Minimizing the Energy Cost of Offshore Wind Farms by Simultaneously Optimizing Wind Turbines and Their Layout

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satisfactory turbine with low COE require suitable physical and operational parameters, which are related to the wind resource at the selected site but have rarely been studied. In Reference [22], the authors presented a framework for the site-specific design optimization of a horizontal-axis onshore wind turbine, in which the blade number, rotor diameter, tower height, rotor rotational speed, rated wind speed, and rated power were optimized to match the wind condition described by the Weibull distribution and parameters. Mirghaed et al. [23] developed an iterative approach to optimize a single turbine with minimum COE, in which the capital cost was a function of all turbine components, such as rotor diameter, hub height and rated power. In the study, it was found that the onshore turbines with capacities of about 1-2 MW lowered their COE by about 45-75 $/MWh. In Reference [24], Luo et al. proposed a mathematical approach to minimize the COE of onshore turbines, in which the COE model was expressed as a function of rated power and rated wind speed. The above references have shown that the site-specific turbine design can achieve a low COE for onshore turbines, but studies of offshore wind turbines are lacking. Furthermore, the WTDO was only performed on an individual wind turbine rather than considering the whole wind farm, thus the achieved results may be unrealistic.
Currently, offshore wind turbines are designed to fit the large-size trend of high capacity and long blades and, thus, the issue of the high cost of energy is increasingly prominent. This paper aims to obtain the parameters of a cost-effective offshore wind farm. That is, a wind farm with a minimum COE. With regards to the literature, the contributions of this paper are summarized as follows. Firstly, this was the first study to address the COE optimization of wind farms by simultaneously optimizing the design parameters and the layout of wind turbines. Secondly, it established a mathematical description of the COE as a function of two design parameters of wind turbines and their layout, which can be extended to include more design variables. Thirdly, it proposed a composite optimization approach comprising an iterative algorithm and an improved particle swarm optimization (IPSO). As the concerned optimization issue involves two types of design variables, turbine design parameters and turbine layout, and the latter variable depends to some extent on the former, a single optimization approach may not be efficient to solve the issue. In the proposed approach, the iterative algorithm was the main optimization loop used to find the optimal wind turbine parameters, while the improved particle swarm optimization was a sub-optimization loop used to help the construction of the optimized layout. By doing so, the optimal solution could be conveniently determined. Finally, the utilization of the proposed approach was illustrated through three real offshore wind sites, which can be used as references for offshore wind farm designers.
The remaining sections are organized as follows: The COE model of offshore wind farms is discussed in Section 2, and Section 3 presents the method for optimizing COE by selecting the optimal wind speed, rotor diameter and the turbine layout. The case studies are described in Section 4 and, finally, Section 5 concludes the study.

Proposed Energy Cost Model for Offshore Wind Farms
The proposed COE of offshore wind farms relates to two factors: the total annual energy production ( total AEP ) and the total annual cost ( total Cost ), and is defined as follows:

Annual Production Cost
On the offshore wind farm, two types of support platforms were applied, namely, the bottom-fixed platform and floating platform. Usually, the bottom-fixed platform is installed when the offshore wind farm is near the coastline, namely, the shallow-water sea, while the floating platform is employed on sites far from the coastline or beyond, namely, the deep-water sea. As the cost is a key factor in determining the COE, the detailed cost model of the bottom-fixed offshore wind turbine developed by the National Renewable Energy Laboratory (NREL) was utilized and is expressed as [25]: where i Cost is the total cost of turbine i, i ICC is the initial capital cost, and i AOE is the annual operating expense.
FCR is the fixed charge rate, which is set to 0.1158 per year [25]. In the NREL cost model, the cost data of turbine components was available for different years. For the purpose of consistency, all cost data were converted to 2002 dollars before the cost and scaling factors were developed. ICCi consists of the turbine system cost and the support platform station cost. The detailed costs are listed below.
The turbine system is divided into four main subsystems, the mechanical system (such as blade, gearbox and so on), electrical system (including generator, power converter and electrical connection), control system (safety system, yaw control, torque control and pitch control), and auxiliary system, such as hydraulic cooling equipment, hub and tower. Since the number of blades is three, the turbine system cost, ICCturb, is given by: (2) The support platform station cost. The ICCBoP involves the cost of infrastructure and offshore engineering, and the detailed mathematical models are shown in Reference [25]. According to the report of NERL, the ICCBoP can be summarized as ) + 0 311325 () .
(3) The annual operating expense. The annual operating expense AOE consists of the land lease, levelized operation and maintenance (OM), and the levelized replacement costs. Its mathematical formula is 0.02108 (5) According to Equation (5), the AOEi is determined by the rated power and the annual energy production. So, referring to Equations (2)-(5), the wind turbine i cost can be expressed as  (6) In the above equation, Equation (6), Costi was estimated based on the empirical model, which is related to the rated power, rotor radius, tower height, and AEP of the wind turbine i. Since the AEP of the wind farm is determined by the parameters and layout of wind turbines, it is possible to Appl. Sci. 2019, 9, 835 5 of 19 minimize the cost of the wind farm by designing the appropriate wind turbine parameters and their layout.

Annual Energy Production
Normally, the AEPi of the offshore turbines is estimated based on the Weibull probability distributions of the wind statistics, a standardized power curve, a physical description of the turbine and physical constants. Although this method can effectively estimate the AEP of single wind turbines, it was simplified in the present context to ignore the wake effects and wind turbine layout when applied to estimate the AEPtotal of the whole offshore wind farm. So, in this paper, the existing AEP model is modified by taking wind turbine's wake effect and layout into consideration.
The AEPi of a turbine can be given by the average power production, avg,i P , of the wind turbine i during one hour and the total hours of one year: where  is the total power generation loss, including power converter loss, electrical grid loss, availability loss and so on. In this study,  is assumed to be a constant of 0.16 [26].
Meanwhile, the mean power production, , avg i P , of a single wind turbine can be calculated as: where v is the wind speed,

()
Pv denotes the power curve model as a function of the wind speed, and () i fv is the Weibull distribution corresponding to the installation site, which is also as a function of the wind speed.  where the wind speed is between vc and the rated speed (vr), the turbine is able to generate partial power. In Region 3, where the wind speed is between vr and the cut-off speed (vf), the wind turbine is limited to the rated power. In Region 4, where the wind speed is above the vf, the turbine is shut down or, better, set aside to avoid component over-loads. Thus, the power curve model can be expressed analytically as: where () f Pv is the active power when the wind passes through a wind turbine and is exploited by it. The mathematical formula of () f Pv is given by: where  and R denote the air density and the rotor radius, respectively; p C denotes the coefficient of power that depends on the wind speed, which is assumed as constant and equal to 0.42 for simplicity [26]. Then, the turbine rated power r P can be calculated as

Weibull Probability Density Distribution of Offshore Wind Statistics
In this study, the Weibull probability density function () i fv was employed to represent the offshore wind statistics of the installation site, which depended on the Weibull scale and the shape parameters i k and i c that determined the shape and intensity of the wind during one year on a site [27].
where  is the gamma function.
The wind speed becomes stronger as the altitude, i.e. the distance from the ground, increases. Based on the relationship between the wind speed and the altitude, the shape factor k and scale factor c at the turbine i hub height H are obtained by: where, α is the Hellmann exponent, which depends on surface properties of the wind field. The typical value of  is 0.1 over water and 0.14 over land [27]. When the Weibull probability distribution of a single wind turbine in its erected site is mainly determined by the wind characteristics of the wind farm, it is also influenced by the wake effects of the wind farm. Generally, the wake effect leads to a reduced wind speed that is faced by the downwind turbines. In this study, the Jensen wake loss model was adopted for its low computational cost. With the Jensen wake loss model, the wind velocity deficit behind an upstream wind turbine is calculated by the following equation: , (16) where m v is the mean wind speed and Ct is the axial induction factor, which is set to 0.88 for simplify [28]. D and w D denote the wind turbine rotor diameter and wake diameter, respectively. The equation of w D is given by: where x is the downstream distance from the wind turbine and w k presents the wake decay coefficient. In this study kw was set at 0.04 [28].
In the actual offshore wind farm, a downstream wind turbine i was affected by multiple wakes of upstream wind turbines. Supposing that the number of upstream wind turbines is M, the wind velocity of wind turbine i can be represented as follows: where xij denotes the horizontal distance between upstream wind turbine j and downstream wind turbine i. Based on Equations (12)

Energy Cost Model of Offshore Wind Farms
By replacing Equation (1) with Equations (6) and (19), the COE can be written as AEPi According to the European Wind Energy Association, there is a relationship between the hub height and the turbine rotor radius, which is expressed by Reference [29] as AEPi In Equation (22), the COE model of offshore wind farms was formulated as a nonlinear function in which the rated wind speed, the rotor radius, and the spacing among the turbines were the design parameters, i.e., the target parameters, the cut-in and cut-out wind speeds were typically known constant parameters, and the wind statistics of the offshore windfarm were the mean wind speed, the shape factor, and the Hellmann exponent. Thus, the final expression of COE after modifications is

Energy Cost Optimization for Offshore Wind Farms
Based on Equation (23), the minimization of the COE of the offshore wind farms can be fulfilled by optimizing the three design parameters: the rated wind speed, the rotor radius and x. As the objective is to minimize the COE, when considering the practical constraints, the objective function and the constraint can be expressed as: where, min x and max x are the minimum and maximum distances between two wind turbines, respectively.
In the following sections, the method to minimize the cost of energy of the offshore wind farm is elaborated, the optimization results are presented, and discussions explain the obtained optimization results.

Optimization Method
In order to minimize the COE, a composite optimization approach was proposed, comprising an IPSO and an iterative algorithm. The first was applied to help the construction of the optimized layout, while the second was used to find the optimal parameters of wind turbines that give the minimum COE. The theory and optimization procedure are presented in the following section.

Improved Particle Swarm Optimization
Particle swarm optimization (PSO), developed by Eberhart and Kennedy, is a population-based stochastic optimization technique and it is recognized as a simple concept algorithm, with easy coding implementation, robustness to control parameters and computational efficiency. When compared with the PSO, the IPSO algorithm has improved the convergence performance [12]. The IPSO equations are given as:   n is the total number of particles and M is the dimension of searching space. In order to avoid going beyond the searching space, the velocity, id v , needs to be set into a limited range, namely w is the inertial weight in this study and its expression is as follows: where t is the current iteration number, and T is the total iteration number. max w and min w are maximum and minimum inertial weights, set to 0.9 and 0.4 in this study, respectively.

Optimization Procedure
As regards the model to minimize the COE model, the composite optimization algorithm is designed into two parts: One is the main optimization algorithm, which uses the iterative algorithm to obtain the optimally designed parameters of the wind turbines, while the other is the sub-optimization algorithm, which employs the IPSO algorithm to obtain an optimal turbine layout by comparing the values of the objective function. The main flow chart of the developed algorithm is presented in Figure 2, and the implemented procedures comprise the following two loops.
(1) The main optimization loop. • Step 1: Initialize the wind statistics of the offshore windfarm, m v , k 0 and  ; • Step 2: Define the ranges of the design parameters r v , R and initialize the minimum design parameters. • Step 3: Update the design parameters through their own iterative intervals.
• Step 4: Turn to the IPSO algorithm and obtain the optimal turbine layout and value of the objective function (COE). • Step 5: Repeat step 3 until all design parameters sets of r v and R have been evaluated; • Step 5: Output, with minimal objective function, the corresponding design parameters and wind turbine layout.

Method Application and Resulting Discussion
To show how to minimize the cost of energy for the offshore wind farms by using the proposed step by step optimization approach, case studies were conducted. Since the proposed COE model depended on the wind statistics of the offshore wind farm, the real wind information from some offshore wind farms was used for this study.

Parameter Settings
In this study, the capacity of planned-installed wind farms was assumed to be 60 MW. The parameters of the wind turbines and the proposed COE are summarized in Table 1, in which the rated wind speed is in the range 10-16 m/s with a step of 1 m/s and the rotor radius is in the range 30-70 m with a step of 5 m. Assuming a regular grid for the offshore wind farm, the grid size was 10 × 10, and the interval between the two wind turbines was 6D (D denotes the rotor diameter).
For the IPSO algorithm, the algorithm convergence speed relied largely on the initial values of the particles at the first iteration. Meanwhile, the two important parameters, including the population size and the maximum iteration number, also significantly influenced the optimal results. With a certain set of initial values of the particles, the selected objective function value was more optimal, that is, more significant, when the population size and the maximum iteration number was bigger. However, more computation time is required. In this study, the population size and the maximum iteration number of IPSO were set to 20 and 200, respectively. By doing so, the IPSO reached convergence and the computation time was about ten days for each case. The acceleration factors 1 c and 2 c were set as 1.49445. Then, the obtained position layout was applied to analyze the developed COE. On the other hand, in order to analyze the relationship between the developed COE and the layout, the optimal wind turbine obtained by the developed COE and the actual installed wind turbines were taken into consideration for comparison.

Method Application in Three Cases
Three real installations were used as case studies: Newport nearshore windpark (NNW) wind site in the USA, Xiangshui intertidal Pilot project (XIPP) offshore wind farm in China, and n R land  offshore wind farm in Denmark. The wind statistical data of the three wind sites is given in References [30][31][32], respectively, and the Weibull distribution parameters of wind speed are summarized in Table 2. The wind direction is assumed to be constant. The distribution curves of the wind speed are plotted in Figure 3.  Figure 3. Weibull probability distribution functions of three cases at reference hub height H 0 . Case 1: NNW offshore wind farm, USA, is located at latitude 39.201 and longitude -75.241 . At this wind farm, the mean wind speed was 8.23 m/s and the shape factor 0 k of the Weibull distribution function was 2.0. Based on the presented cost of energy and the optimization approach, the results about the relationships among the rated speed, rotor radius and COE were obtained and are shown in Figure 4a,b. As seen in Figure 4, the results show that the wind farm has a minimum COE of 0.077 $/kWh for turbines with a rated speed of 11 m/s and a rotor radius of 50 m. By calculation, the rated power of the turbine is about 2.7 MW, and the hub height is 95.23 m. The actual wind turbines installed in NNW wind farm include the SWT-3.6-120 for which the main parameters are given in Table 3. It is shown that the optimization results are close to the values of the actual wind turbine. However, compared to the wind turbine with the minimum COE, the actual wind turbine has a higher speed and rotor radius.  In order to prove the effectiveness of the proposed COE method, an analytical comparison about the minimum COE of an optimal turbine and SWT-3.6-120 was performed. It considered the optimal wind turbine layout obtained by the IPSO algorithm. As the planned-installed capacity of offshore wind farms is 60 MW, 17 wind turbines were expected to be installed, N = 17. The optimization results and the optimal layout obtained by the IPSO algorithm are shown in Figure 5a,b, respectively. In Figure 5a, when the value of the minimum COE of the SWT-3.6-120 (COEmin = 0.0787 $/kWh) is higher than the COE value of the optimal wind turbine, of which the value is 0.0774 $/kWh, the minimum COE of the actual wind turbine is very close to the minimum COE value of the optimal wind turbine. In Figure 5b, as expected, the layout of the wind turbines has a tendency to locate the turbines in the outermost zone of the grid and the distance between two wind turbines is higher in the wind direction. As such, the layout of the wind turbines can reduce the influence of the wake-effect and, accordingly, the effectiveness of the layout optimization is confirmed. The IPSO was used to search for the optimal layout of wind turbines and, thus, the convergence of the IPSO could be judged by checking the final layout of the wind turbines. Since the optimized layout in Figure 5b is consistent with the expected results, it can be determined that the IPSO converged after 200 iterations. At this wind farm, the mean wind speed was 6.94 m/s and the shape factor, 0 k , of the Weibull distribution function was equal to 2.0. Based on the cost of energy presented and the two optimization methods, the results about the relationships between the speed, rotor radius and COE were obtained, as shown in Figure 6a,b. The results show that the wind farm has a minimum COE of 0.089 $/kwh at a rated speed of 10 m/s and a rotor radius of 60 m. The power of the turbine was about 2.9 MW and the hub height was 109.5 m. The actual wind turbines installed in the XIPP offshore wind farm included the GW 109/2500 (Goldwind, Beijing, China) and the W2000/93 (Sewind, Shanghai, China). The main parameters of these wind turbines are detailed in Table 4. Compared with the optimal turbine, the actual wind turbines had slightly higher speeds and lower rotor radii. When comparing the two actual turbines, the GW 109/2500 was the optimal turbine due to its lower COE. The analytical comparison of the minimum COE of the optimal turbine, GW 109/2500 and W2000/93 was adopted in terms of the optimal wind turbine layout obtained by the IPSO algorithm. As the planned-installed capacity of offshore wind farms was 60 MW, 24 GW 109/2500 wind turbines were expected to be installed, N = 24. The number of W2000/93 wind turbine was 30. These optimization results and the optimal layouts of the two wind turbines obtained by the IPSO algorithm are shown in Figure 7. It can be seen from Figure 7a that the GW 109/2500 had a smaller COE than the W2000/93. The former had the same result as that of the developed optimization method. In Figure 7b,c, the layouts of the wind turbines are arranged into three rows: Two rows are located in the outermost zone of the grid and one row is located in the medial grid. In this way, the spacing between the two wind turbines is biggest in the wind direction. Consequently, the influence of the wake-effect on the energy production was reduced and, again, the effectiveness of the layout optimization was confirmed.  Table 5. It shows that the Vestas v80-2000 has lower speed, lower power and the same rotor radius, and the SWT-2.3-93 has lower speed and higher rotor radius compared to the optimal turbine. Besides, from the Figure 8b, the minimum COE of the two actual wind turbines was approximately the same.  In order to prove the effectiveness of the proposed COE method, an analytical comparison of the minimum COE of the optimal turbines, SWT-2.3-93 and Vestas v80-2.0 MW was proposed. This considered the optimal wind turbine layout obtained by IPSO algorithm. According to the planned-installed capacity of the offshore wind farm, 27 wind turbines were expected to be installed, N = 27. Analogously, the planned-installed number of Vestas v80 was 30. The optimization results and the optimal layouts obtained by the IPSO algorithm are shown in Figure 9. It can be seen from Figure 9a that when the Vestas v80-2.0 MW had a COE slightly smaller than the SWT-2.3-93, both of their COE were higher than the optimization result. Figures 9b and c shows the optimal layouts of the wind farm with the two actual wind turbines. Similar to Figures 7b and c, there are three rows of the layout of the wind turbines, among which two rows are located in the outermost zone of the grid, while one row is located in the medial grid. The biggest spacing was maintained between two wind turbines in the wind direction and, thus, the influence of the wake-effect on the energy production was most diminished.

Discussion
In order to check the advantages of the proposed optimization method, the differences between the optimal COE of the three optimized wind farms and the actual wind turbines were calculated and are shown in Table 6. The results show that these differences are positive and their values are in a range of 0-0.001 $/kWh, and the reduced ratio of the COE is in the range 0-1.27%. Thus, it seems that the actual wind turbines may have been optimally designed before being installed in the offshore wind farms [33][34]. On the other hand, the positive differences mean that it is necessary to simultaneously optimize the wind turbines and their layout to achieve the minimal energy cost of the offshore wind farms.  In order to check the optimization rules of the designed parameters of the wind turbines in the offshore wind farms, considering the optimal value of the COE varying in the range of 0.002 $/kWh, the range of the optimally designed parameters are shown in Table 7. From Table 7, it is clear that when the mean wind speed is increased, the optimally designed rotors slightly decrease their size, while the rate wind speeds are higher. Although it seems contradictory to the current trend of offshore wind turbines towards having long blades, it is acceptable as the optimization results actually confirm the fact that the COE is more sensitive to the variation of the wind speed rather than the rotor radius. Furthermore, the final capacities of the optimally designed wind turbines are calculated by using the optimal rotor radius and wind speed, and their results are 2.0-3.9, 2.4-4.5, and 1.7-5.6 MW for the annual mean wind speed with 6.94, 8.23, and 10.2 m/s, respectively. These results reveal that the wind farm with a high mean wind speed can have a wider range of the turbine capacities than one with a low wind speed. Thus, there is freedom for designers to design the offshore wind turbines, which can be seen as another advantage in the construction of wind farms at wind sites with rich wind resources, besides the favorable COE. Furthermore, by summarizing the optimal layouts of the three cases studied, it is clear that the results show a similar tendency, that the biggest spacing between two wind turbines is kept in the wind direction, which is consistent with the expected results. By doing so, the impact of the wake loss effect on the energy production was farthest and, accordingly, the results confirmed the usefulness of optimization in layout determination.

Conclusions and Future Work
This paper proposed a systematic optimization method to minimize the energy cost of offshore wind farms through a search for the optimal wind speed and rotor radius of individual turbines and the optimal turbine layout. To do this, the energy cost minimization problem was formulated and a composite optimization approach was used, as presented in this paper. The proposed method was applied to three case studies using real wind resources. The results showed that the energy cost was reduced by 0-1.27%, confirming the effectiveness of the conducted optimization. More importantly, exploring the obtained optimal results provided some deep insights into the optimization design of offshore wind farms, including that the energy cost of wind turbines is more sensitive to the variation of the wind speed than the rotor radius, and that wind sites with rich wind resources can have a wider range of turbine capacity than those with poor wind resources. In future work, more design parameters and new objective functions can be considered within the presented optimization framework. However, it is worth pointing out that the optimization algorithm and the proposed energy model may be improved to save time during the optimization procedure.