Resilience analysis of the nexus across water supply, power generation and environmental systems from a stochastic perspective

Highlights

• The trajectories of multi-dimensional WPE model were projected onto the coordinate axes.

• Resilience measures were quantified with the stationary probability distribution functions.

• Environmental system was most vulnerable because of its long convergence time after a disturbance.

Abstract

Effects of external disturbances such as the population change on dynamics of water supply, power generation and environmental (WPE) systems have seldom been investigated. Following the WPE nexus profiled in the study of Feng et al. (2016), this study incorporated stochasticity of population, water supply and power generation into the modeling of the dynamical system in the Hehuang region of China, and further quantified resilience measures to understand the system's ability to withstand stochastic disturbances. First, the stochastic differential equations were used to improve the simulation of stochasticity in the WPE nexus. Next, the transient probability distribution functions (pdfs) of system variables, obtained by Monte Carlo simulation, were used to describe the evolutionary process of the system. Finally, the stationary pdfs of variables which reflect stable states of the system were derived to calculate four resilience measures. It is shown that: (1) The system approached a stable state after Year 2400 by calculating the L2 norm of the difference between transient and stationary pdfs. (2) The environmental system was identified as the most vulnerable subsystem because of its long convergence time. (3) The water supply system did not change greatly and it would remain stable at its current low level, i.e., water consumption per capita would be less than 80m³. The method adopted in this study is conducive to avoiding risk and the results provide valuable insights for regional management of a WPE nexus.

Introduction

Understanding the resilience of an interconnected system is important to maintain the system within safe and normal states. Generally, an interconnected system is investigated within the framework of a nexus (Hoff, 2011; Liang et al., 2020; Naderi et al., 2021; Nie et al., 2019; Pahl-Wostl, 2019; Rasul and Sharma, 2016; van den Heuvel et al., 2020; Zhang et al., 2018), where the complex interdependencies and interactions between system components are induced by natural processes and socioeconomic connections (Cai et al., 2018). The resilience of an interconnected system represents the system's ability to withstand potential disturbances. In particular, resilience can indicate the maximum disturbance that can be withstood by a system, or the rate of system recovery from disturbed to safe states (Holling, 1996; Srinivasan and Kumar, 2015; Walker et al., 2004). Thus, quantifying the resilience of an interconnected system provides scientific support for system design and control.

The water supply, power generation and environmental (WPE) nexus (Feng et al., 2016) is developed to improve understanding of the evolutionary mechanisms in interconnected socio-hydrological systems (Kapsalis et al., 2019; Sivapalan et al., 2012, 2014; Tsiantikoudis et al., 2019; Zamparas et al., 2020). In studies of the WPE nexus, the system dynamics model has been developed to simulate various interconnected processes (Chen et al., 2016; Di Baldassarre et al., 2015; van Emmerik et al., 2014). It has become a common practice to describe system dynamics through coupled ordinary differential equations, which can be considered deterministic if they provide a single response for the system variables, model parameters and initial conditions (Bakhshianlamouki et al., 2020). However, in most cases, such research has focused on how a system might respond to change in the mean values of the environmental parameters and variables, while the effects of change in the variance have seldom been studied (Ridolfi et al., 2011). The stochasticity is caused by disturbance, for example, a sudden decrease in generating capacity or sudden increase of water supply (Barnett and Pierce, 2008; Verma et al., 2012). A deterministic model can reflect the interaction between variables and describe the reality of processes effectively, but it fails to recognize the stochasticity of model structure and parameters. Conversely, a stochastic dynamical model can provide statistics of the stochastic model outcome through simulation of the stochasticity in environmental dynamics (Stijnen et al., 2003).

Resilience reflects the capability of recovering from a disturbance or shifting to a new state (Carpenter et al., 2001), and it can help improve understanding of regime shift phenomena (Feng et al., 2019a), e.g. tidal morphological dynamics (Marani et al., 2010), lake eutrophication (C Lisa et al., 2002) and insect outbreaks (Strogatz, 2015). The system resilience can be quantified within the context of a stochastic dynamical model. For instance, Srinivasan and Kumar (2015) proposed a set of resilience measures under two types of disturbance (i.e., instantaneous shock and continuous stochasticity). Among them, common measures are “engineering resilience” and “ecological resilience”, which respectively emphasize the rate of recovery and the amount of resistance to disturbance of a system (Holling, 1996; Srinivasan and Kumar, 2015; Walker et al., 2004). Moreover, specific measures, such as “latitude”, “precariousness” and “resistance” have been developed to enhance understanding of resilience (Gunderson, 2000; Menck et al., 2013; Scheffer et al., 2001; Srinivasan and Kumar, 2015; Walker et al., 2004).

The dynamics of an n-dimensional stochastic system can be visualized as a trajectory in n-dimensional phase space. The trajectory that arises from any single sequence of random perturbations is called a realization. By averaging many realizations together, one can obtain a probability density function (pdf) for the system, which gives the relative probability that the system will occupy a particular region of phase space at a particular time.

One useful tool for studying resilience related to pdfs is the concept of a potential function (Nolting and Abbott, 2016). This is easily visualized in the two-dimensional case, where the state of the system is represented as a ball rolling on a surface specified by the potential function. In a deterministic system, the ball will roll to a local minimum (called a stable state) on the surface. In a stochastic system, the ball is subject to small random perturbations. Resilience measures the ability of the system to withstand such perturbations, so that the ball is not pushed too far from the stable state. Hence the shape of the surface specified by the potential function is key to determining the resilience of the stable state. The potential function is intimately connected to the probability density, with local minima of the potential function corresponding to local maxima in the probability density (Srinivasan and Kumar, 2015).

Various resilience measures have been developed in relation to potential functions (and more general quasi-potential functions) (Nolting and Abbott, 2016). Unfortunately, when the dimensionality of the system becomes large (>2), it is much more difficult to calculate and visualize these functions, so it is desirable to find alternative ways of measuring resilience. In this paper we offer a new approach, which consists of obtaining the marginal pdf (Xie et al., 2019) for each state variable from the multi-dimensional pdf of the system. Existing resilience measures are applied in a one-dimensional context to each of these marginal pdfs. Resilience of the system can then be studied by considering the resilience measures for each state variable. Our method thus represents a novel approach to tractably analyze the resilience of high-dimensional systems.

The stochastic dynamical model can illustrate how resilience measures change with the properties of a system and external stochasticity by analyzing the stationary pdfs of the state variables (Srinivasan and Kumar, 2015). This study aimed to construct a stochastic dynamical system to revisit the WPE nexus in the Hehuang region of China. The main objectives are: (1) to consider how to describe a stochastic WPE system; (2) to investigate how the system responds to external disturbance; (3) to determine what resilience the system exhibits to resist disturbance.