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Integrated Solutions and Distributed Models' Linkage Procedures for Food-Energy-Water-Environmental Nexus Security Modeling and Management

This chapter discusses a new modeling approach enabling the linkage of distributed individual food, energy, water optimization models under joint (e.g. water and land) resource constraints, uncertainty and asymmetric information (ASI). The approach is based on an iterative stochastic quasigradient (SQG) solution procedure of, in general, non-smooth, non-differentiable optimization. The SQG procedure organizes an iterative computerized negotiation between individual food-energy-water (FEW) systems (models) representing intelligent agents. The convergence of the procedure to the socially optimal solution is based on the results of non-differentiable optimization providing a new type of machine learning algorithm. The linkage problem can be viewed as a general endogenous reinforced learning problem. The models act as "agents" that communicate with a "central hub" (a regulator) and take decisions in order to maximize the "cumulative reward". In this way, they continue to be the same individual models and different modeling teams do not need to exchange information about their models - instead, they only need to harmonize the inputs and outputs that are part of the joint resource constraints. In this way, the agents link their models into an integrated model (IM) under ASI.

The convergence of the solution of the linked models to the solution of the hard-IM is discussed. Application of the approach is illustrated with a case study linking distributed agricultural, water and energy sector models. This approach can be effectively used for decentralized, deregulated planning of interdependent agricultural, energy and water systems.

1.1. Introduction

The increasing interdependencies among food-energy-water-environmental (FEWE) sectors require integrated coherent planning and coordinated policies for sustainable development and nexus security. The sectors become more interconnected because they utilize common, often rather limited, resources, both natural (e.g. land, water and air quality) and socioeconomic (e.g. investments and labor force). For example, land and water are needed not only for agricultural production but also for hydropower generation, coal mining and processing, power plants cooling, renewable energy and hydrogen production.

Detailed sectoral and regional models have traditionally been used to anticipate and plan desirable developments of respective sectors and regions. These models operate with a set of feasible decisions and aim to select a solution optimizing a sector- or region-specific objective function. When interdependencies between sectors and regions are increasing, such an independent analysis without accounting for interconnections can become misleading and even dangerous. Hence, it is necessary to link the sectoral (regional) models together to derive truly integrated solutions. Interdependent FEWE security goals contribute immensely to signifying the nexus security between sectors and regions (Zagorodny et al. 2020, 2024; Ermolieva et al. 2021a, 2021b).

In this chapter, we consider the problem of linking individual sectoral and/or regional linear programming (LP) models into a cross-sectoral IM in the presence of joint constraints, when "private" information about the models is not available or it cannot be shared by modeling teams (sectoral agencies), that is, under ASI. This approach provides a means of decentralized cross-sectoral coordination and enables us to investigate policies in interdependent systems in a "decentralized" fashion. This enables more stable and resilient systems' performance and resource allocation as compared to the independent policies designed by separate models without accounting for interdependencies.

Limited resources can be allocated between systems (sectors/regions) in many ways. For example, Böhringer and Rutherford (2009) consider integrating mathematical programming models of the energy system into a general equilibrium (GE) model of the overall economy. Unfortunately, the convergence of the iterative procedure integrating the models could not be proven. For resource and production allocation with the generalized Nash equilibrium (GNE) approach, the existence, uniqueness and stability of the GNE, and a realistic large-scale implementation of this concept cannot be guaranteed. Ermoliev and von Winterfeldt (2012) demonstrate the complexity of the game-theoretic approaches, for example, the Stackelberg leadership model, emerging due to quite unrealistic assumptions that each player (sector/region) possesses knowledge about other players.

Traditional integrated deterministic optimization modeling incorporates goals, constraints and data of all models into a single code (hard integration), which can be considered as a multi-criteria optimization problem (Ogryczak 2000). In the case of separate distributed models and ASI, the linkage requires (see sections 1.2.2-1.2.4) specific non-smooth optimization methods. Problems under ASI are addressed with the agency theory (Gaivoronski and Werner 2012), in particular, on how to motivate information exchange. In this chapter, we consider, in a sense, the opposite and minimize the necessity to exchange information.

Our approach for linking separate optimization models under ASI is based on the parallel solving of equivalent non-smooth optimization models by an iterative stochastic quasigradient (SQG) procedure (Ermoliev 1976, 2009), based on subgradients or generalized gradients (Ermoliev 1976; Rockafeller 1981; Ermoliev and Norkin 1997) converging to an optimal welfare maximizing linkage solution, that is, to the solution of a "hard-integrated" model (Ermoliev et al. 2022). This approach does not require us to share details about models' specification. We can assume that there is a network of distributed computers connecting individual computer models with the computer of a "social planner" (decision-makers or regulatory agencies), who attempts to achieve the best result for all sectors/regions (parties) involved. The linkage procedure can be interpreted as a kind of a "decentralized market system" (Ermoliev et al. 2015, and references therein). According to this procedure, the optimization of sectoral/regional goals under individual constraints is performed independently and in parallel, without considering joint constraints. Joint constraints can be imposed on total production, natural and financial resource use, emissions, pollution and joint FEWE security targets. The constraints can establish supply-demand relationships between the systems enabling us to estimate optimal production, resource use and emission quotas for each system. The balance between the total energy (including biofuels) production and demand defines energy security, agricultural production and consumption reflect food security, and total emissions and pollution constraints correspond to environmental security. The joint FEWE constraints satisfaction establishes the FEWE security nexus (Zagorodny and Ermoliev 2013; Ermolieva et al. 2016, 2021a; Zagorodny et al. 2020, 2024). After the independent optimization using initial approximations of various (e.g. production, resource use and emission) quotas, the sectors/regions provide a social planner with the information on their actual production, resource use and respective shadow prices. The planner checks if the joint constraints are fulfilled. If not, that is, there is "excess demand" or "excess supply" (i.e. total resource use, production and emissions by all systems are higher/lower than required), the planner revises the individual systems' quotas via shifting their current approximation in the direction defined by the corresponding dual variables. Thus, shadow prices signal systems to adjust their activities accordingly. Formally, the procedure is described in sections 1.2.3-1.2.5.

In this way, the linkage allows us to avoid "hard linking" of the models in a single code, which is not possible because the systems (agencies) may not want to share the information or because the individual models are too detailed and complex to be "hard-linked". The approach saves reprogramming efforts and enables parallel distributed (decentralized) computations of sectoral models instead of a large-scale integrated (centralized) model. This also preserves the original models in their initial state for other linkages. The use of detailed sectoral and regional models instead of their aggregated simplified versions enables us to also account for critically important local details. Similar computerized, decentralized "negotiation" processes between distributed models (agents) have been developed for the design of robust carbon trading markets (e.g. Ermoliev et al. 2015, and references therein). The linkage procedure can be considered as a new machine learning algorithm, namely, as a general endogenous reinforced learning problem of how software agents (models) take decisions in order to maximize the cumulative reward (total welfare) (Ermolieva et al. 2021b)

This chapter is organized as follows. Section 1.2 discusses the problem of models' linkage under joint constraints. Section...