Economies of Scale and Efficiency in the Wastewater Treatment Sector: A Decision Tree Approach

Abstract

In this study, we use the efficiency analysis trees (EAT) model to evaluate the efficiency of wastewater treatment plants (WWTPs), focusing on identifying the key variables that enhance their performance. While traditional methods consider factors such as the plant size, age, and technology, the EAT model improves the precision of and reduces errors in the efficiency estimation process. The results highlight the significance of facility size, particularly in areas with smaller populations, suggesting that economies of scale can play a crucial role in optimizing treatment processes. Centralizing the sector could lead to cost savings through ensuring better resource allocation and more effective management strategies. This study provides insights into how facility dimensions impact efficiency, aiding in strategic planning for wastewater treatment operations. The novelty of this study pertains to its implementation of the EAT model to assess efficiency from an economic point of view. Specifically, this makes it possible to identify which facility dimensions ensure better performance and, consequently, will help operators to establish criteria for intervention and geographical planning, both technically and economically.

1. Introduction

In the last decade, water availability—in terms of both quantity and quality—has become threatened by prolonged periods of drought and the pollution of water bodies [1,2]. During the 21st century, together with population growth, industrial development has increased demand for water. While water demand increases, the quality of water bodies are worsening due to the wastewater that is returned to the environment [3]. Wastewater water treatment plants (WWTPs) have therefore become a key element in the urban water cycle, as they represent an alternative source of water that is capable of guaranteeing economic development while protecting and conserving water resources. Although water reuse does not reduce the demand for water, it relieves the pressure on water resources by decreasing the abstraction of water and minimizing the environmental impacts that treated wastewater can cause to the environment once it is returned to the ecosystem [4,5]. In this context, rethinking the urban water cycle should become a priority to extend the potential opportunities that WWTPs can offer. Not only does circular water management ensure the availability of water, but valuable products could also be obtained, such as energy (biogas) and nutrients including nitrogen and phosphorous [6,7,8]. In this sense, these facilities require large amounts of energy for the correct operation of treatment processes; energy requirements are strictly related to the technology adopted, which depends on the characteristics of the wastewater and the quality requirements of the effluent. For example, the aeration process, which is necessary for the removal of pollutants in the water, can account for up to 70% of the total energy consumption [9]. Similarly, depending on the technology used, sludge treatment also generally has high energy demands. However, these facilities have the potential to contribute to the decarbonization of the energy system. This could be achieved through utilizing the thermal and chemical energy present in wastewater and sewage sludge. These processes, combined with other alternatives—such as the use of renewable resources and energy optimization of the facility itself—can significantly increase the balance of clean energy used, thus reducing the consumption of non-renewable energy sources [10]. Beyond self-consumption, it is also possible to integrate the energy production of these facilities as an agent of local energy supplies, thus reducing the consumption of energy from non-renewable sources, contributing to decarbonization and enhancing the circular economy.

As WWTPs are an essential part of guaranteeing the circularity of water resources in urban water management, it is also necessary that wastewater treatment plants fulfill the principles of the circular economy; that is, improving the efficiency of the processes by reducing the amount of resources needed to perform optimally, keeping the products and materials in the loop as long as possible while reducing the amount of residue generated [11,12]. In terms of WWTP processes, the treatment involves using a combination of physical, chemical, and biological technologies to reduce wastewater pollutants to meet the quality standards set by Directive 91/271/EEC. The performance of WWTPs is influenced by numerous factors such as the technology [13,14], the size of the facilities [15,16], their age [17,18], and maintenance management [19]. However, the optimum size of wastewater treatment systems is an area that has received a great deal of attention in the literature; there is significant debate regarding the cost-efficiency of centralized and decentralized sewage solutions [20,21,22].

Economies of scale are well-known in urban water cycle management; numerous studies have shown the influence of the network (supply and/or sanitation) on the efficiency of the process [23,24,25,26,27]. This suggests that the most efficient way to achieve a reduction in wastewater treatment costs is to increase the size of treatment facilities. However, the influence of Directive 91/271/EEC and the environmental need to treat as much wastewater as possible has led to an increase in the number of treatment facilities and a reduction in their size [28,29,30]. It is therefore necessary to analyze this phenomenon in order to solve the dilemma of whether it is appropriate (economically and environmentally speaking) to centralize or decentralize urban wastewater treatment. In other words, we must consider whether it is better to build large WWTPs or to create a network of smaller WWTPs disaggregated by territory and adapted to the conditions of the territory and the pollutant loads they receive.

The most widely recognized methodology for efficiency analysis is Data Envelopment Analysis (DEA) [31], which allows researchers to analyze the behavior of the units thanks to the construction of an efficient operating frontier. This frontier represents the optimum efficiency of the sample and is used to locate the units of the analysis with respect to it; the closer they are to the frontier, the more efficient they will be. Within DEA, a wide variety of models allow for the analysis of different aspects of the production processes. This study focuses on the efficiency analysis trees (EAT) model [32]. The EAT model focuses on efficiency analysis through combining machine learning and linear programming techniques to solve the problems derived from overfitting in the efficiency calculations and to help improve the robustness of the results obtained. This generates reliable efficiency values that facilitate a better comparative analysis of the selected units of analysis and provides guarantees when using these results in decision-making processes. The application of the EAT model is recent; it has been applied in the education sector with the aim of assessing efficiency in higher education institutions, as well as in industrial assembly processes [33,34]. In the water sector, it has only recently been adopted; studies have focused on the energy efficiency of WWTPs and the influence that greenhouse gas emissions have on treatment costs and their eco-efficiency [35]. In contrast, the novelty of this study lies in its application of the EAT model to a sample of WWTPs in order to determine the influence of economies of scale on the treatment efficiency of WWTPs.

Based on the above assumptions, the aim of this study is to analyze the economies of scale in the wastewater treatment sector in the Valencian Community from economic and environmental points of view through the application of the EAT model. This analysis seeks to determine the behaviors of differently sized WWTPs with respect to the existing quality criteria and the possible implementation of reuse models in the territory. The novelty of this research is twofold. On one hand, it applies the EAT model to determine the effect of economies of scale on the treatment efficiency of WWTPs. On the other hand, it includes water quality criteria (i.e., lower quantities of nutrients such as nitrogen and phosphorus) as a starting point for the future implementation of reclaimed water in Valencia. This analysis highlights the potential for minimizing the economic cost of treatment; by extension, operators are encouraged to implement changes in the structuring of the WWTP network in order to take maximum advantage of the existence of economies of scale and the circular economy.

2. Materials and Methods

The term “efficiency” generally refers to the optimal utilization of production factors within a production process, based on current technology. Farrell pioneered the study of frontier functions [36], which are used as benchmarks to assess efficiency for each productive unit. Farrell’s model establishes a benchmark or frontier of best practices that comprises the most efficient units within a given sample. Consequently, if a firm maximizes output for a specified input vector or minimizes inputs to achieve a certain output, it is considered to be on the “production frontier”. In such cases, a firm’s technical efficiency is determined by calculating the maximum possible proportional reduction in factor usage that remains compatible with its level of output. Efficiency assessments are widely applied across productive sectors [37], with two primary approaches to calculating efficiency: (i) the parametric approach, exemplified by stochastic frontier (SF) analysis [38], and (ii) the nonparametric approach, such as data envelopment analysis (DEA) [39]. SF methods distinguish inefficiency from heterogeneity in terms of costs [40] and are frequently applied in water sector studies [41,42]. DEA, however, is particularly suited for the study of wastewater treatment efficiency, as it benchmarks without assuming specific behaviors for treatment plants [43]. DEA, a linear programming technique, assesses efficiency by comparing sample units based on their respective inputs and outputs. The flexibility of DEA has fueled its use in benchmarking studies within the water sector, offering insight into scale economies and highlighting process improvements [44]. Some nonparametric approaches for estimating production frontiers rely on using envelopment techniques to assess the efficiency of decision-making units (see, e.g., [32,45,46]). These techniques model production frontiers by enveloping the observed data points, creating a benchmark that distinguishes efficient units from inefficient ones. However, a limitation of the DEA and FDH models is their tendency to produce estimated frontiers that lie below the actual or theoretical frontier, often resulting in an underestimation of technical inefficiency [47].

This limitation creates challenges in generating a generic model, as the DEA and FDH methods may perform well for specific datasets but fail to accurately represent broader patterns. This is similar to the contrast between descriptive statistics, which summarize specific data, and statistical inference, which seeks generalization. In statistical learning, generalization is critical: according to Vapnik [48], a model that finds an appropriate balance between fitting the training data while remaining adaptable to new data achieves the best performance. This is especially relevant as operations research increasingly adopts data analytics and machine learning methods, where the issue of overfitting often arises.

For instance, Breiman [49] developed the classification and regression trees (CART) method, which constructs decision trees by iteratively splitting nodes to increase predictive accuracy. While a deep tree can capture intricate patterns within the data, it risks overfitting, yielding overly optimistic estimates for the training set but lacking accuracy for new data. Conversely, a shallow tree may fail to capture important patterns, leading to poor model performance. To address these issues, Breiman [49] introduced a cost–complexity pruning technique, which refines the model by removing excess complexity based on cross-validation. This approach optimizes the tree’s accuracy and enhances its ability to generalize by preventing overfitting.

In this study, we employ a regression tree technique based on CART to estimate production frontiers that adhere to the free disposability property [50]. This method partitions the input space into terminal nodes through a series of binary splits, where each terminal node represents a constant predicted output. Graphically, this produces a step-function-like predictor that shares certain traits with FDH but differs in that it constructs deeper trees. To mitigate overfitting, we implement a cross-validated pruning procedure, which helps evaluate out-of-sample efficiency for the decision-making units (DMUs).

While this non-parametric technique does not provide statistical significance for each input variable, it does enable us to rank their relative importance. From a data visualization perspective, this method also allows for the representation of graphically complex, multivariate situations that would be challenging to visualize using a simpler tree model. This article represents one of the first contributions that connects two key topics: machine learning and frontier analysis applied to the urban water cycle, specifically with regard to the study of economies of scale in the wastewater treatment sector.

Following Esteve [51], we assume a vector of predictor variables, representing the amount of each pollutant that the WWTP is capable of removing in the treatment process (suspended solids, biological oxygen demand, chemical oxygen demand, nitrogen, and phosphorus), denoted as $x_1\dots ,x_m$ with $x_i\in {\mathfrak{R}}_{+}^m$. This set of predictors is used to estimate a response variable vector, representing different cost usages in the process (energy, staff, maintenance, waste, and other costs), denoted as $y\dots ,y_n$ with $y_i\in {\mathfrak{R}}_{+}^n$. In the EAT approach, a predictor variable $j$ and an optimal threshold $s_j\in S_j$, where $S_j$ consists of a vector of possible thresholds for the variable $j$ to partition the dataset into right and left nodes $t_R \mathrm{a}\mathrm{n}\mathrm{d} t_L$, respectively [51]. The mean squared error (MSE) criterion determines the threshold, defining the resulting right and left nodes. Mathematically, this can be expressed as follows:

$$R(t_L)+R(t_R)=1/n\sum _{(x_i,y_i)\in t_L}(y_i-y(t_L){)}^2+1/n\sum _{(x_i,y_i)\in t_R}(y_i-y(t_R){)}^2$$

(1)

where $t$ represents the node of the tree, $R_t$ is the MSE of each node t, n denotes the sample size, and $y\left(t_L\right)$ and $y{(t}_R)$ represent the predicted value of the response variable, which is derived based on the data that belongs to the nodes $t_L$ and $t_R$, respectively. Nodes $t_L$ and $t_R$ denote the left and right nodes of the tree, respectively.

The EAT algorithm’s regression tree stops when further meaningful data splits are impossible, specifically, when n (t) ≤ n_min = 5 [49,52]. EAT extends CART by introducing two features: estimating the frontier (maximum) variable instead of the response variable’s average and splitting data in each node to satisfy the free disposability assumption while minimizing Equation (1). These features are achieved through Pareto-dominant nodes [32,53].

As a result, the predicted (estimated) values of the response variable for a given node $t$, $y$(t), must be equal to or greater than those associated with the Pareto-dominant nodes (or, equivalently, must not be lower). Mathematically, the fulfillment of the free disposability assumption in the regression tree that is built based on the EAT algorithm is expressed as follows:

$$y\left(t_L\right)=\max \left\{\max \left\{y_i:\left(x_i,yi\right)\in t_L\right\},y\left(I_{T(\left.k\right|t*\to t_L,t_R)}\left(t_L\right)\right)\right\}$$

(2)

$$y\left(t_R\right)=y\left(t\right)$$

where T denotes the sub-tree that is produced by applying the EAT algorithm, k is the number of splits, and $y\left(I_{T(\left.k\right|t*\to t_L,t_R)}\left(t_L\right)\right)$ and $y\left(I_{T(\left.k\right|t*\to t_L,t_R)}\left(t_R\right)\right)$ represent the highest estimate of variable y at Pareto-dominance nodes of $t_L$ and $t_R$, respectively. Thus, the predictor function is non-decreasing, and the estimated production frontier looks like a step function [40]. Thus, the production technology estimated using EAT is defined as follows:

$$P{T}_{T_k}=\left\{\left(x,y\right)\in {\mathbb{R}}_{+}^{m+1}:y\le d_{T_k}\left(x\right)\right\}$$

(3)

where $d_{T_k}\left(x\right)$ denotes the predictor estimator related to sub-tree $T_k$.

Cross-validation techniques, such as determining the optimal number of leaf nodes or the minimum number of DMUs required for a split, can be used to obtain the best regression tree [54]. The tree constructed using the EAT algorithm uses a vector of inputs and outputs, with each node working on a sub-matrix of these data. At each node, a predictor and a set of thresholds are applied to generate a predicted value of the response variable, and each split is determined by minimizing the MSE. The EAT algorithm includes the free disposability assumption, so the estimated frontier resembles a step function, with the predicted value representing the frontier rather than the average. Since the EAT algorithm produces a sequence of trees of increasing size, it ensures that each tree satisfies the free disposability condition. The efficiency score for EAT is then obtained by solving the following linear equation:

$${\theta }^{EAT}\left(x_k,y_k\right)=\mathit{min}\theta$$

(4)

This is subject to

$${\sum }_{t\in {\tilde{T}}^{*}}{\lambda }_t{a}_j^t\le \theta x_{jk},j=1,\dots , m$$

$${\sum }_{t\in {\tilde{T}}^{*}}{\lambda }_t{d}_{rT}^t\left(a^t\right)\le y_{rk},r=1,\dots , s$$

$${\sum }_{t\in {\tilde{T}}^{*}}{\lambda }_t=1$$

$${\lambda }_t\in \left\{\mathrm{0,1}\right\};L=1,\dots ,n$$

where θ is the efficiency score, ($a^t$, $d_{T^{*}}$ ($a^t$)) are points in the input–output space for all $t\in T^{*}$, in which * denotes the final sub-tree, and λ values are intensity variables used to construct the efficient frontier. A value of one indicates that the unit under evaluation (WWTP in this study) is fully efficient.

Finally, based on the cost efficiency scores estimated using Equation (5), the potential cost saving if WWTP were efficient can be estimated using the following equation:

$$\mathit{cost}{t}_s=\mathit{cost}{t}_c\ast \left(1-\theta \right)$$

(5)

where cost t_s is the potential saving in costs, and cost t_c is the actual level of cost of the evaluated WWTP.

Based on the results of the application of these models and the subsequent analysis of the potential savings, we analyze the impact of economies of scale on the efficiency of WWTPs. The facilities are classified into groups according to their population equivalent (P.E.), and their efficiency levels are compared between these groups. To test for statistically significant differences between groups, we use the Kruskal–Wallis test [55], a non-parametric method that extends the Mann–Whitney test to compare three or more groups.

3. Sample Description

The study uses a sample consisting of 362 WWTPs located in the Valencia Region (Spanish Mediterranean coast), which treats an average annual volume of 301,516 m³/year. A basic requirement for the application of this methodology and the analysis of the economies of scale is that the influence of other factors should be isolated; for this reason, the sample should be as homogeneous as possible in terms of the technology of the wastewater treatment process. Accordingly, the WWTPs of the sample consist of a primary treatment followed by a secondary treatment based on extended aeration. None of the facilities are equipped with a tertiary treatment. The information to be used in the analysis, which describes the performance of these facilities, is shown in

Table 1. Sample description.

		Average
P.E.		13,320
Flow (m3/year)		301,516
Inputs (EUR/year)	Energy	19,988
	Staff	58,107
	Maintenance	13,154
	Residues	6832
	Others	13,431
Outputs (kg/year)	TSS	267,815
	COD	507,945
	BOD5	283,287
	TN	41,203
	TP	5525

; it was provided by the Valencian Wastewater Treatment Agency (EPSAR). The parameters included in this study are different, such that we can differentiate between economic and technical variables. The first group includes variables related to the operation and maintenance costs of the WWTPs (EUR/year): energy (including the cost of the fixed part, power, and the variable part, energy consumed); staff (defined by salaries, fees, taxes, and social insurance); maintenance (the revision, repair, or replacement of the equipment and facility); residues (expenses associated with WWTP management); and others (reagents and consumables). The second group refers to the information related to the amount of pollutant material removed in the process (kg/year): total suspended solids (TSS), chemical oxygen demand (COD), biochemical oxygen demand (BOD₅), total nitrogen (TN), and total phosphorous (TP).

To analyze the effect of economies of scale, the sample is divided into four groups based on the P.E. treated by the facilities. It should be noted that the disaggregation of the sample into these four groups maintains the proportionality according to the characteristics of the territory. The design of a WWTP is influenced by different factors such as the climate, the receiving peak flows, the P.E., and pollutant loads [56]. Additionally, the geography of the region could become another influential factor, affecting the construction or the operation of the process. As shown in

Table 2. Sample groups considered, accounting for P.E.

Group	P.E.	Number of WWTPs	%
1	<10,000	295	81.4
2	10,000–20,000	23	6.2
3	20,000–50,000	21	6.3
4	>50,000	23	6.1
		362	100

, a large part of the Valencian territory is composed of small municipalities; in accordance with Directive 91/271/EEC, which mandates that municipalities with more than 2000 P.E. must treat their wastewater through a biological wastewater treatment system, these municipalities have been equipped with WWTPs. Therefore, small WWTPs, represented by Group 1, which includes those facilities that treat less than 10,000 P.E., make up the majority of the sample, accounting for 81.4% of the WWTPs included in the study.

Table 3. Description of the annual costs of contaminants removed by wastewater treatment for each one of the groups (average values).

		Group 1	Group 2	Group 3	Group 4
Total cost	EUR/m3	1.3615	0.3195	0.2757	0.1963
Energy	EUR/m3	0.1342	0.0647	0.0663	0.0497
Staff		0.8705	0.1595	0.1322	0.0815
Maintenance		0.1440	0.0323	0.0274	0.0241
Residues		0.0627	0.0206	0.0202	0.0152
Others		0.1502	0.0424	0.0296	0.0258
TSS	kg/m3	0.194	0.245	0.253	0.267
COD		0.431	0.500	0.538	0.511
BOD5		0.235	0.270	0.275	0.296
TN		0.043	0.044	0.047	0.040
TP		0.004	0.005	0.004	0.005

summarizes the economic and environmental variables (pollutants removed) for each of the groups. Considering the four groups into which the sample has been divided, it is evident that an increase in the P.E. leads to a reduction in process costs. Thus, from an economic point of view, the differences in costs (EUR/m³) in each group analyzed reveal a reduction in unit costs as the size of the facility increases. Similarly, the pollutants removed generally increase as the size of the facility increases. In the case of suspended solids, the increase from Group 1 to Group 3 offers an increase of approximately 30%. For COD, there is an increase of 24% with respect to the smallest size group; for BOD₅, this figure is approximately 17%. The variability in nutrients, nitrogen, and phosphorus is lower.

4. Results

This section is structured in three parts: Section 4.1 presents the results of the EAT model for the entire sample (divided into four groups). Section 4.2 presents the results of the individualized analysis for the different WWTP groups in relation to the existing economies of scale. The efficiency indices, according to the methodology explained above, range from 0 to 1; the value 1 represents the efficient WWTPs, which constitute the efficient production frontier. Finally, Section 4.3 analyzes the potential savings that each of the groups would obtain if they were to increase in size based on the P.E. treated.

4.1. Efficiency Analysis of the Global Sample with the EAT Approach

First, the efficiency scores for the 362 WWTPs were calculated following the EAT model using variable returns to scale [57], which allowed for a comparison of each unit analyzed, with the units having a similar size (see

Table 4. Average efficiency scores for the four groups with variable returns to the scale model (EAT). Kruskal–Wallis test results for the four groups.

Group	Number of WWTPs	Average Efficiency Score		Score
1	295	0.1509
2	23	0.2995	Chi-square	90.3972347
3	21	0.3980	df	3
4	23	0.5393	Asymp. Sig	0.000
Total	362

).

The EAT model results show that Group 4 (treating wastewater above 50,000 P.E.), which includes 23 WWTPs, obtained the greatest average efficiency score, with an efficiency index of 0.54. This value is followed by the one obtained by Group 3 (treating wastewater between 20,000 and 50,000 P.E.), which consists of 21 WWTPs, with an average efficiency score of 0.40. Then, Group 2 (treating wastewater between 10,000 and 20,000 P.E.), which comprises 23 WWTPs, has an average efficiency score of 0.30. Finally, Group 1 (treating fewer than 10,000 P.E.), including 295 WWTPs (the largest group), is the least efficient group, with an average efficiency score of 0.15. These results demonstrate that the increase in the efficiency of WWTPs is directly linked to the P.E. that the WWTPs treat. Therefore, as the P.E. of the WWTP increases, the unit economic costs for wastewater treatment decrease. These findings are consistent with the results obtained in other studies [16,28,58,59,60], in that bigger WWTPs work more efficiently than smaller ones; an increase in WWTP size incurs lower unitary production costs. In view of these results, the wastewater treatment process presents opportunities for saving by reducing the average cost (EUR/P.E.). This information can be highly useful in the design of more efficient WWTPs in terms of reducing costs and obtaining treated water of a suitable quality.

The novelty of the current study, compared to the studies mentioned previously, lies in its more exhaustive analysis, since a greater number of output variables were considered.

Extending the study to include the elimination of nutrients (TN and TP) allows us to obtain more complete results that confirm that economies of scale are not only obtained for the main pollutants. However, it is necessary to study subgroups in order to identify the possible existence of an optimal minimum size.

Although the results obtained show that there is a clear relationship between the efficiency score of the WWTP groups and the costs per P.E., it is worth checking that the differences between them are statistically significant. The results of the Kruskal–Wallis test are shown in Table 4. With a significance of 5%, the null hypothesis can be rejected. Therefore, we can accept that the differences between the four groups are significant. To become more efficient, the WWTPs belonging to Group 1 need an average reduction in all costs. In terms of efficiency, the score obtained is 0.15, which implies a high capacity for improvement. However, an increase in the size of the plants (from Group 1 to Group 2) would increase the score obtained and allow them to achieve an average of 0.29. Group 2 increases the score to 0.29: this result implies a decrease in the economic costs of the treatment, as well as the improved elimination of pollutants. Again, Group 2 could increase the obtained efficiency to 0.39 (Group 3) if the facilities were increased in size according to the treated P.E. Group 3 shows similar behavior, obtaining an efficiency score of 0.39, which implies a reduction in economic costs as well as an increase in pollutant removal, but this score is lower than that of Group 4, which obtains 0.52, indicating a much better performance. With these results, we can confirm the existence of a direct relationship between the size of the facilities and the efficient performance of the inputs in the wastewater treatment processes. In other words, bigger WWTPs show better efficiency indicators in terms of the use of the inputs. Due to their relevance for the operation costs, energy management and staff are the most representative cases, in line with the literature [60,61,62,63]. In this sense, the results obtained by Lorenzo-Toja [64] show a cost inefficiency associated with the use of energy in plants < 20,000 P.E.

As reflected in the study conducted by Hernández-Sancho [52], WWTPs with secondary treatments are influenced by scale economies, so that smaller WWTPs are less efficient. Other work [39] shows that only 3% of the small WWTPs analyzed operate efficiently, while the remaining WWTPs are far from efficient in their performance. The overlap of these results with those obtained in our study demonstrates that scale economies have to be considered, and that energy use represents a key factor in the efficient operation of WWTPs.

Table 5. Coefficient correlation between variables and score (non-parametric).

EAT Input	Energy	Staff	Maintenance	Residues	Others
Global results	−0.452 **	−0.634 **	−0.680 **	−0.252 **	−0.947 **
Group 1	−0.421 **	−0.573 **	−0.636 **	−0.241 **	−0.948 **
Group 2	−0.359 *	−0.584 **	−0.593 **	−0.446 **	−0.939 **
Group 3	−0.610 **	−0.524 **	−0.657 **	−0.505 **	−0.981 **
Group 4	−0.414 **	−0.533 **	−0.429 **	−0.103	−0.636 **

** The correlation is significant at the 0.01 level (two sided). * The correlation is significant at the 0.05 level (two sided).

shows the correlations between the different costs (energy, staff, maintenance, residues, and others) and efficiency (as measured using the efficiency analysis tree (EAT) model) for each group. It allows us to analyze the behavior of each input in relation to the score obtained.

At a general level, significant negative correlations are observed in all cost categories, implying an inverse relationship between efficiency and higher economic costs. A higher result obtained in each input could imply that operators need to make a greater effort to reduce specific costs. In general, the four groups show a consistent negative correlation, although the inputs dedicated to energy, staff, maintenance, and others are the most strongly correlated. However, in economic terms, this last group (others) does not involve as much investment as the energy, staff, and maintenance groups. The correlation with energy costs, staff, and maintenance offers negative results of −0.452, −0.634, and −0.680, respectively. It is important to note that the behavior of maintenance costs in relation to the scores obtained by each group decreases as the group includes larger facilities, ranging from −0.636 for Group 1 to −0.429 for Group 4. The correlation observed for staff follows a similar pattern: Group 1 presents a stronger correlation than Group 4 in this regard.

4.2. Individualized Efficiency Analysis

Based on the results obtained in the EAT analysis (Table 4), the sample was subdivided into different smaller groups to determine whether the presence of economies of scale follows a pattern (Figure 1).

The mean values of the inputs of each subgroup are shown in

Table 6. Average values of the input and output of each subdivision of Group 1.

Group		1.A (<3000)	1.B (3000–6000)	1.C (6000–10,000)	1.A.1 (<1000)	1.A.2 (1000–2000)	1.A.3 (2000–3000)
P.E.		631	4413	8375	335	1442	2625
Energy	EUR/m3	0.116	0.082	0.093	0.141	0.083	0.069
Staff		0.737	0.214	0.208	1.016	0.323	0.228
Maintenance		0.143	0.042	0.047	0.191	0.074	0.050
Residues		0.056	0.025	0.024	0.061	0.058	0.031
Others		0.135	0.049	0.050	0.177	0.065	0.062
Volume		58,448	221,111	315,154	45,285	98,650	139,155
TSS	kg/m3	0.202	0.397	0.054	0.129	0.266	0.401
COD		0.424	0.806	1.198	0.284	0.553	0.789
BOD5		0.225	0.421	0.612	0.154	0.300	0.398
TN		0.045	0.082	0.102	0.031	0.063	0.070
TP		0.004	0.009	0.012	0.003	0.006	0.008

. This table breaks down the data for the various subgroups by range, showing various indicators such as P.E., costs (energy, staff, maintenance, waste, and other), and pollutants removed (TSS, COD, BOD₅, TN, and TP).

The objective is to analyze the behavior of smaller structures in order to corroborate the existence of economies of scale. The sample was divided into subgroups by specific volume ranges (Figure 1): 1.A (under 3000), 1.B (3000–6000), and 1.C (6000–10,000), with further breakdowns under 1.A into 1.A.1 (<1000), 1.A.2 (1000–2000), and 1.A.3 (2000–3000).

Regarding P.E., Table 6 shows the P.E. treated by each group. Group 1.A treats a total of 630 P.E., group 1.B 4400 P.E., Group 1.C approximately 8300 P.E. Subgroups 1.A.1, 1.A.2, and 1.A.3 break down the first group into smaller subgroups, treating 335, 1400, and 2600 P.E., respectively.

Table 6 shows the values per unit for each input (costs). For instance, energy costs per unit range between 0.116 in 1.A and 0.093 in Group 1.C. This decrease in costs implies that the facilities acquire larger dimensions according to the P.E. treatedand, consequently, achieve lower energy costs. Likewise, staff and maintenance costs follow the same trend; when the size of the WWTP is increased, a significant reduction is achieved in the different costs required by the wastewater treatment process.

As for the volumes of pollutants removed, these generally increase with the size of the group. Focusing on nutrients (TN and TP), WWTPs can increase the quantities of these pollutants removed. Regarding Group 1.A, it removes 0.045 and 0.004 kg/m³ of nitrogen and phosphorus, respectively. These values increase in Group 1.C to 0.102 and 0.012 kg/m³, which implies significant growth. Focusing on the group of smaller WWTPs, the nutrient removal quantities show a similar trend, where larger facilities not only reduce operating costs but also enhance the amount removed per cubic meter treated. In summary, larger groups (those with the largest volume) tend to have higher initial values, generally lower resource costs per unit, and higher production volumes.

The results of the EAT analysis for each of the subgroups are shown in Figure 2. In the case of the first block (Groups 1.A; 1.B; 1.C), the presence of economies of scale cannot be confirmed for those facilities that treat under 10,000 P.E. Group 1.A (<3000 P.E.) obtains a score of 0.12. Group 1.B (3000–6000 P.E.) achieves, on average, a slightly higher score of 0.30. Finally, Group 1.C (6000–10,000 P.E.) records a lower score of 0.22. The results show that there is no clear trend regarding the use of lower inputs in the process and higher quantities of pollutants being removed. These findings are corroborated by the Kruskal–Wallis test, showing that the differences between the scores are significant.

Continuing with the analysis of the smaller WWTPs, as shown in Figure 2, the following efficiencies are obtained: Group 1.A.1 (<1000 P.E.) obtains a score of 0.09; Group 1.A.2 (1000–2000 P.E.) records a score of 0.22, and, finally, Group 1.A.3 (2000–3000 P.E.) obtains a score of 0.24. However, the Kruskal–Wallis test corroborates that, with a significance of 0.479, the null hypothesis is accepted. This implies that there are no significant differences between Groups 1.A.1, 1.A.2, and 1.A.3: that is, for smaller WWTPs, the pattern of economies of scale is not met and, therefore, there is no clear relationship between the size of the WWTP and its efficiency.

The results reveal that the optimal size at which economies of scale are obtained in the wastewater treatment sector, taking into account the different types of pollutants removed, is at 10,000 P.E. In addition to obtaining lower efficiencies, smaller-scale facilities (according to P.E.) do not demonstrate any improvements in efficiency related to the size of the facility. In contrast, the largest sample analyzed (from 10,000 P.E.) shows a clear positive trend in terms of the efficiency obtained on average in each of the main groups (Groups 1, 2, 3, and 4). These findings suggest that, with regard to the design of and investment in WWTPs, small towns or dispersed settlements may consider centralized wastewater treatment services. Nevertheless, the economic and environmental impact of the centralization of these services should be assessed to determine the feasibility and the efficiency of the system. In this sense, the following section (Section 4.3) quantifies the potential savings for municipalities that might centralize these services.

4.3. Potential Operational Cost Savings in WWTP by Group

We assessed the cost-saving potential of WWTPs, considering the impact that an increase in the size of the WWTP would have on the wastewater treatment process. In this sense, using Equation (5) and the current operating costs (EUR/m³) of each WWTP group, the savings that would be generated by belonging to the next group (next size) were assessed. This involves an assessment of the increasing influence of size, according to the P.E. treated. The results obtained are shown in Figure 3, which presents the costs generated in the wastewater treatment processes by each group. Considering the influence of scale economies, the centralization of services enables a reduction in operational costs, with potential savings estimated by applying the cost structure of larger facilities. As a result, the WWTPs in Group 1 would achieve a cost reduction similar to the WWTPs in Group 2, while Group 2 would achieve further reductions, similar to the performance of Group 3, and so forth.

As shown in Figure 3, Group 1 (WWTPs treating less than 10,000 P.E.) incurs operating costs of around EUR 1.36/m³, since this group of WWTPs treats the wastewater of small populations. Group 2 is made up of facilities that treat between 10,000 and 20,000 P.E. Their operating costs are EUR 0.32/m³, on average. Group 3 includes those WWTPs that treat between 20,000 and 50,000 P.E., with lower than average operating costs, around 0.27 EUR/m³. This trend continues to decline, reaching EUR 0.19/m³ in Group 4 (>50,000 P.E.). This reduction in the operational costs, along with the increased amounts of pollutants removed (Kg/m³), allows for the quantification of the direct economic benefits of implementing economies of scale in the wastewater treatment sector. The reduction in operational costs would be possible if, during the design and investment stage, the sewage collection of multiple populations were centralized. In this regard, larger facilities would lower wastewater treatment costs, increasing plant efficiency through lower costs and higher pollutant removal rates.

Based on the results obtained, centralizing services in smaller-scale facilities (Group 1) would achieve an average cost of EUR 0.32/m³. Group 2 would achieve a reduction of approximately 14% and Group 3 of around 29%. These potential savings in operational costs must be quantified to assess the feasibility of centralizing the service. Some of the variables that ultimately justify sharing services depend on the distance between urban agglomerations and the topography; it is therefore necessary to analyze both the investment and the subsequent maintenance of the sewer networks that would enable their connection. The potential savings that could be obtained by using centralized sewage networks open up a new line of research. Specifically, research should consider the construction of new sewerage networks in areas where municipalities or urban agglomerations are close to each other.

Table 7. Potential cost savings for Group 1.

WWTP I.D.	P.E.	Flow Treated (m3/Annual)	Current Operational Costs (EUR/m3)	Total Costs (EUR/Year)	Potential Operational Costs (EUR/m3)	Potential Operational Costs (EUR/Year)
1	650	57,786	0.81	46,807	0.320	18,492
2	652	165,973	0.75	124,480		53,111
3	655	76,240	0.78	59,475		24,397
4	674	50,005	0.65	32,503		16,002
5	683	63,721	0.81	51,614		20,391
6	719	199,265	0.71	141,478		63,765
7	754	88,260	0.69	60,899		28,243
8	787	76,143	1.15	87,564		24,366
9	789	104,157	0.77	80,201		33,330
10	790	115,293	0.69	79,552		36,894
11	868	79,247	0.85	67,360		25,359
12	872	114,875	0.63	72,371		36,760
				904,305		369,708

shows an example of the potential savings that the facilities would obtain in the case of sharing services.

The operational costs associated with each of the facilities fluctuate between EUR 0.63/m³ and EUR 1.15/m³. The heterogeneity of the costs may depend on the flow rates and organic loads, as well as variables related to the facilities themselves. However, the geographical proximity of these facilities would make it possible to share services and take advantage of the economies of scale generated by a larger facility. In the case analyzed, all of the WWTPs treat approximately 10,000 P.E., so they would reach the operational costs of Group 2, of EUR 0.320/m³. More specifically, WWTP1 treats a total of 650 P.E. and has an annual flow of 57,786 m³. The operational costs amount to EUR 46,807/year, which implies approximately EUR 0.81/m³. WWTP1 treats a total of 652 P.E. and has an annual flow of 165,973 m³ at a cost of EUR 0.75/m³. The rest of the facilities have similar characteristics; however, by sharing services, they could achieve lower operational costs, around EUR 0.320/m³, which is the cost associated with larger facilities (Group 2). This represents a potential saving of more than 50%.

This example can be extended to larger facilities (Group 2).

Table 8. Potential cost saving for Group 2.

WWTP I.D.	P.E.	Flow Treated (m3/Annual)	Current Operational Costs (EUR/m3)	Total Costs (EUR/Year)	Potential Operational Costs (EUR/m3)	Potential Operational Costs (EUR/Year)
68	12,115	732,958	0.30	219,887	0.276	202,296
69	13,436	1,573,368	0.31	487,774		434,250
70	14,319	2,449,930	0.41	1,004,471		676,181
71	11,584	1,874,452	0.37	693,547		517,349
				2,405,650		1,830,075

shows four facilities that treat between 10,000 and 20,000 P.E., with operational costs between EUR 0.30 and 0.41/m³. The total costs amount to EUR 2,405,650/year, showing the potential to share services, integrating these facilities into a larger WWTP (Group 3). This situation would achieve lower operational costs on average, of approximately EUR 0.276/m³.

WWTP 68 treats a total of 732,958 m³ annually, with associated costs of EUR 0.30/m³; the total annual costs amount to EUR 219,887. WWTP 69 treats a total of 1,573,368 m³ with an operational cost of EUR 0.31/m³. Similarly, WWTPs 70 and 71 incur costs of EUR 0.41/m³ and EUR 0.37/m³, respectively. The possibility of integrating these services into a larger facility (Group 3) would achieve potential savings ranging between 8% and 32%, depending on the WWTP evaluated, placing the cost at around EUR 0.276/m³.

In this regard, a larger facility that consolidates geographically close populations would reduce the operating costs of smaller WWTPs. The economic quantification of these benefits allows decision makers to plan new investments while considering their cost-related impact. However, with any investment aimed at centralizing services, the construction of pipelines to transport wastewater to a single point must be considered. In this context, geographic proximity can justify such investments, prioritizing areas that require lower investment costs.

5. Conclusions

The existence of economies of scale in the wastewater treatment sector is closely related to efficiency assessments. This study assessed 362 WWTPs in the Valencian Community and analyzed their efficiency according to the plant size, measured in terms of P.E. In order to evaluate the plants’ efficiency, the efficiency analysis trees (EAT) model was used, which overcomes some limitations of previously used models. The main results obtained confirm that larger WWTPs tend to be more efficient than smaller ones, which is in line with previous studies; the results also indicated a strong correlation between efficiency and WWTP size. This study also included other variables that affect the economies of scale, such as the economic costs and the most common pollutants removed, as a proxy for the environmental benefits associated with avoiding the discharge of nitrogen and phosphorus into receiving water bodies. Including these variables in a scale economy assessments suggests that installing a WWTP for each municipality may not be the most efficient solution, due to the high operating costs of smaller WWTPs.

Furthermore, including the importance of pollutants into the assessment revealed the high influence that economies of scale have on the three groups of WWTPs analyzed. According to these results, the minimum optimal size for WWTPs is the capacity to treat 10,000 P.E., achieving high levels of pollutant removal and lower operational costs. The detailed disaggregation of smaller WWTPs confirmed that there is no significant economic benefit when smaller facilities are installed. These results represent a valuable contribution to the literature, as they provide a more accurate assessment of economies of scale in the wastewater treatment sector.

A direct consequence of these results is the modification of existing legislation. Specifically, there should be changes in the requirements for small municipalities, as defined in Directive 91/271/EEC. According to the results obtained for smaller communities, we recommend designing WWTPs according to the minimum size proposed here, thus ensuring their technical and economic efficiency. In the case of disaggregated WWTPs, we advise using joint facilities to ensure the optimal performance of the wastewater treatment process. In this regard, this study quantifies the benefits of implementing larger facilities to take advantage of lower operational costs. Thus, based on a sample of WWTPs with specific characteristics, decision makers can calculate the benefits of centralizing services in terms of the increased investment in pipelines to bring wastewater to a specific point for treatment and subsequent discharge. While it may be difficult to modify the existing small WWTPs, sustainable management strategies related to the use of soft treatment can be applied to help increase the efficiency of discharges while ensuring their efficacy. On the other hand, the possible self-generation of energy can help to reduce the economic costs of smaller facilities, ensuring their economic, environmental, and social sustainability.