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A biomimetic approach for modeling cloud shading with dynamic behavior
Renewable Energy 96 (2016) 157e166
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Renewable Energy journal homepage: www.elsevier.com/locate/renene
A biomimetic approach for modeling cloud shading with dynamic behavior Jesús M. García a, Ricardo Vasquez Padilla b, Marco E. Sanjuan a, * a b
Department of Mechanical Engineering, Universidad del Norte, Barranquilla, Colombia School of Environment, Science and Engineering, Southern Cross University, Lismore, NSW, 2480, Australia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 1 February 2016 Received in revised form 6 April 2016 Accepted 24 April 2016
Clouds are a complex phenomenon which is the result of a strong interaction between multiple variables. Modeling its behavior through physical principles is a task that requires time and is computationally demanding. One of the main effects caused by clouds are the shadows produced over the earth's surface, a phenomenon inherently complex due its origin. This paper proposes a computationally lowdemanding model for imitating (not predicting) the behavior of cloud shading by applying a biomimetic approach. This analogy relays on using a bacterial colony growth behavior. The aim of this paper is to establish a methodology to develop a cloud-shading model useful for transient analysis in solar fields. The proposed model is evaluated qualitative and quantitative by comparing it with a model based on fractal surfaces and with real sky images. The qualitative evaluation indicates that shadows created by the proposed model change its shapes and move as seen in the real phenomenon. On the other hand, the quantitative assessment is accomplished through the Fast Fourier Transform analysis. This analysis indicates that the proposed model is able to achieve the performance shown by real cloud images. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Bacteria colony Agent based modeling Dynamic shading Fuzzy
1. Introduction Clouds are a complex phenomenon which is the result of the interaction of several parameters. These are defined as a type of hydrometeor, that is, an ensemble of liquid or solid water particles suspended in or falling through air. According to the World Meteorological Organization (WMO) [1], clouds classification has ten different groups depending on the height of the cloud. Names of these groups come from Latin roots such as stratus (layer), cumulus (heap or puffy), cirrus (curl of hair) [2]. Modeling this phenomenon using physical principles has been proposed by several research based on momentum, continuity, and energy balances. This kind of approach is useful for studying the impact of variables such as temperature, pressure, and velocity on cloud formation In [2e5]. In this way, after implementing the model, it is possible to determine the shading patterns over the surface. However, this complex model calculates several physical variables, and therefore solving these partial differential equations is computationally demanding. For this reason, there are several approaches using graphic tools for
* Corresponding author. E-mail addresses: [email protected] (J.M. García), ricardo.vasquez. [email protected] (R.V. Padilla), [email protected] (M.E. Sanjuan). http://dx.doi.org/10.1016/j.renene.2016.04.070 0960-1481/© 2016 Elsevier Ltd. All rights reserved.
modeling clouds and that consequently are useful for developing cloud shading. Among these approaches, there are some graphic tools which use the same principles developed in computer games. Examples of these methods are polygon, procedural noise, textures sprite, metaball, particle system, or voxel volumes. These methods can be combined to obtain better results In [6]. There are several impacts of clouds on the earth. One of these is the attenuation of solar radiation, mainly the direct normal irradiance [7,8]. Several approaches for taking into account clouds in solar radiation analysis have been proposed. Some of them are built in order to match data from specific places and then develop solar radiation maps [9,10]. In Ref. [11], the changes in solar radiation were analyzed through the representation of clouds as a group of circles. These circles are spread in a two dimensional space following the homogeneous spatial Poisson process. Another approach is shown by Ref. [12]; in which clouds are represented by fractal surfaces truncated at different heights. There are also models whose core consists of statistical analysis of time data series or all-sky images in order to predict solar radiation changes [13,14]. In general, most of the above mentioned models follow the variable-based approach for representing the phenomenon. However, there are several ways for developing such representation in some complex cases. One approach that has received special
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Nomenclature M s v Vmax Km Ymax mr r d1e2 m C
Cell dry mass Substrate concentration Rate of substrate uptake Constant for tuning the model Constant for tuning the model Maximum efficiency of substrate conversion Maintenance rate Radius of cells Distance between two cells Slope of a straight line Constant for the direction of movement between two cells
attention is the Agent-based modeling. Simulations under this perspective allows representing complex structures in multiple scales of analysis through adaptation of explicit rules for the individuals within the model [15]. This type of modeling describes the behavior seen in biological systems, nevertheless it may be extrapolated to depict the performance of a third system. This extrapolation is called biomimetics, which is a term created in 1969 by Schmidt [16]. As it is well know, direct solar radiation changes throughout the day, mainly due to atmospheric disturbances such as clouds (excluding earth movements). These changes are commonly sudden and highly heterogeneous. Some models have proposed to study the effect of clouds over a solar field by using a plane that completely covers the field, therefore radiation goes from its clearday value to zero and it is maintained in this lower limit value [17]. However, clouds are dynamics and in some cases they might not impact the entire solar field, therefore this kind of approach may not be suitable. In order to study the transient effect of clouds movement on solar field, it is necessary to develop a flexible model which can emulate the dynamic of this type of phenomena. The cloud shadows model allows studying the dynamic performance of solar plants while there are heterogeneous changes in solar radiation over the solar field. This paper proposes a cloud-shading Agent Based Model (AbM) through a biomimetic approach, specifically based on a bacterial growth colony. In this kind of models, agents are taken into either single-member or multiple-member groups. AbMs do not require specifying a global growth law such as the exponential, instead, according to the rules given to the agents for interacting with its environment, and then the global growing law will emerge [15]. The proposed model is evaluated both qualitative and quantitative by comparing it with real cloud images and the fractal model proposed by Ref. [12]. In large-scale solar systems, transient behavior is a critical subject that requires special attention. Once the proposed model is appropriately tuned, it can be connected to any solar system to study its performance under this dominant disturbance. Studies such as the presented by Ref. [17] can be done in a broader manner. A graphical explanation of the model proposed by this paper is shown in Fig. 1. In this figure, it can be seen the movement of a cloud in two different times (t1 and t2). In time t2 the cloud is located in another place and has a different shape as compared with the same cloud in time t1. Likewise, the individuals of a bacteria colony are initially located in a place and then they move and reproduce according to the characteristics of their surroundings. Therefore, the dynamic of the bacterial growth colony has the
Dy Dx w1
sx sy r
εi dmax k C3 Vwind
Movement of the cell along y Movement of the cell along x Maximum amount of R1 consumption Coefficients for maximum movement in x direction Coefficients for maximum movement in y direction Normally distributed random noise source Energy of the cell under study Distance from the cell to the maximum substrate concentration Constant used for tuning the model Cell position respect to the maximum substrate point Wind velocity
potential to match the general behavior of cloud shading over a period of time by using a simple mathematical structure which requires a low computational usage.
2. General notions for the bacterial analogy Bacteria are microscopic organisms formed by single cells that are very different than the cell structures shown by other life forms in earth [18]. Initially, it was thought bacteria survival strategy was focused on finding nutrients and then multiply themselves. However, it has been revealed that exists intercellular communication among bacteria. This fact has led to the conclusion that bacteria are capable to carry out group behavior similar to multicellular organisms. This group behavior allows some advantages such as migration to either a more suitable environment or better nutrient supply. It also allows new modes of growth as sporulation or biofilm formation, which can be a protection from toxic environments. This communication cell-to-cell is called quorum sensing. It is based on self-generated signal molecules called autoinducers [19]. The first quorum sensing mechanism was seen in Vibrio fischeri, which is bioluminescent bacteria that resides in light-producing organs of the squid for reproduction. Therefore, Vibrio fischeri produces AI-1 synthase, then it diffuses and triggers the growth of the cell-population concentration. Once it exceeds a threshold level, luciferase operons are activated, which then generates light [20].
3. Proposed model As defined by Ref. [15]; agent-based models may fall in four main categories: scale models, ideal-type models, analogical models, and equation-based models. Hence the proposed model fits two of these categories analogical and ideal-type. First, because it accomplishes an analogy between the bacterial-growth system and the cloud-shading phenomenon, and second due to some of the characteristics of the model are simplified. The model implemented in this paper is based on the model developed by Ref. [21]; and its rules are dominated by interactions such as substrate consumption, metabolism and maintenance behavior of each agent. Although it is well known that bacteria colonies use quorum sensing mechanisms for growing and moving, this model does not include either the representation of the selfinducing communication molecules or its effect on the colony. Consequently, the main features to be accomplished are:
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Fig. 1. Graphical representation of the analogy between bacterial colony growth and cloud-shading dynamics used in the model proposed at this research.
1. Dynamic behavior (movement and shape) for representing the cloud-shading 2. Easy implementation 3. Uses low computational resources.
3.1. Established interactions for agents in the model In order to maintain a more general and appealing terminology for an easier comprehension, the interactions for the bacterial analogy model are explained in a broader way. Thus, the behavior of agents is represented through the following characteristics: Modification of its surroundings: Agents are spread over a heterogeneous surface, this characteristic gives them the ability to change that surface. It can be seen as a representation of “food” consumption. This feature allows cloud shades to change its size and shape over time and space. The rate of change for this feature depends on: e Replication of the agent within the simulation. e Keeping the agent “alive”. Displacement: This component enables agents to move in the spatial domain. As explained in following subsections, it allows linking the model to real parameters such as wind patterns. Additionally, in conjunction with the surrounding-modification feature, it gives to the clouds shades a spatio-temporal thickness distribution, which is related to the transmissivity of the clouds. Those elements are explained in following subsections. Fig. 2 shows a scheme about the integration of all the features added to the model. As seen, each agent has inside several aspects such as growing and division, then its displacement depend on the interaction among agents and its surroundings. 3.1.1. Modification of agent's surrounding environment In order to allow agents to increase its size and quantity, it is required to supply some kind of “fuel”. The rate of change for this feature consists of a first order differential equation proposed by Refs. [22,21] and is derived from the bacterial analogy. This expression is given by:
ds M$Vmax $s ¼v¼ dt Km þ s
(1)
where s represents the value assigned to the surface in the spatial domain (environment), M is the size of the agent, and v is the rate of
change of s. Vmax and Km are constants used for tuning the agent's growth rate in the model. Eq. (1) assumes that the rate of change in the surrounding environment is proportional to the size of agents (M). 3.1.2. Replication of agents This process allows converting the resources taken from the environment to create new agents or grow the old ones. The expression used for modeling this process is as follows [21,23]:
dM v ¼ dt Ymax
(2)
where Ymax represents the maximum efficiency of resource conversion to agent constituents. 3.1.3. Preserving the agent “alive” This term allows estimating how much of the resources taken from environment are used for maintaining the agent running during the simulation. It is also derived from the bacterial analogy proposed by Ref. [24]. The expression used for modeling the variation due this feature is given by Ref. [21]:
dM ¼ M$mr $Ymax dt
(3)
where mr is the maintenance rate, and Ymax is the maximum yield, which are used for tuning the model. Adding Eq. (2) and Eq. (3) give the net growing of each agent as shown in the following expression:
dM v ¼ M$mr $Ymax dt Ymax
(4)
Therefore, the total effect is given by the equation system defined through Eq. (1) and Eq. (4). 3.1.4. Displacement The following four main characteristics define the movement of each agent: 1. Avoiding overlap among agents. 2. Local maximum search over the surface at the neighborhood of the agent. 3. Global maximum search over the surface in the space domain. 4. Reacting to the effect of an external disturbance.
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Fig. 2. Features integrated into the proposed model. Each agent changes according the features implemented while interacting with its surroundings.
The description of each characteristic is presented below. Displacement to avoid overlap among agents. An important issue that must be addressed consists of avoiding -as much as possible-agents to occupy the same physical space in the domain, but it is also required not to separate each one permanently. This movement is useful for arranging the old-agents once new-ones are born. Thus, for adding this characteristic, it was set a movement proportional to the distance between pair of agents minus the radius of both agents. The movement is calculated as:
DD ¼ k1 $d12 k2 $ðr1 þ r2 Þ
(5)
where k1 and k2 are constants used for setting how fast the movement is, r1 and r2 are the radius of the agents, and d1e2 is the distance between two agents. Using this distance, and the equation of the straight line formed between the centers of each agent, the movement of each agent along axes x and y is calculated as:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DD2 Dx ¼ 1 þ m2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DD2 Dy ¼ m$ 1 þ m2
(6)
(7)
The magnitude and the direction of the movement vector for each agent have been previously defined without sign. The sign is included by applying the following Equations: For agent 1:
Dxagent1 ¼ C1 $Dx$signðDDÞ
(8)
Dyagent1 ¼ C1 $Dy$signðDDÞ
(9)
meaning that if C1 is þ1 then C2 is 1. Once the movement is defined along the x axis, there are only two possible movements along y, meaning that C1 and C2 can be used for defining Dy. There are multiple pairs of agents, then for the same agent is possible to obtain several values for changes in x and y. Therefore, for obtaining a single value in each axis, the average among all the values was used. Displacement for finding high-substrate concentration at the neighborhood of the agent. In addition to allow the agent to modify its environment for growing and replication, it is required to give the agent the ability of moving from its original position to another one. This displacement allows the agent to find places -in the space domain-with better environmental conditions. This characteristic is given by two methods: Local maximum search near the agent and for finding the overall maximum in the domain. The first approach consists of treating each agent as a processing unit that interprets some input parameters using predefined rules which results in some behavior of the unit. The rule based system used here was developed by Ref. [25]. This system uses the fuzzy logic theory (Fig. 3) for interpreting input signals into some outputs through applying pre-established rules. Therefore in this case, there is only one kind of environmental resource, then the input signal is a vector composed by the environmental resource (R1), the first derivative of the environmental resource, and the energy of the agent under study (εi). This input vector is given by:
Inputs ¼ R1
Dxagent2 ¼ C2 $Dx$signðDDÞ
(10)
Dyagent2 ¼ C2 $Dy$signðDDÞ
(11)
In these equations, m stands for the slope of the straight line, C1 is a constant that is þ1 if agent 1 is on the left of agent 2, and 1 if agent 1 is on the right of agent 2. Moreover, C2 is opposed to C1,
vR1 vy
εi
(12)
As output parameters, a vector with three values is obtained:
ai ¼ ½ A1
For agent 2:
vR1 vx
A2
A3
(13)
These three values are related to the position and the energy of each agent as:
Inputs
Fuzzy Inference System
Fig. 3. Fuzzy inference system.
ai
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εi ðt þ 1Þ ¼ εi ðtÞ þ ð2$A1 1Þ$w1 $R1 ðtÞ xi ðt þ 1Þ ¼ xi ðtÞ þ ð2$A2 1Þ$sx þ rx yi ðt þ 1Þ ¼ yi ðtÞ þ ð2$A3 1Þ$sy þ ry
(14)
where w1 is a positive coefficient that determine the maximum amount of R1 that can be converted into energy at any time. sx and sy are positive integer coefficients that determine the maximum movement step size in the x and y directions, respectively, while rx and ry are normally distributed random noise sources. Table 1 shows the membership functions used in the fuzzy inference system and a graphical representation is shown in Fig. 4. The rules that connect each input and output variables are shown in Table 2. Displacement to find the overall maximum environmental resource. In addition to allowing the agent to find higher environmental resources at its surrounding, it was also set that each agent should follow the location of the maximum environmental resources in the domain space. For adding this characteristic, the maximum displacement along the straight line between the center of the agent and the maximum point of the environmental resource, dmax, is determined as:
dmax ¼ k3 $
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 xmax xagent þ ymax yagent
Ddxmax
Ddymax
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2max ¼ mmax $ 1 þ m2max
Fig. 4. Graphical example for membership functions of R1,
(16)
(17)
where mmax is the slope of the line where the movement takes place. The displacements on each axes are determined as:
Dxmax ¼ C3 $Ddxmax $signðdmax Þ
(18)
Dymax ¼ C3 $Ddymax $signðdmax Þ
(19)
vR1 vR1 vx , vy ,
and εi.
Table 2 Rules used in the fuzzy inference system. R1
vR1 vx
vR1 vy
ε1
A1
A2
A3
L H L L L L L L L L H H H H H H H H
L H L L L L H H H H L L L L H H H H
L H L L H H L L H H L L H H L L H H
L H L H L H L H L H L H L H L H L H
L H e e e e e e e e e e e e e e e e
e e Z L Z L Z H Z H Z L Z L Z H Z H
e e Z L Z H Z L Z H Z L Z H Z L Z H
(15)
where k3 is a constant used for tuning how fast the displacement is. The movements along each axes are calculated as:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2max ¼ 1 þ m2max
161
where C3 can take the value of ±1 depending on the position of the agent respect to the point with the maximum environmental resource.
Displacement due to the effect of an external disturbance. An important parameter that must be taken into account in the model is the wind velocity [26]. This effect is added to the model as an external force that is acting over the agents. It is implemented by assuming that wind follows a certain pattern. For simplicity, the wind velocity is assumed to behave according the following mathematical expression:
Table 1 Membership functions used in the fuzzy inference system. The representation of L and H is shown in Fig. 4.
Vwind ¼ sinðk4 $yÞ k5 $x
Variable R1 vR1 vx vR1 vy εi A1
A2 A3
Function
Type
L
Trapezoidal
H
Trapezoidal
L
Trapezoidal
H
Trapezoidal
L
Triangular
Z
Triangular
H
Triangular
Values x: 0 y: 1
0 1
0:1 1
x: y:
0:1 0
x: y:
0 1
x: y:
0:1 0
x: y:
0 1
0 0
x: y:
0 0
0:5 1
1 0
x: y:
0:4 0
1 1
1 1
0:9 1 0 1
0:9 0 1 1
0:2 1 0:8 1 0:6 0
1 1 0:9 0
1 1
1 1
(20)
where x and y represents the axes of the space domain. It is important to mention this expression can be changed to follow any required pattern. Additionally, it was not used a simple expression in order to show the flexibility of the proposed model. Then, the displacement of each agent is obtained from the first derivative of this function as:
Dx ¼ k6 $
vVwind vx
(21)
vVwind vy
(22)
Dy ¼ k7 $
where k4, k5, k6, and k7 are constants used for tuning the model. The vector field created by this wind pattern is shown in Fig. 5.
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sagent ¼ k9 $ragent ragent ¼
Vagent ¼
Fig. 5. Vector field of the external disturbance created by the wind pattern in Eq. (20).
Each agent has a displacement with respect to the other agents and the environmental resource. It means that there are several displacements along x and y for each agent. Therefore, for each agent all these displacements are added and a single movement along x and y takes place.
3.2. Additional subjects about the model There are some aspects that are important to mention in order to give a better idea about the way the model works. A brief description of these subjects is presented below.
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3 $Vagent 4,p M
ragent
(25)
(26)
(27)
3.2.3. Reproduction and death of the agents Once an agent modifies its surroundings, immediately start to grow. However, it does not keep growing unrestrictedly, instead once it reaches a predefine limit value then the agent is divided. This division process consists of creating a new agent with an initial volume (Vu), then the original agent reduces its size until it has a volume equal to the volume before division (V) minus the volume of the new agent (Vu). Now the death process of the agents is relatively simple. A timer in installed on each agent and is used to register its age. Then, after the timer reaches a predefined value the agent is deactivated. It is worth mentioning that not all the agents are born at the same time, then they all do not suddenly disappear.
3.2.4. Coding features Finally, it is important to be aware about some issues can arise while the proposed model is implemented. First of all, there is a considerable amount of agents so that it is not suitable to run a routine for each one of them. Thus, The code was developed by using 2D or 3D arrays rather than for-loops. Therefore, taking advantage of the matrix operations and some functions of Matlab, it was possible to create a routine with a single for-loop used for each time step.
4. Preliminary results 3.2.1. Initial distribution of the environmental resource Given that the agents follow the maximum environmental resource, it is important then to set some kind of environmental resource gradient. In this paper, the environmental resource distribution proposed by Ref. [20] is used. The expression of this distribution is given by:
S¼e
ðxxcenter Þ2 þðyycenter Þ2 k8
Initially, the results are focused on the versatility of the proposed model. Two configurations are proposed (Fig. 6): Configuration 1:
(23)
where xcenter and ycenter are the coordinates where the maximum concentration is located, and k8 is a constant to adjust the magnitude of the substrate gradient.
3.2.2. Agent shape Agents can take many forms, however in this study is assumed that each agent has a circular shape. In order to represent this shape, a bell shape function was used rather than a circumference expression. A two dimensional representation of this bell function is a circumference that is not homogeneous inside. This feature allows having cloud shadows with different covering tones. The following expression is used to simulate the agent shape:
Agent ¼ e
ðxxagent Þ2 þðyyagent Þ2 sagent
(24)
where sagent is used for determining the size of the agent. Hence, this size is related to the growth of the agent by:
Fig. 6. Substrate distribution used for Configurations 1 and 2.
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S¼e
ðxþ2:8Þ2 þðyþ2:5Þ2 k8
163
(28)
Configuration 2:
S¼e
x2 þðyþ2:5Þ2 k8
ðxþ2:8Þ2 þðy2:5Þ2 k8
þe
(29)
Table 3 shows the parameters used for obtaining two different responses from the proposed biomimetic model. It is worth mentioning that some of this values were taken from the references mentioned in this paper, and other values were set while constructing the model and therefore, the proposed model is not constrained to work using those values. The idea behind this comparison is to show the potential and flexibility of the model. Other investigations might be focused on optimizing these parameters according to their own requirements. Fig. 7 and Fig. 8 show how sensitive is the proposed model to the input parameters, so that changing a few of them lead to completely different responses. This shows that the proposed model is able to adapt and emulate different situations and scenarios. Fig. 7. Response obtained using parameters of configuration 1.
5. Evaluation of results Once the model is implemented, it is important to validate the results obtained with the real phenomenon data. In this paper, the comparison between data sets is accomplished in two parts. The first one consists of a qualitative comparison among real images (Fig. 9a), clouds using the fractal method developed by Ref. [12] (Fig. 9b), and the proposed biomimetic model (Fig. 9c). Real images are data published by the National Renewable Energy Laboratory (NREL), which were taken in August 05, 2014 and reported by Ref. [27]. As shown in Fig. 9a, real clouds behavior is a very complex phenomenon, whose shapes are continuously varying. Therefore, the cloud shadows have also complex behavior over time as well. These images also show that it is possible to have sudden changes of the cloud shadows in a small period of time and the light passing through the clouds can vary even though they have the same shape. This is an important feature that our model can replicate, specially for solar applications. The fractal surface model showed similar
Table 3 Set of parameters used for showing versatility of the model.
Fig. 8. Response obtained using parameters of configuration 2.
Parameter
Config. 1
Config. 2
Vmax Km Ymax mr Vu Vd
0.0180 0.07 0.4 6 104 6.897 104 2.3 103 0.1 0.1 [0.05 þ 0.05] [0.05 þ 0.05] 0.1 4.8 33.6 0.06 2 1 1.5 1.5 50 0.43 Eq. (28) Random
0.0270 0.07 0.4 6 104 6.897 104 2.3 103 0.1 0.1 [0.05 þ 0.05] [0.05 þ 0.05] 0.1 4.8 33.6 0.1 2 1 1.5 1.5 10 0.43 Eq. (29) Random
sx sy rx ry
w1 k1 k2 k3 k4 k5 k6 k7 k8 k9 Initial Subs. Distr. Initial agent xey position
cloud patterns to those presented by the real clouds (Black regions in Fig. 9b). However, the simulations showed that this model lacks of a proper transient behavior, since only shows a static image that moves over a fix frame, which is an assumption that may not be suitable to simulate some kind of clouds. The proposed model showed shadows whose shapes are continuously changing and moving, a behavior that is closer to the dynamic seen in the real phenomenon. Besides, different cloud's attenuations of direct solar radiation can be obtained by optimizing the parameters previously described in order to follow the responses shown in real direct radiation curves. Due to its simplicity, this approach allows having a flexible model which can follow some desired characteristics. For example, two different sets of clouds shadows can be overlapped to create a more complex shadow pattern. The second approach, quantitative comparison, is accomplished by analyzing the Fourier Descriptors (FD) of the images. The Fast Fourier Transform (FFT) is an analysis employed to convert a signal to the frequency domain. The discrete form of this transform is
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Fig. 9. Comparison between real cloud patterns, and patterns created by the fractal model and the proposed model. (a) Real images of clouds for a time series every 10 min obtained from Ref. [27]. (b) Fractal methodology proposed by Ref. [12]. (c) Proposed biomimetic model.
expressed as follows [28].
Fðp; qÞ ¼
M1 X N1 X
2pmp i M
f ðm; nÞe
2pnq i N
e
(30)
m¼0 n¼0
Where p ¼ 0,1,/, M 1; q ¼ 0,1,/, N 1. N and M are the dimensions of the image. A direct application of the 2D FFT in any image is an option. However, it is not practical due the captured features are not rotation invariant. For this reason, the approach developed by Ref. [29] is used. This approach uses the Polar Fourier Transform, and a variation of the Zernike Moment Descriptors (VZM) for obtaining the normalized Generic Fourier Descriptors (GFD) [29]. A Matlab code for obtaining the GFDs is available in the MathWorks official web page by Ref. [30]. In consequence, for analyzing the differences among cloud shapes, the procedure proposed includes four stages. These steps are described below.
of each cloud from the images must be extracted. Fig. 10aec show examples of image conditioning for real images, for the fractal model, and for the biomimetic model respectively. As it respect to the size of the images, each one is fixed to have a width and a height of 300 pixels. Provided all the images are set under the same reference, then the shape of the cloud is the main source of variation. 5.2. Stage 2: obtaining the GFDs At this point the Generic Fourier Descriptors are obtained using m and n radial and angular divisions. For this study m and n are 30 and 100 respectively and a total of 3101 Fourier Descriptors are obtained (m n þ n þ 1). Fig. 11 shows that the number of proposed Fourier Descriptors are enough to take into account even very small values. 5.3. Stage 3: overall response variable
5.1. Stage 1: image conditioning For the FFT analysis, it is required to have binary images with a single object centered at its centroid. In consequence, the silhouette
As stated by Ref. [29]; the similarity of two images represented by two sets of Fourier Descriptors can be measured through the Euclidean distance (a scalar value) between the two vectors.
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Fig. 10. Silhouette extraction from cloud pictures for shape analysis. (a) Processed real image of clouds from Ref. [27]. (b) Image from fractal methodology proposed by Ref. [12]. (c) Image from proposed biomimetic model.
analysis images were divided in four groups: two sets of real images, one group for fractal images, and another group for biomimetic images. In order to guarantee a statistical power of 0.9, each group has a sample of 50 images, therefore a total of 200 images were analyzed. Parameters used for the biomimetic model are shown in Table 4. These values were obtained from an optimization procedure carried out by manipulating some of its tuning inputs. Main goal of this optimization consists of obtaining a behavior near the patterns seen in the real phenomenon. The results obtained for the Least Significance Difference (LSD) are shown in Fig. 12. The analysis shows that there is no statistical difference between the real images groups, as expected. With regards to the proposed model, the statistics analysis shows there that is no statistical difference between the real images and the proposed model. However, there is a significant difference between the fractal model and real images/the proposed model. The results demonstrated that the biomimetic model is flexible enough to be tuned for closely matching the real responses.
Fig. 11. Generic Fourier Descriptors for (a) Real image of cloud from Ref. [27]. (b) Image from fractal methodology proposed by Ref. [12]. (c) Image from proposed biomimetic model. Table 4 Set of parameters used for obtaining the images for the quantitative evaluation.
However, in this paper the difference between pairs of images is not directly measured. Instead, each image is compared with a reference image. In this case, the standard image is a circle of the same size as the conditioned images. This approach allows measuring and comparing the general performance among the groups of images through a single value. 5.4. Stage 4: statistical analysis Once the images are prepared, and the response variable is defined, an Analysis Of Variance (ANOVA) is executed. For this
Parameter
Value
Parameter
Value
Vmax Km Ymax mr Vu Vd
0.0165 0.1606 0.8235 3.9719 104 0.5778 0.0039 0.1184 0.1493 [0.05 þ 0.05] [0.05 þ 0.05] 0.0320
k1 k2 k3 k4 k5 k6 k7 k8 k9 Initial Distr. Initial agent xey position
2.5451 12.5773 0.0047 2 1 0.2515 0.6430 50 0.43 Eq. (28) Random
sx sy rx ry
w1
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Fig. 12. Graphical LSD results among the studied groups.
6. Conclusions A biomimetic model to represent the dynamic behavior of cloud shadows based on a bacterial colony growth was proposed. The model took into account some important parameters such as: surroundings modification, replication of the agents, maintenance of agents, and displacement. Inside this features, the model uses several parameters that can be set in order to tune the model to a specific desired behavior. The analysis of variance indicated that there are no statistical differences between the proposed model and the real phenomenon. This means that using proper tuning parameters the proposed model can reach the desired performance, which in this paper has been measured from the cloud-shape point of view. Additionally, through using the analogy related to the transmissivity through clouds, next step consists of developing a statistical validation and tuning in order to obtain solar radiation responses as shown in real data. References [1] Secretariat-WMO, W.M.O., International Cloud Atlas, in: Volume 1: Manual on the Observation of Clouds and Other Meteors (Annex 1 to WMO Technical Regulations), 1975. WMO (Series). [2] M.Z. Jacobson, Fundamentals of Atmospheric Modeling, Cambridge University Press, 2005. [3] R. Miyazaki, S. Yoshida, Y. Dobashi, T. Nishita, A method for modeling clouds based on atmospheric fluid dynamics, in: Computer Graphics and Applications, 2001. Proceedings. Ninth Pacific Conference on, IEEE, 2001, pp. 363e372. [4] D.O. Starr, S.K. Cox, Cirrus clouds. Part i: a cirrus cloud model, J. Atmos. Sci. 42
(1985a) 2663e2681. [5] D.O. Starr, S.K. Cox, Cirrus clouds. Part ii: numerical experiments on the formation and maintenance of cirrus, J. Atmos. Sci. 42 (1985b) 2682e2694. [6] B. Wang, Techniques for Efficient Cloud Modeling, Simulation and Rendering, ProQuest, 2007. [7] R.C. de Andrade, C. Tiba, Extreme global solar irradiance due to cloud enhancement in northeastern brazil, Renew. Energy 86 (2016) 1433e1441. [8] R. Tapakis, A. Charalambides, Enhanced values of global irradiance due to the presence of clouds in eastern mediterranean, Renew. Energy 62 (2014) 459e467. [9] V. Badescu, A. Dumitrescu, Simple models to compute solar global irradiance from the cmsaf product cloud fractional coverage, Renew. Energy 66 (2014) 118e131. rez-Burgos, R. Romn, J. Bilbao, A. de Miguel, P. Oteiza, Reconstruction of [10] A. Pe long-term direct solar irradiance data series using a model based on the cloud modification factor, Renew. Energy 77 (2015) 115e124. [11] E. Arias-Castro, J. Kleissl, M. Lave, A Poisson model for anisotropic solar ramp rate correlations, Sol. Energy 101 (2014) 192e202. [12] C. Cai, D. Aliprantis, Cumulus cloud shadow model for analysis of power systems with photovoltaics, in: PES General Meeting d Conference Exposition, 2014 IEEE, 2014, p. 1. [13] M. Brabec, V. Badescu, M. Paulescu, Cloud shade by dynamic logistic modeling, J. Appl. Stat. 41 (2014) 1174e1188. [14] H.Y. Cheng, C.C. Yu, Multi-model solar irradiance prediction based on automatic cloud classification, Energy 91 (2015) 579e587. [15] G.N. Gilbert, Agent-based Models, 153, Sage, 2008. [16] J.F. Vincent, O.A. Bogatyreva, N.R. Bogatyrev, A. Bowyer, A.K. Pahl, Biomimetics: its practice and theory, J. R. Soc. Interface 3 (2006) 471e482. [17] G. Augsburger, D. Favrat, Modelling of the receiver transient flux distribution due to cloud passages on a solar tower thermal power plant, Sol. Energy 87 (2013) 42e52. [18] J. Wearing, Bacteria: Staph, Strep, Clostridium, and Other Bacteria, Crabtree Publishing Company, 2010. [19] T.R. De Kievit, B.H. Iglewski, Bacterial quorum sensing in pathogenic relationships, Infect. Immun. 68 (2000) 4839e4849. [20] B. Niu, H. Wang, Q. Duan, L. Li, Biomimicry of quorum sensing using bacterial lifecycle model, BMC Bioinforma. 14 (Suppl 8) (2013) S8. [21] J.U. Kreft, G. Booth, J.W. Wimpenny, BacSim, a simulator for individual-based modelling of bacterial colony growth, Microbiol. Read. Engl. 144 (Pt 1) (1998) 3275e3287. [22] A.L. Koch, Microbial physiology and ecology of slow growth, Microbiol. Mol. Biol. Rev. 61 (1997) 305e318. [23] T.D. Paustian, Microbiology Textbook: The World of Microbes, Department of Bacteriology, University of Wisconsin-Madison, 2001. [24] S.J. Pirt, The maintenance energy of bacteria in growing cultures, Proc. R. Soc. Lond. Ser. B, Biol. Sci. 163 (1965) 224e231. [25] C. Vlachos, R. Paton, J. Saunders, Q. Wu, A rule-based approach to the modelling of bacterial ecosystems, Biosystems 84 (2006) 49e72. [26] M. Li, Y. Chu, H.T. Pedro, C.F. Coimbra, Quantitative evaluation of the impact of cloud transmittance and cloud velocity on the accuracy of short-term {DNI} forecasts, Renew. Energy 86 (2016) 1362e1371. [27] A. Andreas, T. Stoffel, NREL Solar Radiation Research Laboratory (SRRL): Baseline Measurement System (BMS); Golden, Colorado (Data), 1981. NREL Report No. DA-5500e56488. [28] Z. Yuan, F. Li, P. Zhang, B. Chen, Description of shape characteristics through fourier and wavelet analysis, Chin. J. Aeronaut. 27 (2014) 160e168. [29] D. Zhang, G. Lu, Shape-based image retrieval using generic fourier descriptor, Signal Process. Image Commun. 17 (2002) 825e848. [30] F. Kratzert, Implementation of the Generic Fourier Descriptors, 2015. URL: http://www.mathworks.com/matlabcentral/fileexchange/52643-fd-¼-gfdbw-m-neimplementation-of-the-generic-fourier-descriptors/all_files.
Contents lists available at ScienceDirect
Renewable Energy journal homepage: www.elsevier.com/locate/renene
A biomimetic approach for modeling cloud shading with dynamic behavior Jesús M. García a, Ricardo Vasquez Padilla b, Marco E. Sanjuan a, * a b
Department of Mechanical Engineering, Universidad del Norte, Barranquilla, Colombia School of Environment, Science and Engineering, Southern Cross University, Lismore, NSW, 2480, Australia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 1 February 2016 Received in revised form 6 April 2016 Accepted 24 April 2016
Clouds are a complex phenomenon which is the result of a strong interaction between multiple variables. Modeling its behavior through physical principles is a task that requires time and is computationally demanding. One of the main effects caused by clouds are the shadows produced over the earth's surface, a phenomenon inherently complex due its origin. This paper proposes a computationally lowdemanding model for imitating (not predicting) the behavior of cloud shading by applying a biomimetic approach. This analogy relays on using a bacterial colony growth behavior. The aim of this paper is to establish a methodology to develop a cloud-shading model useful for transient analysis in solar fields. The proposed model is evaluated qualitative and quantitative by comparing it with a model based on fractal surfaces and with real sky images. The qualitative evaluation indicates that shadows created by the proposed model change its shapes and move as seen in the real phenomenon. On the other hand, the quantitative assessment is accomplished through the Fast Fourier Transform analysis. This analysis indicates that the proposed model is able to achieve the performance shown by real cloud images. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Bacteria colony Agent based modeling Dynamic shading Fuzzy
1. Introduction Clouds are a complex phenomenon which is the result of the interaction of several parameters. These are defined as a type of hydrometeor, that is, an ensemble of liquid or solid water particles suspended in or falling through air. According to the World Meteorological Organization (WMO) [1], clouds classification has ten different groups depending on the height of the cloud. Names of these groups come from Latin roots such as stratus (layer), cumulus (heap or puffy), cirrus (curl of hair) [2]. Modeling this phenomenon using physical principles has been proposed by several research based on momentum, continuity, and energy balances. This kind of approach is useful for studying the impact of variables such as temperature, pressure, and velocity on cloud formation In [2e5]. In this way, after implementing the model, it is possible to determine the shading patterns over the surface. However, this complex model calculates several physical variables, and therefore solving these partial differential equations is computationally demanding. For this reason, there are several approaches using graphic tools for
* Corresponding author. E-mail addresses: [email protected] (J.M. García), ricardo.vasquez. [email protected] (R.V. Padilla), [email protected] (M.E. Sanjuan). http://dx.doi.org/10.1016/j.renene.2016.04.070 0960-1481/© 2016 Elsevier Ltd. All rights reserved.
modeling clouds and that consequently are useful for developing cloud shading. Among these approaches, there are some graphic tools which use the same principles developed in computer games. Examples of these methods are polygon, procedural noise, textures sprite, metaball, particle system, or voxel volumes. These methods can be combined to obtain better results In [6]. There are several impacts of clouds on the earth. One of these is the attenuation of solar radiation, mainly the direct normal irradiance [7,8]. Several approaches for taking into account clouds in solar radiation analysis have been proposed. Some of them are built in order to match data from specific places and then develop solar radiation maps [9,10]. In Ref. [11], the changes in solar radiation were analyzed through the representation of clouds as a group of circles. These circles are spread in a two dimensional space following the homogeneous spatial Poisson process. Another approach is shown by Ref. [12]; in which clouds are represented by fractal surfaces truncated at different heights. There are also models whose core consists of statistical analysis of time data series or all-sky images in order to predict solar radiation changes [13,14]. In general, most of the above mentioned models follow the variable-based approach for representing the phenomenon. However, there are several ways for developing such representation in some complex cases. One approach that has received special
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Nomenclature M s v Vmax Km Ymax mr r d1e2 m C
Cell dry mass Substrate concentration Rate of substrate uptake Constant for tuning the model Constant for tuning the model Maximum efficiency of substrate conversion Maintenance rate Radius of cells Distance between two cells Slope of a straight line Constant for the direction of movement between two cells
attention is the Agent-based modeling. Simulations under this perspective allows representing complex structures in multiple scales of analysis through adaptation of explicit rules for the individuals within the model [15]. This type of modeling describes the behavior seen in biological systems, nevertheless it may be extrapolated to depict the performance of a third system. This extrapolation is called biomimetics, which is a term created in 1969 by Schmidt [16]. As it is well know, direct solar radiation changes throughout the day, mainly due to atmospheric disturbances such as clouds (excluding earth movements). These changes are commonly sudden and highly heterogeneous. Some models have proposed to study the effect of clouds over a solar field by using a plane that completely covers the field, therefore radiation goes from its clearday value to zero and it is maintained in this lower limit value [17]. However, clouds are dynamics and in some cases they might not impact the entire solar field, therefore this kind of approach may not be suitable. In order to study the transient effect of clouds movement on solar field, it is necessary to develop a flexible model which can emulate the dynamic of this type of phenomena. The cloud shadows model allows studying the dynamic performance of solar plants while there are heterogeneous changes in solar radiation over the solar field. This paper proposes a cloud-shading Agent Based Model (AbM) through a biomimetic approach, specifically based on a bacterial growth colony. In this kind of models, agents are taken into either single-member or multiple-member groups. AbMs do not require specifying a global growth law such as the exponential, instead, according to the rules given to the agents for interacting with its environment, and then the global growing law will emerge [15]. The proposed model is evaluated both qualitative and quantitative by comparing it with real cloud images and the fractal model proposed by Ref. [12]. In large-scale solar systems, transient behavior is a critical subject that requires special attention. Once the proposed model is appropriately tuned, it can be connected to any solar system to study its performance under this dominant disturbance. Studies such as the presented by Ref. [17] can be done in a broader manner. A graphical explanation of the model proposed by this paper is shown in Fig. 1. In this figure, it can be seen the movement of a cloud in two different times (t1 and t2). In time t2 the cloud is located in another place and has a different shape as compared with the same cloud in time t1. Likewise, the individuals of a bacteria colony are initially located in a place and then they move and reproduce according to the characteristics of their surroundings. Therefore, the dynamic of the bacterial growth colony has the
Dy Dx w1
sx sy r
εi dmax k C3 Vwind
Movement of the cell along y Movement of the cell along x Maximum amount of R1 consumption Coefficients for maximum movement in x direction Coefficients for maximum movement in y direction Normally distributed random noise source Energy of the cell under study Distance from the cell to the maximum substrate concentration Constant used for tuning the model Cell position respect to the maximum substrate point Wind velocity
potential to match the general behavior of cloud shading over a period of time by using a simple mathematical structure which requires a low computational usage.
2. General notions for the bacterial analogy Bacteria are microscopic organisms formed by single cells that are very different than the cell structures shown by other life forms in earth [18]. Initially, it was thought bacteria survival strategy was focused on finding nutrients and then multiply themselves. However, it has been revealed that exists intercellular communication among bacteria. This fact has led to the conclusion that bacteria are capable to carry out group behavior similar to multicellular organisms. This group behavior allows some advantages such as migration to either a more suitable environment or better nutrient supply. It also allows new modes of growth as sporulation or biofilm formation, which can be a protection from toxic environments. This communication cell-to-cell is called quorum sensing. It is based on self-generated signal molecules called autoinducers [19]. The first quorum sensing mechanism was seen in Vibrio fischeri, which is bioluminescent bacteria that resides in light-producing organs of the squid for reproduction. Therefore, Vibrio fischeri produces AI-1 synthase, then it diffuses and triggers the growth of the cell-population concentration. Once it exceeds a threshold level, luciferase operons are activated, which then generates light [20].
3. Proposed model As defined by Ref. [15]; agent-based models may fall in four main categories: scale models, ideal-type models, analogical models, and equation-based models. Hence the proposed model fits two of these categories analogical and ideal-type. First, because it accomplishes an analogy between the bacterial-growth system and the cloud-shading phenomenon, and second due to some of the characteristics of the model are simplified. The model implemented in this paper is based on the model developed by Ref. [21]; and its rules are dominated by interactions such as substrate consumption, metabolism and maintenance behavior of each agent. Although it is well known that bacteria colonies use quorum sensing mechanisms for growing and moving, this model does not include either the representation of the selfinducing communication molecules or its effect on the colony. Consequently, the main features to be accomplished are:
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Fig. 1. Graphical representation of the analogy between bacterial colony growth and cloud-shading dynamics used in the model proposed at this research.
1. Dynamic behavior (movement and shape) for representing the cloud-shading 2. Easy implementation 3. Uses low computational resources.
3.1. Established interactions for agents in the model In order to maintain a more general and appealing terminology for an easier comprehension, the interactions for the bacterial analogy model are explained in a broader way. Thus, the behavior of agents is represented through the following characteristics: Modification of its surroundings: Agents are spread over a heterogeneous surface, this characteristic gives them the ability to change that surface. It can be seen as a representation of “food” consumption. This feature allows cloud shades to change its size and shape over time and space. The rate of change for this feature depends on: e Replication of the agent within the simulation. e Keeping the agent “alive”. Displacement: This component enables agents to move in the spatial domain. As explained in following subsections, it allows linking the model to real parameters such as wind patterns. Additionally, in conjunction with the surrounding-modification feature, it gives to the clouds shades a spatio-temporal thickness distribution, which is related to the transmissivity of the clouds. Those elements are explained in following subsections. Fig. 2 shows a scheme about the integration of all the features added to the model. As seen, each agent has inside several aspects such as growing and division, then its displacement depend on the interaction among agents and its surroundings. 3.1.1. Modification of agent's surrounding environment In order to allow agents to increase its size and quantity, it is required to supply some kind of “fuel”. The rate of change for this feature consists of a first order differential equation proposed by Refs. [22,21] and is derived from the bacterial analogy. This expression is given by:
ds M$Vmax $s ¼v¼ dt Km þ s
(1)
where s represents the value assigned to the surface in the spatial domain (environment), M is the size of the agent, and v is the rate of
change of s. Vmax and Km are constants used for tuning the agent's growth rate in the model. Eq. (1) assumes that the rate of change in the surrounding environment is proportional to the size of agents (M). 3.1.2. Replication of agents This process allows converting the resources taken from the environment to create new agents or grow the old ones. The expression used for modeling this process is as follows [21,23]:
dM v ¼ dt Ymax
(2)
where Ymax represents the maximum efficiency of resource conversion to agent constituents. 3.1.3. Preserving the agent “alive” This term allows estimating how much of the resources taken from environment are used for maintaining the agent running during the simulation. It is also derived from the bacterial analogy proposed by Ref. [24]. The expression used for modeling the variation due this feature is given by Ref. [21]:
dM ¼ M$mr $Ymax dt
(3)
where mr is the maintenance rate, and Ymax is the maximum yield, which are used for tuning the model. Adding Eq. (2) and Eq. (3) give the net growing of each agent as shown in the following expression:
dM v ¼ M$mr $Ymax dt Ymax
(4)
Therefore, the total effect is given by the equation system defined through Eq. (1) and Eq. (4). 3.1.4. Displacement The following four main characteristics define the movement of each agent: 1. Avoiding overlap among agents. 2. Local maximum search over the surface at the neighborhood of the agent. 3. Global maximum search over the surface in the space domain. 4. Reacting to the effect of an external disturbance.
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Fig. 2. Features integrated into the proposed model. Each agent changes according the features implemented while interacting with its surroundings.
The description of each characteristic is presented below. Displacement to avoid overlap among agents. An important issue that must be addressed consists of avoiding -as much as possible-agents to occupy the same physical space in the domain, but it is also required not to separate each one permanently. This movement is useful for arranging the old-agents once new-ones are born. Thus, for adding this characteristic, it was set a movement proportional to the distance between pair of agents minus the radius of both agents. The movement is calculated as:
DD ¼ k1 $d12 k2 $ðr1 þ r2 Þ
(5)
where k1 and k2 are constants used for setting how fast the movement is, r1 and r2 are the radius of the agents, and d1e2 is the distance between two agents. Using this distance, and the equation of the straight line formed between the centers of each agent, the movement of each agent along axes x and y is calculated as:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DD2 Dx ¼ 1 þ m2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DD2 Dy ¼ m$ 1 þ m2
(6)
(7)
The magnitude and the direction of the movement vector for each agent have been previously defined without sign. The sign is included by applying the following Equations: For agent 1:
Dxagent1 ¼ C1 $Dx$signðDDÞ
(8)
Dyagent1 ¼ C1 $Dy$signðDDÞ
(9)
meaning that if C1 is þ1 then C2 is 1. Once the movement is defined along the x axis, there are only two possible movements along y, meaning that C1 and C2 can be used for defining Dy. There are multiple pairs of agents, then for the same agent is possible to obtain several values for changes in x and y. Therefore, for obtaining a single value in each axis, the average among all the values was used. Displacement for finding high-substrate concentration at the neighborhood of the agent. In addition to allow the agent to modify its environment for growing and replication, it is required to give the agent the ability of moving from its original position to another one. This displacement allows the agent to find places -in the space domain-with better environmental conditions. This characteristic is given by two methods: Local maximum search near the agent and for finding the overall maximum in the domain. The first approach consists of treating each agent as a processing unit that interprets some input parameters using predefined rules which results in some behavior of the unit. The rule based system used here was developed by Ref. [25]. This system uses the fuzzy logic theory (Fig. 3) for interpreting input signals into some outputs through applying pre-established rules. Therefore in this case, there is only one kind of environmental resource, then the input signal is a vector composed by the environmental resource (R1), the first derivative of the environmental resource, and the energy of the agent under study (εi). This input vector is given by:
Inputs ¼ R1
Dxagent2 ¼ C2 $Dx$signðDDÞ
(10)
Dyagent2 ¼ C2 $Dy$signðDDÞ
(11)
In these equations, m stands for the slope of the straight line, C1 is a constant that is þ1 if agent 1 is on the left of agent 2, and 1 if agent 1 is on the right of agent 2. Moreover, C2 is opposed to C1,
vR1 vy
εi
(12)
As output parameters, a vector with three values is obtained:
ai ¼ ½ A1
For agent 2:
vR1 vx
A2
A3
(13)
These three values are related to the position and the energy of each agent as:
Inputs
Fuzzy Inference System
Fig. 3. Fuzzy inference system.
ai
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εi ðt þ 1Þ ¼ εi ðtÞ þ ð2$A1 1Þ$w1 $R1 ðtÞ xi ðt þ 1Þ ¼ xi ðtÞ þ ð2$A2 1Þ$sx þ rx yi ðt þ 1Þ ¼ yi ðtÞ þ ð2$A3 1Þ$sy þ ry
(14)
where w1 is a positive coefficient that determine the maximum amount of R1 that can be converted into energy at any time. sx and sy are positive integer coefficients that determine the maximum movement step size in the x and y directions, respectively, while rx and ry are normally distributed random noise sources. Table 1 shows the membership functions used in the fuzzy inference system and a graphical representation is shown in Fig. 4. The rules that connect each input and output variables are shown in Table 2. Displacement to find the overall maximum environmental resource. In addition to allowing the agent to find higher environmental resources at its surrounding, it was also set that each agent should follow the location of the maximum environmental resources in the domain space. For adding this characteristic, the maximum displacement along the straight line between the center of the agent and the maximum point of the environmental resource, dmax, is determined as:
dmax ¼ k3 $
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 xmax xagent þ ymax yagent
Ddxmax
Ddymax
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2max ¼ mmax $ 1 þ m2max
Fig. 4. Graphical example for membership functions of R1,
(16)
(17)
where mmax is the slope of the line where the movement takes place. The displacements on each axes are determined as:
Dxmax ¼ C3 $Ddxmax $signðdmax Þ
(18)
Dymax ¼ C3 $Ddymax $signðdmax Þ
(19)
vR1 vR1 vx , vy ,
and εi.
Table 2 Rules used in the fuzzy inference system. R1
vR1 vx
vR1 vy
ε1
A1
A2
A3
L H L L L L L L L L H H H H H H H H
L H L L L L H H H H L L L L H H H H
L H L L H H L L H H L L H H L L H H
L H L H L H L H L H L H L H L H L H
L H e e e e e e e e e e e e e e e e
e e Z L Z L Z H Z H Z L Z L Z H Z H
e e Z L Z H Z L Z H Z L Z H Z L Z H
(15)
where k3 is a constant used for tuning how fast the displacement is. The movements along each axes are calculated as:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2max ¼ 1 þ m2max
161
where C3 can take the value of ±1 depending on the position of the agent respect to the point with the maximum environmental resource.
Displacement due to the effect of an external disturbance. An important parameter that must be taken into account in the model is the wind velocity [26]. This effect is added to the model as an external force that is acting over the agents. It is implemented by assuming that wind follows a certain pattern. For simplicity, the wind velocity is assumed to behave according the following mathematical expression:
Table 1 Membership functions used in the fuzzy inference system. The representation of L and H is shown in Fig. 4.
Vwind ¼ sinðk4 $yÞ k5 $x
Variable R1 vR1 vx vR1 vy εi A1
A2 A3
Function
Type
L
Trapezoidal
H
Trapezoidal
L
Trapezoidal
H
Trapezoidal
L
Triangular
Z
Triangular
H
Triangular
Values x: 0 y: 1
0 1
0:1 1
x: y:
0:1 0
x: y:
0 1
x: y:
0:1 0
x: y:
0 1
0 0
x: y:
0 0
0:5 1
1 0
x: y:
0:4 0
1 1
1 1
0:9 1 0 1
0:9 0 1 1
0:2 1 0:8 1 0:6 0
1 1 0:9 0
1 1
1 1
(20)
where x and y represents the axes of the space domain. It is important to mention this expression can be changed to follow any required pattern. Additionally, it was not used a simple expression in order to show the flexibility of the proposed model. Then, the displacement of each agent is obtained from the first derivative of this function as:
Dx ¼ k6 $
vVwind vx
(21)
vVwind vy
(22)
Dy ¼ k7 $
where k4, k5, k6, and k7 are constants used for tuning the model. The vector field created by this wind pattern is shown in Fig. 5.
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sagent ¼ k9 $ragent ragent ¼
Vagent ¼
Fig. 5. Vector field of the external disturbance created by the wind pattern in Eq. (20).
Each agent has a displacement with respect to the other agents and the environmental resource. It means that there are several displacements along x and y for each agent. Therefore, for each agent all these displacements are added and a single movement along x and y takes place.
3.2. Additional subjects about the model There are some aspects that are important to mention in order to give a better idea about the way the model works. A brief description of these subjects is presented below.
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3 $Vagent 4,p M
ragent
(25)
(26)
(27)
3.2.3. Reproduction and death of the agents Once an agent modifies its surroundings, immediately start to grow. However, it does not keep growing unrestrictedly, instead once it reaches a predefine limit value then the agent is divided. This division process consists of creating a new agent with an initial volume (Vu), then the original agent reduces its size until it has a volume equal to the volume before division (V) minus the volume of the new agent (Vu). Now the death process of the agents is relatively simple. A timer in installed on each agent and is used to register its age. Then, after the timer reaches a predefined value the agent is deactivated. It is worth mentioning that not all the agents are born at the same time, then they all do not suddenly disappear.
3.2.4. Coding features Finally, it is important to be aware about some issues can arise while the proposed model is implemented. First of all, there is a considerable amount of agents so that it is not suitable to run a routine for each one of them. Thus, The code was developed by using 2D or 3D arrays rather than for-loops. Therefore, taking advantage of the matrix operations and some functions of Matlab, it was possible to create a routine with a single for-loop used for each time step.
4. Preliminary results 3.2.1. Initial distribution of the environmental resource Given that the agents follow the maximum environmental resource, it is important then to set some kind of environmental resource gradient. In this paper, the environmental resource distribution proposed by Ref. [20] is used. The expression of this distribution is given by:
S¼e
ðxxcenter Þ2 þðyycenter Þ2 k8
Initially, the results are focused on the versatility of the proposed model. Two configurations are proposed (Fig. 6): Configuration 1:
(23)
where xcenter and ycenter are the coordinates where the maximum concentration is located, and k8 is a constant to adjust the magnitude of the substrate gradient.
3.2.2. Agent shape Agents can take many forms, however in this study is assumed that each agent has a circular shape. In order to represent this shape, a bell shape function was used rather than a circumference expression. A two dimensional representation of this bell function is a circumference that is not homogeneous inside. This feature allows having cloud shadows with different covering tones. The following expression is used to simulate the agent shape:
Agent ¼ e
ðxxagent Þ2 þðyyagent Þ2 sagent
(24)
where sagent is used for determining the size of the agent. Hence, this size is related to the growth of the agent by:
Fig. 6. Substrate distribution used for Configurations 1 and 2.
J.M. García et al. / Renewable Energy 96 (2016) 157e166
S¼e
ðxþ2:8Þ2 þðyþ2:5Þ2 k8
163
(28)
Configuration 2:
S¼e
x2 þðyþ2:5Þ2 k8
ðxþ2:8Þ2 þðy2:5Þ2 k8
þe
(29)
Table 3 shows the parameters used for obtaining two different responses from the proposed biomimetic model. It is worth mentioning that some of this values were taken from the references mentioned in this paper, and other values were set while constructing the model and therefore, the proposed model is not constrained to work using those values. The idea behind this comparison is to show the potential and flexibility of the model. Other investigations might be focused on optimizing these parameters according to their own requirements. Fig. 7 and Fig. 8 show how sensitive is the proposed model to the input parameters, so that changing a few of them lead to completely different responses. This shows that the proposed model is able to adapt and emulate different situations and scenarios. Fig. 7. Response obtained using parameters of configuration 1.
5. Evaluation of results Once the model is implemented, it is important to validate the results obtained with the real phenomenon data. In this paper, the comparison between data sets is accomplished in two parts. The first one consists of a qualitative comparison among real images (Fig. 9a), clouds using the fractal method developed by Ref. [12] (Fig. 9b), and the proposed biomimetic model (Fig. 9c). Real images are data published by the National Renewable Energy Laboratory (NREL), which were taken in August 05, 2014 and reported by Ref. [27]. As shown in Fig. 9a, real clouds behavior is a very complex phenomenon, whose shapes are continuously varying. Therefore, the cloud shadows have also complex behavior over time as well. These images also show that it is possible to have sudden changes of the cloud shadows in a small period of time and the light passing through the clouds can vary even though they have the same shape. This is an important feature that our model can replicate, specially for solar applications. The fractal surface model showed similar
Table 3 Set of parameters used for showing versatility of the model.
Fig. 8. Response obtained using parameters of configuration 2.
Parameter
Config. 1
Config. 2
Vmax Km Ymax mr Vu Vd
0.0180 0.07 0.4 6 104 6.897 104 2.3 103 0.1 0.1 [0.05 þ 0.05] [0.05 þ 0.05] 0.1 4.8 33.6 0.06 2 1 1.5 1.5 50 0.43 Eq. (28) Random
0.0270 0.07 0.4 6 104 6.897 104 2.3 103 0.1 0.1 [0.05 þ 0.05] [0.05 þ 0.05] 0.1 4.8 33.6 0.1 2 1 1.5 1.5 10 0.43 Eq. (29) Random
sx sy rx ry
w1 k1 k2 k3 k4 k5 k6 k7 k8 k9 Initial Subs. Distr. Initial agent xey position
cloud patterns to those presented by the real clouds (Black regions in Fig. 9b). However, the simulations showed that this model lacks of a proper transient behavior, since only shows a static image that moves over a fix frame, which is an assumption that may not be suitable to simulate some kind of clouds. The proposed model showed shadows whose shapes are continuously changing and moving, a behavior that is closer to the dynamic seen in the real phenomenon. Besides, different cloud's attenuations of direct solar radiation can be obtained by optimizing the parameters previously described in order to follow the responses shown in real direct radiation curves. Due to its simplicity, this approach allows having a flexible model which can follow some desired characteristics. For example, two different sets of clouds shadows can be overlapped to create a more complex shadow pattern. The second approach, quantitative comparison, is accomplished by analyzing the Fourier Descriptors (FD) of the images. The Fast Fourier Transform (FFT) is an analysis employed to convert a signal to the frequency domain. The discrete form of this transform is
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Fig. 9. Comparison between real cloud patterns, and patterns created by the fractal model and the proposed model. (a) Real images of clouds for a time series every 10 min obtained from Ref. [27]. (b) Fractal methodology proposed by Ref. [12]. (c) Proposed biomimetic model.
expressed as follows [28].
Fðp; qÞ ¼
M1 X N1 X
2pmp i M
f ðm; nÞe
2pnq i N
e
(30)
m¼0 n¼0
Where p ¼ 0,1,/, M 1; q ¼ 0,1,/, N 1. N and M are the dimensions of the image. A direct application of the 2D FFT in any image is an option. However, it is not practical due the captured features are not rotation invariant. For this reason, the approach developed by Ref. [29] is used. This approach uses the Polar Fourier Transform, and a variation of the Zernike Moment Descriptors (VZM) for obtaining the normalized Generic Fourier Descriptors (GFD) [29]. A Matlab code for obtaining the GFDs is available in the MathWorks official web page by Ref. [30]. In consequence, for analyzing the differences among cloud shapes, the procedure proposed includes four stages. These steps are described below.
of each cloud from the images must be extracted. Fig. 10aec show examples of image conditioning for real images, for the fractal model, and for the biomimetic model respectively. As it respect to the size of the images, each one is fixed to have a width and a height of 300 pixels. Provided all the images are set under the same reference, then the shape of the cloud is the main source of variation. 5.2. Stage 2: obtaining the GFDs At this point the Generic Fourier Descriptors are obtained using m and n radial and angular divisions. For this study m and n are 30 and 100 respectively and a total of 3101 Fourier Descriptors are obtained (m n þ n þ 1). Fig. 11 shows that the number of proposed Fourier Descriptors are enough to take into account even very small values. 5.3. Stage 3: overall response variable
5.1. Stage 1: image conditioning For the FFT analysis, it is required to have binary images with a single object centered at its centroid. In consequence, the silhouette
As stated by Ref. [29]; the similarity of two images represented by two sets of Fourier Descriptors can be measured through the Euclidean distance (a scalar value) between the two vectors.
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Fig. 10. Silhouette extraction from cloud pictures for shape analysis. (a) Processed real image of clouds from Ref. [27]. (b) Image from fractal methodology proposed by Ref. [12]. (c) Image from proposed biomimetic model.
analysis images were divided in four groups: two sets of real images, one group for fractal images, and another group for biomimetic images. In order to guarantee a statistical power of 0.9, each group has a sample of 50 images, therefore a total of 200 images were analyzed. Parameters used for the biomimetic model are shown in Table 4. These values were obtained from an optimization procedure carried out by manipulating some of its tuning inputs. Main goal of this optimization consists of obtaining a behavior near the patterns seen in the real phenomenon. The results obtained for the Least Significance Difference (LSD) are shown in Fig. 12. The analysis shows that there is no statistical difference between the real images groups, as expected. With regards to the proposed model, the statistics analysis shows there that is no statistical difference between the real images and the proposed model. However, there is a significant difference between the fractal model and real images/the proposed model. The results demonstrated that the biomimetic model is flexible enough to be tuned for closely matching the real responses.
Fig. 11. Generic Fourier Descriptors for (a) Real image of cloud from Ref. [27]. (b) Image from fractal methodology proposed by Ref. [12]. (c) Image from proposed biomimetic model. Table 4 Set of parameters used for obtaining the images for the quantitative evaluation.
However, in this paper the difference between pairs of images is not directly measured. Instead, each image is compared with a reference image. In this case, the standard image is a circle of the same size as the conditioned images. This approach allows measuring and comparing the general performance among the groups of images through a single value. 5.4. Stage 4: statistical analysis Once the images are prepared, and the response variable is defined, an Analysis Of Variance (ANOVA) is executed. For this
Parameter
Value
Parameter
Value
Vmax Km Ymax mr Vu Vd
0.0165 0.1606 0.8235 3.9719 104 0.5778 0.0039 0.1184 0.1493 [0.05 þ 0.05] [0.05 þ 0.05] 0.0320
k1 k2 k3 k4 k5 k6 k7 k8 k9 Initial Distr. Initial agent xey position
2.5451 12.5773 0.0047 2 1 0.2515 0.6430 50 0.43 Eq. (28) Random
sx sy rx ry
w1
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Fig. 12. Graphical LSD results among the studied groups.
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