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Barely three months into the new year and we are happy to announce a monumental milestone reached - 150 million downloads.
\n\nThis achievement solidifies IntechOpen’s place as a pioneer in Open Access publishing and the home to some of the most relevant scientific research available through Open Access.
\n\nWe are so proud to have worked with so many bright minds throughout the years who have helped us spread knowledge through the power of Open Access and we look forward to continuing to support some of the greatest thinkers of our day.
\n\nThank you for making IntechOpen your place of learning, sharing, and discovery, and here’s to 150 million more!
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His research areas and published articles focus on endocrinology, especially the thyroid, parathyroid and diabetic disorders.",institutionString:"University of Valladolid",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"2",totalChapterViews:"0",totalEditedBooks:"2",institution:{name:"Hospital Clínico Universitario de Valladolid",institutionURL:null,country:{name:"Spain"}}}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,coeditorOne:{id:"153844",title:"Dr.",name:"Manuel",middleName:null,surname:"Puig-Domingo",slug:"manuel-puig-domingo",fullName:"Manuel Puig-Domingo",profilePictureURL:"https://mts.intechopen.com/storage/users/153844/images/5498_n.jpg",biography:"Manuel Puig Domingo is currently Head of the Service of Endocrinology and Nutrition at Germans Trias i Pujol University Hospital and Associate Professor of Endocrinology at the Autonomous University of Barcelona. 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Global EV sales are increasing; annually, these global vehicle unit sales reached approximately 100,000 in 2012, 200,000 in 2013, and 300,000 in 2014, being the most important market in the USA, followed by Japan, Europe, and China [1]. Several automotive companies have wide programs to manufacture and sell electric vehicles; additionally, media reflects that the government, industries, as well as non-governmental organizations are pushing to have policies and benefits to adopt electric vehicle as a symbol of an eco-friendly transport technology. In particular, the actions of the governments of China, France, Germany, Japan, the Netherlands, Norway, the UK, and the USA are leading with policy incentives and infrastructure investments, and these countries make up over 90% of the world’s electric vehicle market [1].
Countries of Northern Europe show important advances in spreading the use of electric vehicles. Countries with potential bigger markets as France, the UK, Japan, and the USA are developing consumer incentives and charging infrastructure support to increase the numbers of EVs in the cities.
India, China, and Latin American countries are rapidly changing to EV production and, in less proportion, to their use, as a result of two motivations: they are low-cost manufacturing countries and their hungriness to have access to the newest technologies. But, it is possible that the current state of this technology is not suitable for those countries, especially for Latin American ones.
In this chapter, the influence of geographic, economic, and social conditions is taken into account to determine the performance of EVs in Latin American countries; a case study of their behavior in Mexico City is also discussed.
Finally, a different scenario of their use and benefits is concluded; at this moment, the use of actual EVs conceived for different geographic, economic, and social conditions is not the best solution to really impact on the environment in Latin American countries.
To quantify the energy consumption in a vehicle, it is important to know three sets of key data:
Vehicle dynamic characteristics
Driving cycle
Geo-environmental conditions
In particular, it is very important to also take into account the interconnected geographic and environmental conditions because, under real-driving conditions, EVs seem to be much more dependent on environmental conditions than fuel vehicles, especially in urban areas [2]. In those areas, the conditions of traffic obligate, sometimes, to make use of ventilation and air conditioning units; these conditions have an impact of approximately 10% of the fuel consumption on conventional vehicles [3], whereas it can be responsible of a range drop up to 40% for EVs [4]. Finally, the altitude of cities over the sea level and even the pavement deterioration are additional conditions that directly impact on the EV performance.
To ensure a good knowledge of energetic requirements of EVs, it is important to assume the range autonomy according to the purpose and the mobility needs of users. A utility vehicle will have a very different energy demand compared with a personal vehicle. Even for the same purpose, the traffic conditions shape different energy configurations.
A classic vehicle dynamic model is shown in Figure 1, where acceleration force (Fa), drag force (Fd), roll force (Fr), mechanic force (Fm), and gravity force (Fg) are represented under an inclination θ (θ is the measured angle over the road).
Vehicle force diagram.
The mechanical power needed at the tire of the vehicle is the product of the speed (
From Eq. (1), it is seen that the mechanic force required moving the vehicle is the sum of the acceleration force, the drag force, the roll force, and the gravity force; these forces are, respectively, depicted by Eqs. (2)–(5) as follows:
where
From Eqs. (1)–(5), it can be seen that the mass and the speed are the most important variables in the calculation of mechanic power for the vehicle. However, looking into each equation, there are other important parameters that usually influence consumption, for instance, the acceleration and the inclination of the road.
An additional parameter that needs to be taken into account for the determination of the energy consumption is the driving cycle; in a few words, it is a data set showing the speed profile in function of driving.
In the driving cycles known as “New” European Driving Cycle (NEDC), the speed variations are considerably reduced [6]. Therefore, under this supposition, the driving cycle does not represent the real driving conditions in real-world vehicle usage as depicted in [7], and therefore the toxic emissions and fuel consumption of the vehicles are underestimated [8].
For this driving cycle, the influence of Eq. (2) over Eq. (1) is underestimated [9, 10]. An example of the NEDC results can be appreciated in Figure 2.
Driving cycle NEDC.
The driving cycle called the World-wide harmonized Light duty driving Test Cycle (WLTC) is a random cycle that proportionates nearest results to real driving conditions compared with those obtained in the driving cycle NEDC.
Several works [11, 12] have shown closer results to real conditions using the WLTC because the consideration of the changes in accelerations and the different speed phases has a wide-range operation of vehicle. This year, WLTC becomes accepted as a better driving cycle [13]. As can be seen in Figure 3, the driving cycle WLTC provides more detailed information than driving cycle NEDC (Figure 2).
Driving cycle WLTC.
Geo-environmental conditions are not, so often, taken into account on consumption analysis. This analysis is strongly influenced by the θ angle, because the elevation could dramatically change during the displacements, but in many cases, the angle is simplified to zero; then the force of gravity (Eq. (5)) is not more taken into account to calculate the power consumption (Eq. (1)). In the next section, the importance of this contribution would be emphasized.
By integrating the power consumption (Eq. (1)), the required energy could be determined; during the cycle, it is possible to determine the maximum, the average, and the required power at each point.
After obtaining energy requirements, another parameter could be also determined, for example, the net energy required on the system during the cycle, the energy required per kilometer traveled, the energy recovered by regenerative brakes during the cycle, and the number of batteries.
A second iteration has to be done considering the new weight of the vehicle with the inclusion of the number of batteries calculated in the initial step. From these results, the real autonomy expected for the driving cycle is determined.
The commercial electric cars have been designed to operate with NEDC driving cycles in the best geographical environmental conditions, ergo low traffic and without altitude changes (θ angle considered as zero) in order to facilitate comparisons between different vehicle companies.
The electric motors of vehicles have around 60 kW of power. The battery stack installed is about 24kWh. The automotive companies try to give to the consumer at least the same characteristic of performance of a combustion engine car, see Table 1.
Car model | Company | Capacity (kWh) | |
---|---|---|---|
1 | Leaf | Nissan | 24 |
2 | iMiEV | Mitsubishi | 16 |
3 | E6 | BYD | 75 |
4 | Tesla model S | Tesla | 85 |
5 | Chevrolet spark | GM | 21 |
6 | Fiat 500e | Chrysler | 24 |
7 | BMW i3 | BMW | 33 |
8 | Focus | Ford | 33 |
The marketing of electric cars uses the traditional features of conventional gasoline cars. These accessories allow maximum comfort, such as air conditioning and electric screens for handling; however, in an electric car, this represents an additional spending of energy and bigger weight of the vehicle, so a greater number of batteries are necessary, and the cost of the vehicle substantially augments.
From these facts, Are then all these features necessary? Maybe EVs must be designed differently.
Latin American cities have different conditions compared with cities of Europe or the USA, first of all, the traffic and road conditions, the elevation of a geographic location in cities, and finally the economic and social aspects of populations of these cities.
In Table 2, main Latin American metropolises are listed in function of their population; also the gross domestic product and the elevation in the cities are shown [17].
Latin American metropolitan areas.
As it can be seen in Table 2, last column, several Latin American metropolises have considerable elevation differences on the same city; in these cases, the vehicles are confronted to important slopes, where the
By the other side, as it can be seen in Table 2, the big populations in these metropolises are related with high traffic density, and then, the average speed is lower, and acceleration changes could be important.
Comparing data reported in Table 2 with European cities as reported in Table 3 [17], it can immediately be remarked that elevation differences in European cities are lower than Latin American ones, considering the population density is also lower in European cities. The energy requirements are then different for the same category and use of vehicles in European or Latin American countries.
European cities.
Besides these geographic and social differences shown in Tables 2 and 3, another notorious difference is the gross domestic product (GDP) between countries; in general, GDP are lower in Latin American cities.
Therefore, the electric vehicles in Latin American cities must have different characteristics making them more expensive. Then, the overcrowding of EVs in Latin American cities could be farther than European cities for economic factors. Since the EV market is smaller in Latin American cities, the infrastructure to have cleaner electric energy is also poorest.
Mexico City’s government through Science and Innovation Development Agency (SECITI) sponsored the development of an electric taxi for the city in order to reduce pollution emissions. The group developing the cab for Mexico City was formed by the companies Moldex, Giant Motors, and the Research Center in Mechatronic Automotive from Tecnológico de Monterrey.
In the frame of this project, during 2016 and 2017, several tests under real driving conditions were running to design a specific electric taxi for Mexico City, taking into account the traffic, slopes, and operation conditions for the main commercial and residential areas.
Table 4 shows some results from these tests. First, it is clear that at a determined cycle, when slopes are not considered, the energy consumption is underestimated of around 16%. Then, the range autonomy will be shorter than that expected if the geographic data is not considered.
Speed average (km/h) | Maximum speed (km/h) | Maximum acceleration (m/s2) | Energy consumption (kWh/km) | Difference between energy consumption (%) | |
---|---|---|---|---|---|
Real cycle (reference) | 30.57 | 95.10 | 2.2 | 186 | |
Real cycle without slopes | 30.57 | 95.10 | 2.2 | 156 | −16% |
Real driving conditions.
The energy consumption was also determined from the study of the same car in a chassis dynamometer under different driving cycles (see Table 5). The real driving cycle, shown in Table 4, has an energy consumption value similar to that obtained from WLTC 3 (Table 4), but concerning parameters as speed average and maximum speed, the values from real driving cycle are nearest to WLTC 2; then the traffic conditions (see Figure 3) shape the energy requirements in an important way. Limiting speed (60–100 km/h) allows energy savings of around 32%.
Speed average (km/h) | Maximum speed (km/h) | Maximum acceleration (m/s2) | Energy consumption (kWh/km) | Difference between energy consumption (%) | |
---|---|---|---|---|---|
WLTC 3 without slopes | 49.34 | 139.18 | 2.38 | 196 | |
NEDC (without slope) | 32 | 120 | 1.04 | 163 | −17% |
WLTC 2 without slope | 35.69 | 85.2 | 1 | 133 | −32% |
Driving conditions in a dynamometer.
WLTC 3 was analyzed at maximum speed superior than 100km/h, and WLTC 2 at maximum speeds comprised between 60 and 100km/h.
Concerning the energy consumption obtained from NEDC, and reported in Table 5, a difference of 17% with WLTC 3 results was founded, indicating that energy consumption is underestimated with NEDC cycles.From Tables 4 and 5, it could be perceived that the quantity of batteries, calculated from NEDC driving cycle, is underestimated, 16% caused by slopes and 17% caused by speed and acceleration.
In general, several commercial vehicles have been modeled using the NEDC driving cycle. Its performance would be far away from the published standards of operation.
In Mexico, the main electric energy comes from thermal methods, then electric vehicles do not considerably reduce the pollutions caused by fossil fuels, and the emissions are only moved from cars to electric production centers.
From the analyses of electric taxi project in Mexico City, photovoltaic solar panel with battery support was not an economically feasible solution because the initial investment is very expensive; from additional calculations it was found that more than 7 years of daily use of vehicle is necessary to equal the electric cost offered by the electric company. The hypothetical electric energy requirements of an important number of EVs (1000 or more) cannot be assured with the actual capacity in Mexico City.
Finally, when the price of electric vehicle for personal use is compared versus engine vehicles, they are three times more expensive.
If the automotive companies kept betting on personal EVs for Latin American cities, then those cars would require more energy (between 16 and 33%) than European ones, and therefore they would be more expensive or with a reduced autonomy. In order to adapt the EVs, they would have to be designed to be smaller and lighter.
Therefore, EVs for Latin American metropolis have to follow a different strategy from Europe looking in order to really obtain an environmental purpose.
The following declarations are proposed as recommendations to be implemented in Latin American cities:
First, EV manufactures have to determine the real cycle for these cities in order to keep the ethical use of the vehicle.
Second, the EVs for personal utilization are not suitable at that moment due to the current state of technology. The utility cars and the public transport could represent a better solution to the EV market, and the green effects could be more important.
Third, more efforts in research to enhance energy systems shall be done with an ecological approach instead of the economic motivation.
The author thanks Moldex, Giant Motors companies, and Research Center in Automotive Mechatronics team for support to realize this study. The tests were also addressed by the sponsorship of SENER and CONACYT funds.
Moldex is a Mexican company manufacturing electric vehicles for utility purposes (http://moldexmexico.com/en/vehiculos-electricos/).
Giant Motors is a Mexican company assembly light trucks (http://www.fawtrucks.mx/).
CIMA is a research center in automotive mechatronics from Tecnológico de Monterrey located in Toluca that focus in forming engineers and applied research for design, manufacturing, and energy issues for automotive industries (http://cima.tol.itesm.mx/).
Many engineering design problems require simultaneous optimization of multiple, often competing, objectives. Unlike in single-objective optimization, a multi-objective problem with competing objectives has no single solution. An optimum solution with respect to only one objective may not be acceptable when measured with respect to the other objectives. Multi-objective problems have a number of solutions called the Pareto-optimal set, named after Vilfred Pareto [1], that represent the range of best possible compromises amongst the objectives. Traditional gradient-based optimization algorithms are capable of addressing the multi-objective problems by converting the problem into a single-objective formulation. On the other hand, evolutionary algorithms (EAs)1 are well suited for the multi-objective problems as they can evolve to a set of designs that represent the Pareto frontier in a single run of the algorithm [2, 3]. As a result, EAs often find application to address multi-objective problems. Despite the popularity of these algorithms to solve a wide range of problems, they, like all non-gradient meta-heuristic searches, have issues with computational cost and rate of convergence to the Pareto frontier. After some number of generations, the candidate solutions may begin to exhibit little or no improvement. Modified versions of these algorithms exist which improve the convergence rate [4]. However, hybridizing EAs with an efficient gradient-based algorithm may significantly improve the convergence rate and has demonstrated the ability to solve multi-objective problems more efficiently than the EA alone [3]. Hybridization of an EA with a gradient-based local search algorithm has started to gain popularity owing to its promising capabilities to address the demerits of many optimization algorithms when used independently.
The genetic algorithm (GA) [5] is a class of EA and is a well-known population-based global search algorithm. Apart from its ability to explore the design space, GA is also capable of handling both discrete and continuous type design variables. This makes the GA an ideal choice to address problems that combine both discrete and continuous variables. However, the GA, like other EAs, does not provide any proof of convergence, and the GA cannot directly enforce constraints. Commonly, constraint handling for a GA search uses a penalty approach such that the fitness function reflects the objective function value and accounts for violated constraints. This generally requires the use of penalty multipliers to adjust the “strength” with which the penalty impacts the fitness function and selecting suitable penalty multipliers is often difficult. Further, for multi-objective problems, the different scaling or magnitude of the objectives can complicate selecting appropriate penalty multipliers.
On the other hand, Sequential Quadratic Programming (SQP) [6], is a well-known gradient-based search algorithm that directly handles constraints and provides proof of convergence to local optima using Karush-Kuhn-Tucker (KKT) optimality criteria [7]. Because SQP uses gradient information, it is a computationally efficient search algorithm. However, SQP cannot handle discrete design variables or discontinuous functions and has difficulty with multi-modal functions. Therefore, both of these (GA and SQP) well-known optimization algorithms have their own pros and cons that limit their individual applicability to fully address constrained multi-objective problems that combine both continuous and discrete type design variables. Combining the GA with SQP creates a hybrid approach that improves the overall optimization process for constrained mixed-discrete nonlinear programming problems (MDNLP).
The chapter presents a combination of the two-branch tournament GA for multi-objective problems with an SQP-based local search implementation of the goal attainment problem formulation allowing an improved information sharing between the two algorithms. To the best of the authors’ knowledge, there exists no work that emphasizes the process of hybridization combining an N-branch tournament selection GA with the goal attainment formulation as the local search in a compatible manner and then demonstrates application of the approach to solve a hard-to-solve constrained multi-objective, mixed-discrete nonlinear optimization problem. Later in the chapter, the hybrid approach is applied to solve a three-bar truss problem, a ten-bar truss problem, and a greener aircraft design optimization problem – all representatives of constrained multi-objective, mixed-discrete nonlinear programming problem. The truss problems have basis in test problems for structural optimization, and the motivation to select a greener aircraft design optimization problem arises from the increased concern about the environmental impact of the growing air transportation system.
The ability of the EAs to evolve to a Pareto-frontier as the generation progresses makes them an ideal choice for several multi-objective optimization problems. Vector Evaluated GA (VEGA), proposed by Schaffer [8] back in 1985, is one of the earlier versions of multi-objective GA. Several multi-objective EAs are developed since then including Multi-Objective Genetic Algorithm (MOGA) [9], Strength Pareto Evolutionary Algorithm [10], Non-dominated Sorted Genetic Algorithm (NSGA) [11] to mention a few popular ones.
Coello [2, 12] has conducted comprehensive literature surveys of various evolutionary multi-objective techniques. Konak et al. [13] compared various multi-objective optimization algorithms and provides a set of guidelines to follow while developing a multi-objective algorithm. Their effort primarily lies in guiding researchers with very little background in MOGA and making them familiar with the ideas and approaches of multi-objective optimization.
One such multi-objective algorithm named Non-dominated Sorting Genetic Algorithm (NSGA), developed by Srinivas and Dev [11] – arguably one of the most widely used multi-objective EAs – uses the concept of non-dominated sets originally proposed by Goldberg in his book on Genetic Algorithm and Machine Learning [14]. The NSGA approach maintains sets of non-dominated individuals, with the first set of individuals not dominated by any other individuals in the population. The second set finds the new set of non-dominated individuals after excluding the individuals from the first set. This step continues until all the individuals in the population are categorized inside the non-dominated sets.
A majority of these multi-objective algorithms, in some form, require an assignment of a scalar measure of a fitness value to the individuals in the population. As an example, MOGA [9] and NSGA [11] assign a fitness value based on a ranking scheme depending on the individual’s levels of domination. The two-branch tournament selection genetic algorithm presented by Crossley et al. [15] uses a tournament selection scheme that chooses parents considering both the objectives directly in the fitness functions. The individuals are evaluated based on their fitness across both the objectives. The overall process remains the same as that of a traditional GA. However, the only difference appears in the tournament selection operator. During the tournament selection step, the algorithm selects 50% of the parents based on the fitness value associated with the first objective, that is, the individuals are evaluated solely with respect to the first objective without consideration of the other objective. These selected parents are by nature strong in objective 1, or
The hybrid approach, presented in this chapter, uses this two-branch tournament selection GA as the global search optimizer and combines with a gradient-based approach to refine the search using a novel information sharing concept in the process of hybridization. The unique tournament selection strategy of the two-branch tournament GA allows to understand the underlying trait of the parents, i.e., if they are
Another challenge with multi-objective EAs is their ability to enforce constraints. Unlike gradient-based methods, which use constraint gradient information to guide the search in the feasible direction, no such constraint gradient is available for EAs. There have been several efforts to handle the constraints for EAs; however, not all of these methods strictly or directly enforce the problem constraints. The penalty function approach is arguably the most widely known of the various approaches to handle constraints in EAs. Assuming a minimization problem, this approach adds a penalty to the objective function when constraints are violated [14].
Another simple approach includes ignoring any infeasible design solution; because this does not differentiate between constraints that are close to the constraint boundaries and those that are far apart, this constraint handling method is inefficient.
Binh and Korn [16] suggested a method to assign fitness to individuals based on combining both the objective function vector as well as the degree to which the individual violates the constraint. Infeasible individuals are categorized into different classes based on how close or how far they are to the constraints boundaries.
Fonseca and Fleming [17] proposed a priority-based constraint handling strategy where search is first driven for feasibility followed by optimality by assigning high priority to constraints and low priority to objective functions. Although there are various techniques to “handle” constraints in EAs, “enforcing” them in a robust way is still an open issue. This is another motivation to pursue the hybrid approach that leverages the efficacy of gradient-based search to enforce the problem constraints.
Further, these population-based searches have issues with computational cost and rate of convergence to the Pareto frontier. After some number of generations, the candidate solutions may begin to exhibit little or no improvement. Modified versions of the algorithms work to improve the convergence rate [4, 18]; however, hybridizing EAs or GAs with an efficient gradient-based algorithm can significantly improve the convergence rate, thereby reducing the computational cost. Hybridization of an EA or GA with a gradient-based local search algorithm is not new. There are numerous references demonstrating how hybridization may improve the quality of the search for both single objective and multi-objective problem formulations; these include, but are not limited to, those appearing in [3, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. The local search can be considered as the local learning that takes place in an individual throughout its lifespan. Some of the approaches apply the local search to the final non-dominated set, while some techniques apply local search to all or many individuals of the population as the generation progresses.
The effort here extends the previous effort by Lehner and Crossley [27] to include a multi-objective formulation and combine the advantage of the hybrid approach with an novel information sharing technique between the global and the local search. The two-branch tournament selection GA algorithm globally explores the design space handling both discrete and continuous type variables, while the gradient-based approach sees only the continuous variables in a goal attainment formulation and seeks to efficiently refine the population based on the information passed on by the top-level GA while enforcing all the problem constraints.
The hybrid approach presented in this chapter combines the two-branch tournament GA (see Figure 1) for the global search [15] and the goal attainment SQP algorithm provided in the function
Original two branch tournament selection GA for two objective problems. Adapted from reference [
The top level of the problem, which the GA sees as its optimization problem, is a bound constrained (i.e., only side constraints on the continuous design variables) multi-objective minimization problem that uses the two-branch tournament selection technique with some modification to include the local search. This level includes both discrete and continuous design variables of the original problem. The continuous variables in this level,
In the original two-branch tournament selection GA, the tournament step selects 50% of the parents based on the fitness value associated with the first objective. These parents are by nature strong in objective 1 or
Modified two branch tournament selection GA and SQP interaction.
With a given goal
Figure 3 demonstrates the parent selection process of the new two-branch tournament selection GA and the goal assignment technique with a simple example. The approach starts with a population size of 8
Selective parent mixing strategy.
The lower-level problem presented to the SQP algorithm refines the population of the GA by searching the continuous variable space and helps the hybrid algorithm converge to the Pareto frontier at a faster rate. The
This goal attainment formulation seeks to attain values for the objectives close to a set of predefined goal values,
The
Goal assignment technique.
To assign the goal point values, the hybrid approach first identifies the local ideal point in each generation. This ideal point is the combination of the lowest
To assign a goal point to an individual, the approach defines a vector that originates from an individual and ends to where the vector intersects with either of the dotted lines. The point of intersection becomes the goal point for that individual. Children of parents from sub-pool 1, the
Referring back to Figure 3, parents 1 and 4 from sub-pool 1 create children
Although the approach seems robust in enforcing constraints via goal attainment formulation, there may be instances when no feasible solution exists to the goal attainment formulation for a given set of discrete variables. In such cases, the local search will not be able to return a feasible solution and the fitness function receives a severe penalty in the GA-level in an effort to discard such discrete design choices from the population. This severe penalty has some resemblance to the approach of ignoring infeasible designs that was criticised above; however, because the situation where no locally-feasible design exists results from a specific combination of discrete variables, there is no analog to having a “nearly feasible” design with a slightly violated constraint. Severely penalizing such infeasible designs for certain combinations of discrete variable choices, in this context, is appropriate.
To demonstrate the efficacy of the hybrid approach in solving constrained multi-objective MDNLP problems, we solve three different engineering test problems with varying difficulties - a three-bar truss, a ten-bar truss, and greener aircraft design problem.
For the three-bar truss problem (see Figure 5), the problem formulation includes the objectives of minimizing the weight of the truss and minimizing the deflection of the free node. The deflection of a node is calculated as the resultant of the deflections in both the x and y directions. The problem consists of six design variables, of which three are continuous and three are discrete. The continuous variables describe the cross-sectional area of the three bars while the discrete variables describe the material selection properties of these bars. The details of the continuous design variables and their design bounds appear in Table 1. For this problem, four discrete material selection choices are available for each element and include aluminum, titanium, steel, and nickel options. The yield stress for every bar acts as a constraint for the problem (total three constraints), not allowing the stress in the bar to go beyond that upper limit. References [35, 36] provide more details about the three-bar truss problem. For the hybrid approach, the GA population is limited to 8 individuals while setting the upper limit for the number of generations to 50. The probability of crossover is set to 0.5 and the mutation rate is fixed at 0.005. The continuous and discrete variables uses 8 and 2 bits respectively in the Gray-coded binary scheme.
Three bar truss problem.
Design variables | Lower bound | Upper bound |
---|---|---|
Cross-sectional area of bar 1 [cm2] | 0 | 5 |
Cross-sectional area of bar 2 [cm2] | 0 | 5 |
Cross-sectional area of bar 3 [cm2] | 0 | 5 |
Continuous variables for three-bar truss problem.
The resulting Pareto frontier for the three-bar truss problem appears in Figure 6(a). The plot shows the Pareto frontier has a good spread, leading to a total of 248 non-dominated points as solutions to the optimization problem. The visible trend in the non-dominated design set indicates that as the weight of the three bar truss system increases, they are accompanied by similar increases in the cross-sectional area of the bars with the material selection choice gradually shifting to steel for all the three bars. Aluminum or nickel never appeared as the material selection choice in the first two bars. The designs visible in the top left corner of the Pareto front in Figure 6(a) correspond to high displacement and low weight designs. The separated cluster of points (six designs) visible at the bottom right corner of the Pareto frontier corresponds to low displacement and high weight designs, with the maximum weight design having a material combination of all steel bars.
Pareto front for the three-bar truss problem and its comparison with the other approaches. (a) Pareto front for the three-bar truss problem using the hybrid approach. (b) Comparison of Pareto frontier obtained using the hybrid approach, a weighted sum approach and the original two-branch tournament GA approach.
For the three-bar truss problem, only 64 possible combinations of discrete design variables exist. Hence, it is possible to perform a complete enumeration of the discrete design space and get a sense of the shape of the true Pareto front and help assess the performance of the hybrid approach. This led the authors to compare the hybrid approach (and the original two-branch tournament selection GA2) with a gradient-based weighted sum approach for this three-bar truss problem. The weighted sum approach converts the multi-objective problem formulation into a single objective problem by assigning weights to both the objectives and solves the single objective problem with the gradient-based approach using MATLAB’s
First, the objectives are normalized using the utopia point. Next, objective 1 is assigned a weight
Figure 7 compares how the Pareto frontier evolved with generations using the original two-branch tournament GA and the proposed hybrid approach. As expected, without the local search feature, the original two-branch tournament selection GA shows distinct improvement in both the quality and the spread of the Pareto front as the generation progresses. That is, the black diamonds (non-dominated set after second generation) are replaced with better non-dominated designs as the generation progresses. However, in the hybrid case, we start to see the shape of the final Pareto front immediately after the second generation. As the generation progresses further, more points get added to the list of non-dominated designs. This is due to the multi-start approach where the top-level GA populates various possible combinations of the discrete material selection choices and the local gradient-based search then improves these designs by varying the continuous design variables. The hybrid approach is able to rapidly get to the final Pareto front at the expense of increased number of function evaluations needed by the gradient-based local search.
Evolution of the non-dominated sets as the generation progresses. (a) Original two-branch tournament selection GA. (b) Proposed hybrid approach.
Next, the hybrid approach solves a more difficult and challenging version of the three-bar truss problem – a ten-bar truss. Similar to the three-bar truss problem, the ten-bar truss has the competing objectives that include minimizing the weight of the ten-bar truss system and minimizing the resultant displacement of any of the free nodes. The displacement is taken as the absolute of the maximum calculated displacement among all the bar elements. This problem consists of twenty design variables – ten continuous type and ten discrete type. The continuous variables describe the cross-sectional diameters of the ten bars, ranging from 0.1 cm2 to 40 cm2, while the discrete variables specify the material selection properties of these bars. Like the three-bar problem, the four discrete material choices available for each bar include aluminum, titanium, steel, and nickel. However, this problem has over one million possible combinations of the discrete choices (
Figure 8(a) compares the Pareto front obtained using the hybrid approach after 20 GA generations with the Pareto frontier obtained using the two-branch tournament selection GA after 100 generations. The figure shows both the approaches performed well for this problem with the two-branch tournament selection GA resulting a better spread in the low weight/high displacement region of the objective space, whereas the hybrid GA has a better spread in the low displacement/high weight region. Figure 8(b) shows how the non-dominated set evolved as the generation progresses using the hybrid approach. We see a similar trend as that of the three-bar truss problem. That is, there is not much significant change in the final shape of the Pareto front other than the increase in the number of non-dominated designs as the generation progresses. However, this time there is slight improvement in the quality of the Pareto front (the red non-dominated set obtained after generation 20 is slightly better than the blue or the black non-dominated designs obtained at generation 5 and 2 respectively).
Ten bar truss problem results. (a) Comparison between the original two-branch tournament GA (after 100 GA generation) and the hybrid approach (after 20 GA generations) for the 10 bar truss problem. (b) Evolution of non-dominated set as the generation progresses for the 10 bar truss problem using the hybrid approach.
For the three-bar example, a majority of the improvements across the objective space are due to the gradient-based local search’s ability to obtain designs with better cross-sectional area. With only 64 possible material selection combinations, there are not many discrete material selection options to explore. On the other hand, for the ten-bar truss problem, a vast majority of the improvement is due to the ability of the GA to find a better material selection combination rather than fine-tuning the cross sectional variables. It is not possible to seek further improvement in the Pareto front just by varying the continuous variables, so the local search saturates as appear in the case of black (diamonds) and blue (squares) non-dominated designs. After few more GA iterations, the algorithm is able to find better combinations of material selection that lead to further improvement in the Pareto front (red dots).
The third application problem solved using the hybrid approach is the greener aircraft design problem. Here, a “greener” aircraft design problem provides an example to demonstrate the efficacy of the hybrid algorithm and its ability to solve such MDNLP problems. The intent is to find aircraft designs that represent the best possible trade-offs among performance, economics, and environmental metrics which essentially makes this a multi-objective problem. Further, with the inclusion of discrete technologies, the problem becomes MDNLP in nature.
The aircraft design optimization problem employs the NASA sizing code FLOPS [37] to evaluate discrete design configurations and perform the sizing and performance calculations of the candidate aircraft designs. The sizing code accepts both continuous and discrete design variables as input and returns the aircraft gross weight along with environmental metrics (fuel weight, which corresponds to CO2 emissions, and NO
The problem includes ten continuous variables that define the wing and the engine parameters of the aircraft. The details of these continuous design variables and their design bounds appear in Table 2.
Design variables | Lower bound | Upper bound |
---|---|---|
Aspect Ratio | 8 | 12 |
Taper Ratio | 0.3 | 0.5 |
Thickness to Chord Ratio | 0.09 | 0.17 |
Wing Area [ft2] | 1,000 | 1,500 |
Wing Sweep at 25 percent [deg] | 0 | 40 |
Thrust per engine [lbs] | 20,000 | 30,000 |
By-Pass Ratio | 5 | 10 |
Turbine Inlet Temperature [R] | 3010 | 3510 |
Overall Pressure Ratio | 35 | 55 |
Fan Pressure Ratio | 1.6 | 1.7 |
Continuous variables for aircraft design problem.
This aircraft design optimization study models three types of discrete technologies. Table 3 lists the set of discrete technologies considered in this study. To model composite material selection choice on various aircraft components, the approach here uses a binary variable for each of the aircraft components that includes wing, fuselage, tail, and nacelle. A value of one represents composites being present while a value of zero represents no composite materials in that structure. The second discrete variable includes the eight possible combinations of the location and the number of engines. Lastly, eight combinations of laminar flow technologies are included for this problem, depending on whether it is natural laminar flow (NLF) or hybrid laminar flow control (HLFC) technology and the number of components on which it is applied (as listed in Table 3). References [38, 39, 40] describe the various discrete technologies used in this study in further detail.
Laminar Flow Technologies | Engine Position | Composite Material Choices | |||
---|---|---|---|---|---|
Wing | Fuselage | Nacelle | Tail | ||
NLF-Wing | 2 wing | Yes | Yes | Yes | Yes |
HLFC-Wing | 2 fuselage | No | No | No | No |
HLFC-Wing + Nacelle | 2 wing +1 fuselage | ||||
HLFC-Wing + Tail | 3 fuselage | ||||
HLFC-Wing + Tail + Nacelle | 4 wing | ||||
NLF-Wing + HLFC-Tail | 2 wing +2 fuselage | ||||
NLF-Wing + HLFC-Nacelle | 1 fuselage | ||||
NLF-Wing + HLFC-Tail + HLFC-Nacelle | 4 wing +1 fuselage |
Discrete technologies for aircraft design problem.
The problem also has four constraints that appear in Table 4. The constraints ensure that the design solution meets the desired field length criteria, has sufficient ground clearance, and sets a maximum limit on the amount of allowable fuel carrying space in the fuselage.
Take-off field length [ft] | |
Landing field length [ft] | |
Landing gear length [in] | |
Fuselage fuel capacity [lbs] |
Problem constraints.
The aircraft design optimization problem considers two different pairs of competing objectives. The first pair involves simultaneous minimization of the aircraft fuel weight (index of CO2 emissions) and the total operating cost of the aircraft, and the second pair involves minimizing the NO
Figure 9 shows the set of 24 non-dominated designs for the competing objective pair – aircraft fuel weight and total operating cost. The aircraft fuel weight, analogous to fuel burn, is directly related to the amount of CO2 produced during the trip. The Pareto frontier consists of designs employing combinations of composite structures, eight different engine position(s), and eight different laminar flow technologies, modeled as a part of the greener technology approaches described in the previous section.
The non-dominated set for objective pair – aircraft fuel weight (index for CO2 production) and the total operating cost.
The design point ND1 (for Non-Dominated design number 1) in Figure 9 corresponds to highest total operating cost (also lowest fuel weight) and makes use of NLF technology on the wing and HLFC technology on the nacelles and tail, along with two wing-mounted engines. This design also features composite wings, fuselage, and nacelles. The use of composite structures leads to a decrease in the fuel consumption (due to the reduction in aircraft empty weight) at the expense of increased total operating cost (due to increase in the manufacturing and maintenance costs associated with composite materials). The design with the lowest total operating cost (ND24) makes use of NLF technology on the wing and HLFC technology on the nacelles and tail, along with two wing-mounted engines as well. But, this design has no composite components and, hence, has the lowest total operating cost according to the models used in this study.
All the non-dominated designs employ NLF technology on the wings and HLFC technology on both the nacelle and tail, along with two wing-mounted engine configuration. The laminar flow technologies tend to reduce the skin friction drag of the aircraft, making the design more aerodynamically efficient, and reducing its fuel consumption for a given mission range. All the non-dominated designs employ these technologies in various forms (NLF or HLFC) to reduce fuel burn, depicting the importance of employing these technologies in near future “greener” aircraft design.
An interesting region in the Pareto frontier from an airline’s standpoint would be near the points ND1 and ND3 (or ND2), where a substantial decrease in total operating cost is possible for a marginal increase in the aircraft fuel weight (index of CO2 production per trip). Considering non-dominated designs ND1 and ND3, a nearly 1% reduction in total operating cost is possible to achieve for only a 0.6% increase in the total fuel weight needed for the mission, as one move from ND1 to ND3. Similar trends for the objective pair in consideration are also observed for designs ND9, ND10, and ND11.
The Pareto front corresponding to the NO
The non-dominated set for objective pair – NOX emissions versus total operating cost.
The design with minimum NO
An interesting region from the airline’s point of view is the near the points ND2, ND3, ND4 and ND5, where a nearly vertical portion is visible in the top left portion of the Pareto frontier (refer to Figure 10). Moving from left to right in this region, a substantial decrease in total operating cost is possible for a marginal increase in the NO
Given there is some degree of randomness associated with the genetic operations in the GA, subsequent runs of the hybrid GA for the two objective pairs find a slightly different number of non-dominated designs points. However, the basic trait of the Pareto frontier, in terms of the discrete choices, did not alter; only the density of points in the Pareto frontier varied with different runs.
This chapter describes a hybrid multi-objective algorithm that makes use of an efficient gradient-based SQP algorithm for fitness evaluation inside a GA in a learning approach. The combination allows the GA to evolve a population of designs in the direction of the Pareto frontier while the SQP algorithm enforces constraints, eliminating the need for penalty multipliers or other special constraint handling methods and refines the values of the continuous design variables. The selective parent mixing and unique sets of goal point assignment to the individual lead to a distinct improvement in convergence and the quality of the Pareto frontier from a previous variation of this approach. When applied to various constrained MDNLP engineering design problems, the hybrid algorithm shows the ability to identify promising designs.
Although the ability of the hybrid approach to solve difficult constrained MDNLP problems is demonstrated in this chapter, the methodology relies heavily on the constraint enforcing ability and efficient searching of the continuous design space via the local gradient-based SQP algorithm that requires some estimates (either numerically or analytically) of the gradients of the objectives and the constraints with respect to the continuous design variables. A major advantage of a gradient-based approach besides being able to enforce the problem constraints (hence, the motivation to hybridize) is that the computational cost needed to compute the gradients is nearly independent of the number of design variables [41] when using adjoint-based methods to estimate the derivatives. This allows the gradient-based approach to efficiently solve problems with a very large number of design variables. However, if the objectives are encapsulated in a black-box function and are computationally very expensive to evaluate, then it may not be possible to directly implement a gradient-based search and may require a surrogate-based design optimization approach [40, 42, 43].
αi | Weight vector for the relative under/over-attainment of objective |
fi(x) | Value of the objective |
fiG | Goal value for objective |
gi(x) | Nonlinear inequality constraints |
γ | Attainment factor |
hi(x) | Nonlinear equality constraints |
n | Population size |
xc | Continuous design variable |
xd | Discrete design variable |
xL | Design variable lower bound |
xU | Design variable upper bound |
EA | Evolutionary algorithm |
GA | Generic algorithm |
HLFC | Hybrid laminar flow control |
MDNLP | Mixed-discrete nonlinear programming |
ND | Non-dominated design |
NLF | Natural laminar flow |
NSGA | Non-dominated Sorted Genetic Algorithm |
SPEA | Strength Pareto Evolutionary Algorithm |
SQP | Sequential Quadratic Programming |
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Pal is Professor of Physics at Mahindra École\nCentrale Hyderabad India since July 1st 2014 after retirement\nas Professor of Physics from IIT Delhi; Ph.D.’1975 from IIT\nDelhi; Fellow of OSA and SPIE; Senior Member IEEE;\nHonorary Foreign Member Royal Norwegian Society for\nScience and Arts; Member OSA Board of Directors (2009-\n11); Distinguished Lecturer IEEE Photonics Society (2005-\n07).",institutionString:null,institution:{name:"Indian Institute of Technology Delhi",country:{name:"India"}}},{id:"69653",title:"Dr.",name:"Chusak",middleName:null,surname:"Limsakul",slug:"chusak-limsakul",fullName:"Chusak Limsakul",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"Prince of Songkla University",country:{name:"Thailand"}}},{id:"23804",title:"Dr.",name:"Hamzah",middleName:null,surname:"Arof",slug:"hamzah-arof",fullName:"Hamzah Arof",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/23804/images/5492_n.jpg",biography:"Hamzah Arof received his BSc from Michigan State University, and PhD from the University of Wales. 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However, in the last 3 years, two studies have deeply characterized food-grade TiO2 and converged to the fact that the size distribution of food-grade TiO2 spans over the nanoparticle range (<100 nm) and the surface is not pure TiO2 but covered by phosphate and eventually silicon species or aluminium species, which modify the surface chemistry of these particles. Until now, this material was considered as safe. However, the toxicological studies later to the last re-evaluation by the European Food Safety Agency reveal some concerns due to the ability of TiO2 particles to alter the intestinal barrier. 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The usual appearance of pesticides in surface water, waste water and groundwater has driven the search for proper methods to remove persistent pesticides. Although typical biological treatments of water offer some advantages such as low cost and operability, many investigations referring to the removal of pesticides have suggested that in many cases they have low effectiveness due to the limited biodegradability of many agrochemicals. In recent years, research for new techniques for water detoxification to avoid these disadvantages has led to processes that involve light, which are called advanced oxidation processes (AOPs). Among the different semiconductor (SC) materials tested as potential photocatalysts, titanium dioxide (TiO2) is the most popular because of its photochemical stability, commercial availability, non-toxic nature and low cost, high photoactivity, ease of preparation in the laboratory, possibility of doping with metals and non-metals and coating on solid support. 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