Energy-Water Nexus: An Optimal Control Model

Geraldo Andrade de Oliveira  and  Fernando Menezes Campello de Souza

One may say that economics is the study of the allocation of scarce resources between competing uses. Scarcity is a characteristic of exhaustible resources.

Most of the world's energy sources known today are finite i.e., exhaustible, or limited.

In the global energy matrix, oil comes first, followed by coal and natural gas. These sources together account for approximately 80% of the world's energy supply. Coal is the source most used to generate electricity. It accounts for generating 41% of the world's electricity supply. The United States and China are examples of countries that are highly dependent on this resource. In Brazil, water is the resource most used to generate electricity and is followed by biomass. These resources, although renewable, are limited.

What oil, coal, natural gas, water and biomass have in common is that they are conventional. Conventional resources, represent stored energy; are found in specific and unchangeable locations; and offer a limited supply. Their scarcity and high demand create a capitalizable commodity and an avid and impatient market, i.e., these sources are goods with high marketability in the international market and thus subject to price variations.

The issue is that these natural resources are allocated both for producing energy and for producing non-energy goods. Therefore, they are used as an input for production and as an input to produce a different kind of input, namely power.

Energy and water are at the heart of any country's economy and way of life. National defense, food production, human health, manufacturing, recreation, tourism, and the daily functioning of households all rely on a clean and affordable supply of one or both of them.

It is known that the production and consumption of energy and water are closely intertwined. They are diversified. Energy includes electric energy, and fuels like gasoline, diesel, nafta, kerosene, alcohol, fuel oil, uranium, and the like. Water includes drinkable water (potable), water for irrigation, water for cooling, water for industrial processes, and so on. The end users of both, energy and water, are many. There are also several producers of both.

Energy and water are, for their turn, intrinsically related to the production and consumption of food and transport.

Keeping electric power plants cool requires lots of water. Keeping water safe takes lots of energy. Apparently this potentially forces a choice between the two.

Water is needed to generate energy. Energy is needed to deliver water. Both resources are limiting the other --- and both may be running short. Is there a way out? What would be the rational to deal with this problem?

In some countries, the two greatest users of freshwater are agriculture and power plants. Thermal power plants --- those that consume coal, oil, natural gas or uranium --- generate more than 90 percent of U.S. electricity, and they are water hogs. The sheer amount required to cool the plants impacts the available supply to everyone else. In other countries, like Brazil, for example, hydroelectric energy plays a major role, and it has been found that water to be used directly as water, as in agriculture, industrial processes, and human consumption, for example, worths twice the economic value it has after it has been transformed into electricity in a hydroelectric power plant.

At the same time, one uses a lot of energy to move and treat water, sometimes across vast distances. Health standards typically get stricter with time, too, so the degree of energy that needs to be spent per gallon will only increase.

A mathematical model is proposed here formulated in terms of an optimal control problem representing an evolving economy; an optimal economic growth model. It is written as a maximization of an intertemporal social welfare function, subject to constraints defined by income and investment identities, production technologies, the reserves consumption dynamics, the labor force growth dynamics, the energy balance and the labor force balance. The policy instruments are the investments in each sector, the consumption rate for the energy resources, the water usage rate, and the labor force growth rate. The model is treated via the Pontryagin maximum principle. The results obtained from the model are useful in the understanding of the sector as a whole, and as a support in establishing integrated policies in the context of the energy-water nexus.