines’ total generating capacity calculation at the maximum speed in the flood tide and ebb tide. Finally, the total generating capacity, regarded as objective function, is calculated in the final simulation, thus the DPSO could guide the individuals to the feasible area and optimal position. The results have been concluded that the optimization algorithm, which increased 6.19% more recourse output than experience method, can be thought as a good tool for engineering design of tidal energy demonstration.


Introduction
With the increasing issues of the environment degradation, the countries in the world one after another adhere to create the energy sustainable development system through the energy-structure adjustment. Because of high predictability in extracting power and little effect on environment, tidal energy is one of most potential resources in ocean energy (Bahaj, 2011). In order to improve the efficiency of the tidal power generation, the array with hundreds of tidal turbines should be arranged at a particular area (Macleod et al., 2002), which leads to the question of how to place the turbines within the area. When the space (between the rows and turbines) is too small, the turbines located in the downstream will be influenced by the wake effect, which results in the power reduction of the downstream turbines. When the space is too large, the tidal resource will be wasted and the economic benefits of the whole farm will be declined.
However, it is difficult to determine the optimal turbine array due to the complicated flow interaction between the turbines.
Then, it is necessary to find out an effective method to optimize and arrange the position of turbines, so that the power generation efficiency will be improved. To increase the tidal energy efficiency, many experimental studies have been conducted. Funke et al. (2014) applied the turbines farm optimization software in the optimization of four idealized scenarios, which is successful in increasing the power extracted by the farm. This software can predict the power extracted by using a two-dimensional nonlinear shallow water model. Lee et al. (2010) studied the reasonable distance between adjacent turbines in an array layout by applying a three-dimensional model. Myers and Bahaj (2005) investigated the energy losses within its layout and impacts because of the interaction of many turbines by optimizing the struc-ture of the array. Bilbao et al. solved the power maximization problem by using a gradient-based optimization algorithm (Jensen, 1983).
However, the array optimization is formulated as a complex nonlinear problem which is restricted to multi-variable and multi-constraint. In this paper, a method has been presented to maximize the power extraction of the array configurations that combines the wake model, elliptic distribution model and Farm/Flux model with DPSO that takes the orders of magnitude iterations. The methodology could be taken as a scientific reference for the optimum arrangement of the tidal power generators.

Theoretical models
Jensen wake model is introduced to analyze the influence of the wake effect between the tidal turbines on the flow distribution (Jensen, 1983;Kiranoudis and Maroulis, 1997). This model is based on the principle of the conservation of the momentum which is considered as conserved inside the wake. The wake has a radius D ij which is the radius of the downstream wake, while D is the upstream turbine radius. X is considered as the distance between the upstream of Turbine j and the downstream of Turbine i, while the relationship between D and D ij is described in the Jensen model as shown in Fig. 1.
The combined wake effect created by the turbines in tidal field may cause a reduction in the energy power output, and also arise unsteady loads on the downstream machines. The short distances between the turbines will make the downstream generators suffer serious influence of wake effect, which leads to low performance of the energy generation. Generally, in order to relief the loads of the downstream turbines, the minimum distances between the turbines are constant in the whole range which regarded as the circular distribution. But this layout which refers to the layout of the wind power generators easily makes the tidal resources to be wasted and the economic benefits of the whole farm to be declined. The directionality of the tidal flow throughout the tidal cycle has important implications for the tidal energy capture. There has been a tendency to infer that energetic sites possess near bi-directional flows or that there are sufficient sites with near bi-directional flows such that more omni-directional flow tidal currents can be neglected (Lin et al., 2017). For this reason, an elliptic distribution model is used to limit the minimum distances between the adjacent rows of machines as shown in Fig. 2, which is better than the circular distribution model.
The elliptical distribution, reducing the influence of the wake effect, can enlarge the longitudinal distance (parallel to the direction of tidal flow) between the adjacent rows of the turbines and shorten the transverse turbine spacing (perpendicular to the direction of tidal flow). Meanwhile, it can avoid the turbine located in the downstream of the adjacent upstream turbines, and also obey the distribution of the wake effect using this distribution. Therefore, based on the research conclusions in reference (Legrand, 2009), the distances considered in the present cases are in the range from 2.5D (Diameter of turbine rotor) to 10D, where 2.5D is the space of the adjacent columns of generators, and 10D is the space of the adjacent rows of generators. Therefore, the following equations of calculating maximum number of turbines are defined in Eq. (1).
where x min , x max , y min , and y max denote the range of research region, N i is an integer.
A key issue of improving the extracted power and reducing the cost of the power generation is to obtain the number of the best installed turbines by the Farm/Flux and empirical method (staggered grid array) (Ammara et al., 2002), where Farm/Flux are reviewed as the analytical models of the resource assessment. The following equations of calculating number of the turbines are used: where P Asite denotes the maximum extraction power, P Edevice is the extraction power of a single turbine, ρ is the sea water density, V is the flow velocity, is the cross-sectional area of the channel, denotes the total tidal reserves, SIF denotes the ratio of P Asite to the total tidal reserves, P m is the power density, A swept =π(D 2 /4) is the area of the turbine rotor swept, D is the turbine radius, η trans is the overall efficiency. Thus, the number of turbines  WU Guo-wei et al. China Ocean Eng., 2018, Vol. 32, No. 3, P. 358-364 359 is defined in Eq. (3).
The number of turbines is assumed to be N i , which is calculated by the empirical method from Eq. (1). In this method, all the turbines are deployed in a staggered array, and the distance between adjacent rows are equal in the array layout. The layout of the full array can be seen in Fig. 9. Then the best installed turbine number is defined in Eq. (4).
(4) Particle swarm optimization algorithm (PSO) has captured great attention in recent years, which was proposed by Kennedy (2010), and it has been successfully applied in many fields. A particle's position is deemed as a potential solution, and the flying trajectory of the particle is regarded as a continuous searching process. However, a number of studies demonstrated that the PSO easily falls into the premature convergence when facing complicated conditions (Lin et al., 2017). And the principle of the PSO is described in the previous study (Xia et al., 2017).
Therefore, considering the serious resource wasted by unreasonable layout scheme of tidal current turbines, which would influence the ratio of cost to output, the particle swarm optimization algorithm is introduced and improved in the paper. By re-defining the updating strategies of the particle's velocity and position, the discrete particle swarm optimization algorithm (DPSO) has been executed to optimize the arrangement of tidal current turbines. Assuming the collection , where n represents the dimensionality of the space, is a random value generated in 0 and 1, and m denotes the number of particles. During the searching process in the rasterized tidal farm where the grids are regarded as the solution space, x i is defined as a candidate position that represented by 1 as illustrated in Fig. 3.
The population (swarm) in the DPSO contains N t candidate solution (particles). The fitness function, defined in Eq. (4) is used to determine whether the particle positions of N t are good. The fitness value is calculated on the basis of the tidal speed of the particle position. If the fitness value is very small, the particle position result will be assumed as a bad state, and it is not used as the critical target during searching process.
where is the gross power generation, is the gross power generation at the maximum speed in the flood tide, is the gross power generation at the maximum speed in the ebb tide, and is the power generation efficiency related to the tidal velocity.
Every particle is represented by two vectors, i.e., a position vector and a velocity vector , where n represents the number of raster. The vector is regarded as a candidate particle while the vector is treated as a searching direction and step size of the particle. Assuming that the present particle position is and the previous particle position is at Iteration t, the new speed of the particle from to can be calculated by Eq.
where E denotes the collection of the optimum layout r i .
x t k,best x t gbest Along with the optimization process, each particle adjusts its trajectory relying on two vectors, namely as the personal historical best position vector and the global best position vector, respectively. Then, according to the speed of the particle, a new particle position can be calculated by Eq. (6). Actually, the new speed is calculated by the present and previous position.
In Eq. (6), T(x) is the convergence function and R(x) is the probabilistic selection function, which are all related to p, a random number generated in the interval [0, 1]. If the function T(x) is selected to update the speed of the particle with the probability of p, the positions where the positive value of 1 or 2 is located in will be randomly selected and assigned to be 1. Similarly, the positions where the negative value of -1 or -2 is located in will be randomly selected and assigned to be -1, and the other positions are assigned as the value of 0. For the complex functions T(x) with high probability of p, the DPSO has a slow speed on the convergence. Conversely, if the function R(x) is selected with the probability of 1-p, the positions where the positive value of 1 is located in will be randomly selected and assigned to be -1, the positions where the value zero is located in will be randomly selected and assigned to be 1, and the other position is assigned to be 0. For the functions R(x) with high probability of p, the DPSO with good randomness is hard to converge and has low optimization precision. After the velocity is updated, the particle moves to a new position from the current position. The framework of the DPSO is shown in Fig. 4.

Selection of turbine
The MCT's SeaGen turbines are selected. The rotor diameter of the turbine is 16 m, rated flow velocity is 2.25 m/s, cut-in speed is 0.7 m/s, cut-out speed is 3.5 m/s, rated power is 1.2 MW, and power coefficient is 0.45. The relationship between the output power (P) and inflow speed (v) is shown in Fig. 5.
The output power and inflow speed are fitted by the least squares method with 28 points, and the relationship is given as follows.
3.2 Experimental data Statistically, the total tidal current energy of the important water course can be 1400 MW in Zhoushan sea area (Hou et al., 2014). Especially in the Putuo Mountain-Hulu Island (P-H) waterway, the technical exploitation amount is 1.98-3.23 MW based on the Flux method, and that is between 5.33 MW and 6.08 MW based on the Garret method (Wu et al., 2017). As shown in Fig. 6, the research region located in the P-H water way has rich and stable tidal current resources, where is identified as the best development environment for tidal current energy. The range of this region is 600 m×1000 m, the average depth of the seabed is 37.03 m, and the grid resolution is 5 m. Ocean tide contains high tide process and low tide process. As shown in Table 1 and Fig. 7, the average annual flow velocity and flow direction at the maximum speed in the flood tide and ebb tide are used to facilitate the calculation.

Algorithm simulation η total
According to the Bryden's study, SIF is selected as 15%, is 40%, and A swept is 200.96. In the simulation experiment, the DPSO algorithm is applied to arranging the turbines, the particle position range is x min =442635.2662,    H=50 m x max =443635.2662, y min =3320478.175, y max =3321078.175, and . The best number of the installed turbines is 55 in this research region from Eq. (4), because N i is 90 from Eq. (1) and N f is 55 from Eq. (3). The higher the probability of the function T(x) in the discrete particle swarm algorithm, the higher the accuracy of the optimization result, but the longer the calculation time. Therefore, in order to ensure the accuracy, the probability of this paper is set to 0.9, and the number of the particles is 20. This paper sets the maximum number of iterations to be 16000, and E total is calculated on the basis of the maximum speed in the flood tide and ebb tide.
Since the DPSO algorithm will be probabilistic to select the particle velocity to update the function, the optimal layout results with the same iteration times may be unstable. Fig. 8 shows the change in the power generation efficiency with the number of iterations. With the increase of the iteration number, the power generation efficiency of the turbine keeps gradually increasing, indicating that the DPSO algorithm can optimize the turbine layout, and can effectively improve the power output and enhance the trend of energy utilization. And after 15000 of the DPSO algorithm iterations, the objective function values all have the indication of convergence.
Owing to different layouts of the turbines, the power output is different. In order to verify the validity and superiority of the model and algorithm, the DPSO algorithm is compared with the empirical method considering the same research region and turbines. In addition, the maximum iteration number of the DPSO is 15000 which can be determined from Fig. 8. As shown in Fig. 9, the empirical layout is interlacedly placed according to the equal spacing between the row and column. The longitudinal spacing is 10 times   the turbine diameter, and the horizontal spacing is 2.5 times the turbine diameter. Table 2 shows the simulation results and empirical res-ults, which are obtained by running 10 times with the maximum number (15000 times) of iterations of the DPSO algorithm.
As shown in Table 2, the power generation is not ideal with the empirical method, which is easy to perform. Compared with the DPSO algorithm, the empirical method could not fully consider the flow field distribution and wake effect on the importance of micro-sitting, which results in the low exploitation rate of tidal energy. the DPSO algorithm can achieve the optimal value of the power generation efficiency of 20231.61 kW, and the layout results are shown in Table 2. The power generation efficiency with the DPSO algorithm increased 6.19% compared with the empirical method. As shown in Fig. 10, according to the flow rate of the flood tide and ebb tide and safe running distance, the DPSO algorithm distributes more turbines in the area with large power density, and less turbines in the area with low power density. The optimized turbines layout by the DPSO is more reasonable than that by the empirical method.

Conclusion
In consideration of the wake effect, the reasonable distance between the turbines, and tidal velocities influencing factors in the tidal farm, the discrete particle swarm optimiz-ation algorithm (DPSO) has been performed in order to solve the problem of optimal deployment of tidal current turbines in rasterized tidal farm. In order to validate the scientificity and practicability of this algorithm, it is investig- ated in this paper by performing the simulation experiments with the actual sea area data. Compared with the power generated by running experience method, the results show that the DPSO algorithm can make the tidal turbine array selfoptimization, thus maximize the production capacity. In summary, it is feasible to use an intelligent optimization algorithm under multi-constraint condition. However, due to the immaturity of the method which determines the optimal number of the turbines, we find that the number of 55 turbines determined in this paper is not enough during the process of the simulation experiments. In addition, the distribution of the power density in the test area is relatively uneven, resulting that part of the study region is short of turbines. Therefore, the algorithm also needs to be further improved. The specific improvement of the DPSO algorithm will be comprehensively analyzed and verified according to the actual power generation after the operation of the power station.