Introduction

As of 2015, all United Nations Member States agreed to approve the 2030 Agenda for Sustainable Development; this was a direct call to immediate action in order to promote social welfare while protecting the health of the planet. The core of this agenda was organized around 17 main objectives—the so-called Sustainable Development Goals (SDGs)—which together included 169 particular targets, all of them intended to be achieved before the year 2030. As a straightforward and expected consequence of this action, these goals have become a very hot topic in the scientific literature, with a very large number of publications addressing the topic through very varied approaches. So, as of November 15, 2023, the Scopus citation database had indexed 5832 articles with the term “Sustainable Development Goal” in the title. However, for unknown reasons, only 116 of these were in the subject area of mathematics (for instance, in the particular field of operations research (OR), an overview of works that focus on the SDG can be found in Smith 2019). This very small percentage clearly shows the widespread lack of understanding of the broad range of ways in which mathematics can aid the rational achievement of these sustainability objectives, beyond the obvious uses of statistical techniques.

Within the framework of mathematics, and specifically in the field of systems optimization, this work focuses on SDG 9, which deals with industries, innovation and infrastructure. To be specific, we focus our attention mainly on the first of its objectives: “Develop quality, reliable, sustainable and resilient infrastructure, including regional and cross-border infrastructure, to support economic development and human well-being, focusing on affordable infrastructure and equitable access for all.” So, in this paper, we will try to show how systems optimization can be a very useful tool to achieve this target. In order to do this, we will first explain what the systems optimization process consists of from an applied mathematical viewpoint, and then we will exemplify its application in three different problems related to SDG 9.1 on which the authors have worked intensively in recent years:

  • The optimal shape design of a vertical slot fishway for fish migration through obstacles in rivers

  • The optimal design of an irrigation channel to prevent malfunctions due to sedimentation

  • The optimal management of an urban road network that prioritizes the environmental perspective.

As is well known, society—especially its decision makers—needs to expand its environmental knowledge and apply it to achieve more sustainable and resilient development that is compatible with a reduction in environmental degradation. We feel confident that a greater awareness of the mathematical tools currently available is a good step in this direction. This article presents some realistic examples—fully developed by the authors—of how the innovative application of non-basic mathematical skills to the resolution of environmental problems can be a novel and effective option. We believe that our approach to these particular problems can be easily extrapolated to many others that cover a wide variety of SDGs related to health, water, energy, industry, climate, etc. In this sense, a greater relationship between mathematicians and other scientists would be desirable in order to exploit the broad possibilities that mathematical sciences can offer to other subjects. In our modest opinion, the inclusion of mathematical experts within the multidisciplinary teams that study these types of problems cannot bring anything but advantages.

Systems optimization

When we are forced to face any real-world problem, regardless of the area in which it arises, in the end it will always be necessary to predict (or simulate) what will happen when a final decision is made. As a well-known example, if an artificial barrier is going to be placed in the sea to contain hydrocarbon spills, the behavior of that barrier in different situations (in terms of waves, temperatures, winds, etc.) must be studied beforehand in order to determine whether it will (or will not) be able to perform the task for which it is designed (Castro et al. 2010).

As is well known, two different techniques can be employed to simulate a specific process, and these should complement each other. On the one hand, a reduced experimental model (for example, a scale model developed in the laboratory) can be used to provide experimental results for the behavior to be analyzed. On the other hand, we can utilize mathematical modeling; that is, we can formulate a system of equations (the so-called model) whose solution gives us the values of the variables to be studied. Solving this model numerically (which is usually known as numerical simulation) provides a set of results which must be compared to those obtained experimentally to analyze whether or not the model is useful for predicting the performance of the phenomenon under study. If the numerical results are not good, the model must be adjusted; it is solved successively until the obtained results agree with the experimental ones. When that happens, the model is said to be validated, and it can already be used to predict the performance in similar situations.

Once we reach this stage we can think about taking another step forward. This is where a new question arises: if we can predict what happens in very different situations, why not use this ability to design the “best” of them—that is, the optimal system? (Obviously, we must bear in mind that the meaning of “best” was previously concretized by the particular interests of the decision makers involved in the problem.) The control of this entire process from the point of view of optimizing one or more criteria is what we understand by systems optimization. In Fig. 1, interested readers can find a diagram of the successive stages performed along the whole process.

Fig. 1
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Detailed flowchart of the different stages of a systems optimization process applied to a real-world problem

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Application to the optimal design of sustainable infrastructures

As stated above, this section is devoted to illustrating how systems optimization can help in the design of sustainable infrastructures. For this, we show how three problems of great environmental interest can fit within this framework: the design of a river fishway for allowing fish to climb over a dam, the design of a sustainable irrigation channel to avoid undesirable sedimentation effects, and the management of an urban road network in order to minimize environmental pollution.

River fishways

A fishway (also known as a fish pass or a fish ladder) is a waterway to allow some fish species to pass by a dam or any other human-made obstacle in a river or stream (Prasetyorini et al. 2020), since stream obstructions block the movement of migrating fish. Usually, this type of structure consists of a sloping channel partitioned by weirs, baffles or vanes with openings for fish to swim through Katopodis (1992). There are many different kinds of fishways, and they can range from sophisticated (left image in Fig. 2) to much simpler (right image in Fig. 2), but the objective is always the same: to achieve a suitable water velocity in the channel that enables fish to overcome the dam and continue upstream towards spawning areas.

Fig. 2
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Two very different examples of fishways: in the Iregua River, Spain (left; credit: iAgua) and in the Ouro River, Spain (right)

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To assure good performance of the fishway, it is necessary to simulate the water velocity into the channel before building the structure. For each fixed channel geometry, this can be done using the well-known shallow water equations (Bermúdez et al. 1991), which provide the water height in the channel and its velocity field over time. This model can be solved by finite-difference, finite-volume or finite-element methods, and it can be validated by comparing the numerical results, for example, with those reported by Teijeiro-Rodríguez et al. (2006). For a fishway like the one studied there, Fig. 3 shows the velocity fields in the central pool obtained by the authors Alvarez-Vázquez et al. (2008) by solving the shallow water equations with a mixed implicit finite-element method for two different configurations of the fishway. In the velocity field on the left in Fig. 3, two large recirculation areas can be observed on both sides of the slot, which were analyzed in the experimental work of Teijeiro-Rodríguez et al. (2006).

Fig. 3
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Velocity field in the central pool of a standard vertical slot fishway: an initial random baffle configuration (left) and the optimal baffle design (right)

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As discussed in previous section, once this model has been validated, it can be used for predicting and designing the optimal fishway. The problem here consists of determining the location and size of the baffles which provide the optimal water velocity field: we seek to minimize the eddy flow (thus preventing the fish from becoming disoriented) and to obtain a velocity convenient for fish to leap and swim in the region near the slots, while the velocity should be as low as possible in the rest of the channel (facilitating the passage of the other fish). Taking into account the previous model (the shallow water equations), this optimal design problem can be formulated and solved in the framework of optimal control theory of partial differential equations (see Alvarez-Vázquez et al. 2013 and references therein). For the standard vertical slot fishway we are working with, the velocity field on the right in Fig. 3 shows the optimal configuration of the baffles in the central pool and the corresponding velocity field. As can be clearly seen, this configuration is much more appropriate for safe fish migration than the initial one shown in the velocity field on the left in Fig. 3.

Irrigation channels

A basic aspect of human subsistence is that crops need water to grow. In arid regions—which occupy a significant part of the earth’s surface—irrigation canals (Fig. 4) are the most common conveyance mechanism to provide the water necessary for the correct development of crops.

Fig. 4
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Irrigation channel in Egypt (credit: Prensa Latina)

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One of the main problems that this infrastructure suffers from is the sedimentation and erosion caused by the solid particles carried by the water, which are responsible for the instability of canals used for surface drainage, a loss of channel capacity, and, in general, the malfunctioning of irrigation systems. The issue here is to determine the geometry of the trapezoidal section of the channel that minimizes the effects of these sedimentation/erosion processes. Specifically, we focus our attention on determining the best angle of inclination of the side walls and the best width of the channel bed (Fig. 5).

Fig. 5
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Schema showing the the design variables for the cross section of an irrigation channel: \(\alpha\) (angle of inclination) and \(\omega\) (bottom width)

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In this case, to simulate the sedimentation dynamics in a channel, it is necessary to couple the shallow water equations (which provide, as we have said above, the height of water and its velocity field) with the equations for sediment transport (which provide the concentration of suspended particles in the water and the height of sediment on the bottom). This coupled system can be solved using very different numerical techniques, most of which have already been implemented in commercial software packages; see, for instance, MIKE21 (DHI 2001), which was chosen by us for our study. Using a numerical simulation tool like this, the optimal design of an irrigation canal can be formulated and solved within the framework of numerical optimization (for the full details, see Alvarez-Vázquez et al. 2018b for example).

A suitable channel design can help us to significantly reduce the sediment accumulation at the bottom of the canal, avoiding the harmful consequences derived from this undesired phenomenon (a decrease in the capability of the irrigation channel, reduced growth of vegetation, and so on). For example, Fig. 6 shows how the effect of erosion in a trapezoidal channel is much lower when the geometry of its cross section is optimized.

Fig. 6
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Erosion at the bottom of a trapezoidal channel with a random initial configuration (top) and the optimized geometry (bottom)

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Urban road networks

To finish the examples of possible applications of mathematical tools to solve real-world environmental problems, we will focus on the worrying problem of air pollution due to road traffic in large cities.

Traffic flow management in cities is a crucial problem for municipal governments, since almost all cities, both large and medium-sized, suffer from traffic congestion problems. Linked to this situation of traffic saturation is the problem of air pollution, which occurs mainly in anticyclonic situations (i.e., when there are weak winds and temperature inversions) and causes the so-called pollution beret (Fig. 7), one of the most serious environmental problems encountered today.

Fig. 7
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Example of a “pollution beret” above Madrid, Spain (credit: lahoradigital.com)

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Given this scenario, the main problem here is to simulate air pollution due to traffic flow in order to design an urban road network that helps minimize these unwanted episodes of air pollution. Recently, Alvarez-Vázquez et al. (2017) proposed a mathematical model to simulate air pollution that combines the classical 1D Lighthill–Whitham–Richards traffic model for road networks with a standard 2D advection–diffusion reaction pollution model for the atmosphere. Using this simulation model, different problems related to the mitigation of atmospheric pollution have been studied within the framework of systems optimization. For example, (1) the management of an urban road network has been approached from cooperative (Vázquez-Méndez et al. 2019) and hierarchical (García-Chan et al. 2022) perspectives, (2) the expansion of the road network has been considered from an environmental perspective (Alvarez-Vázquez et al. 2018a), and (3) the design of an ecological road corridor in which a new highway will be planned has been addressed (García-Chan et al. 2021). A numerical simulation of air pollution in the Guadalajara Metropolitan Zone in Mexico (corresponding to the first case above) can be seen in Fig. 8. Two optimized designs of an ecological road corridor in the same zone (corresponding to the third case above) can be found in Fig. 9.

Fig. 8
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Example of the numerical simulation of traffic-related air pollution on the main roads in the Guadalajara Metropolitan Zone (Mexico), taking into account traffic intensities and wind conditions

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Fig. 9
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Two examples of an optimized design of a road corridor for a new highway, showing pollution concentrations. Forbidden zones are in red, sensitive areas to protect are in gray, the road network is in black and the optimal road corridor is in magenta. A suboptimal design is shown on the left, and an optimal one is shown on the right

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These types of environmental problems are very complex, given that they involve a high number of variables and parameters that are not always completely determined, but systems optimization appears to be a very useful tool to help in the decision-making processes. However, there are still many other aspects that can influence the simulation step and the optimization one, so the results obtained so far should be approached with great caution since there is still a long way to go.

Conclusions

Systems optimization, which has been widely used for a long time to minimize economic costs in industrial processes, can also be very useful in the design of sustainable infrastructures, where it is also necessary to take ecological and environmental objectives into consideration. To apply these techniques to real-world problems, it is first necessary to have a reliable model that allows the phenomena to be simulated under very different scenarios. Depending on the phenomenon to be studied, this is a task that requires further research. Secondly, satisfactory management of optimization algorithms is also essential in order to achieve the optimal environmental situation. Fortunately, optimization has undergone extensive development in recent decades, mainly in aspects related to its application to realistic problems.

In this article, the authors, after a motivated and detailed presentation of systems optimization techniques, presented three different application cases (studied by them in recent years) that verified the advantages of using these methods to achieve sustainable and resilient infrastructures. The main objective here has been to show how mathematical tools—the great ones that were forgotten until now—can play a significant role in the appropriate evolution of the SDGs. After more than a decade of positive experiences in the optimal design of sustainable infrastructures, the authors are sure that the environmental integration of mathematical skills can contribute to solving a wide range of complex environmental problems by helping decision makers develop well-informed responses and strategies. The authors believe that the innovative use of sophisticated mathematical tools can be a good complement to the intuitive (and necessary) use of knowledge from previous experiences of stakeholders.

Our future interests include the application of our methodology to other SDGs. In some of them (Goal 6: clean water and sanitation, Goal 7: affordable and clean energy, Goal 11: sustainable cities and communities, Goal 13: climate action, etc.), systems optimization seems to be applicable in a straightforward way (for example, to the optimization of water quality, research into cleaner energy, improving human settlement safety, and minimizing the impact of climate change). But, in our opinion, systems optimization could also help with other goals in which mathematical tools do not seem to be so directly applicable (Goal 2: zero hunger, Goal 3: good health and well-being, Goal 14: life below water, Goal 15: life on earth, etc.), relating to topics like improving food security, reducing mortality, or optimizing the sustainable use of terrestrial and marine resources.