Risk management of wind farm micro-siting using an enhanced genetic algorithm with simulation optimization

Renewable Energy 107 (2017) 508e521

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Risk management of wind farm micro-siting using an enhanced genetic algorithm with simulation optimization Peng-Yeng Yin*, Tsai-Hung Wu, Ping-Yi Hsu Department of Information Management, National Chi Nan University, Nantou 54561, Taiwan

a r t i c l e i n f o

a b s t r a c t

Article history: Received 24 May 2016 Received in revised form 28 October 2016 Accepted 14 February 2017

Wind farm micro-siting is the decision problem for determining the optimal placement of wind turbines in consideration of the wake effect. Existing micro-siting models seek to minimize the cost of energy (COE). However, little literature addresses the production risk under wind uncertainty. To this end, we develop several versions of the simulation optimization based risk management (SORM) model which embeds the Monte Carlo simulation component for obtaining a large number of samples from the wind probability density function. Our SORM model is flexible and allowing the decision makers to conduct various forms of what-if analysis trading profit, cost and risk according to their business value. Then we propose an enhanced genetic algorithm (EGA) which is customized to the properties of wind farm dimensions. The experimental results show that the EGA can obtain the SORM decision both effectively and efficiently as compared to other metaheuristic approaches. We demonstrate how the risk under wind uncertainty can be effectively handled with our SORM models. The simulations with what-if analyses are conducted to disclose important characteristics of the risky micro-siting problem. © 2017 Elsevier Ltd. All rights reserved.

Keywords: Enhanced genetic algorithm Micro-siting Risk management Simulation optimization What-if analysis Wind farm

1. Introduction The reduction on the reliance of main fossil fuels (oil, coal and gas) has been included in the long-term energy policy by many countries. The reason is two-fold. First, the oil and gas was predicted to be exhausted by 2042 and the depletion time for coal is 2112 [1]. For the nation’s safety and maintaining a stable economic growth, it is prudent to invest renewable energy as alternative sources. Second, the social awareness of environmental protection for restraining the emission of greenhouse gases has pushed the government toward the use of various sources of renewable energy. Among others, wind energy generation is fast growing in recent years. The leading Big Five countries, namely, China, United States, Germany, Spain and India have respectively installed from 114 to 22 GW wind power capacity as of the end of 2014 [2]. The investment for new wind farms involves three phases: the planning phase, the operation and maintenance phase, and the decommissioning phase. Several factors affecting the investment return may be uncertain such as the wind directions and speeds, turbine failure, personnel and material cost, and energy price.

* Corresponding author. E-mail address: [email protected] (P.-Y. Yin). http://dx.doi.org/10.1016/j.renene.2017.02.036 0960-1481/© 2017 Elsevier Ltd. All rights reserved.

When there is uncertainty, there exists risk. Some researches [3e9] have addressed the risk issue for wind farm investment under the uncertainty of wind conditions. The focus of risk management for different phases of wind farm investment varies significantly. For the risk management in the planning phase, the focus is to seek the optimal configuration of micro-siting which determines the number and positions of wind turbines considering the varying wake effect under wind uncertainty. While in the operation and maintenance phase, as the layout of the wind farm micro-siting has been determined, the aim is to find the best control configuration for each individual wind turbine in order to stabilize the power generation against the wind variations. And for the decommissioning phase, apparently, there is no risk induced by wind uncertainty. Most of the existing works were devoted to technology improvement in the wind-farm operation and maintenance phase. To the best of our knowledge, only the following three works [7e9] have considered the risk management in the wind-farm planning phase. Messac et al. [7] has noted the ill formulation of generation prediction adopted by classic micro-siting methods which calculate the mean generation from the wind distribution, overlooking the risk that the generation could be greatly deviated from the mean with non-negligible probabilities. The expected return of investment is thus vulnerable by significant risk. Therefore, Messac et al. [7] proposed a sophisticated wind uncertainty model considering

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wind variations characterized by short-term and long-term factors, respectively. Both parametric and nonparametric models were formulated. A PSO is used for uncertainty minimization of the cost of energy (COE) while keeping the COE value within a satisfactory lez et al. [8] sought to find the optimal microrange. Serrano-Gonza siting configuration by modeling the wind uncertainly based on a set of input scenarios and its probability of occurrence. The expected value of the net present value (NPV) is calculated for trial micro-siting configurations. A GA is thus developed for searching for a near-optimal micro-siting to mitigate the profitability risk by maximizing the expected NPV. Both [7] and [8] identify risk as the variation of the production benefit and this paper follows the same line. Nevertheless, a different angle for identifying risk has been presented in Ref. [9] which proposed the power deficiency risk management (PDRM) model considering power deficiency risk under weekly electricity demand. A cyber swarm algorithm is developed to approximate the optimal micro-siting solution of the PDRM model. The strategy employed in the planning phase may have greater impact on the effectiveness of risk management than that practiced in the operation and maintenance phase because the former is a one-time decision (the micro-siting is unchangeable once it was implemented) and it deals with the wind uncertainty with a longer time horizon. The turbulence range in power generation produced by implementing different micro-siting varies more significantly than by tuning individual turbine control settings. However, the importance of risk management in the wind-farm planning phase has been long neglected in the related research. Although [7] and [8] provided initial attempted solutions, these two works are not suitable for the decision makers who require a decision support system which is flexible enough to be able to provide what-if analyses through iterative interactions with different scenario settings. The inadequacy of [7] and [8] includes the following. First, the best strategy for risk management could vary for different decision makers. For instance, some decision makers may strive to risk minimization on generation variations with the COE upper-bound constraint, and some others would focus on generation maximization while constraining the risk within a tolerance level. Secondly, the risk can be gauged by various statistical measures such as mean, standard deviation, minimum, maximum, or percentile. The choice for the most suitable measure depends on the business value determining the attributes of the project investment. Thirdly, the quality of decision is really affected by the applied values for the model parameters. The decision makers prefer to see the analysis results obtained by a pool of alternative values for each parameter. However, none of [7] and [8] provided such flexible interfaces for conducting what-if analyses. In this paper, we propose new micro-siting strategies to be employed in the wind farm planning phase under the risk of varying wind condition. To handle wind uncertainty, we develop several versions of the simulation optimization based risk management (SORM) model which embeds the Monte Carlo simulation component for obtaining a large number of samples from the wind probability density function. Our SORM mode is flexible and allowing the decision makers to conduct various forms of what-if analysis trading profit, cost and risk according to their business value. Most existing works for wind farm planning adopt some forms of evolutionary algorithm (EA) [10], revealing that EA is a reliable method for tackling the micro-siting problem. So we propose an enhanced genetic algorithm (EGA) which is customized to the properties of wind farm dimensions to obtain the planning result effectively and efficiently. The remainder of this paper is organized as follows. Section 2 presents the literature review for micro-siting and risk management. Section 3 articulates the proposed SORM models and the EGA

509

approach for obtaining the optimal micro-siting decision. We report in Section 4 the experimental results and comparative performances. Finally, Section 5 concludes this work. 2. Literature review Wind farm micro-siting planning involves a number of factors including wind potentials, wake effect, wind energy extraction, cost analysis, optimization planning, etc. In this section, a generic micrositing model which is broadly used in the literature is presented. Then, we review main existing approaches for obtaining optimal micro-siting. Finally, several important applications for risk management with the simulation optimization technique are presented. 2.1. Micro-siting problem A generic model for wind farm micro-siting planning consists of several mandatory components including the representation and calculation for wind data (direction and speed of wind and the probability of occurrence), wake effect incurred by wind turbines in the implemented micro-siting, the cost and production of the established wind farm, and the applied optimization methods to obtain the micro-siting planning. Multiple methods exist in the literature for describing each of the noted components. The reader is referred to a most updated survey [10]. 2.1.1. Generic model For the wind data component, wind rose and Weibull distribution are two alternative representations for describing the proportion of occurrence for every instance of wind speed and direction. Wind rose model has been deployed by many existing works such as [11e16]. A wind rose (see Fig. 1) is a circular histogram having a number of bins, each of which indicates a particular range of wind-direction degrees. The occurrence proportion for

Fig. 1. The wind rose for indicating occurrence proportion for each wind speed in various wind-direction bins.

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each wind speed in a wind-direction bin is shown in different colors. The maximal power that can be extracted from the wind depends on the significance of the wake effect. The wake effect is a wind flow disturbance phenomenon due to the flow passing of a solid object. In the case of a wind farm, the speed of the wind flow at a downstream turbine may decay and the significance of the decay depends on the number of and distance to the upstream turbines. We employ the wake effect model proposed by Jensen [17], which has been intensively used in the literature [11e16]. Considering a wind farm discretized as d
d grids at the center of which can be placed a wind turbine for energy extraction. Let the wind turbines be of the same model and let the rotor blade radius be r. According to Jensen’s model, the wake radius (r1) is a constant which only depends on the wind turbine model and is determined by

1 1a 2 r1 ¼ r 1  2a 1

(2)

where a is the axial induction factor and TC is the turbine thrust coefficient. While the wake zone radius (r2) is a variable which grows with the deviation distance (y) between the upstream turbine and the observed downstream point. The wake zone radius is given by the following expression.

r2 ¼ r þ l y

(3)

where parameter l is the wake effect constant and its empirical value is 0.075 for an onshore wind farm. For a downstream turbine i in the wake zone affected by exactly one upstream turbine, the decay ratio (xi) on wind speed is estimated by the following expression.

Wi 2a ¼ W0 ð1 þ aðy=r1 ÞÞ2

(4)

where W0 and Wi are the wind speed in free air flow and in the wake zone at downstream turbine i, respectively. Parameter a is the entrainment constant which depends on the hub height (h) and the surface roughness (z),

a¼

0:5 lnðh=zÞ

(5)

Finally, for the case with n upstream turbines, the mean downstream wind speed (Wi ) at wind turbine i can be estimated by considering all the decay ratios. We thus obtain

0

0

B W i ¼ W0 @1  @

n X

11 1 2 x2j A C A

(6)

j¼1

The total energy extracted by a turbine at a candidate position in the wind farm can be derived by calculating the integral of the theoretic power curve over the entire range of observed wind speed, direction, and planning time horizon, as follows.

Z Z Z

3

d
d Z Z Z X 3 xi pSDT kWi dS dD dT i¼1

(8)

S D T

where the decision variable vector X denotes the micro-siting planning result and X ¼ {xi}i¼1, …,d
d where xi ¼ 1 indicates the center of the ith grid is placed a wind turbine, and xi ¼ 0 otherwise. The turbine installation cost C(X) is derived by the following expression,





  2 2 1 þ e0:00174 N 3 3

(9)

(1)

1  ð1  TCÞ2 2

xi ¼ 1 

EðXÞ ¼

CðXÞ ¼ N




a¼

every instance of S, D and T, and k is the production efficiency coefficient related to the turbine model. Considering a wind farm consisting of d
d grids, the overall energy generation is given by the following expression.

pSDT kWi dS dD dT

(7)

S D T

where S, D and T are the variable for the range of wind speed, direction, and operation time, pSDT is the probability of occurrence for

where N is the number of installed turbines. The cost function consists of a fixed cost and a variable cost. The variable cost serves as an incentive for installing more turbines. Hence, the mean cost for installing a turbine is cheaper if the number of installed turbines increases. Various objectives for micro-siting models can be pursued, for instance, the maximization of energy extraction, the maximization of production efficiency, and the minimization of the COE (defined as CðXÞ=EðXÞ). 2.1.2. Micro-siting optimization Let us again consider a wind farm discretized as d
d grids. The computational complexity incurred by an enumerative exact method for determining the optimal turbine placement would be Oð2d
d Þ which grows exponentially with the dimensions of the available turbine installation positions. Hence, most researches resort to metaheuristics approaches for obtaining a near-optimal micro-siting as a tradeoff between computation efficiency and solution quality [10]. Mosetti et al. [11] gave the first proposal for optimal micro-siting which applies a genetic algorithm (GA) to determine the optimal turbine installation positions under the COE minimization objective. Grady et al. [12] proposed another GA validated with three cases of different wind scenarios where the authors showed the COE obtained is more economic than that obtained by the GA proposed in Ref. [11]. Yin and Wang [15] proposed a hybrid algorithm combining GRASP and VNS for finding optimal wind turbine layout. In Ref. [15], better micro-siting in terms of COE compared to previous studies has been reported on several benchmark test cases and a new case with land constraint. Chen et al. [18] determined optimal wind farm layout using a GA considering wind turbines having different hub height. Pookpunt and Ongsakul [19] proposed a binary PSO for optimal placement of wind turbines under the COE criterion. Various scenarios having full or partial wake effect combination due to single or multiple upstream turbines were lu and Seçkiner [20] used the particle filtering considered. Erog approach to estimate the probability density function of the wind conditions. The maximum energy extraction is pursued under the wind farm boundary and the wind turbine inter-distance constraints. 2.2. Risk management Classic optimization approaches usually seek to optimize the average-case system performance, but the planning result is volatile to uncertain scenarios with various occurrence probabilities. On the contrary, risk management methods are used for assuring the performance in most cases when uncertain scenarios exist in the

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system, by constraining the probability of producing undesired result. Risk management involves risk identification, critical analysis, and actions taken for mitigating or transferring the risk [21]. To take necessary actions, classical risk-mitigating methods, such as scenario optimization [22] or robust optimization [23], provide mechanisms to obtain a compromise solution by referring to the optimal solution for each separate scenario and the probability that it will occur. However, these methods have to go through each scenario and they are ill-suited when the number of scenarios is great. To handle uncertainty with many scenarios, simulation optimization (SO) methods have emerged [24]. The SO methods estimate the performance measure of a solution by carrying out a system simulation instead of directly dealing with each scenario. The SO is a useful tool for risk management because it can handle a large number of scenarios, circumvent the difficulty for deriving an analytic form of modeling for the uncertain system, and is highly flexible in analyses due to the availability of approximate distributions obtained from repetitive simulations. The SO has shown several successful applications. Among others, Better et al. [25] studied the applications in portfolio optimization and business process reengineering and applied the simulation optimization to build up several models which not only provide optimal solutions under risky factors but is also able to interpret the merits of the obtained solutions. Kumar et al. [26] identified the operational risk factors in supply chains. Through simulations, the expected value, probability of occurrence, and additional cost of the risk factors are estimated. As previously noted, very little literature on micro-siting planning has taken risk management into consideration. The only related works [7,8] resort to the scenario optimization with the expected-value mechanism for risk control, lacking the flexibility of using various mechanisms in response to the great number of scenarios in risk control and the diversity among business values. In this paper, we propose a risk management scheme based on the simulation optimization technique to approximate the nonparametric probability density function of wind data. This flexible scheme allows us to propose several risk management models using different mechanisms, so as to provide flexibility allowing the decision makers to conduct various forms of interactive what-if analysis and to obtain a solution fitting to their individual needs. 3. Proposed approaches 3.1. Proposed risk-management models 3.1.1. Simulation-based solution fitness evaluation The previous works, which estimate the produced energy by the expected-value scheme, overlook the great variations in wind data that could cause significant deviation between the expected energy and that actually produced. In order to handle weather uncertainty, we attain the probability distributions of wind data by simulations. The produced energy distributions allow us to be able to formulate several useful models capable of handling risky decisions, as will be noted in Section 3.1.3. Given a historical dataset of hourly wind records, we aggregate the multi-year data by projecting the records contained in the same time window. The aggregation is based on monthly wind patterns, such that the wind data of the same time projection across multiple years are collected to form a probability distribution. After the projection, we obtain a time series of multiple probability distributions PSDpj , j ¼ 1, …, j where j is the number of time projections. Each distribution PSDpj describes the probability of occurrence for all wind speeds S and directions D observed during the time projection pj. Clearly, the energy produced by the wind farm with the wind distributions can be calculated by substituting PSDpj for pSDT in the

511

classic production model (Eq. (8)) and adding up to all j time projections as follows.

EðXÞ ¼

Z Z Z d
d X J X 3 xi pSDpj kWi dS dD dpj i¼1 j¼1

(10)

S D pj

The wind rose model is employed in this study, the integrals on wind speed S and direction D are replaced by summations over prevailing wind instances, as by the following expression.

EðXÞ ¼

d
d X J X X X i¼1 j¼1 k2S l2D

Z xi

3

ppj ðk; lÞkWi dpj

(11)

pj

But in order to handle production risk incurred by wind uncertainty, we apply the Monte Carlo simulation where the wind values are not estimated by the mean-value scheme using integrals but are chosen randomly from the projection probability distribution Ppj ðk; lÞ. The set of wind sample values are used to activate the energy extraction simulation with micro-siting solution X using the wake model. The simulation is repeated until all Ppj ðk; lÞ, j ¼ 1, …, j distributions have been processed. We thus are able to obtain a probability distribution PE ðXÞ for the energy production with micro-siting solution X. The benefit and the production risk involved by the micro-siting solution X can be analyzed by using various statistics calculated from PE ðXÞ such as the mean, the standard deviation, a percentile, or a cumulative probability for a given range of energy production. Hence, the fitness evaluation for micro-siting solution X will be designed as a function of simulationbased statistics for the benefit and the risk. 3.1.2. SO-based metaheuristics The framework of SO-based metaheuristics is particularly useful for our problem because the wind energy simulation is very computation-intensive and metaheuristic algorithms can obtain quality solutions more quickly by comparing to dynamic programming or branch-and-bound methods which require to evaluate many solutions. Metaheuristic algorithms are very effective in providing a feedback with an improved solution based on a performance estimate for a previous solution. The performance estimate can be either obtained from an analytic function or a blackbox. The SO-based metaheuristics applies the simulation model for obtaining the performance estimate of each feedback solution produced by the metaheuristic algorithm. The feedback-andsimulation cycle is repeated until a stopping criterion is reached. We design an SO-based enhanced GA for dealing with optimal micro-siting under risk of wind uncertainty. The reason is two-fold. First, the analytical modeling of risk management for micro-siting is extremely complicated because the uncertainty interacts with several nonlinear models (wind condition, wake effect, energy generation, installation cost, etc.) Using simulation for obtaining the performance estimate of a solution to this complicate model is a viable and robust scheme to circumvent the difficulty of deriving a closed-form system formulation. Secondly, boosted by many applications of GA in solving micro-siting problems, an enhanced GA with customized features is potentially more appropriate than other metaheuristics. In Section 3.1.3, we propose four riskmanagement models based on the SO-based metaheuristics. The details of the proposed SO-based enhanced GA for micro-siting are described in Section 3.2. 3.1.3. Risk-management models for micro-siting under wind uncertainty We propose several alternative simulation optimization based risk management (SORM) models which enable fast, inexpensive

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and non-disruptive what-if analyses through iterative interactions with various scenario settings. The SORM models are based on analytics in terms of benefit, cost, and risk. The benefit and risk are derived from the energy production distribution PE ðXÞ through Monte Carlo simulation with micro-siting solution X. The cost considered in this paper is the turbine installation cost C(X) given by equation (9). Model 1. s-Constrained Cost/Benefit Analysis

.

Minimize

f ðXÞ ¼ CðXÞ mPE ðXÞ

(12)

subject to

g1 ðXÞ ¼ sPE ðXÞ  t1  0

(13)

where f(X) and g1 ðxÞ are the objective and constraint functions, mPE ðXÞ and sPE ðXÞ are the mean and standard deviation of the energy production distribution PE ðXÞ generated by Monte Carlo simulations. Model 1 aims to minimize the simulated COE based on cost/ benefit analysis and facilitate the risk management by bounding the variations of energy production. By specifying different values for t1 in a possible range, the decision-makers can practice risk control to various extents. Thus, repetitive and cheap what-if analyses with Model 1 can be quickly conducted until a satisfied decision is obtained. The COE value would be larger with a stricter level of risk control as will be shown in the experiments. Model 2. Percentile-Constrained Cost/Benefit Analysis

Minimize

. f ðXÞ ¼ CðXÞ mPE ðXÞ

(14)

subject to

g2 ðXÞ ¼ t2  PP5E ðXÞ  0

(15)

where PP5E ðXÞ is the fifth-percentile of PE ðXÞ. Model 2 is deployed to minimize the simulated COE and facilitate the lower-bound limit on the uncertainty of the fifth-percentile power value. In other words, Model 2 has the ability to only accept a micro-siting decision solution which produces power greater than t2 with 95% confidence. By specifying different values for t2, the decision-makers can conduct interactive what-if analyses. It should be noticed that the two alternatives as using standard deviation or percentile value have different physical meanings in gauging uncertainties. The standard deviation constrains the data variation of the entire uncertainty distributions (but the production outcomes may be of bad quality), while the percentile value focuses on the lower bound of the possible worse energy production (but the variation of overall distributions could be great). Both of the two alternative risk measures are prevalent because each individual may fit better to particular business diversity or decision-making behaviors. Model 3. s-and-Cost Constrained Benefit Maximization

Minimize

. f ðXÞ ¼ 1 mPE ðXÞ

(16)

subject to

g1 ðXÞ ¼ sPE ðXÞ  t1  0

(17)

g3 ðXÞ ¼ CðXÞ  t3  0

(18)

Model 3 is used when the decision makers have a hard constraint on both production risk and investment budget, while still trying to maximize the benefit in terms of mean production. Multiple-constraint model such as Model 3 is useful when decision-makers intend to quantize the sensitivity of the objective

by relaxing the constraints to various degrees, which is a commonly practiced technique in mathematical programming domain, such as εeconstraint programming and goal programming. By specifying different values for t1 and t3, the decision-makers can find the most suitable tradeoff solution. Model 4. s-Constrained Benefit Maximization

Minimize

. f ðXÞ ¼ 1 mPE ðXÞ

(19)

subject to

g1 ðXÞ ¼ sPE ðXÞ  t1  0

(20)

Model 4 is a simplified version of Model 3. As will be explained in our experimental analysis, the distribution of objective value f(X) of Model 3 is not sensitive to variations of the cost threshold value t3 when a s-constraint is specified. So we propose Model 4 with only the s-constraint for obtaining the summarized information. 3.2. Proposed enhanced GA To obtain a quality and feasible solution for the proposed SORM models, we develop an EGA adopting specific features for resolving the problem. We name our algorithm the Binary Matrix Genetic Algorithm (BMGA) and articulate its major components in the following. 3.2.1. Solution representation and fitness evaluation The BMGA deploys a binary matrix for representation of the solution of the SORM models. It has been shown in Ref. [27] that binary solution representation may have potential advantages in search with inequality constraints. Considering a wind farm with d
d grids, an arbitrary turbine placement can be represented by a binary matrix where each entry corresponds to a binary decision variable xi as defined in Eq. (8). Fig. 2(a) shows an arbitrary turbine placement in a 10
10 wind farm and Fig. 2(b) depicts the corresponding binary matrix solution for the micro-siting. The binary matrix is a natural solution representation for the wind turbine placement problem and the sub-matrix operations are particularly suitable for tackling the wake effect which strongly depends on the turbine placement decision of adjacent grids contained in a sub-matrix. We thus will devise a set of lower level sub-matrix operations to facilitate crossover and mutation elements of our BMGA. For the fitness evaluation of a micro-siting solution X, we propose a relaxation function considering the respective objective and constraints of the SORM models. The relaxation function R(X) is defined as follows.

RðXÞ ¼ f ðXÞ þ M

X

maxf0; gb ðXÞg

(21)

gb 2Model

The second term on the right-hand side is the overall penalty used to discredit the solution which violates any constraints of the corresponding model, and M is a great constant which avoids the constraint penalty to be neglected by the objective value. As all the SORM models are formulated as minimization problems, the less the value of R(X) is, the better the quality of the solution X is. Our relaxation function guarantees that the relaxation value of every feasible solution is less than that of any infeasible solutions. 3.2.2. Sub-matrix based genetic operations To efficiently explore the space of feasible micro-siting solutions, we design sub-matrix based genetic operations to evolve the building blocks as sub-matrix patterns which are robust under the wake effect.

P.-Y. Yin et al. / Renewable Energy 107 (2017) 508e521

(a)

513

(b)

Fig. 2. The solution representation scheme for wind turbine placement. (a) Wind turbine placement. (b) Binary matrix representation.

3.2.2.1. Sub-matrix crossover. The crossover operator produces offspring solutions by exchanging gene information between two parent solutions randomly selected from the population. The submatrix crossover randomly selects a sub-matrix of size 3
3 from each of the parents, then swaps the two sub-matrices to produce two new offspring solutions. Fig. 3(a) shows two parent micro-siting solutions in a 10
10 wind farm where two submatrices of the same size are arbitrarily selected. Fig. 3(b) illustrates the produced offspring micro-siting solutions after performing the sub-matrix crossover on the parent solutions. 3.2.2.2. Sub-matrix mutation. The mutation operator produces a new solution by altering gene information contained in an arbitrarily solution. The sub-matrix mutation randomly selects two sub-matrices of size 3
3 in an arbitrary solution, then replaces the first sub-matrix by a copy of the second sub-matrix. Fig. 4(a) shows

an arbitrary micro-siting solution with two marked sub-matrices. Fig. 4(b) illustrates the resultant new micro-siting solution after performing the sub-matrix mutation. It is noted that the performing of sub-matrix crossover and mutation should respect the wind farm border. When the area of generated sub-matrix exceeds the wind farm border, only the submatrix layout within the wind farm is considered. Moreover, the sub-matrix genetic operations can be performed with variable submatrix sizes. However, our preliminary experiments show that the performances with larger sub-matrix sizes are not significantly different to that with 3
3 sub-matrix, but only induce more computational time. In this paper, we perform the crossover and mutation operations in the 3
3 sub-matrix fashion.

4. Experimental results and comparative performances The experiments are conducted as follow. First, the proposed BMGA algorithm is compared to several existing approaches in the literature with the classic Mosetti’s model. Then the proposed SORM models are experimented with a real wind dataset to show how the risk under wind uncertainty can be handled with our models. Finally, deliberated what-if analyses are conducted to disclose important problem characteristics. All tested models adopt the following settings for the wind farm and turbine parameters, d ¼ 10, TC ¼ 0.88, h ¼ 60 m, z ¼ 0.3 m, and r ¼ 40 m. The experimental platform for the implemented methods is a personal computer with a 3.2 GHz CPU and 4 GB RAM, the programs were codified using C# language.

(a)

(b) Fig. 3. The sub-matrix crossover. (a) Parent solutions with arbitrarily selected submatrices. (b) Produced offspring solutions.

(a)

(b)

Fig. 4. The sub-matrix mutation. (a) The original solution with two arbitrarily selected sub-matrices. (b) The produced solution by the sub-matrix mutation.

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4.1. Performance with Mosetti’s model

Table 2 Performance comparison with Mosetti’s model: Case (c).

We first evaluate the performance of our BMGA algorithm on the Mosetti’s model benchmark dataset which was broadly experimented in the literature [11e16]. The Mosetti’s model benchmark dataset contains three problem cases (a)-(c). Case (a) has a fixed wind direction (to the south of the wind farm) and a constant wind speed of 12 m/s. Case (b) involves multi-directional wind with a constant wind speed of 12 m/s. Case (c) describes the situation of multi-directional wind with variable speeds of 8, 12, and 17 m/s. The performance of the BMGA is compared to that of Mosetti’s method [11], Grady’s method [12], and Yin’s method [15]. As Case (a) is the simplest case where the wake downstream only affects the turbines positioned in the same column, the problem can be easily conquered by dividing it to d sub-problems. Each subproblem only contains a column of d grids. Our BMGA algorithm obtains the same optimal solution as reported in the compared methods [11,12,15]. In particular, the optimal solution for Case (a) installs 30 wind turbines on three rows (the first, the sixth, and the tenth rows). The best numerical results are: C(X) ¼ 22.0889, E(X) ¼ 15091, and COE ¼ 0.0014637. The numerical results with Case (b) and Case (c) are listed in Tables 1 and 2, respectively. It is observed in both cases that Mosetti’s method is prone to place fewer wind turbines in the farm than other methods in order to reduce the wake effect, leading to a low energy production. On contrary, Grady’s method produces the maximal volume of energy by installing the greatest number of turbines, nevertheless, it also incurs the highest cost. Yin’s method and the BMGA method place a moderate number of turbines, but seeking an optimal wind farm layout to reduce the COE. For both problem cases, the ranking with COE performance is consistent: the BMGA method produces the minimal COE (shown in bold), Yin’s method yields the second least COE, followed by Grady’s method and then by Mosetti’s method. This experiment justifies that the proposed BMGA algorithm is an effective alternative approach for the classic Mosetti’s model.

4.2. Performance with SORM models 4.2.1. Wind data acquisition The wind data were collected from the Taiwan Central Weather Bureau (http://www.cwb.gov.tw/V7/index.htm) during August 1, 2011 to July 31, 2014. The weather repository records the hourly wind speeds and directions. To avoid intensive computations in simulation, the wind speed is quantized into three levels: 8, 12, and 17 m/s, respectively. The wind direction is represented using a 16section wind rose and each section represents 22.5 increments from 0 to 360 . As the wind data of Taiwan manifest typical seasonal wind patterns, we aggregate the hourly wind data to generate a probability density function (pdf) of wind occurrence for each month. So for each month in a year, we construct a wind pdf Ppj constituting 2160 (¼3
30
24) observed weather data. With all Ppj , j ¼ 1, …, 12, we can simulate the yearly energy production distribution PE ðXÞ for a micro-siting solution X, as previously described in Section 3.1.1.

Mosetti et al. [11] Grady et al. [12] Yin & Wang [15] BMGA

Mosetti et al. [11] Grady et al. [12] Yin & Wang [15] BMGA

C(X)

COE

9245 17220 15520 15937

19 39 33 34

16.0460 26.9217 23.6537 24.1830

0.001737 0.001567 0.001529 0.001517

C(X)

COE

13.3802 26.9217 24.1830 24.1830

0.000994 0.000803 0.000744 0.000741

Table 3 Comparative performances of competing methods with SORM Model 1. t1

Method

C

mPE ðXÞ

sPE ðXÞ

N

f ¼ COE

500

BMGA GRASP PSO BMGA GRASP PSO BMGA GRASP PSO BMGA GRASP PSO BMGA GRASP PSO

4.929054 4.929054 1.995376 11.11346 11.89600 11.11346 15.41439 16.65717 17.82571 20.00645 20.53138 22.60773 23.12898 23.65373 26.3601

1631 1622 605 3835 3975 3413 5689 5903 5937 7857 7741 7968 9439 9297 9451

446.7135 405.4234 158.8629 999.0939 995.4063 855.1737 1493.336 1481.633 1499.341 1999.45 1925.525 1988.183 2415.387 2411.347 2423.429

5 5 2 12 13 12 18 20 22 26 27 31 32 33 38

0.003021 0.003038 0.003299 0.002898 0.002993 0.003256 0.00271 0.002822 0.003003 0.002546 0.002652 0.002839 0.002450 0.002544 0.002789

1000

1500

N

N 15 39 34 34

4.2.2. Performance comparison We now perform BMGA with various SORM models and compare its performance with those obtained by two state-of-theart metaheuristics, GRASP [15] and PSO [19]. Model 1 is deployed to obtain the least-COE micro-siting subject to an upper limit t1 on the standard deviation of the power distribution generated by simulation. Various values for t1 have been specified. For each t1 value, all the compared algorithms are executed for 30 independent runs. The simulation evaluation of every trial solution picks 1000 random samples from the probability distribution. The best result obtained by each algorithm is shown in Table 3 and the best COE value for each t1 group is shown in bold. It is seen that all compared algorithms are able to obtain a feasible micro-siting solution which satisfies the s-constraint by meeting sPE ðXÞ  t1  0. For each t1 value group, the BMGA algorithm consistently outperforms the other competing algorithms in terms of the COE which is the objective value (f) of Model 1. The GRASP consistently ranks at the second place followed by the PSO. It is worth noting that the BMGA algorithm results in a smaller number of installed wind turbines compared to the other two methods, especially when t1 value is increased. This phenomenon indicates that when the constraint on power standard deviation is relaxed by increasing t1, the solution space also grows in size with more feasible parameter values. The BMGA algorithm is able to explore the entire solution space considering both wake effect and wind uncertainty, and produce a solution with a better COE value than those obtained by GRASP and PSO, both of which get trapped by local optimal solutions resulting in a larger number of installed wind turbines. In other words, the optimal COE value is not obtainable by installing a maximal number of wind turbines, but is discovered with a best number of wind turbines. Model 2 is also seeking to minimize the COE but respecting to a lower limit t2 on the fifth-percentile of the power distribution generated by simulation. Table 4 shows the best result of each algorithm and the best COE value for each t2 group is shown in bold. BMGA, again, successfully generates a feasible micro-siting solution for each t2 value by satisfying the percentile-constraint t2  PP5E ðXÞ  0. However, both GRASP and PSO fail to tackle the

Table 1 Performance comparison with Mosetti’s model: Case (b). E(X)

E(X) 13460 32038 32485 32609

2000

2500

P.-Y. Yin et al. / Renewable Energy 107 (2017) 508e521 Table 4 Comparative performances of competing methods with SORM Model 2. t2

Method

C

mPE ðXÞ

PP5E ðXÞ

N

f ¼ COE

6000

BMGA GRASP PSO BMGA GRASP PSO BMGA GRASP PSO BMGA GRASP PSO BMGA GRASP PSO

24.71769 24.18302 26.92165 26.3601 28.06679 30.44244 28.65031 29.8384 34.82354 32.29044 36.11265 50.66776 39.37938 NFa NF

10168 9390 9411 10605 10596 10141 11271 10922 11111 11984 11912 12145 12681 NF NF

6351.32 6001.214 6011.566 6755.13 6609.823 6590.985 7239.802 7036.974 7090.406 7626.057 7527.061 7517.693 8016.226 NF NF

35 34 39 38 41 45 42 44 52 48 54 76 59 NF NF

0.002431 0.002575 0.002861 0.002486 0.002649 0.003002 0.002542 0.002732 0.003134 0.002694 0.003032 0.004172 0.003105 NF NF

6500

7000

7500

8000

a

NF: no feasible solution.

hardest case with t2 ¼ 8000 and they cannot report a feasible solution when the programs terminate. For those cases that GRASP

515

and PSO can derive a feasible solution, the performance ranking in terms of the COE (f of Model 2) is consistently as follows: BMGA, GRASP, and PSO. BMGA results in a smaller number of installed wind turbines compared to GRASP and PSO when t2 value is increased. However, the reason differs to that indicated for Model 1. As t2 value is increased, the constraint on power distribution is stricter for requirement of producing large enough power with 95% confidence. BMGA is very effective in satisfying the strict constraint with a reasonable number of wind turbines and obtains the best COE value. While GRASP and PSO has to inevitably use a large number of installed wind turbines to satisfy the lower bound for the fifth-percentile power distribution. This reveals that both GRASP and PSO are inferior to BMGA in searching for the optimal solution under stricter problem settings. Model 3 is seeking to maximize the mean power production subject to the s and cost constraints. Various values for t1 and t3 have been specified. It is observed in Table 5 that the three compared algorithms can satisfy both constraints in all 16 cases. For the performance rank in terms of objective value f, each of the three compared algorithms performs best in some cases as shown in bold values in the f and the rank columns. We summarize in Table 6 the

Table 5 Comparative performances of competing methods with SORM Model 3. t1

t3

Method

mPE ðXÞ

C

sPE ðXÞ

N

f ¼ 1/mPE ðXÞ

rank value

500

22.1

BMGA GRASP PSO BMGA GRASP PSO BMGA GRASP PSO BMGA GRASP PSO BMGA GRASP PSO BMGA GRASP PSO BMGA GRASP PSO BMGA GRASP PSO BMGA GRASP PSO BMGA GRASP PSO BMGA GRASP PSO BMGA GRASP PSO BMGA GRASP PSO BMGA GRASP PSO BMGA GRASP PSO BMGA GRASP PSO

2449 2389 2480 2470 2365 2477 2428 2344 2473 2479 2403 2493 4778 4700 4863 4880 4684 4868 4832 4718 4885 4795 4716 4805 6871 6777 6794 6942 6832 6945 6998 6867 6830 6893 6812 6940 8225 8223 7612 8734 8736 8547 8754 8757 8677 8836 8710 8768

11.11346 11.11346 11.896 11.11346 11.11346 11.896 10.30387 9.467656 11.896 10.30387 10.30387 11.11346 17.82571 16.65717 20.00645 17.82571 17.82571 18.93645 17.24972 17.82571 19.47548 17.82571 16.65717 18.93645 22.08879 21.5708 22.08879 23.12898 23.12898 24.71769 23.12898 22.60773 24.18302 22.60773 22.08879 25.25843 22.08879 22.08879 22.08879 26.92165 27.49054 27.49054 28.65031 29.24093 29.8384 26.92165 26.92165 29.24093

488.0524 498.3367 496.4327 496.5392 487.4087 497.3112 494.2429 495.1821 496.3248 499.4553 497.0085 493.8071 991.8964 997.7146 993.6884 997.1492 994.9971 994.9391 996.6774 997.5476 998.4576 997.3453 995.983 974.5853 1488.454 1495.816 1489.357 1477.392 1489.218 1495.891 1490.013 1472.565 1491.706 1489.219 1488.982 1490.511 1990.846 1994.253 1948.389 1983.627 1993.939 1997.02 1972.812 1981.628 1998.447 1972.403 1981.17 1985.396

12 12 13 12 12 13 11 10 13 11 11 12 22 20 26 22 22 24 21 22 25 22 20 24 30 29 30 32 32 35 32 31 34 31 30 36 30 30 30 39 40 40 42 43 44 39 39 43

0.000408 0.000419 0.000403 0.000405 0.000423 0.000404 0.000412 0.000427 0.000404 0.000403 0.000416 0.000401 0.000209 0.000213 0.000206 0.000205 0.000213 0.000205 0.000207 0.000212 0.000205 0.000209 0.000212 0.000208 0.000146 0.000148 0.000147 0.000144 0.000146 0.000144 0.000143 0.000146 0.000146 0.000145 0.000147 0.000144 0.000122 0.000122 0.000131 0.000114 0.000114 0.000117 0.000114 0.000114 0.000115 0.000113 0.000115 0.000114

2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 1 3 2 2 3 1 2 3 1 1 3 2 2 3 1 1 2 3 2 3 1 1 2 3 2 1 3 2 1 3 1 3 2

27.5

33.5

40.1

1000

22.1

27.5

33.5

40.1

1500

22.1

27.5

33.5

40.1

2000

22.1

27.5

33.5

40.1

516

P.-Y. Yin et al. / Renewable Energy 107 (2017) 508e521

Table 6 Count of obtained rank value and the overall weighted rank value of each method with SORM Model 3. Method

Rank#1

Rank#2

Rank#3

Weighted Rank

BMGA GRASP PSO

5 2 9

11 2 3

0 12 4

27 47 27

Table 7 Comparative performances of competing methods with SORM Model 4. t1

Method

mPE ðXÞ

C

sPE ðXÞ

N

f ¼ 1/mPE ðXÞ

rank value

500

BMGA GRASP PSO BMGA GRASP PSO BMGA GRASP PSO BMGA GRASP PSO BMGA GRASP PSO

2438 2362 2475 4861 4655 4859 6964 6799 6911 8830 8838 8671 10366 10392 10119

11.896 9.467656 11.896 18.38724 16.65717 18.38724 22.60773 21.5708 24.71769 28.06679 26.3601 27.49054 30.44244 31.66884 33.54845

498.0433 495.6782 496.1312 987.9215 990.8699 999.3212 1489.887 1490.39 1482.693 1990.193 1999.566 1981.93 2474.676 2495.54 2475.452

13 10 13 23 20 23 31 29 35 41 38 40 45 47 50

0.000410 0.000423 0.000404 0.0002057 0.000215 0.0002058 0.000144 0.000147 0.000145 0.0001133 0.0001132 0.000115 0.0000965 0.0000962 0.000099

2 3 1 1 3 2 1 3 2 2 1 3 2 1 3

1000

1500

2000

2500

Table 8 Count of obtained rank value and the overall weighted rank value of each method with SORM Model 4. Method

Rank#1

Rank#2

Rank#3

Weighted Rank

BMGA GRASP PSO

2 2 1

3 0 2

0 3 2

8 11 11

number of times that each algorithm ranks at every different place and obtain a weighted rank by multiplying the count with the corresponding rank value. It is seen that BMGA and PSO perform best overall, and GRASP is inferior to the other two algorithms. If we put stronger weight on the number of times of obtaining rank #1 than other rank values, PSO is better than BMGA by obtaining four more times rank #1. Model 3 is useful to analyze the sensitivity of the mean power production to the variations of parameters. For each s value group in Table 5, the mean power production does not

(a)

vary much for a great range of cost values (between 22.1 and 40.1). On the other hand, the mean power production is very sensitive to the variations of s value. The reason is due to that the power production is significantly affected by the wake effect which is determined by the turbine layout and the wind distribution. As the wind distribution is not regulated, the decision for turbine layout, i.e., micro-siting, dominates the possible range of power production. Given a s-value upper limit, the turbine layout would be optimized by Model 3 to produce a power distribution governed by the s-bounded standard deviation with the allowable number of turbines which costs less than the budget limit. However, if a costvalue upper limit is specified, the power distribution still vary a lot with variable standard deviation s values. As a result, Model 3 is more sensitive to the s-constraint rather than the cost constraint, leading us to the development of a simplified version of Model 3. Model 4 simplifies Model 3 by removing the cost constraint. Various values for t1 have been specified. It is observed in Table 7 that for each t1 value the numerical result of every variable is very close to the mean value for the same variable under each method in the corresponding t1 group in Table 5. Hence, Model 4 is able to provide a summarization for Model 3. Again, the three compared algorithms have different rank values in the cases. The best f and rank values are shown in bold. We summarize in Table 8 the count of obtained rank value and the overall weighted rank value. We see that BMGA performs the best, and GRASP and PSO have comparable performances. The comparative performances of the three algorithms seem to be inconsistent for Model 3 and Model 4. We conjecture that PSO is prone to reach a micro-siting with more turbines and a higher cost than the other two algorithms as in the result of Model 1, Model 2, and Model 4, and PSO performs worse in these cases. But in Model 3 which constrains the solution with a cost upper limit, PSO produces similar cost values with those obtained by the other two algorithms, and the performance of PSO improves. The previous performance evaluations with various SORM models show that the overall best algorithm is the BMGA which is more effective than GRASP and PSO in terms of the obtained objective value for each SORM model. Here, we illustrate the BMGA is also efficient in conducting the evolution of the best-so-far solution. It is crucial to analyze the convergence behavior of the BMGA by statistical tests, in order to guarantee that the success in obtaining a quality solution is not resulted from a lucky random run. We conduct the convergence analysis with 95% confidence interval as follows. For a given problem instance, the BMGA is executed independently for 30 times. For each independent run,

(b)

Fig. 5. Convergence curve of the best objective value along the evolution. (a) The convergence curve before 500 OFEs. (b) The convergence curve between 500 and 20000 OFEs.

P.-Y. Yin et al. / Renewable Energy 107 (2017) 508e521

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the best objective value (OV) observed after every objective function evaluation (OFE) is recorded. By assuming that the best OV values of the 30 runs observed at every OFE form a Gaussian distribution, we are able to plot the 95% confidence interval of the best OV by the estimated range [m-1.96s, mþ1.96s] where m and s are the mean and the standard deviation of the 30 OV values. Then we plot the best OV curve and its associated 95% confidence interval as the number of OFEs increases. Fig. 5 shows the convergence curve with 95% confidence interval of the best OV for Model 1 obtained at various numbers of OFEs. The analyses for the remaining models have similar results and are omitted here. It is seen that the confidence interval is very thick at early evolution stage before 500 OFEs as shown in Fig. 5(a). This is because the BMGA algorithm explores the large solution space at this stage in order to identify promising regions, however, the great diversity in solution quality causes a large COE interval. After 500 OFEs, the BMGA algorithm exploits promising regions containing quality solutions and the confidence interval of COE becomes thinner and converges to a lower bound value as shown in Fig. 5(b). The convergence analysis implies that the BMGA algorithm is constantly efficient over repetitive runs in improving the best OV.

(a)

4.3. What-if analysis with SORM models

(b) Fig. 6. What-if analyses with Model 1 given various degrees of s-constraint. (a) Power distribution. (b) COE distribution.

The previous sections show that the proposed BMGA algorithm is both effective and efficient in micro-siting optimization with both the classic Mosetti’s model and various SORM models. We now conduct what-if analyses by using BMGA with SORM models such that the risk of the decision problem can be identified and mitigated. Fig. 6(a) shows the power distribution obtained with Model 1 by specification of various t1 values. We also execute BMGA with the unconstrained case of Model 1 which corresponds to the classic Mosettis’s model with the addition of simulation optimization. It is seen that as the value of t1 increases, the distribution of the produced power is shifted to the right. This phenomenon has several implications. (1) A higher value of t1 for the s-constraint would relax the risk control and result in a distribution with a broader power range but with a higher mean power. This is a typical tradeoff between risk and benefit. When t1 is equal to or larger than 2500, the obtained power distribution resembles to that obtained by Mosetti’s Model which has no mechanism for risk

Fig. 7. Risk-benefit analysis of Model 1 by showing COE variations as t1 value increases.

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management. (2) Decreasing the t1 value has the effect of practicing stricter risk management, which lead to a conservative investment with fewer installed wind turbines (see Table 3) to reduce the variation of produced power. (3) There exists a compromise between the obtained COE (minimization objective of Model 1) and the desired degree of risk management (s-constraint of Model 1). Fig. 6(b) shows the distributions of the COE with the same settings of t1 value. It is seen that the distribution of the COE is shifted to the left as the value of t1 increases. The COE distribution with the unconstrained model can be considered as the lower bound of the simulation outcomes that can be obtained by Model 1. The previous analyses are useful when the decision maker varies the risk tolerance level t1 until the desired power or COE distribution is obtained. In the case that the decision maker has no idea of

the ideal distribution, the following risk-benefit analysis may be applied. By varying t1 value with a fixed increment, the risk tolerance is incrementally relaxed. Repeating this process until the increase of t1 has no effect on the obtained power or COE. The plot of the benefit versus risk is able to disclose the maximal reward from every relaxed risk unit. Fig. 7 shows the variation of COE as t1 value increases, the unconstrained case is also shown for comparison. It is seen that the COE curve has the least slope when the t1 value varies from 1000 to 1500. So the recommended t1 value is 1500 where we can obtain the most COE reduction by relaxing risk constraint according to the risk-benefit analysis. Further breakdown details of the simulation such as monthly production can be used to identify risky periods. Fig. 8(a) shows the 95% confidence interval with 1000 simulations for the monthly

(a)

(b) Fig. 8. Monthly power production with 95% confidence interval obtained by the BMGA with various models. (a) SORM Model 1. (b) Mosetti’s model.

P.-Y. Yin et al. / Renewable Energy 107 (2017) 508e521

519

June and August in every year is the most risky period where the variation of produced energy is the greatest due to large wind uncertainty. In February and November, on the other hand, we observe a more stable energy production. The risk identification analysis is useful for further risk control such as planning the demand response to better match the power demand with the supply or managing alternative energy sources. Fig. 8(b) shows the 95% confidence interval for the monthly power production obtained by the BMGA with Mosetti’s model. It is seen that the variation of monthly production with Mosetti’s model is nearly double of that with SORM Model 1. The analysis justifies the effective risk management mechanism facilitated by our proposed model. Next, we conduct what-if analyses by using BMGA with SORM Model 2. Fig. 9(a)e(b) show the power and COE distributions obtained with Model 2 by specification of various t2 values for the lower-bound fifth-percentile limit and those obtained with the unconstrained version of Model 2. It is seen that as t2 increases, both of the power and COE distributions are shifted to the right. We explicate the reasons and their implications as follows. (1) Model 2 would only accept produced power that is greater than t2 with 95% confidence, so the overall power distribution is shifted to the highpower range if a higher value of t2 is specified. (2) Increasing the t2 value also tightens the Model’s constraint, many micro-siting solutions would be rejected. Therefore, the possibly obtained COE (the objective value of Model 2) is higher than those with a less t2 value. (3) A higher value of t2 would lead to an investment with more installed wind turbines (see Table 4) to raise up the value of the fifth-percentile of possibly produced power, however, the COE is inevitably increased simultaneously. There is a tradeoff between the fifth-percentile power and the obtained COE. Similarly, we apply the risk-benefit analysis by varying t2 value with a fixed increment in order to obtain a recommended value for t2. Fig. 10 shows the variation of COE as t2 value increases. It is seen that when t2 ¼ 7500, the COE curve has the greatest curvature, and the COE deterioration is more significant after this critical t2 value. So the t2 value could be set to 7500 such that we can obtain a highenough fifth-percentile power but also with a reasonable COE. Instead of showing the monthly breakdown production confidence analysis which can be similarly conducted as in Model 1, here we present another interesting analysis with the worst cases. As Model 2 stipulates that the wind farm produces power greater than t2 with 95% confidence, we can investigate the patterns of the worst wind conditions where the production is lower than t2. Fig. 11 shows the monthly wind rose for the worst 5% of production simulations obtained by Model 2 with the recommended threshold

(a)

(b) Fig. 9. What-if analyses with Model 2 given various degrees of fifth-percentile constraint. (a) Power distribution. (b) COE distribution.

power production obtained by the BMGA with SORM Model 1 and t1 set to 1500. On the one hand, we observe that the season during

.

Fig. 10. Risk-benefit analysis of Model 2 by showing COE variations as t2 value increases.

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P.-Y. Yin et al. / Renewable Energy 107 (2017) 508e521

January

February

March

April

May

June

July

August

September

October

November

December

Fig. 11. Monthly wind rose for the worst 5% of production obtained by Model 2.

value t2 ¼ 7500. We find that the occurrence percentage of the strongest wind (17 m/s) in the typical strong-wind seasons (including January, June, July, August, October and December) is the main factor contributing to the low production. Among the worst cases, the occurrence percentage of 17 m/s wind is only 2% and 8% in January and June, respectively. However, these percentages are around 12%e20% in usual cases. The worst case analysis provides a risk identification mechanism which in particular carefully monitors the wind condition in January and June, and may take advance risk control if the worst case patterns are identified.

Model 3 is designed for decision makers who are given limited budget but still want to enforce risk control. We specify different values for t1 (power standard deviation limit) and t3 (budget limit) and conduct the simulation optimization to obtain the power production distribution for each instance of t1 and t3 as shown in Fig. 12. It is seen that the distribution is not sensitive to the variation of t3 given a particular value for t1. The reason of this phenomenon is that the constraint on standard deviation of production would have constrained the maximal number of installed turbines as previously seen in Model 1 simulations. Hence, the budget is

Fig. 12. What-if analyses of power distribution with Model 3 given various instances of t1 and t3.

Fig. 13. What-if analyses of power distribution with Model 4 given various instances of t1.

P.-Y. Yin et al. / Renewable Energy 107 (2017) 508e521

bounded by allowable turbine number no matter how high t3 is. The inverse implication is not true because the constraint on budget only confines the number of turbines but the variation of production could still be great due to bad positioning of turbines. The sensitivity analysis lead us to apply Model 4 for summarizing Model 3 simulations. Fig. 13 shows the production distribution by Model 4 with the same settings of t1 as in previous simulations. It is seen that the distributions resemble to those observed in the simulation outcomes with Model 3. 5. Conclusions In this paper we have proposed the simulation optimization based risk management (SORM) model which embeds the Monte Carlo simulation for evaluating the probabilistic fitness of a solution to the complex wind farm micro-siting system under wind uncertainty. The SORM model allows the decision makers to conduct fast, inexpensive and non-disruptive what-if analyses with various scenario settings. The SORM model is based on analytics in terms of benefit, cost, and risk, drawn from the energy production distribution. To derive a near-optimal micro-siting solution to the SORM model, an enhanced genetic algorithm (EGA) has been developed. The EGA is customized to the micro-siting dimensions and it deploys a binary matrix solution representation and conducts submatrix forms of crossover and mutation. The experimental results show that the EGA is an effective approach to both the classic Mosetti’s model and the proposed SORM model. The results of what-if analyses with simulations show that the risk with wind uncertainty can be effectively controlled by executing various forms of SORM model. We have also conducted risk identification with SORM models by using monthly breakdown production analysis and worst-case analysis, indicating that the season during June and August is the most risky wind power production period for the considered wind farm site, and that the wind condition in January and June is the most critical factor among the worst simulation cases. Acknowledgments This research is partially supported by Ministry of Science and Technology of ROC, under Grant MOST 105-2410-H-260 -018 -MY2. References [1] S. Shafiee, E. Topal, When will fossil fuel reserves be diminished, Energy Policy 37 (2009) 181e189. [2] annual report, World Wind Energy, World Wind Energy Association (WWEA), 2015, http://www.wwindea.org/. downloaded on March 31, 2016. [3] A. Kusiak, Z. Zhang, Adaptive control of a wind turbine with data mining and swarm intelligence, IEEE Trans. Sustain. Energy 2 (2011) 28e36.

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