When mathematics nurtures innovation in agronomy

According to the United Nations, 10% of the world's population lives in areas under high or critical water stress; yet irrigation is a crucial need. In a forward-planning exercise, France Stratégie has estimated that, according to the most pessimistic climate projection, water consumption for irrigation in France could double between 2020 and 2050.

Against this alarming backdrop, reusing wastewater in agriculture is seen as a promising way to optimise irrigation. “Experiments have been carried out in this field, but they remain empirical and ignore the reality of soils and plant nutrient requirements,” points out Mahugnon Gildas Dadjo, a PhD student and member of the Valse project team at the University of Lille Inria centre and the MISTEA Mixed Research Unit (UMR) at INRAE Montpellier. “And without treatment, this irrigation may have harmful effects on plants, soil and even the water table. However, the treatments required are often expensive!” 

Researchers from Inria and INRAE have therefore decided to tackle the problem together, using mathematical control methods, with the aim of supplying high-quality water that is adapted to agricultural requirements.

Integrating the presence of nitrogen in irrigation water

Their project originated at the LBE, INRAE's Environmental Biotechnology Laboratory in Narbonne. “For the last ten years or so, I've been leading INRAE's REUSE network on water reuse," explains Jérôme Harmand, Senior Researcher at the LBE. “Because all the water we consume is potentially reusable water.” His conclusion? From an agronomic standpoint, while models describing soil-plant interactions according to nutrient availability do exist, nothing has been proposed regarding the optimisation of wastewater quality according to the requirements of plants. 

“We adopted a systemic approach to this problem, with inputs (including water flow and initial nutrient levels) and outputs (such as soil condition and crop growth),” explains Alain Rapaport, Senior Researcher at the MISTEA (Mathematics, Computer Science and Statistics for the Environment) Mixed Research Unit at INRAE in Montpellier, who is also a member of the REUSE network. 

In particular, Valse “specialises in the analysis and control of non-linear systems, which reflect the full complexity of the living world and are therefore particularly suitable for agricultural issues,” says Denis Efimov, Senior Researcher at the University of Lille Inria centre and head of the Valse project team. 

Applying optimal control theory to wastewater irrigation

The partnership came to fruition in 2022 with Mahugnon Gildas Dadjo's thesis, proposed by the Valse project team and co-supervised by Alain Rapaport, Rosane Ushirobira, Denis Efimov and Jérôme Harmand. It aims to: develop mathematical control laws to reduce the volume of treated wastewater while optimising its reuse in agriculture, based on the specific water and nitrogen requirements of plants.

One of the major challenges is posed by the dynamic coupling between wastewater treatment models and crop growth models, to which little attention has been paid to date. Mahugnon Gildas Dadjo began by applying viability theory (which analyses the evolution of dynamic systems under state constraints). This is one way to demonstrate that the following statement is true: avoiding water stress and nitrogen stress (in the event of fluctuating nitrogen levels in the soil) is a prerequisite for obtaining maximum crop production. 

“My research was then based on optimal control theory, which sets out to determine controllers to minimise or maximise a given criterion,” explains the PhD student. “In this case, the aim was to minimise the amount of water used over the course of an agricultural season, while maximising the plant biomass harvested. In practical terms, the aim is to avoid both water stress and nitrogen stress in crops, but without adding too much water.” Ultimately, this mathematical optimisation method should lead to frugal irrigation management strategies that can be applied in the field.

Using software “observers” to overcome the lack of sensors in agriculture

However, Mahugnon Gildas Dadjo's thesis ran into a snag: the lack of reliable and affordable physical sensors in the agricultural world. Measuring nitrogen levels at the start of the season is essential to understanding the initial state of the soil, which is a key parameter for choosing the right irrigation strategy. 

“We therefore had to develop software sensors called ‘observers’”, explains the researcher. “These algorithms can be used to estimate unmeasured state variables – in this case, the quantity of nitrogen in the soil – from available measurements such as moisture content or the biomass produced.” The aim is for these estimates, provided by the “observers”, to converge over time towards the actual values. “My work consisted in making these algorithms robust to real-world uncertainties and extending them to realistic scenarios in which crops can withstand a certain level of water or nitrogen stress,” explains the PhD student.

Innovating by revealing optimal irrigation strategies 

At the end of this process, Mahugnon Gildas Dadjo's thesis led to a major innovation: determining the quantity and quality of water to be added to the crops, as well as the ideal time for irrigation. 

In fact, these strategies depend on the initial amount of nitrogen in the soil and the composition of the water supplied. Therefore, when the soil contains sufficient nitrogen and the moisture content is at its highest, irrigation is not necessary at the start of the season. It should be carried out when the system reaches the water stress limit, and then maintained at this limit until the end of the season. 

However, when the initial quantity of nitrogen is lower and the only source of nitrogen is that available in irrigation water, it is preferable to start watering at the beginning of the season to supply the crops with nitrogen, even if the soil is already wet. Finally, when the initial amount of nitrogen is very low or even non-existent, it is advisable to irrigate from the outset at the maximum rate until a certain point, and then stop the irrigation and maintain the system at the limit of water and nitrogen stress until the end of the season.

"This thesis has produced good theoretical results. We would now like to continue our research on this subject with a view to developing more sophisticated models for estimation and control, and to compare them with simulations and experiments, says Rosane Ushirobira, a researcher at the University of Lille Inria centre and a member of the Valse project team. Other sources of nitrogen should also be taken into account, which can be added on an ad hoc basis to supplement the nitrogen present in the water.” Indeed, Inria and INRAE have no intention of stopping there. They are now planning to launch new theses, but this time on a more experimental basis to enable the direct application of the mathematics to the realities of farming.