A CAS framework was developed to simulate non-linear feedbacks among consumers, policy makers, and the water resources system [
10]. The framework consists of three major components: an ABM of residential consumers, an ABM of a policy maker, and a mechanistic water resources model. Consumer demands vary based on seasonal needs for watering lawns, adoption of water conservation technologies, and compliance with water use restrictions. A policy maker agent implements drought management plans, which include outdoor water use restrictions based on the current reservoir storage, and fixture incentive programs, such as replacing appliances with low-flow toilets, low-flow showers, and efficient washing machines. Historic climatic data that include the drought-of-record are simulated to explore the dynamic interactions among conservation, drought management, and inter-basin transfers and to evaluate adaptive water management strategies. The CAS framework is developed using AnyLogic 6.5 simulation software [
33]. A detailed description can be found in Kanta and Zechman [
10]; however, a brief description of the model is provided here.
3.1. Agent-Based Model of Residential Consumers
Each household in a water district is represented as an agent who demonstrates heterogeneous and adaptive water use behavior. Agent’s heterogeneity is introduced through month specific demand, year of house built, and lot size. The monthly demand for each household is estimated based on the water district’s billing data which contains customer specific water usage volume per month for a long time horizon. From the statistical analysis of the water consumption data, a gamma distribution is chosen as a best fit to model the water use of agents [
10] which stochastically satisfies the following relation:
where
Em(
Demand) = expected demand for month
m (m
3/month);
Em{Usage |
Pr (
X ≤
x)} = expected value of water usage for month
m from gamma fitted distribution with a given probability of non-exceedance (m
3/month);
X = a uniform random variable [0, 1]; and
x = a possible value of
X. The gamma fitted stochastic model ensures agent’s heterogeneity in terms of water use at the beginning of the simulation period which may change if an agent adopts a water efficient technology or complies with any water use restrictions imposed by the water district.
Each agent is also associated with three indoor end-uses, including toilet, shower head, and washing machine, and one outdoor end-use, the sprinkler system. The build year of the house determines if the existing appliances are old (water-inefficient) or new (relatively water-efficient). The lot size determines the possible size of gardens and lawns for each household. Both the indoor end-uses and outdoor sprinkler demand are modeled after Jacobs and Haarhoff [
34]:
where
AMDDi,e,m = average monthly daily demand for indoor appliance
e during month
m (L/day);
ae,m = volume parameter of indoor appliance
e during month
m (L/event);
be = frequency parameter of indoor appliance
e (number of events/person/day);
n = household size;
AMDDo,m = average monthly daily outdoor demand during month
m (L/day);
fm = a model parameter representing a household’s behavioral choice of over- or under-watering the lawn with respect to the ideal water requirement (unitless);
s = surface area of garden and lawn (
m2), which is a function of lot size;
km = crop factor during month
m (unitless);
EVm = pan evaporation during month
m (mm/month);
Pm = the portion of rainfall that penetrates into the soil during month
m (mm/month);
dm = number of days in month
m (days/month); and
Rm = rainfall during month
m (mm/month). The factor
represents evapotranspiration from the lawn or garden. The volume parameter (
ae,m), frequency parameter (
be), and household’s behavioral choice (
fm) values are adopted from Jacobs and Haarhoff [
34].
A social communication model, or word-of-mouth framework [
10], is developed based on Watts-Strogatz model [
35] to simulate agents’ communication about technology adoption. The parameters of this model are cluster size, degree of connection, and the technology adoption threshold. Each agent is initialized randomly as a member of a cluster at the beginning of simulation period. Each Agent also has a user defined number of connections. When a policy maker agent offers incentive based programs, such as toilet/shower head/washing machine replacement rebate, the consumer agents adopts the new water efficient appliances and send information about technology adoption to their corresponding social network through the connections. The initial adoption of technology is determined randomly, without incorporation of a consumer price model. A threshold based decision rule is used to model voluntary adoption of water efficient technology, regardless of incentives offered by the policy maker agent. An agent adopts voluntarily once a certain portion of its social network has adopted. A detailed sensitivity analysis of word-of-mouth framework was done by Kanta and Zechman [
10], which concludes that the model is more sensitive to cluster size and threshold value; the number of connections, on the other hand, does not have any significant effect on the rate of voluntary adoption.
All consumers begin with old or new appliances based on the age of their corresponding houses. Each year, the policy maker agent offers a rebate to a specified number of randomly selected agents, which accept the rebate and replace existing appliances with water-efficient appliances. The reduction in demand for each appliance is modeled using end-use model (Equation (2)) with a reduced value of the volume parameter, ae,m. Agents who reduce demands through adopting water efficient technologies, send information to their peers. Based on the information received from a social network, consumers may voluntarily adopt water efficient appliances, an aggregation of which may also result in demand reduction.
The outdoor demand is a function of climatic data (EVm and Rm in Equations (3) and (4), respectively), size of lawns and gardens (s in Equation (3)), and affinity of a household to over- or under-water in maintaining gardens and lawns (fm in Equation (3)). Agents reduce outdoor demands based on outdoor watering restrictions. All agents are initialized with fm = 2.0 in Equation (3), which represents a frequency of outdoor watering at five times per week. It is assumed that all outdoor watering restrictions are mandatory. When a restriction is implemented, the outdoor demand is reduced based on the watering restrictions as twice per week (corresponding to drought stage 1 and fm = 1.0), once per week (drought stage 2 and fm = 0.5), and banned completely (drought stage 3 and fm = 0.0). The total demand for each agent is calculated as a sum of indoor and outdoor demand with reduction at the above four end-uses.
3.2. Agent-Based Model of Policy Makers
The utility manager is modeled as a policy maker agent that offers incentive-based conservation programs, imposes water use restrictions, and allocates inter-basin transfers. Both the incentive-based appliance replacement program and outdoor watering restriction options are adopted from the water district’s current conservation plan, and data is not included describing the water district’s price data and consumer specific income data. At the beginning of each month, the policy agent evaluates the level of the reservoir compared to the conservation pool, selects inter-basin transfer volumes, and imposes conservation and restrictions accordingly. The conservation pool of a reservoir represents the water surface elevation whose equivalent storage is allocated for the purposes of water supply, irrigation, recreation, and hydropower generation. The policy maker agent implements drought stages based on the volume of water stored in a reservoir at each month. The policy maker agent also determines the volume of water supplied through inter-basin transfers based on the reservoir storage. For months when the total demand is greater than the supply, and the demand cannot be met, the policy agent restricts outdoor water use completely. Each year, the policy agent implements monthly toilet, shower, and washing machine replacement rebate programs for the residential consumers.
3.3. Water Resources Model
The water resources model simulates the hydrologic processes of a watershed and the reservoir. Runoff from the watershed is calculated using the Rational Method [
36] to generate inflow to the reservoir. The reservoir receives additional inflows from inter-basin transfers, and outflows include customer withdraws, evaporation, and reservoir spills. Due to changes in both inflows and outflows throughout the year, the reservoir storage and surface level vary with time. The water balance equation is used to calculate reservoir storage as follows:
where
St = reservoir storage volume at current month
t (m
3);
St-1 = reservoir storage volume at the previous month
t − 1 (m
3);
ROt = runoff from the watershed into the reservoir at current month
t (m
3);
IBTt = inter-basin transfer volume to the reservoir at current month
t (m
3);
Dt = residential demand at current month
t (m
3);
LEt = lake evaporation at current month
t (m
3); and
Rt = reservoir release at current month
t (m
3). Reservoir releases are set at the required instream flow volume if storage volume does not exceed reservoir capacity. The instream flow for environmental/ecological purposes can be estimated as 10% to 15% of average annual runoff from the watershed, as suggested by Cai and Rosegrant [
37]. When storage volume exceeds reservoir capacity, reservoir releases are calculated as the excess volume of water beyond the reservoir capacity.