Pumping schedule assignment using chromatic graphs to reduce groundwater pumping energy consumption Open Access

Groundwater is the source of one third of all freshwater withdrawals (Döll et al. 2012) and its importance for global water and food security will probably increase under climate change (Taylor et al. 2013). In Africa, for instance, the widespread development of groundwater is the most affordable and sustainable way of improving access to secure water for the rural poor on the scale required to achieve coverage targets (MacDonald & Calow 2009). Nonetheless, groundwater in not invulnerable to degradation (MacDonald et al. 2012), while its use presupposes energy consumption.

Meeting increasing water, food and energy demand in the best possible way, entails innovative approaches, such as combination of food and energy production (Miskin et al. 2019), increased use of renewable energy resources (Bassi 2018; Shah et al. 2018) and design – implementation of smart irrigation systems (Zaier et al. 2015). Moreover, it entails optimization of the engineering aspects of irrigation systems, taking into account environmental restrictions (Singh 2016); but it also entails consideration of social factors (Vörösmarty et al. 2005; Sivapalan et al. 2012; Johnson et al. 2014). We now know that it is often social factors and not physical ones that have the greatest influence on per capita water use (Cole et al. 2017). Moreover, groundwater conservation policies, detached from the wider socio-environmental framework, can have disastrous results (Sarma et al. 2016; Balwinder-Singh et al. 2019).

In this paper, we use optimization techniques, to investigate the role of a social factor. Specifically, we examine how a change in well users' habits (pumping during more time phases) can result in reduction of energy consumption for groundwater pumping.

Proper assignment of pumping schedules to a group of well users, presupposes cooperation among them, which is not an easy task (Madani & Hipel 2011; Alamanos et al. 2022). Moreover, it presupposes optimization of pumping schemes, using different criteria (Pisinaras et al. 2013; Alqahtani & Sale 2020). In this paper we propose simple cooperation rules that can be applied by well users. Following these simple rules, farmers can create local groups, and reduce their energy consumption (therefore their pumping cost) by pumping alternately to their neighbors. Allocating pumping schedules to reduce pumping cost is a unique opportunity for small farmers, as usually the economic benefits of new technologies are more easily harnessed by larger farms that can spread their fixed costs over many acres, and that can reduce labour costs through automation (Basso & Antle 2020).

From the mathematical point of view, we have to study hydraulic head drawdowns due to pulsed pumping from n wells. Pulsed pumping has been mostly studied in terms of aquifer recharge (Wang et al. 2012), and pollutants' removal (Harvey et al. 1994). Recently, Nagkoulis (2020) examined the connection between pulsed pumping, hydraulic head drawdowns and pumping cost, considering two wells. In this paper, the results of the aforementioned paper are expanded in order to examine more pumping schedules and systems of any number of wells. We suppose that the pumping flow rates of the users are equal, which can be a sensible assumption when considering small farmers. Specifically, we have created an algorithm using R language to calculate the hydraulic drawdowns, using Theis equation (Theis 1935) and the pumping energy consumption. Each day is divided into four pumping periods with 6 h duration, and each well is allowed to operate exclusively in one pumping period with constant flow rate. Finding the optimal pumping schedules' allocation among the wells, in order to minimize the overall energy consumption, is a problem falling under the general class of constrained, nonlinear, optimization problems and can be handled using a number of methods.

Assuming there are four pumping schedule options available to each well user, we get 4ⁿ possible pumping schedule combinations for the whole system, some of which are identical, where n indicates the number of wells. As the number of combinations could be quite large (e.g. over 10⁹ for n = 15), to find the optimum we have resorted to genetic algorithms (GA), which are widely used in water resources management problems (Etsias & Katsifarakis 2017; Ayvaz & Elçi 2018; Han et al. 2020; Vali et al. 2020; Seyedpour et al. 2021; Tanyimboh 2021). We have used a GA package in R language (Scrucca 2013). The variables inserted in the objective function of the binary version of GA used (Randy & Haupt 2003) represent pumping schedules of well users. Then, each chromosome represents a possible combination of individual pumping schedules. Moreover, we have investigated another approach to schedule groundwater pumping, which is probably conceptually easier for end-users of our results: use of nearest neighbor graphs (NNGs) (Eppstein et al. 1997) and chromatic graphs (Harary 1994; Lewis 2016) to represent interactions of well users. Graph theory is increasingly used in water resources. Its applications derive from the similarities between graphs and water distribution networks (Tianwei et al. 2020) and from graph theory's ability to map social interactions and represent social conflicts (Sharifian et al. 2022). A graph G with a chromatic number γ(G) = x is called a x-chromatic graph. The chromatic number of a graph is the minimum number of colors needed to color the nodes so that there are no adjacent nodes colored identically (Eppstein et al. 1997). K-NNGs are the graphs which result by connecting each node to the k closest neighbors. The idea stemmed from a previous study (Nagkoulis & Katsifarakis 2022) dealing with Nash Equilibria that appear when examining the scheduling preferences of the well users. There, we noticed that some chromatic graphs appear. In this paper, we show that for using graph theory, we can deduce simple directions to the well users that can replace GA, with minor loss in accuracy. Specifically, a variety of cases is tested, using hypothetical aquifers with different transmissivity and storativity values and different well settings, in order to get indicative results for many cases that might appear in practice.

We examine the following five pumping scenarios in terms of pumping energy consumption, using two optimization approaches:

• (I)

  All wells operate simultaneously during the morning (06:00–12:00) or any other period (monochromatic graph).

• (II)

  GA are used to choose one of two pumping schedules (i.e. 06:00–12:00 or 18:00–24:00) for each well, so that the overall energy consumption is minimized.

• (III)

  GA are used to assign one of four pumping schedules (0:00–6:00, 6:00–12:00, 12:00–18:00, 18:00–24:00) to each well, so that the overall consumption is minimized.

• (IV)

  2-Chromatic Graphs are used to assign one of two pumping schedules to each well.

• (V)

  4-Chromatic Graphs are used to assign one of four pumping schedules to each well.

In Equations (5) and (6), γ is the specific weight of water (γ = 9.81 kN/m³), Q (m³/s) is the pumping flow rate of well i at moment t (s) and s (m) is the hydraulic drawdown at moment t (s) at well i (no matter which wells contribute to this drawdown). K is counted in kWsec, or kWh/3,600, which is used to get kWh in the results section. Finally, the diameter of the well r_i,i is considered to be r₀ = 0.250 m. To calculate numerically the total energy consumption, the integral is divided in a number of sum terms. The more the sum terms are, the more accurate the energy consumption calculation is. Same applies to the sum term in Equation (3).

To calculate the hydraulic drawdowns, the principle of superposition is used as follows: supposing that a number of wells operates continuously for a time period Δt, the resulting hydraulic drawdown at each well is added to the hydraulic drawdown of the previous period. The same applies to the shutdown periods that have the reverse effect on hydraulic drawdowns. Finally, the average drawdowns for each well, for each pumping period, are used to calculate the energy consumption for that period, using Equation (6). For visualization purposes, ‘ggplot2’ (Wickham 2009) and ‘plotly’ (Sievert 2020) packages are used in R.

The variable ‘t_qi’ is used to indicate the pumping time period of well ‘i’. This means that when t_qi = 1, well i operates for the first 6 of each day (00:00 to 06:00), when t_qi = 2 the well operates from 6am to 12am, etc.

Since the time periods are set as variables, all other parameters have to be set as constant for each test, namely: Q, T, S, total pumping duration (D), well distances, number of wells. The first four parameters are going to be tested independently. To generalize the results, so that they can be applied to any number and layout of wells, four cases are investigated.

Two of the most important parameters that affect pumping energy consumption are the number of wells and their locations. To create random well locations ‘rnorm’ function is used in R, resulting in random coordinates (x, y).

GA are used first, to minimize the total energy consumption, which serves as the the objective function. Its calculation, for different sets of variables, has been described in previous sections (Equations (2)–(7)), namely it includes use of the hydraulic drawdown algorithm. The lower the pumping energy consumption, the better the solution.

Binary representation of the chromosomes is adopted; therefore, values can be 0 or 1 for two pumping schedules and 00, 01, 10, 11 for four pumping schedules. These values are converted to the decimal system as ⁠. It has turned out that 300 generations and a population size equal to 20 were sufficient for this problem, as increasing these values did not have any real impact on the result. The crossover and the mutation probabilities were set to 0.8 and 0.1 respectively and elitism value was set to 5.

In this section the subroutines used to create the Chromatic Graphs are briefly described (2-Chromatic and 4-Chromatic Graphs). An undirected simple graph G = (V, E, w), where |V| is the number of wells, |E| is the number of edges between the wells and w: E → ℝ ≥ 0 is the edge-weight function that associates a positive weight to each edge, is k-NNG when each node is connected to its k nearest neighbors. Moreover, an edge between the nodes V_i and V_j is considered monochromatic when the color of node V_i is identical to the color of node V_j (c_i = c_j). The weight of each edge is w = 1/ ⁠, where r is the distance between V_i and V_j.

Specifically, 150 generations and population size equal to 40 are chosen. Elitism value is set to 5, which means that the five best chromosomes always ‘survive’ to the next generation. The crossover and the mutation probabilities are set as 0.8 and 0.1, respectively. The GAs are used to find individuals with minimum f value. A chromatic graph is created when f = 0, which means that no wells with identical colour are connected to each other.

We have preferred to use GAs for the following three reasons. At first, they have been used extensively in water resources management literature, then they can be used to create both 2-chromatic and 4-chromatic graphs. Moreover, GAs are reliable, because we definitely know that a chromatic graph has been created as long as f = 0.

Finally, it should be mentioned that the computational time needed for the GA to construct a chromatic graph is more than 100 times lower, than the time needed to find directly the pumping schedule that minimizes energy consumption. This is because to create chromatic graphs, we use the distance of the wells in the objective function, whereas to find the optimum, we use the energy consumption, which increases the computational cost.

In some cases, the DE values increase, but BoC doesn't. Examining BoC can be helpful in order to generalize the results of this research, increasing the understanding of the concepts presented.

The layouts tested are those of Figure 4, named as follows: Layout 1: 5 wells, short distances. Layout 2: 5 wells, large distances. Layout 3: 15 wells, short distances. Layout 4: 15 wells, large distances (this Layout is the basic scenario). In the following figures, the blue lines represent the results of GAs and the red lines represent the results of the chromatic graphs. Moreover, solid lines represent scenarios III and V, namely with four pumping periods, while broken lines represent scenarios II and IV, namely with two pumping periods.

The first parameter examined is the overall duration of pumping. The parameters T, S and Q are kept constant. We present four plots, one for each well layout, testing how the total duration of pumping (D) affects the ‘Decrease of Energy consumption’.

From Figure 7 we can see that the BoC remains constant (or nearly constant after a minor decrease, when examining duration of pumping), even though that DE increases. For example, from Figure 6, Layout 4, we can see that when four pumping schedules are used, the system's energy consumption is reduced by DE = 62,500 kWh and by DE = 250,000 kWh, for Q = 0.025 m³/sec and Q = 0.05 m³/sec, respectively. From Figure 7 we can see that in both cases the BoC is 47%. This means that when four pumping schedules are used, the energy consumption is 47% lower than using one pumping schedule. The reason for that is that even though DE increases, the overall pumping energy consumption increases as well with time and pumping flow rates, which results in a constant BoC.

For the rest of the tests, we have chosen D = 5 days, as BoC stabilizes close to 5 days. Moreover, as Q does not affect BoC, we have chosen Q = 0.01 m³/s, as a characteristic value. The main reason that BoC needs some days to stabilize is that during the first days, the beginning of pumping (initial condition) plays an important role. For example, if the total pumping duration scheme lasts only for two days, pumping during the last time phase (night) will result in lower pumping costs than pumping during the morning, because it results in lower overall pumping duration. However, as duration increases (e.g. 15 days) this energy reduction becomes insignificant, compared to alternate pumping benefit. To improve computational speed, we have chosen D = 5 days, because stabilization is partially achieved at the fifth day in these examples.

For instance, from Figure 10 up we can see that for S = 1*10⁻⁶ and T = 2*10⁻⁴ we get DE5,000 kWh and from Figure 10 down we can see that this corresponds to a 35% reduction. For S = 1*10⁻⁶ and T = 1*10⁻⁴ we get a higher DE8,000 kWh, but a lower percentage reduction (33%). This is because as T decreases, the overall pumping energy consumption increases more sharply than the economic benefit.

In our research, some assumptions were made, which were tested for their accuracy before conducting the main experiments. First, it needs to be mentioned that there can be more than one chromatic graphs created through the previous process. In Figure 1 for example, coloring at least one NNG reversely will result in different energy consumption than the initial one. However, these chromatic graphs were found to have minor differences to each other and therefore did not affect the results of this analysis. To test this hypothesis, we conducted 100 tests in the typical scenario, described in the results section. The average consumption for the 2-chromatic graph has been found to be 14,877 kWh and the average consumption for the 4- chromatic graph has been found to be 12,080 kWh. The standard deviation for the first case is 63 kWh and 38 kWh for the second one. It is clear, then, that these differences do not affect the results of this paper. Examining the chromatic graphs that lead to the lowest pumping energy consumptions exceeds the scopes of this research.

Moreover, it should be mentioned that this research can be used to emphasize the role of storativity. In order to have high percentages of energy reduction, it is necessary to have high interference between the wells, which means low storativity values. Accurate estimation of storativity (which is often not done in practice (Younger 1993)) is necessary to calculate the benefit of alternate pumping.

Finally, considering that some chromatic graphs might be Nash Equilibria (Nagkoulis & Katsifarakis 2022), it is interesting to examine in future research if chromatic graphs can be used in order to create coalitions that can be stable in terms of creating stable alternate pumping schemes.

In this paper, a new method is proposed to reduce groundwater pumping energy consumption, by assigning alternate pumping schedules to a system of wells, pumping from an infinite aquifer. We suggest local cooperation, as we prove that when well users cooperate locally in order to pump alternately with their nearest neighbors, the overall pumping energy consumption is reduced. To test if cooperation among neighbors can result in overall energy reduction, we compared the energy that is consumed when chromatic graphs are used to assign pumping schedules to the energy that is consumed when GAs are used to assign pumping schedules and to the energy consumed when everyone pumps at the same time (no cooperation at all). The results indicate that:

• (I)

  The energy consumption when all users pump simultaneously, is generally higher than when they pump alternately during two (and even better four) time periods.

• (II)

  The chromatic graphs can be considered as simple accurate alternatives to GAs to assign pumping schedules.

• (III)

  Low S and T values result in high values of DE (energy savings).

• (IV)

  Low S and high T values result in high values of BoC (energy savings (%)).

Finally, the formation of 2-chromatic graphs is simple in practice, just by letting each well user pump alternately with their nearest neighbor. Same applies for 4-chromatic graphs, but in this case, it is necessary for each well user to pump alternately with their three nearest neighbors.

The main practical conclusion is that local cooperation between well users, in order to pump alternately leads always to energy savings. The magnitude of these savings depends on the aquifer features, the distances between wells, the duration of pumping and the flowrates that are pumped.

In future research, the results of this paper can be expanded to include more complicated aquifers, unequal pumping flow rates and well users' willingness to cooperate.

The results presented in this work have been produced using the Aristotle University of Thessaloniki (AUTH) High Performance Computing Infrastructure and Resources.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.