Towards Regulating Consumption in a Socio-hydrological Model for Groundwater Extraction

1st IFAC Workshop on Control Methods for Water Resource 1st IFAC 1st IFAC Workshop Workshop on on Control Control Methods Methods for for Water Water Resource Resource Systems 1st IFAC Workshop on Control Methods for Water Resource Systems Available online at www.sciencedirect.com Systems Delft, Netherlands, September 19-20, 2019 1st IFAC WorkshopSeptember on Control 19-20, Methods for Water Resource Systems Delft, Netherlands, 2019 Delft, Netherlands, September 19-20, 2019 Systems Delft, Netherlands, September 19-20, 2019 Delft, Netherlands, September 19-20, 2019

ScienceDirect

IFAC PapersOnLine 52-23 (2019) 94–100

Towards Regulating Consumption in a Towards Regulating Consumption in a Towards Regulating Consumption in a Socio-hydrological Model for Groundwater Towards Regulating Consumption in a Socio-hydrological Model for Groundwater Socio-hydrological Model for Groundwater Extraction Socio-hydrological Model for Groundwater Extraction Extraction Extraction Ansir Ilyas ∗∗∗ Wasim Hassan ∗∗∗ Talha Manzoor ∗∗ ∗∗ ∗∗

Ansir Ilyas ∗ Wasim Hassan ∗ Talha Ansir Wasim Talha Manzoor ∗∗ ∗ Manzoor Muhammad Ansir Ilyas Ilyas Abubakr Wasim Hassan Hassan Talha ∗ ∗ Manzoor ∗∗ Muhammad ∗Abubakr ∗ Abubakr Muhammad ∗ Manzoor Ansir Ilyas Abubakr Wasim Hassan Talha Muhammad ∗ Abubakr Muhammad ∗Center for Water of Electrical Engineering ∗ ∗ Department Department of of Electrical Electrical Engineering Engineering & & Center Center for Water Department & Water ∗ Informatics and Technology (WIT), Lahore University of Management Department of Electrical Engineering & Center for for Water Informatics and Technology (WIT), Lahore University of Management ∗ Informatics and Technology (WIT), Lahore University of Management Department of Electrical Engineering & Center for Water Sciences (LUMS), Lahore, Pakistan Informatics and Technology (WIT), Lahore University of Management Sciences (LUMS), Lahore, Pakistan Sciences (LUMS), Lahore, Pakistan ∗∗ Informatics and of Technology (WIT), LahoreNamal University of Management Electrical Engineering, Institute, Mianwali, Sciences (LUMS), Lahore, Pakistan ∗∗ ∗∗ Department of Electrical Engineering, Namal Institute, Mianwali, Department Electrical Engineering, Institute, Mianwali, ∗∗ Department Sciences (LUMS), Lahore,Namal Pakistan Pakistan Department of of Electrical Engineering, Namal Institute, Mianwali, Pakistan ∗∗ Pakistan Department of Electrical Engineering, Pakistan Namal Institute, Mianwali, Pakistan Abstract: Abstract: Abstract: In this paper, we study the regulation problem for consumption in aa mathematical model Abstract: In this paper, we study the regulation problem for consumption in model In this paper, we study the regulation problem for consumption in aa mathematical mathematical model Abstract: of groundwater extraction. The model combines the dynamics of the groundwater table In this paper, we study the regulation problem for consumption in mathematical of groundwater extraction. The The model model combines combines the the dynamics dynamics of of the the groundwater groundwater model table of groundwater extraction. table In this paper, weextraction. study the model regulation problem fortheconsumption infrom a mathematical model with a recently introduced of consumer behavior inspired relevant studies in of groundwater The model combines dynamics of the groundwater table with a recently introduced model of consumer behavior inspired from relevant studies in with a recently introduced model of consumer behavior inspired studies in of groundwater extraction. The model combines the dynamics offrom the relevant groundwater table social psychology. After introducing the model, we formulate the moderation of groundwater with a recently introduced model of consumer behavior inspired from relevant studies in social psychology. After introducing the model, we formulate the moderation of groundwater social psychology. After introducing the model, we formulate the moderation of groundwater with a recently introduced model of consumer behavior inspired from relevant studies in withdrawal as an optimal control-theoretic problem. Different possibilities for the control social psychology. After introducing the model, we formulate the moderation of groundwater withdrawal as an optimal control-theoretic problem. Different possibilities for the control withdrawal as an optimal control-theoretic problem. Different possibilities for the control social psychology. After introducing the model, wecontext formulate the moderation of groundwater objective are discussed and interpreted in the of policy making sustainable withdrawal as an optimal control-theoretic problem. Different possibilities for the control objective are discussed and interpreted in the context of policy making for sustainable objective discussed and interpreted in the context of policy making for sustainable withdrawal an optimal control-theoretic Different possibilities the control groundwater In the process, we also controllability of process objective are areasmanagement. discussed and interpreted in problem. the investigate context of the policy making for sustainable groundwater management. In the process, we also investigate the controllability of the process groundwater management. In the process, we also investigate the controllability of the process objective are discussed and interpreted in the context of policy making for sustainable for an arbitrary number of agents under a fixed network topology. In the end, we present groundwater management. In the process, we also investigate the controllability of the process for an arbitrary number of agents under aa fixed network topology. In the end, we present for number of under network topology. In the end, we present groundwater management. thetoprocess, also investigate the controllability of the simulations of the LQR problem demonstrate conformity model the optimal control for an an arbitrary arbitrary number ofInagents agents under we a fixed fixed network of topology. Into the end, we process present simulations of the LQR problem to demonstrate conformity of the model to the optimal control simulations of the LQR problem to demonstrate conformity of the model to the optimal control for an arbitrary number of agents under a fixed network oftopology. Intothe end, we present framework. simulations of the LQR problem to demonstrate conformity the model the optimal control framework. framework. simulations framework. of the LQR problem to demonstrate conformity of the model to the optimal control Copyright © 2019. The Authors.Control Publishedinbyagriculture, Elsevier Ltd.Socio-ecological All rights reserved.systems, Control in social framework. Keywords: Groundwater, Keywords: Groundwater, Control in agriculture, Socio-ecological systems, Control in social Keywords: Groundwater, Control in agriculture, Socio-ecological systems, Control Control in in social social systems Keywords: Groundwater, Control in agriculture, Socio-ecological systems, systems systems Keywords: Groundwater, Control in agriculture, Socio-ecological systems, Control in social systems systems 1. Up till the recent turn of the century it has been widely 1. INTRODUCTION INTRODUCTION Up till till the the recent recent turn turn of of the the century century it it has has been been widely widely 1. Up advocated that policy interventions in groundwater con1. INTRODUCTION INTRODUCTION Up till the recent turn of the century it has been widely advocated that that policy policy interventions interventions in in groundwater groundwater conconadvocated 1. INTRODUCTION Up till theyields recent turnmarginally of the century itgroundwater has been widely sumption only better outcomes when advocated that policy interventions in consumption yields yields only only marginally marginally better better outcomes outcomes when when sumption advocated with that policy interventions in groundwater concompared no at all (see the Gissersumption onlyintervention marginally better outcomes when compared yields with no no intervention at all (see (see the GisserGissercompared with intervention at all the The importance of regulating groundwater usage is unsumption yields only marginally better outcomes when S` a nchez effect by (Gisser and S` a nchez, 1980)). However compared with no intervention at all (see the GisserThe importance of regulating groundwater usage is unThe importance of groundwater usage is only unS` nchez effect effect by by (Gisser (Gisser and and S` S`anchez, nchez, 1980)). 1980)). However However aaanchez deniable in world where water resources are not The importance of regulating regulating groundwater usage un- S` compared withby no(Gisser intervention allthat (see theHowever Gisserit is now being frequently highlighted consideration S` nchez effect and S`aaat nchez, 1980)). deniable in aaa world world where water water resources are are notis only only deniable in where resources not it is now being frequently highlighted that consideration The importance of regulating groundwater usage is unit is now being frequently highlighted that consideration depleting at an alarming rate, but their management is deniable in a world where water resources are not only S` a nchez effect by (Gisser and S` a nchez, 1980)). However of various environmental externalities and socio-economic it is now being frequently highlighted that consideration depleting at at an an alarming alarming rate, rate, but but their their management is is of various environmental externalities and socio-economic depleting deniable inincreasingly aanworld where water areal., not2010). only various environmental externalities and socio-economic becoming complicated (Siebert et depleting alarming rate, butresources their management management is of it is now effects being frequently highlighted thatconclusion: consideration spill-over yields quite the opposite the of various environmental externalities and socio-economic becoming at increasingly complicated (Siebert et al., al., 2010). 2010). becoming increasingly complicated (Siebert et spill-over effects yields quite quite the opposite opposite conclusion: the depleting at an alarming rate, but (Siebert their management is spill-over effects yields the conclusion: the The cross-disciplinary and convoluted nature of groundbecoming increasingly complicated et al., 2010). of various environmental externalities and socio-economic unchecked regulation model often leads to precariously spill-over effects yields quite the opposite conclusion: the The cross-disciplinary and convoluted nature of groundThe cross-disciplinary and convoluted nature of groundunchecked regulation model often leads to precariously becoming increasingly complicated (Siebert et al., 2010). unchecked regulation model often leads to precariously water management highlights the need for models that The and convoluted of groundspill-over effects yields quite the opposite conclusion: the sub-optimal aquifer levels that is causing water scarcity model often leads to precariously watercross-disciplinary management highlights highlights the need neednature for models models that unchecked water management the for that sub-optimalregulation aquifer levels levels that is causing causing water scarcity The only cross-disciplinary andecological convoluted nature of also groundsub-optimal aquifer that is water scarcity not incorporate the aspect, but the water management highlights the need for models that unchecked regulation model often leads to precariously and quality degradation at a global scale (Esteban and sub-optimal aquifer levels that is causing water scarcity not only only incorporate incorporate the the ecological ecological aspect, aspect, but but also also the the and quality degradation at a global scale (Esteban and Alnot Alwater management highlights the need for but models that quality degradation aathat global scale (Esteban and socio-economic aspects of hydrological systems and not only incorporate the ecological aspect, alsotheir the and sub-optimal aquifer levels is ofcausing water scarcity biac, 2011). As aa result, aaat studies have emerged and quality degradation atnumber global scale (Esteban and AlAlsocio-economic aspects of hydrological systems and their socio-economic aspects of hydrological systems and their biac, 2011). As result, number of studies have emerged not only incorporate aspect, but alsotheir the biac, 2011). As aaatnumber of studies have emerged consuming populations (Gorelick and Zheng, 2015). socio-economic aspectsthe of ecological hydrological systems and andoptimal quality degradation a global (Esteban and Alon policies for sustainable management biac, 2011).extraction As aa result, result, number ofscale studies have emerged consuming populations populations (Gorelick and Zheng, Zheng, 2015). consuming and 2015). on optimal optimal extraction policies for sustainable sustainable management socio-economic aspects (Gorelick of hydrological systems and their on extraction policies for management consuming populations (Gorelick and Zheng, 2015). biac, 2011). As a result, a number of studies have emerged of groundwater resources, see Mulligan et al. (2014); Broon optimal extraction policies sustainable In this paper, we examine aa social psychology based model of groundwater groundwater resources, see for Mulligan et al. al. management (2014); BroBroconsuming populations (Gorelick Zheng, based 2015).model resources, see Mulligan et (2014); In this paper, paper, we examine examine social and psychology based model of In this we a social psychology on optimal extraction policies for sustainable management zovi´ c et al. (2006); Lin Lawell (2016), and included referof groundwater resources, see Mulligan et al. (2014); Broof resource consumption introduced by Manzoor et al. In this paper, we examinefirst a social psychology based model zovi´ c et al. (2006); Lin Lawell (2016), and included referzovi´ c et al. (2006); Lin Lawell (2016), and included referof resource consumption first introduced by Manzoor et al. of consumption introduced by et al. of groundwater resources, seerecent Mulligan etisal. (2014); Broences. much the work focused on the c etHowever al. (2006); Linof (2016), and included referIn resource this paper, we examinefirst astudied social psychology based model (2016) and subsequently in association with of resource introduced by Manzoor Manzoor et real. zovi´ ences. However much ofLawell the recent recent work is focused focused on the ences. However much of the work is on the (2016) andconsumption subsequentlyfirst studied in association association with re(2016) and subsequently studied in with rezovi´ c et al. (2006); Lin Lawell (2016), and included refereconomic aspect of socio-hydrological systems whereas aa ences. However much of the recent work is focused on the of resource consumption first introduced by Manzoor et al. newable resource dynamics governed by standard logistic (2016) and subsequently studied in association with reeconomic aspect of socio-hydrological systems whereas economic aspect of socio-hydrological systems whereas a newable resource resource dynamics dynamics governed governed by by standard standard logistic logistic economic newable ences. However much ofincorporate the recent work is focused on the study should the behavioral aspects aspect of socio-hydrological systems whereas a (2016) and subsequently studied in association with re- complete growth (see the work by Manzoor et al. (2017, 2018) and newable resource dynamics governed by standard logistic complete study study should should incorporate incorporate the the behavioral behavioral aspects aspects growth (see (see the the work by by Manzoor et et al. (2017, (2017, 2018) 2018) and and complete growth economic aspectshould of well, socio-hydrological whereas a of the consumers as see for instance the exposition of study incorporate the systems behavioral aspects newable resource dynamics governed by the standard logistic Ruf et al. (2018)). Here, we consider groundwater growth (see the work work by Manzoor Manzoor et al. al. (2017, 2018) and complete of the consumers as well, see for instance the exposition of of the consumers as well, see for instance the exposition of Ruf et al. (2018)). Here, we consider the groundwater Ruf et al. (2018)). Here, we consider the groundwater complete study should incorporate the behavioral aspects Gorelick and Zheng (2015), MacEwan et al. (2017), and of the consumers as well, see for instance the exposition of growth (see the work by Manzoor et al. (2017, 2018) and extraction model given by Gisser and Mercado (1973) for Ruf et al. model (2018)). Here, we consider the groundwater Gorelick and and Zheng (2015), (2015), MacEwan MacEwan et et al. al. (2017), (2017), and and extraction given by Gisser Gisser and Mercado (1973) for for Gorelick extraction given by and (1973) of the consumers as well, see behavior for instance of the rejection of free-riding presented by Koch Gorelick and Zheng Zheng (2015), MacEwan etthe al.exposition (2017), and Ruf resource et al. model (2018)). Here, we assumed consider the groundwater the dynamics. The setting is that of extraction model given by Gisser and Mercado Mercado (1973) for the rejection of free-riding behavior presented by Koch the rejection of free-riding behavior presented by Koch the resource dynamics. The assumed setting is that of the resource dynamics. The assumed setting is that of Gorelick and Zheng (2015), MacEwan et al. (2017), and and Nax (2017). In this spirit, we investigate optimal of free-riding behavior presented by Koch extraction model given bybeing Gisser and Mercado (1973) groundwater aquifer harvested by consuming the resource dynamics. setting is that for of the and rejection Nax (2017). In this this spirit, spirit, we investigate optimal and Nax In we optimal the groundwater groundwater aquifer The beingassumed harvested by aaa consuming consuming the aquifer being harvested by the rejection of free-riding behavior presented by Koch consumption schemes our social-psychology based socioNax (2017). (2017). In in this spirit, we investigate investigate optimal resource dynamics. The assumed setting is that of and population of n agents. The decision making process of the the groundwater aquifer being harvested by a consuming consumption schemes in our social-psychology based socioschemes in our social-psychology based sociopopulation of of n n agents. agents. The The decision decision making making process process of of the the consumption population and Nax (2017). In this spirit, we investigate optimal hydrological model of groundwater extraction. We give consumption schemes in our social-psychology based sociothe groundwater aquifer being harvested by a consuming agents is influenced not only by the state of the aquifer but population of n agents. making process of but the hydrological hydrological model model of of groundwater groundwater extraction. extraction. We We give give agents is is influenced influenced not The onlydecision by the the state state of the the aquifer agents not only by of aquifer but consumption schemes ingroundwater our social-psychology based sociothree variants of the optimal control problem by defining model of extraction. We give population of n agents. The decision making process of the hydrological also by the consumption of other neighboring agents. Thus agents is influenced not only by the state of the aquifer but three variants of the optimal control problem by defining three variants of the optimal control problem by defining also by the consumption of other neighboring agents. Thus also by consumption of neighboring agents. hydrological model ofoptimal groundwater extraction. give different objective functions. We present simulations of the variants of the control problem by We defining agents influenced not only by the state of the aquiferThus but three the state of the socio-hydrological system depends both also byisthe the of other other neighboring agents. Thus different objective functions. We present simulations of the the objective functions. We present simulations of the state state ofconsumption the socio-hydrological socio-hydrological system depends depends both different the of the system both three variants of the optimal control problem by defining most basic variant in order to demonstrate the working of different objective functions. We present simulations of the alsothe by properties theofconsumption of other and neighboring agents. Thus on of the aquifer on the psychological the state the socio-hydrological system depends both most basic variant in order to demonstrate the working of basic variant in order to demonstrate the working of on the the properties properties of of the the aquifer aquifer and and on on the the psychological psychological most on different objective functions. We present simulations of the the model and as an incentive to solve the remaining varimost basic variant in order to demonstrate the working of the state of the socio-hydrological system depends both characteristics of the population, which include individual on the properties of the aquifer and on the psychological the model and as an incentive to solve the remaining varithe model and as an incentive to solve the remaining varicharacteristics of of the population, population, which which include include individual individual the characteristics most as basicstep variant inincentive order to demonstrate the working of ants towards gaining more insights and as an solve meaningful the remaining varion the properties of the aquifer and onto the psychological information preferences, susceptibility change and the the ants characteristics of the the population, which include individual antsmodel as aaa step step towards gainingtomore more meaningful insights as towards gaining meaningful insights information preferences, preferences, susceptibility to change and information susceptibility to change and the the model and as an incentive to solve the remaining varifor optimal regulation of groundwater extraction. ants as a step towards gaining more meaningful insights characteristics of the population, which include individual for optimal regulation of groundwater extraction. level of cohesion. information regulation groundwater extraction. level of of social socialpreferences, cohesion. susceptibility to change and the for level cohesion. antsoptimal as a step towardsof more meaningful optimal regulation ofgaining groundwater extraction. insights information level of social socialpreferences, cohesion. susceptibility to change and the for for optimal regulation of groundwater extraction. level of social cohesion. 2405-8963 Copyright ©IFAC 2019. The Authors. Published by Elsevier Ltd. All 1 rights reserved. Copyright © 2019 1 1 Control. Copyright © 2019 IFAC Peer review © under responsibility of International Federation of Automatic Copyright 2019 IFAC 1 Copyright © 2019 IFAC 10.1016/j.ifacol.2019.11.015 1 Copyright © 2019 IFAC

2019 IFAC CMWRS Delft, Netherlands, September 19-20, 2019

Ansir Ilyas et al. / IFAC PapersOnLine 52-23 (2019) 94–100

95

The rest of the paper is organized as follows. Section 2 presents the groundwater model and the coupled sociohydrological system. In the same section we also present the state-space formulation of the model and investigate its controllability. Section 3 specifies the optimal control problem along with different variants of the appropriate cost function. Simulations and discussion follow in Sections 4 and 5 respectively. 2. THE SOCIO-HYDROLOGICAL MODEL This section present the linear hydrological groundwater model of an aquifer with the change in consumption pattern. The complete model is composed into two parts: the hydrological sub-model and the social sub-model. The hydrological sub-model describes the dynamics of the height of the water table, which we consider as the quantity of the resource stock. The social sub-model describes the dynamics of pumping effort of each individual in the society.

Fig. 1. A model of an aquifer, describe the relationship between the water flows, and the height of the water table.

the context of natural resource consumption, the objective information (or the ecological factor) represents the available resource quantity, whereas social information (or the social factor) corresponds to the resource usage of other connected agents (Mosler and Brucks, 2003). The final change in consumption is a weighted sum of both ecological and social factors where the psychological characteristics of agent determines the relative weighing of both factors in the decision making process. We provide only a brief description of the process below and refer the interested reader to (Manzoor et al., 2016) who give the complete theoretical background of the model.

2.1 The Hydrological Sub-model The hydrological groundwater model presented here was first developed by Gisser and Mercado (1973). Consider the single-cell model of an aquifer described in Fig. 1, which illustrates the relationship between the water flow, the height of the water table and the aquifer. We assume that H(t) represents the height of the water table at time t. The height is dependent on the natural recharge R, artificial recharge Ra , and natural discharge D. The natural discharge D is approximated as a linear function of H, and can be written as D = a + bH, Where a and b are the parameters of natural discharge. The groundwater pumping Wi (t), is the consumption effort of individual i ∈ {1 . . . n}, in a society of consisting of n individuals. Some portion of pumped water returns to the water table and its quantity depends upon the coefficient of return flow α. Over the period of time, the aquifer gains water from natural recharge, artificial recharge, and the return flow: R + Ra + αWi (t). The loss of water in the aquifer is the sum of both consumption and natural discharge: Wi (t) + D. The change in height of the water table as a function of time is therefore given by n  dH(t) = R + Ra + (α − 1) Wi (t) − (a + bH(t)), (1) AS dt i=1

The ecological factor for consumer i is defined by H(t) − Hi , where Hi ∈ R is i’s perceived scarcity threshold of the water table height, below which i considers the groundwater to be scarce and above which she considers it to be abundant. Rutte et al. (1987) find that consumers use more resource (in our case pump more groundwater) when, according to their perception the resource is in abundance, than when it is scarce. The ecological factor is weighed by ai ∈ (0, +∞), the attribution of i, where ai → 0 represents a consumer who associates responsible for the scarcity of the resource completely with society while increasing values of ai corresponds to a consumer who associates more responsibility with nature. Thus the weighed ecological factor for a single individual is given as ai (H(t) − Hi ).

The social factor is defined as the difference between i’s consumption and the other socially connected consumers n in the network, and it is given by i=1 ωij (Wj (t) − Wi (t)), where ωij ≥ 0 is the n tie strength between i and j. It is assumed that j=1 ωij = 1 and ωii = 0 ∀ i. The social factor is weighed by si ∈ (0, ∞), the social value orientation of i. si → 0 represents an extremely non-cooperative individual, whereas increasing values of si correspond to an increasingly cooperative individual.  The weighed social factor is given by the n product si i=1 ωij (Wj (t) − Wi (t)).

where A is the area of an aquifer and S is the storage coefficient. The product A S represents the storativity of the aquifer. Equation (1) is simply the Gisser-Mercado model (Gisser and Mercado, 1973) with n individual consumers. 2.2 The Social Sub-model

Based on the above, combining the social and ecological factors gives the dynamics of the consumption effort of groundwater pumping for consumer i as

Here we define the dynamics of the consumer’s efforts Wi (t) over time. The change in consumption of each group is based upon Festinger’s theory of social comparison processes (Festinger, 1954). The theory postulates that human beings evaluate their decisions, against both objective information (non-social) and social information. In

n  dWi (t) = ai (H(t) − Hi ) + si ωij (Wj (t) − Wi (t)). (2) dt j=1

2

2019 IFAC CMWRS 96 Netherlands, September 19-20, 2019 Delft,

Ansir Ilyas et al. / IFAC PapersOnLine 52-23 (2019) 94–100

importance that i attaches to social information relative to ecological information, while making decisions about change in consumption.

Manzoor et al. (2016) describe in detail, the cognitive parameters of the model, i.e., the attributions, social value orientations and scarcity thresholds. Individuals that attribute blame to natural causes (drought, climate variability, etc) tend to give more importance to ecological factors. On the other hands, cooperative individuals in a society are more inclined towards improving equality in the consumption of groundwater than non-cooperative individuals and thus give more weight to the social factor (see Manzoor et al. (2016) for more details).

2.5 Controllability in the State Space In what follows, we consider the control problem of regulating both aquifer level x and the consumption efforts Wi by exerting influence on the environmentalism ρi for each agent i. In this context, the ρi ’s may be considered as control inputs. Given this, we now see that the system described by (5) is not linear but affine. In order to convert it to a linear system, we transform the state variables as follows z1 = x, 1 zi+1 = yi − , n which leads us to the following linear system n+1  z˙1 = −z1 − zj ,

2.3 The Coupled Socio-Hydrological System Combining the ecological and social sub-models, we arrive at the following socio-hydrological system n   1  dH(t) = R+Ra −a−bH(t)+(α−1) Wj (t) , dt AS j=1 (3) n  dWi (t) = ai (H(t)−Hi )+si ωij (Wj (t)−Wi (t)), dt j=1

j=2

where i ∈ {1 . . . n}. Thus the complete system is n + 1 dimensional and is linear in the height and consumption variables.



z˙i+1 = bi (1 − vi )(z1 − ρi ) + vi

Here we formulate the non-dimensionalized model through some simple transformations. The non-dimensionlization process enables direct comparisons between system parameters, reduces the parameter space and yields variables that have clearer interpretations in context of the results obtained. The transformation is carried out as follows. Let x(t) = b H(t)/ (R + Ra − a) be the normalized height of the water table at time t. Furthermore define yi (t) = βWi (t)/ (R + Ra − a) as the normalized consumption effort exerted by i. Transforming (3) accordingly, gives us the following system, n    dx =d 1−x− yj , dt j=1 (4) n  ai  si  d yi = β (x − ρi ) + ωij (yj − yi ) , dt b β j=1

y˙i = bi (1 − vi )(x − ρi ) + vi

n  j=1



vn bn vn bn . . . −vn bn n−1 n−1    0 0 ... 0 ρ1 b (v − 1) 0 . . . 0 1 1   ρ2    0 b2 (v2 − 1) . . . 0 + .   .. .. .. ..    ..  . . . . ρn 0 0 . . . bn (vn − 1)

+

(7)

The complete state controllability matrix of system (7), is defined as follows, Ω = [B AB A2 B . . . An B] The controllability matrix Ω can easily be calculated for a n agent system and is given in Fig. 2 for arbitrary n. It can be seen by inspection that the first n + 1 columns of Ω are linearly independent over the entire defined ranges of the parameters bi and vi . Thus we may conclude that (7) is indeed controllable. This implies that by affecting the environmentalisms alone, a central planner may obtain any desired level of the aquifer and individual consumptions for the considered socio-hydrological model.

(5)

ωij (yj − yi ) ,

where i ∈ {1, ..., n} and the derivative is taken w.r.t the non-dimensional time τ .s Here we have defined bi = 1   β αi si and vi =  αi β si  . The parameter bi is d b + β b

(6)

ωij (zj − zi+1 ) ,

bn (1−vn )

where d = b/ (AS), β = 1−α. Here ρi = b Hi / (R + Ra − a) is the normalized scarcity threshold of i and is called the environmentalism of i (Manzoor et al., 2016). Considering the non-dimensional time τ = d t, the model can be further simplified as follows n  yj , x˙ = 1 − x − j=1

j=2



where i ∈ {1, . . . , n}. Next, we assume a complete, regular graph topology for the network of system (6). This implies that ωij = 1/(n − 1) for all i = j and ωii = 0 for all i. For the state space model, we describe the dynamics as z˙ = Az + Bρ where z = [z1 . . . zn+1 ]T is the n + 1 dimensional state vector and ρ = [ρ1 . . . ρn ]T is the n dimensional control vector. The matrices A and B are defined in the following representation of the system  −1  −1 −1 . . . −1     b1 (1−v1 ) −v1 b1 v1 b1 . . . v1 b1   z˙1  n−1 n − 1  z1 v2 b2 v2 b 2   z 2   z˙2    b2 (1−v2 ) −v2 b2 . . .  . =   n − 1 n − 1   ..   ..    . . . . . .  .. .. .. .. ..  zn˙+1   zn+1

2.4 The Non-dimensionalized System



n+1 

3. THE OPTIMAL CONTROL PROBLEM

β

called the sensitivity of i, and corresponds to i’s openness to change in her consumption. The parameter vi is the socio-ecological relevance of i and corresponds to the

In the previous section, we have presented the dynamical system governing the behavior of the resource stock and 3

2019 IFAC CMWRS Delft, Netherlands, September 19-20, 2019



0

Ansir Ilyas et al. / IFAC PapersOnLine 52-23 (2019) 94–100

0

b1 (v1 − 1) 0    0 b2 (v2 − 1) Ω=  .. ..  . .  0

0

... 0

0

... 0

0

... 0

0

. . ..

.. .

..

. . . 0 bn (vn − 1)

(1 − v1 )b1

(1 − v1 )v1 b21

(v1 − 1)b1 b2 v2 n−1 .. .

97

(1 − v2 )b2 ... (1 − vn )bn (v2 − 1)b2 v1 b1 (vn − 1)bn v1 b1 ... n−1 n−1 (vn − 1)bn v2 b2 (1 − v2 )v2 b22 . . . n−1 .. .. .. . . .

(v1 − 1)b1 vn bn (v2 − 1)b2 vn bn ... n−1 n−1

(1 − vn )vn b2n

...



...

     ... ...

...

Fig. 2. Controllability matrix for (7) ρ˙ in order to keep the required opinion change as small as possible. One such performance index that accomplishes this may be defined as follows,  ∞  T J(z(·), ρ(·), ρ(·)) ˙ = z (t) Q z(t)+ρT (t) R ρ(t) 0  + ρ˙ T (t) R2 ρ(t) ˙ dt → min, where Q is non-negative definite symmetric, R and R2 are positive definite symmetric matrices. This cost function consists of the quadratic term not only involving the state z(t) and the control vector ρ(t), but also the derivative of the control vector ρ(t). ˙ In this case, the eigenvalue of matrix R should be much less than relative to R2 , which give less weight to the environmentalism relative to the change in environmentalism or ρ. ˙ To optimize the given objective the original system represented in (7) is augmented with the derivative of the control vector. The state space model of the augmented system can be written as x˙ 2 = F x2 +Gu2 where the matrices F and G are defined in the following system        z(t) ˙ A B z(t) 0 = + ρ(t) ˙ ρ(t) ˙ 0 0 ρ(t) I

individual consumptions in a socio-hydrological system with a single aquifer and n consumers. We also formulated the system as a state space model by considering the aquifer level z1 and individual consumptions z2 , . . . , zn+1 as states and the environmentalisms ρ1 , . . . , ρn as the control inputs. We found that the system is controllable and thus, any desired equilibrium may be achieved by exertion of the appropriate control by a central planner. In this section we consider the control vector that optimizes a given objective. We pose three different optimal control problems, each consisting of a different objective function. 3.1 Regulation with minimized environmentalism Here we consider the standard LQR problem for a presented socio-hydrological model in order to address the question of obtaining the control vector ρ(t) that minimizes the following objectives.  ∞  T  J= z (t) Q z(t) + ρT (t) R ρ(t) dt, (8) 0

where Q is non-negative definite symmetric, and R is positive definite symmetric matrices that reflect the precedence of the planner. It is now well known (see (Anderson and Moore, 2007)) that the control input ρ(t) that minimizes the cost function J is given by the linear feedback law ρ = −K z, where K is given by K = R−1 B P where P is given by the solution of the following Algebraic Riccati Equation (ARE)

Rewriting the cost function in terms of the augmented state leads us to the following optimal control problem x˙ 2 = F x2 + Gu2  ∞  T  x2 (t) Q2 x2 (t)+ ρ˙ T (t) R2 ρ(t) J(x2 (·), ρ(·)) ˙ = ˙ → min, 0





(P2)

Q 0 and is non negative-definite0 R symmetric. The existence and uniqueness of the optimal control follows from controllability of the the pair (F, G), has been demonstrated by Moore and Anderson (1967). The control law of the augmented system (P2) is given by

AT P − P A + Q − P B R−1 B T P = 0. We thus consider the following optimal control problem z(t) ˙ = A z(t) + B ρ(t),  ∞   T J(z(·), ρ(·)) = z (t) Q z(t)+ρT (t) R ρ(t) dt → min,

where Q2

0

=

u∗2 = −R2−1 GT P x2 , where P is solution of the following Riccati equation given below F T P − P F + Q2 − P G R−1 GT P = −P˙ (t). (9) These results can be interpreted in term of the original system (7)

(P1) where A and B are as defined in (7). The existence and uniqueness of the optimal control follows from controllability of the the pair (A, B), as demonstrated in Section 2.5 (see Anderson and Moore (2007) and included references for further detail).

ρ˙ ∗ = −R2−1 P21 z − R2−1 P22 ρ∗ (10) where P21 and P22 are the matrices obtained by the partitioning P , the solution of (9).

3.2 Regulation with minimized opinion change The objective function in (P1) minimizes a quadratic of the environmentalism vector ρ. Environmentalism is a psychological variable that represents the level of resource stock below which an individual considers it to be scarce. Control over this variable may be exercised through means such as planned education or targeted information propagation and so on. Thus it would make sense to minimize

3.3 Regulation with maximized revenue This section provides the optimal groundwater pumping patterns to reduce the change in the aquifer height. In the above control problem, we considered the ρi ’s as the inputs 4

2019 IFAC CMWRS 98 Netherlands, September 19-20, 2019 Delft,

Ansir Ilyas et al. / IFAC PapersOnLine 52-23 (2019) 94–100

for the state space model. Here we assume that a central planner maximizes total revenue obtained from groundwater extraction. The revenue that individual i extracts is the area under the demand curve for groundwater use (Gisser and Mercado, 1973; Sanval and Helfand, 2016). For the revenue function, we assume that the demand function for the groundwater is defined as follows, zi+1 = g + kP,

z˙1 = −z1 −

zi+1

i=1

z˙i+1 = −γk(z1 +

n  i=1

zi+1 ) − (γk − 1)zi − kγ(ρi − z1 ) + g

4. SIMULATION

(11)

where P is the price of water, and zi+1 represents the groundwater pumping and g & k are parameters of the demand function. We may express P in term of zi+1 and integrate from zero to zi+1 to obtain the total revenue. The total cost of extraction of groundwater is a function of the depth from which the groundwater has to be extracted (Sanval and Helfand, 2016) and is given as follows,   C = γ ρi − z1 (t) , (12)

This section investigates the behavior of the system under the LQR problem presented in Section 3.1. We consider a consumer network with 40 nodes under a complete graph topology. Two different communities are explored • The pro-ecological community with socio-ecological relevance vi ∈ (0.1, 0.11). This community attaches more importance to ecological factors while making decisions regarding consumption. • The pro-social community with vi ∈ (0.9, 0.99). This community attaches more importance to social factors while making decisions regarding consumption.

where γ is the marginal cost of extraction of groundwater. We may now express i s net income as the total revenue minus the total cost: 1    g 2 π= zi+1 − zi+1 − γ ρi − z1 (t) zi+1 . (13) 2k k This leads us to the following optimal control problem n  zi+1 , z(0) = z0 (P3) z˙1 = −z1 − 

n 

In each network, the socio-ecological relevance vi for every node is randomly sampled from the respective intervals given above. Furthermore, the node sensitivities bi are also randomly sampled from the interval (0, 1) for both networks. Figure 3 gives the trajectories of the groundwater consumption for each node. We see that in both cases, the consumers converge to values close to zero consumption rates. In the pro-ecological network there is a significant undershoot which is not exhibited by the pro-social network. In both cases, the consumers settle on negative consumption efforts which imply that all of them recharge the aquifer instead of harvesting from it, see Manzoor et al. (2016) for physical interpretations of negative consumption. The trajectories of each element of the optimal control vector ρ are given in Fig. 4. We see that in both cases the environmentalisms fluctuate significantly, with the magnitude of fluctuations more significant for the pro-social community. Moreover in both cases, the environmentalisms remain in the negative region, corresponding to negative scarcity thresholds. This corresponds to every

i=1

    1  g 2 zi+1 − zi+1 − γ ρi − z1 (t) zi+1 dt, max zi+1 t=0 2k k To solve this problem, we may use Pontryagins Maximum Principle (Kopp, 1962). For this, the Hamiltonian is defined as, n      1  g 2 zi+1 zi+1 , − zi+1 − γ ρi − z1 zi+1 −λ z1 + H= 2k k i=1 ∞

with the following first order necessary conditions, ∂H 1 g = ( zi+1 − − γρi + γz1 ) − λ ∂zi+1 k k n  ∂H = −z1 − zi+1 = z˙1 ∂λ i=1 ∂H = γzi+1 − λ = −λ˙ ∂z1

Next, substituting λ from the above expression for ∂H ∂λ , and further differentiating λ with respect to time, and we get, 1 1 g −(γzi+1 − zi+1 + + γ(ρi − z1 )) = z˙i+1 + γ z˙1 (14) k k k Now, substituting z˙1 in Eq (14), we get, 1 g − (γzi+1 − zi+1 + + γ(ρi − z1 )) k k n  1 = z˙i+1 − γ(z1 + zi+1 ), k i=1 z˙i = −γk(z +

n  i=i

zi ) − (γk − 1)zi − kγ(ρi − z) + g. Fig. 3. Optimal pumping patterns for pro-social and proecological societies under Problem (P1)

The system that governs the dynamics of the optimal state trajectory is given by the following system 5

2019 IFAC CMWRS Delft, Netherlands, September 19-20, 2019

Ansir Ilyas et al. / IFAC PapersOnLine 52-23 (2019) 94–100

Fig. 4. Optimal environmentalisms ρi for pro-social and pro-ecological societies under Problem (P1)

99

Fig. 6. Comparison of the trajectories of the groundwater table height between the optimally controlled system (P1) and the system (7) with all ρi ’s equal to zero.

consumer perceiving the groundwater as abundant for all positive values of the table height, no matter how low the actual height is.

5. DISCUSSION In this paper we have presented a socio-hydrological model for groundwater extraction that includes the dynamics for both the aquifer level and pumping behavior of the consumers. While the social sub-model has already been presented previously by Manzoor et al. (2016) in context of natural resources following the logistic growth model, here it has been considered in conjunction with the level dynamics of a groundwater table. Many physical insights may be gained by the model from treatment under various frameworks (see (Manzoor et al., 2017, 2018; Ruf et al., 2018) for similar work). Here we have studied the treatment of the model under the optimal control framework and have posed three different optimal control problems given by (P1), (P2) and (P3). We have then simulated the behavior of the model under (P1), an instance of the basic linear quadratic regulator under the assumption that control is exerted via the consumer scarcity thresholds.

We next compare the behavior of the system (P1), with that of the free system (7) with constant control vector ρ equal to zero. The result is shown in Fig. 5. We see that the graphs mimic the behavior of the optimally controlled system (P1) shown in Fig. 3, with very similar magnitudes of undershoot and steady state consumption levels. This is as expected since the objective function as formulated in Problem (P1) encourages the control to stay close to zero. The similarity in behavior between the optimally controlled system (P1) and the free system (7) under zero environmentalism may also be observed in Fig. 6, where the trajectories of the resource stock exhibit very similar transient and steady state behaviors.

Simulations from Section 4 of the LQR problem (P1) indicate that both the groundwater table height and individual consumptions converge to zero in the steady state. This is to be expected, given that the objective function in (P1) penalizes values of both z and ρ that deviate from the zero vector. Many different notions for sustainability exist in the literature, see Perman et al. (2003), and each has a different judgment on the relative balance between the environment and society in a sustainable socio-ecological community. However very few of them (if any) would predict that sustainability entails exhaustion of both the resource and consumption for obvious reasons. Thus while the simulations of Section 4 do demonstrate the viability of the model under the LQR, the results admittedly carry very little meaning in the context of policy for groundwater sustainability. This is due in major part to an injudiciously selected cost function, which should have been grounded firmly in context of the original objective i.e., sustainable resource governance (see Weber et al. (2011) for various examples). The cost functions in Problems (P2) and (P3)

Fig. 5. Consumption patterns for system (7) with a constantly zero control vector ρ. 6

2019 IFAC CMWRS 100 Netherlands, September 19-20, 2019 Delft,

Ansir Ilyas et al. / IFAC PapersOnLine 52-23 (2019) 94–100

ternalities. Annual Review of Resource Economics, 8, 247–259. MacEwan, D., Cayar, M., Taghavi, A., Mitchell, D., Hatchett, S., and Howitt, R. (2017). Hydroeconomic modeling of sustainable groundwater management. Water Resources Research, 53(3), 2384–2403. Manzoor, T., Rovenskaya, E., Davydov, A., and Muhammad, A. (2018). Learning through fictitious play in a game-theoretic model of natural resource consumption. IEEE Control Systems Letters, 2(1), 163–168. Manzoor, T., Rovenskaya, E., and Muhammad, A. (2016). Game-theoretic insights into the role of environmentalism and social-ecological relevance: A cognitive model of resource consumption. Ecological modelling, 340, 74–85. Manzoor, T., Rovenskaya, E., and Muhammad, A. (2017). Structural effects and aggregation in a socialnetwork model of natural resource consumption. IFACPapersOnLine, 50(1), 7675–7680. Moore, J. and Anderson, B. (1967). Optimal linear control systems with input derivative constraints. In Proceedings of the Institution of Electrical Engineers, volume 114, 1987–1990. IET. Mosler, H.J. and Brucks, W.M. (2003). Integrating commons dilemma findings in a general dynamic model of cooperative behavior in resource crises. European Journal of Social Psychology, 33(1), 119–133. Mulligan, K.B., Brown, C., Yang, Y.C.E., and Ahlfeld, D.P. (2014). Assessing groundwater policy with coupled economic-groundwater hydrologic modeling. Water Resources Research, 50(3), 2257–2275. Nunes, D.S., Zhang, P., and Silva, J.S. (2015). A survey on human-in-the-loop applications towards an internet of all. IEEE Communications Surveys & Tutorials, 17(2), 944–965. Perman, R., Ma, Y., McGilvray, J., and Common, M. (2003). Natural resource and environmental economics. Pearson Education. Ruf, S.F., Hale, M.T., Manzoor, T., and Muhammad, A. (2018). Stability of leaderless resource consumption networks. arXiv preprint arXiv:1804.04217. Rutte, C.G., Wilke, H.A., and Messick, D.M. (1987). Scarcity or abundance caused by people or the environment as determinants of behavior in the resource dilemma. Journal of Experimental Social Psychology, 23(3), 208–216. Sanval, N. and Helfand, S. (2016). Optimal groundwater management in Pakistans Indus Water Basin, volume 34. Intl Food Policy Res Inst. Siebert, S., Burke, J., Faures, J., Frenken, K., Hoogeveen, J., Dll, P., and Portmann, F. (2010). Groundwater use for irrigation a global inventory. Hydrology and Earth Systems Sciences, 14(10), 1863–1880. Wang, F.Y. (2010). The emergence of intelligent enterprises: From cps to cpss. IEEE Intelligent Systems, 25(4), 85–88. Weber, T.A. et al. (2011). Optimal control theory with applications in economics. MIT Press Books, 1.

have been formulated in exactly this manner. Due to their mathematical nature, the solution of both these problems is more involved than (P1). In Section 3 we have provided some headway towards solving these problems and present both of them as open problems that, in our opinion, merit the attention of the controls sub-community working in related areas. On a concluding note, the socio-hydrological system presented herein represents a class of systems that lie on the interface of environmental, social and economic systems. While scientists from different communities have been studying these systems from there own perspectives, there is an increasing need to integrate these worldviews due to the interdisciplinary nature of these systems (Wang, 2010; Nunes et al., 2015). We hope that the exposition of this paper serves as a motivational tool for more members of the controls community to show further interest in such systems. ACKNOWLEDGEMENTS The authors acknowledge the financial and logistical support provided by the Center for Water Informatics & Technology (WIT) at LUMS, to carry out this work. REFERENCES Anderson, B.D. and Moore, J.B. (2007). Optimal control: linear quadratic methods. Courier Corporation. Brozovi´c, N., Sunding, D., and Zilberman, D. (2006). Optimal management of groundwater over space and time. In Frontiers in water resource economics, 109– 135. Springer. Esteban, E. and Albiac, J. (2011). Groundwater and ecosystems damages: Questioning the gisser–s´anchez effect. Ecological Economics, 70(11), 2062–2069. Festinger, L. (1954). A theory of social comparison processes. Human relations, 7(2), 117–140. Gisser, M. and Mercado, A. (1973). Economic aspects of ground water resources and replacement flows in semiarid agricultural areas. American Journal of Agricultural Economics, 55(3), 461–466. Gisser, M. and S` anchez, D.A. (1980). Competition versus optimal control in groundwater pumping. Water resources research, 16(4), 638–642. Gorelick, S.M. and Zheng, C. (2015). Global change and the groundwater management challenge. Water Resources Research, 51(5), 3031–3051. Koch, C.M. and Nax, H.H. (2017). Rethinking freeriding and tragedy of the commons. Available at SSRN 3075935. Kopp, R.E. (1962). Pontryagin maximum principle. In Mathematics in Science and Engineering, volume 5, 255–279. Elsevier. Lin Lawell, C.Y.C. (2016). The management of groundwater: Irrigation efficiency, policy, institutions, and ex-

7