Capacity Optimization of Rainwater Harvesting Systems Based on a Cost–Benefit Analysis: A Financial Support Program Review and Parametric Sensitivity Analysis

Abstract

Water risk has been continuously rising due to climate change and ownership disputes of water resources. Dam construction to secure water resources may lead to environmental problems and upstream immersion. On the other hand, rainwater harvesting systems can effectively supply water at a low cost, although economic efficiency of these systems is still debatable. This study evaluates financial support programs to promote installation of rainwater harvesting systems, increasing economic feasibility. Based on a cost–benefit analysis, capacity optimization methods are further suggested. A sensitivity analysis is performed to determine the relative importance among uncertain parameters such as inflation and discount rates. In doing so, priority factors to consider in the design of rainwater harvesting systems are ultimately identified. A net present value, although it is sensitive to the inflation rate, is shown to be more appropriate to estimate the economic efficiency of rainwater harvesting system, compared to the typical cost–benefit ratio. Because the high future value overestimates the economic feasibility of rainwater harvesting systems, proper inflation rates should be applied. All in all, a funding program to promote rainwater harvesting systems significantly increases the benefits. Thus, national financial support policies are recommended to ensure economic feasibility of rainwater harvesting systems.

1. Introduction

Many countries across the world have focused on improving water resource management to cope with the water stress caused by climate change, rapid urbanization, and ownership disputes over water resources. The easiest way to mitigate water stress is to construct dams. Dams have been constructed for centuries for various purposes, such as water security, flood control, and power generation. However, as the negative impacts of dams on the environment and hydrological cycles have been discovered, new strategies have been sought to secure water resources. Rainwater harvesting (RWH) systems offer a practical way to mitigate water stress. RWH systems provide direct runoff containment while simultaneously storing water, which can then be used for irrigation, household water supplies, etc. [1]. This favorable function of RWH systems can reduce public water demand concentrated around water sources and mitigate water stress [2].

Various factors influence the performance of RWH systems, including catchment area, rainfall patterns, and demand. However, as shown in previous studies that have focused on determining optimal size, the performance of RWH systems significantly depends on their capacity [3,4,5,6,7]. In general, the capacity of an RWH system is determined by the expected demand. However, when designed with the capacity to supply all of the expected demand, the installation and operation costs of RWH systems can be irrecoverable.

The capacity of RWH systems can be constrained by the reliability or runoff reduction rate, which is a representative index for evaluating the performance of RWH systems, to avoid over-design. Reliability refers the ratio of the number of periods that satisfy the demand to a given total period. The appropriate capacity for RWH systems can be determined by restricting the reliability. On the other hand, various other conditions also need to be considered to determine the appropriate capacity for RWH systems, which means that the optimization problem to determine the optimal capacity becomes more complicated. The following optimization methods have been used to solve the optimization problem to determine the optimal capacity of RWH systems: linear programming [8,9,10], non-linear programming [11,12], and the heuristic method [13,14,15,16,17,18].

The goal of avoiding over-designing the capacity of RWH systems is to maximize the revenue that they generate, i.e., the capacity can be determined using optimization models with objective functions that maximize the benefits. Typical methods for analyzing the economic feasibility of RWH systems include the net present value (NPV) and the benefit–cost ratio (BCR). The NPV is an indicator used in cost–benefit analyses, which represents the difference between the present value of benefits and the present value of costs over a period of time. The BCR refers to the ratio between the present value of benefits and the present value of costs over a period of time. In this regard, the authors of [19] found that the BCR has the limitation of not reflecting the size of a project’s funds. In contrast, the NPV has the limitation of missing investment-efficient opportunities.

The social discount rate and the inflation rate are key parameters in cost–benefit analysis because they represent the potential costs of resource use over time and they compare projected revenues and costs over different time periods [20]. The higher the social discount rate, the smaller the present value of any future benefit or cost. Accordingly, many previous studies have defined the social discount rate as the subject of sensitivity analysis [19,20,21,22,23,24,25,26,27]. At the same time, for the purpose of sustainable water resource management, Korean local governments have implemented financial support programs to promote the installation and operation of RWH systems. For example, in Incheon City, Korea, some of the installation costs for RWH systems are supported, and water utility billing relief is also provided. Therefore, these national or social support policies can affect the cost–benefit analysis of RWH systems.

Over the past decade, RWH systems have been recognized as inefficient facilities because of their low benefits relative to costs [24,28]. Accordingly, recent studies considered social benefits such as constant production cost reduction, dam-related cost reduction, excellent outflow reduction, environmental pollution reduction, heat wave reduction, and fine dust reduction as factors of benefits [29,30,31,32,33]. However, the social benefits are ambiguous to evaluate and are not real economic benefits that users who use RWH systems can actually feel.

The purpose of this study is to determine the capacity of the economic RWH systems, considering the real economic benefits that users who use rainwater utilization facilities can actually feel. At the same time, it is difficult to find cases in previous studies on the cost benefits of RWH systems [8,13,27,34,35,36,37,38,39,40,41] where BCR and NPV have been compared and evaluated or the effects of social discount rate and inflation on cost–benefit analysis have been examined. Thus, in this study, we explored which indicator would be the most appropriate for the cost–benefit analysis of RWH systems based on the results from a comparative evaluation of the BCR and NPV. In addition, via sensitivity analysis, we identified the priority factors to be considered when determining the capacity of RWH systems. We also analyzed the economic feasibility of RWH systems while considering financial support programs to review the necessity of national financial support policies. Because the sensitivity analysis needed a lot of computational time, an optimization model that could determine the economically feasible capacity of RWH systems using a particle swarm algorithm was further developed.

2. Methods

2.1. Financial Support Policies for Rainwater Harvesting Systems in South Korea

The annual precipitation in South Korea is 1.2 times (1207 mm) higher than the world’s average (about 1000 mm), but most precipitation is concentrated in the monsoon season (from 25 June to 25 September), and more than 60% of the annual precipitation occurs over the sea. Thus, the actual available water resources per person do not amount to much. Additionally, the authors found that Korea is among the few OECD countries that experience medium-high water stress [42].

The Korean Ministry of Environment enacted the Act for the Promotion and Support of Water Reuse (APSWR) to utilize water resources more efficiently and promote the sustainable use of water resources. In the APSWR, water reuse is defined as the recycling of treated wastewater (rainwater, sewage, etc.) for beneficial purposes, such as household water supplies and stream or lake maintenance. In addition, the APSWR states that local governments should make efforts to provide technical and financial support for the installation and operation of RWH systems to promote water reuse. The local governments in South Korea provide various financial support programs to promote the installation of RWH systems in accordance with the APSWR.

Table 1. The costs and benefits of the installation and operation of rainwater harvesting systems (USD 1 = KRW 1300).

City	Financial Support and Billing Relief
	Installation Costs	Water Utility Bill
		Water Supply Charge	Sewage Charge	Water Usage Charge
Seoul	90% of installation costs up to KRW 20 million (USD 15,384)	—	—	—
Incheon	Full or partial support	10% of RWU 1	10% of RWU	10% of RWU
Suwon	90% of installation costs up to KRW 10 million (USD 7692)	Some RWU	Some RWU	Some RWU
Sejong	Full or partial support	10% of RWU	30% of RWU	—
Busan	90% of installation costs up to KRW 10 million (USD 7692)	10% of RWU	—	10% of RWU

Note(s): ¹ RWU, rainwater usage.

shows representative local governments that implement financial support programs for the installation and operation of RWH systems. As shown in Table 1, local governments in South Korea cover most of the costs of installing RWH systems, although the scale of support varies depending on the financial situation. In South Korea, water utility bills consist of water supply charges, sewage charges, and water usage charges.

Table 2. A water utility bill for 1000 ${\mathrm{m}}^3$ of water in Incheon, South Korea (USD 1 = KRW 1300).

Classification	Pricing Bracket	Unit Charge	Calculation Details
	(${\mathbf{m}}^3$)	(KRW/${\mathbf{m}}^3$)
Water Supply Charge	1~300	870 (USD 0.67)	300 ${\mathrm{m}}^3\times$ 870 KRW/${\mathrm{m}}^3$ = KRW 261,000 (USD 201)
	More than 300	1120 (USD 0.86)	700 ${\mathrm{m}}^3\times$ 1120 KRW/${\mathrm{m}}^3$ = KRW 784,000 (USD 603)
Sewage Charge	1~50	490 (USD 0.38)	50 ${\mathrm{m}}^3\times$ 490 KRW/${\mathrm{m}}^3$ = KRW 24,500 (USD 18.8)
	51~100	510 (USD 0.39)	50 ${\mathrm{m}}^3\times$ 510 KRW/${\mathrm{m}}^3$ = KRW 25,500 (USD 19.6)
	101~300	1010 (USD 0.78)	200 ${\mathrm{m}}^3\times$ 1010 KRW/${\mathrm{m}}^3$ = KRW 202,000 (USD 155)
	301~500	1100 (USD 0.85)	200 ${\mathrm{m}}^3\times$ 1100 KRW/${\mathrm{m}}^3$ = KRW 220,000 (USD 169)
	501~1000	1130 (USD 0.87)	500 ${\mathrm{m}}^3\times$ 1130 KRW/${\mathrm{m}}^3$ = KRW 565,000 (USD 435)
	More than 1000	1160 (USD 0.89)	-
Water Usage Charge	Whole range	170 (USD 0.13)	1000 ${\mathrm{m}}^3\times$ 170 KRW/${\mathrm{m}}^3$ = KRW 170,000 (USD 131)
Total Water Utility Bill			KRW 2,432,000 (USD 1871)

presents a water bill for 1000 ${\mathrm{m}}^3$ of water over 1 month in Incheon, Korea. In the case of Incheon, 10% of rainwater usage is deducted from the total water consumption as a benefit for the operation of RWH systems. For example, if rainwater usage is 1000 ${\mathrm{m}}^3$ and the total water consumption is 1000 ${\mathrm{m}}^3$, the water utility bill would only charge for 900 ${\mathrm{m}}^3$.

2.2. Study Area

In 2007, Cheongna International City, Incheon, South Korea, planned to supply rainwater to public facilities using RWH systems to enhance its image as an eco-friendly city. The Korea Land and Housing Corporation divided Cheongna-dong into five drainage areas and designed a large-scale RWH system to supply household water for schools and parks, irrigation water for public parks, and water for road cleaning. In this study, a cost–benefit analysis was conducted on the RWH system that was planned for District 1 (Cheongna 1-dong, CN1-D), and the analysis results were compared and reviewed using design data [43].

The watershed area of the study area (CN1-D) is 73.92 ha and the catchment area of the RWH system is 19.24 ha (Figure 1). The Korea Land Corporation simulated the daily inflow to the RWH system over 10 years (1995–2004) via the long-term simulation of watershed runoff using the storm water management model from the EPA (EPA SWMM) and daily precipitation data from Incheon Weather Station (Figure 2) [43].

Figure 3 presents the average daily demand by month, with the calculation method as follows: the daily inflow of waterways was averaged with the daily evaporation over 30 years (1971–2000). The household water for parks was calculated by multiplying the area of the park, the green area per capita, and the water usage per capita per day. The household water for parks was designed to supply 10% of the target draft during winter (January, February, and December) and the monsoon season (from 16 June to 15 July). The household water for schools was calculated by considering the number of schools, grades, classes, students per class, teachers, and the water usage per capita per day. During the winter and summer vacations, the water usage of schools only corresponded to the number of teachers. The water for road cleaning was calculated by considering the road lengths and the water usage per 1 km.

2.3. Economic Analysis of Rainwater Harvesting Systems

2.3.1. Cost–Benefit Analysis

Cost–benefit analyses are conducted to establish the economic feasibility of public projects by comparing the investment costs to the expected benefits. In terms of the cost–benefit analysis results, a project is economically feasible when the expected benefits are greater than the investment costs. Several criteria are used in cost–benefit analyses to determine the economic efficiency of particular investment projects and whether they should be undertaken. Representative criteria for cost–benefit analysis include net present value (NPV) and the benefit–cost ratio (BCR).

The NPV measures the present value of the net benefits of the considered project. The NPV is formulated as follows:

$$\mathrm{NPV}=\sum _{t=0}^T\frac{B_t}{{(1+r)}^t}-\sum _{t=0}^T\frac{C_t}{{(1+r)}^t}$$

(1)

where $B_t$ is the benefit at time t, r is the discount rate at time t, $C_t$ represents the costs at time t, and T is the planning horizon year. A positive NPV means that the considered project could be implemented based on economic feasibility.

The BCR is the ratio of the present value of benefits to the present value of costs. The BCR is formulated as follows:

$$\mathrm{BCR}=\sum _{t=0}^T\frac{B_t}{{(1+r)}^t}/\sum _{t=0}^T\frac{C_t}{{(1+r)}^t}$$

(2)

When the BCR of the considered project exceeds 1, it means that the project is acceptable in terms of economic efficiency, i.e., the present value of the benefits of the project is greater than the present value of the costs.

The BCR should not be used to rank mutually exclusive options, as it can lead to rankings that are inconsistent with those obtained using the NPV [19]. For example, when the initial investment costs of project A are 50 and the present value of the total benefits is 100, the NPV and BCR of project A are calculated as 50 and 2, respectively. Conversely, when the initial investment costs of project B are 100 and the present value of the total benefits is 200, the NPV and BCR of project B are 100 (which is twice that of project A) and 2 (which is the same as that of project A), respectively. Therefore, the BCR has a limitation in terms of not reflecting the size of a project’s funds, whereas the NPV has a limitation in terms of missing investment-efficient opportunities. Thus, the appropriate criterion for each individual project should be selected after a comparative evaluation of these two criteria.

In order to economically determine the capacity of rainwater harvesting systems, the costs and benefits must be quantitatively determined. Costs are inevitably incurred throughout the operation of rainwater harvesting systems, including installation costs, labor expenses for the operation and maintenance of facilities, and electricity charges for the use of pumps. Benefits can include reduced water utility charges due to replacing water usage with rainwater usage.

Table 3. The costs and benefits of the installation and operation of a rainwater harvesting system (USD 1 = KRW 1300).

Classification	Category		Content	Remark
Costs	Installation, construction, equipment, etc.		350,000–450,000 KRW/${\mathrm{m}}^3$ (269–346 USD/${\mathrm{m}}^3$)	[44,45]
	Maintenance expenses (labor, electricity, etc.)		2% of installation costs	[44,45,46]
Benefits	Savings on water utility bills by replacing water usage with rainwater usage		Equivalent to the amount of rainwater usage	See Table 1
	Subsidies for installation costs		Up to KRW 10 million (UDS 7692)	Full or partial support in Incheon, South Korea
	Water Utility Bill Concessions	Water supply charge concessions	10% of water supply charges	Incheon, South Korea
		Sewage charge concessions	10% of sewage charges
		Water usage charge concessions	10% of water usage charges

shows the costs and benefits that were considered in the cost–benefit analysis of the RWH system in this study. In the cost–benefit analysis, the costs included the installation and maintenance costs of the RWH system. It was not easy to estimate the installation and maintenance costs of the RWH system because the installation and maintenance costs of rainwater facilities vary greatly depending on the area, local environment, operation method, and materials and structure. In accordance with the APSWR, which recommends the installation of RWH systems in densely populated areas and facilities with many users, large-scale rainwater facilities with concrete tanks are mostly installed underground in urban areas. Based on previous studies [44,45,46], construction of rainwater harvesting systems with concrete tanks costs 450,000 KRW/${\mathrm{m}}^3$ (346 USD/${\mathrm{m}}^3$), and the maintenance/operation costs are 2% of the construction costs (Table 3). In addition, Article 19 (1) 1 of the Enforcement Rules of the Local Public Enterprises Act of South Korea stipulates that the life span of water supply facilities (water intake, water supply, water purification, and drainage facilities) is 30 years; accordingly, the cost–benefit analysis period in this study was set as 30 years.

The benefits of the operation of the RWH system in this study included reductions in water utility bills by replacing water usage with rainwater usage, financial support for installation costs, and water utility billing relief (Table 3). The study area (CH1-D) is located in Incheon (Figure 1), and the considered benefits of the installation/operation of the RWH system were determined based on Incheon’s water billing standards (Table 2) and financial support programs (Table 1). The benefits of replacing water usage with rainwater usage were calculated as monthly benefits by applying rainwater usage to the water utility billing regulations. Incheon fully or partially supported the installation costs for the RWH system. Because the designed RWH system in CH1-D is a large-scale facility that supplies up to 282 ${\mathrm{m}}^3$/day, the local government was not able to support all of the installation costs. Thus, the financial support for the installation costs was determined to be KRW 10 million (USD 7692) by referring to the financial support from other local governments (e.g., Busan and Suwon offer up to KRW 10 million). Incheon reduces water supply charges, sewage charges, and water usage charges by 10% each when rainwater is utilized. In order to determine the reduced charges, the total water usage had to be determined. As it was not easy to quantify the total water usage in the design stage, we determined the total water usage as the difference between the rainwater supply and the target supply.

2.3.2. Discount Rate and Inflation Rate

The discount rate is the amount of returns used to discount future cash flows back to their present value [47]. The discount rate used in the cost–benefit analysis of public projects is the social discount rate.

Table A1. Estimates for social discount rates in countries across the world.

Institution/Country	Social Discount Rate	Remark
World Bank	Projects for developing countries: 10–12%
United States Environmental Protection Agency	Intergenerational discount rate: 2–3% (subject to sensitivity analysis)	[56]
European Union	Long-term projects/policies: 3%	[57]
United Kingdom	Standard: 3.5% Long-term projects of 30–125 years: 3%; 125–200 years: 2%; 200+ years: 1.5%	[58]
France	Standard: 4% Long-term projects/policies: 2%	[59]
Netherlands	Standard: 5.5% Projects/policies for climate change: 4%	[60]
Germany	Long-term projects/policies: 1%
Japan	Projects within 50 years: 4%
Australia	Standard: 7% Subject to sensitivity analysis: 3% and 10%	[20]
China	Short- and mid-term projects: 8% Long-term projects: less than 8%	[61]
India	12%
Republic of the Philippines	15%
South Korea	Standard: 4.5% Water resources projects of 0–30 years: 4.5%; 30+ years: 3.5%	[62]

shows estimates for the social discount rate in a number of countries across the world. In addition, the cost–benefit analysis results depend on the adopted social discount rate (i.e., r in Equations (1) and (2)). The social discount rate is a key parameter in the cost–benefit analysis because it represents the potential costs of resource use over time and compares projected revenues and costs (net cash flows) over different periods of time [20]. The higher the social discount rate, the smaller the present value of any future benefits or costs. Accordingly, many studies in the literature have defined the social discount rate as the subject of sensitivity analysis [19,20,21].

In the cost–benefit analysis of the RWH system, the inflation rate was also used as one of the uncertain parameters because the RWH system constantly incurred operating and maintenance costs after installation. Considering the inflation rate, the benefits $B_n$ and the costs $C_n$ in Equations (1) and (2) were determined as follows:

$$B_t=B_{install}+(B_{replace}+B_{relief})\times {(1+IR)}^t$$

(3)

$$C_t=C_{install}+C_{oper}\times {(1+IR)}^t$$

(4)

where $IR$ is the inflation rate, $C_{install}$ is the initial installation costs, $B_{install}$ is the subsidies for $C_{install}$. As B and C are incurred at the present point, they do not reflect the inflation rate. The expression $B_{replace}$ is the benefit of replacing water usage with rainwater usage. It is not easy to determine water usage in designing RWH systems. The rainwater supply should be replaced with a water supply to meet demand when rainwater is insufficient. Thus, this study assumed that the benefit of replacing water usage with rainwater usage is the value obtained by subtracting the utility for water usage from the utility when the water supply meets all demand. The expression $B_{relief}$ is the water utility billing relief, and $C_{oper}$ represents the operation and maintenance costs. Hence, Equations (1) and (2) can be modified as follows:

$$\mathrm{NPV}=\left(B_{install}+\sum _{t=0}^T\frac{(B_{replace}+B_{relief}){(1+IR)}^t}{{(1+r)}^t}\right)-\left(C_{install}+\sum _{t=0}^T\frac{C_{oper}{(1+IR)}^t}{{(1+r)}^t}\right)$$

(5)

$$\mathrm{BCR}=\left(B_{install}+\sum _{t=0}^T\frac{(B_{replace}+B_{relief}){(1+IR)}^t}{{(1+r)}^t}\right)/\left(C_{install}+\sum _{t=0}^T\frac{C_{oper}{(1+IR)}^t}{{(1+r)}^t}\right)$$

(6)

Thus, we analyzed the sensitivity of the social discount rate and the inflation rate to review the effects of changes in those rates on the cost–benefit analysis results for the RWH system.

2.4. Optimum Capacity of Rainwater Harvesting Systems Considering Benefit–Cost Analysis

2.4.1. Simulation Model for Rainwater Harvesting Systems

The economically feasible capacity of rainwater harvesting systems can be determined using cost–benefit analysis and mass balance analysis. Mass balance analysis considers the target draft, inflow (caught rainwater), and capacity of rainwater harvesting systems. We used the following reservoir mass balance equation for the mass balance analysis of the rainwater harvesting system [48]:

$$S{T}_t=S{T}_{t-1}+P{P}_t+Q{F}_t-R_t-E{V}_t$$

(7)

where $S_t$ is the reservoir storage at the end of time period t, $P{P}_t$ is the precipitation amount on the reservoir surface, $Q{F}_t$ is the reservoir inflow over period t, $R_t$ is the reservoir release, and $E{V}_t$ is the evaporation.

Unlike reservoirs, underground rainwater harvesting systems are sealed, so there is little precipitation or evaporation on the surface. Thus, $P{P}_t$ and $E{V}_t$ could be ignored in Equation (7). However, because rainwater harvesting systems involve rainwater usage (yield, $Y_t$), it was necessary to divide the release ($R_t$) in Equation (7) into yield ($Y_t$) and spill ($EX{R}_t$).

Table 4. The equations for the simulation model for rainwater harvesting systems.

Classification	Simulation Model
	Condition	Equation
Mass Balance Equation	—	$S{T}_t=S{T}_{t-1}+Q{F}_t-Y_t-EX{R}_t$
Yield Determination	$S{T}_{t-1}+Q{F}_t\ge T{D}_t$	$Y_t=T{D}_t$
	$S{T}_{t-1}+Q{F}_t\le T{D}_t$, and $S{T}_{t-1}\ne 0$	$Y_t=S_{t-1}$
	$S{T}_{t-1}+Q{F}_t\le T{D}_t$, and $S{T}_{t-1}=0$	$Y_t=0$
Spill Determination	$S_{t-1}+Q{F}_t-Y_t\ge Cap$	$EX{R}_t=S{T}_{t-1}+Q{F}_t-Y_t-Cap$
	$S_{t-1}+Q{F}_t-Y_t<Cap$	$EX{R}_t=0$

presents the RWH system mass balance equation that was derived by modifying the reservoir mass balance equation, as well as the conditions and equations for determining the yield and spill of the RWH system. In Table 4, $T{D}_t$ is the target draft in period t.

2.4.2. Optimization Model to Determine the Capacity of Rainwater Harvesting Systems Considering Benefit–Cost Analysis

In this study, we aimed to analyze the sensitivity of the parameters (discount rate and inflation rate) that were determined in the cost–benefit analysis and evaluate the Korean policies that promote the installation/operation of rainwater harvesting systems. In the cost–benefit analysis of RWH systems, the NPV and BCR are calculated differently depending on the capacity of the selected RWH system, even when constant discount rates and inflation rates are applied. As it was inefficient to determine the optimum capacity to maximize the NPV or BCR by generating various capacity scenarios for the considered RWH system, we designed a simple optimization model that could determine the optimum economically feasible capacity of the RWH system using a PSO.

PSO algorithms are meta-heuristic methods that are based on a stochastic optimization technique [49,50]. Particle swarm algorithms are developed by mimicking the social behavioral modalities of the migration of bird populations. The concept of PSO algorithms involves accelerating each particle toward its $Pbest$ and $Gbest$ locations in each iteration, as shown in Figure 4 [51].

In Figure 4, $Pbest$ means that particle i memorized the best position that it has ever found, $Gbest$ is the best position of the swarm (particle group), and $x_i^k$ is the current position of the the ith particle x in iteration k. The next position of the $x_i^{k+1}$ particle is formulated as follows:

$$x_i^{k+1}=\omega x_i^k+c_1(Pbest-x_i^k)+c_2(Gbest-x_i^k)$$

(8)

where $\omega$, $c_1$, and $c_2$ are weighting factors. The best static parameters are recommended as $\omega =0.72984$ and $c_1=c_2=2.05$ [52].

Figure 5 shows a schematic diagram of a PSO searching for the economically feasible capacity of an RWH system. The objective function of the optimization problem is to maximize the NPV or BCR and is formulated as follows:

$$Maximizez=\mathrm{NPV}\mathrm{or}\mathrm{BCR}$$

(9)

where the NPV/BCR is calculated from the simulation results of an RWH system using the capacity determined by the PSO. Thus, in the optimization problem, the decision variable is the capacity of the selected RWH system. In this study, the PSO evaluated the NPV/BCR that was calculated using the simulation model to search for the optimum capacity of the RWH system to maximize the NPV/BCR.

In our RWH system simulation model, the NPV and BCR were determined as follows. The RWH system was simulated on a daily basis from 1995 to 2004 (10 years). The water utility bills were calculated on a monthly basis for those 10 years using the simulation results of the RWH system to determine the benefits. The NPV and BCR were calculated for 30 years, i.e., the durable life span of RWH systems with concrete tanks, using the previously determined average monthly water utility bills.

3. Results

3.1. Water Balance Analysis of the Rainwater Harvesting System

In this study, we performed a water balance analysis for the capacity of the considered RWH system using the equations in Table 4. The capacity of the RWH system used in the water balance analysis was 1961 cases, comprising 5 ${\mathrm{m}}^3$ units from 200 ${\mathrm{m}}^3$ to 10,000 ${\mathrm{m}}^3$.

Figure 6 shows the results for the temporal and volumetric reliability of the capacity of the RWH system. Reliability has been applied in numerous previous studies because it is easy to use and shows the interrelationship between yield and target draft. Reliability is divided into temporal reliability ($Re{l}_{temp}$) and volumetric reliability ($Re{l}_{vol}$). Temporal reliability ($Re{l}_{temp}$) represents the ratio of the period when the yield is satisfied with the target draft to the total period T. Temporal reliability is formulated as follows:

$$Re{l}_{temp}=\frac{1}{T}\sum _{t=1}^{T}B{V}_t$$

(10)

where $B{V}_t$ is a binary variable, which is 1 or 0. If the yield ($Y_t$) satisfies the target draft ($T{D}_t$) for period t, $B{V}_t$ is 1; otherwise, it is 0. Volumetric reliability is expressed as:

$$Re{l}_{vol}=\sum _{t=1}^T\frac{Y_t}{T{D}_t}$$

(11)

As shown in Figure 6, the temporal and volumetric reliability increased as the size of the RWH system increased, whereas the uplift rate for the reliability decreased as the capacity increased. These results were representative of the meteorological characteristics of South Korea, where most of the annual precipitation is concentrated in the summer. Further research on the water rationing strategies of RWH systems is needed to increase reliability.

3.2. Comparative Evaluation of Two Cost–Benefit Analysis Methods

To compare the cost–benefit analysis results using the NPV and BCR, we assumed that the RWH system operated for 30 years, which is the durable life span of water resource facilities. The NPV and BCR were determined under the following conditions: an inflation rate of 4.5% and a discount rate of 3.4%.

Figure 7 shows the NPV and BCR of the capacity of the considered RWH system. The BCR in Figure 7a peaked at a capacity of 285 ${\mathrm{m}}^3$ and continued to decrease as the capacity increased. As shown in Figure 7b, the net present value increased as the capacity increased, peaked at a capacity of 1105 ${\mathrm{m}}^3$, and then gradually decreased as the capacity continued to increase. According to the cost–benefit analysis of the RWH system using BCR, the return on investment was 2.89 times the initial investment when the capacity of the RWH system was 285 ${\mathrm{m}}^3$. However, as shown in Figure 7, when the capacity was 285 ${\mathrm{m}}^3$, the temporal reliability was 16.7% and the volumetric reliability was 7.3%, indicating that the operational efficiency of the RWH system was significantly unreliable. On the other hand, in the case of NPV, the maximum profit of KRW 581 million (USD 446,923) was generated when the capacity was 1105 ${\mathrm{m}}^3$. The capacity of 285 ${\mathrm{m}}^3$, which was found to generate the maximum return in the cost–benefit analysis using BCR, generated KRW 366 million (USD 281,538) in the cost–benefit analysis using NPV, which was a much lower amount of profit.

These results showed that the BCR had significant drawbacks. The BCR can be misleading when two compared projects incur different costs (i.e., when the costs vary depending on the capacity of the selected RWH system). The BCR having a value greater than 1 indicates that a project is worthwhile in the absolute but does not provide a basis for comparison to other projects [53], i.e., the BCR has the disadvantage of not reflecting the scale of investment. For example, a project that costs USD 100 could generate a more significant increase in real wealth than a project that costs USD 10 but the benefit–cost ratio may be lower. In general, in the case of public works, because the scale of the projects is relatively large, it would be more appropriate to conduct cost–benefit analysis using NPV rather than BCR.

3.3. Analysis of the Effectiveness of Financial Support Programs

Unlike existing long-distance water supply systems, which purify and supply water from rivers or reservoirs, RWH systems have the advantage of reducing energy consumption from water transportation. In addition, RWH systems can reduce water stress by reducing water intake from existing sources of water, such as rivers, and can be used as tools to cope with drought. Thus, in countries with high water stress, it is necessary to promote the installation of RWH systems. This section presents the effects of governmental financial support policies that promote the installation and operation of RWH systems.

Figure 7 shows the results from the cost–benefit analysis reflecting Incheon’s financial support programs, as presented in Table 1. In contrast, Figure 8 shows the results of our cost–benefit analysis excluding governmental financial support programs. As shown in Figure 8b, the maximum profit generated without financial support programs was KRW 279 million (USD 214,615), which was about 52% smaller than the profit generated with help from financial support programs, as shown in Figure 7b. It can be seen in Figure 7b that the increase in the NPV with the increase in capacity was greater than the increase in the NPV with the increase in capacity (Figure 8b). This result meant that the benefit of reducing water utility bills by 10% of rainwater usage generated a greater profit than was gained from governmental financial support for the installation costs of the RWH system.

In Figure 7b, the maximum profit was generated when temporal reliability was 32.0% and volumetric reliability was 25.1%. This result may not be enough to indicate that it perfectly corresponds with the purpose of RWH systems. The fact that NPV is larger than zero means that a project has economic feasibility. In other words, RWH systems may be designed as the capacity analyzed that NPV exceeds 0. In excluding financial support, the marginal capacity, which is the largest capacity that can be indicated as having economic feasibility, is 2770 ${\mathrm{m}}^3$, reliability was 48.6%, and volumetric reliability was 46.0% (Figure 8b). In the subsidized case, the marginal capacity is 3385 ${\mathrm{m}}^3$, reliability was 52.9%, and volumetric reliability was 51.4% (Figure 7b). Therefore, governmental financial support for the installation and billing relief system may contribute to the economic feasibility and operational efficiency of RWH systems.

3.4. Sensitivity Analysis of the Discount Rate and Inflation Rate

The NPV evaluates the economic feasibility of a project by converting future value into present value. As shown in Equation (1), the general NPV formula includes the discount rate as the parameter with uncertainty. In the case of the NPV formula for RWH, not only the discount rate but also the inflation rate for water and electric utility bills should be considered (Equation (5)). Thus, this study conducted scenario-based uncertainty modeling to evaluate how the inflation and discount rates uncertainties affect the optimization model output, i.e., the maximum net present value and corresponding capacity.

In order to assess the sensitivity and uncertainty of the model to inflation and discount rates using scenario-based uncertainty modeling, the probability distribution to be sampled should be determined. This study determined the probability distribution of the inflation and discount rate based on the collected data from the literature [54] and the Korean Statistical Information Service [55]. The probability distribution of the inflation rate was determined and fitted using the historical records for the inflation rate of utility prices in South Korea from 1989 to 2020. Figure 9 presents the inflation rates of consumer and utility prices in South Korea from 1989 to 2020 [55]. A set of distributions is tested to find the best one fitted to the historical inflation rate by year. This study considered the following probability distributions: normal, lognormal, gamma, t, and beta distributions. To find the best distribution fitted to the inflation rate, the Kolmogorov–Smirnov (K–S) test is applied. Figure 10a shows a histogram and five probability distribution curves calculated from the parameters estimated by the maximum likelihood estimation from the historical inflation rate. In Figure 10a, the p-value shown in the legend is a significant probability. The higher the p-value, the stronger the evidence that the null hypothesis should be adopted. The t distribution was found to be the best fit for the inflation rate of the water, electric, and gas utility prices. The light red region represents the 90% confidence interval to the fitted t distribution. This study set the range of the inflation rate for our sensitivity analysis as −1.1% to 7.7%, with a 90% confidence interval.

The appropriate social discount rate for public environmental projects in Korea is suggested based on the results of a survey among 114 experts in the economic field [54]. The survey period was from 26 October to 3 November 2015, and according to the survey, the appropriate social discount rate for environmental public projects in Korea was 3.26% on average (standard deviation = 1.06%). As the available data were limited to the mean and standard deviation [54], this study assumed that the discount rate followed a normal distribution and determined the range for the sensitivity analysis. Figure 10b presents the normal distribution with a mean of 3.26 and a variance of 1.06, and the light blue region represents the 90% confidence interval. Thus, this study set the range of the discount rate for the sensitivity analysis as 1.5% to 5.0%, with a 90% confidence interval.

Figure 11 illustrates the effect of the maximum NPV and capacity due to uncertainty in the inflation rate. The maximum NPV and corresponding capacity were derived by the optimization model presented in Section 2.4.2. In Figure 11a, the 90% confidence interval (light blue shaded region) represents the maximum NPV derived by inputting the inflation rate in the range from −1.1% to 7.7%. The 50% confidence interval (deep blue shaded region) was derived from the inflation rate in the range from 1.7% to 4.9%.

As shown in Figure 11a, it can be seen that the uncertainty of the derived maximum NPV is more affected by the uncertainty of the inflation rate as the discount rate decreases. In particular, at an inflation rate of more than 4.9%, which corresponds to the 75th–90th percentile, the volatility of maximum NPV is more significant than the less than 75th percentile. These results can be figured out via Equation (5). The higher inflation rate and lower discount rate contribute to the benefit growth in maximizing NPV. In other words, as the present value of the future cash flow increases, RWH can be evaluated as a profitable facility.

Figure 12 presents the relationship between the two variables with uncertainty and the dependent variable. Figure 12a presents the maximum NPV figures determined by the optimization model according to changes in the social discount rate and inflation rate. Figure 12b shows the different capacities of the RWH system that correspond to the maximum NPV figures in Figure 12a. In order to present the results in Figure 12 more clearly, they are also presented numerically in

Table A2. The results of our sensitivity analysis of inflation and discount rates: the maximum net present value.

Classification		Inflation Rate
		−1.1%	−0.2%	0.8%	1.7%	2.6%	3.5%	4.5%	5.4%	6.3%	7.2%
Discount Rate	1.5%	447	550	677	837	1051	1318	1648	2053	2545	3144
	1.9%	412	508	623	769	958	1204	1504	1877	2328	2878
	2.2%	380	468	575	707	876	1099	1374	1715	2131	2636
	2.6%	352	432	531	651	803	1002	1256	1566	1950	2414
	3.0%	326	398	490	601	739	915	1148	1432	1784	2211
	3.3%	302	369	453	555	680	839	1048	1310	1631	2025
	3.7%	280	342	418	513	627	771	957	1198	1491	1854
	4.1%	260	317	386	474	580	711	877	1096	1365	1697
	4.4%	242	294	358	438	536	655	805	1001	1250	1552
	4.8%	225	272	332	404	496	605	742	915	1144	1421

(Figure 12a) and

Table A3. The results of the sensitivity analysis of inflation and discount rates: capacity.

Classification		Inflation Rate
		−1.1%	−0.2%	0.8%	1.7%	2.6%	3.5%	4.5%	5.4%	6.3%	7.2%
Discount Rate	1.5%	800	837	1015	1272	1602	1681	1895	1992	2093	2235
	1.9%	771	840	913	1054	1474	1611	1801	1942	2054	2222
	2.2%	669	825	841	1032	1280	1597	1682	1917	1997	2096
	2.6%	641	804	836	957	1122	1578	1638	1871	1951	2090
	3.0%	633	720	823	843	1051	1344	1632	1732	1935	1997
	3.3%	617	674	823	844	1014	1267	1607	1669	1897	1986
	3.7%	624	635	773	826	894	1057	1470	1633	1786	1929
	4.1%	537	628	668	823	840	1045	1280	1605	1692	1916
	4.4%	502	622	680	820	858	987	1102	1579	1647	1864
	4.8%	490	611	631	774	832	847	1043	1346	1612	1704

(Figure 12b).

In Figure 12, it can be seen that the maximum NPV and capacity increased as the inflation rate increased. Conversely, the maximum NPV and capacity increased as the discount rate decreased. As the future value increased, the maximum NPV occurred at larger capacities. As an example, as shown in Figure 7b, the inflection point of the NPV curve occurred because the operation and maintenance costs of the RWH system were greater than the benefits. Installation costs are fixed costs that do not increase or decrease at the present point, and maintenance and operation costs depend on installation costs. As the benefits of using the RWH may be increased by the future value increases, the maximum NPV may appear at a larger capacity. On the other hand, as shown in Figure 6b, the uplift rate for reliability decreased as the capacity increased. That is, the maximum NPV may not be increased indefinitely because the efficiency of rainwater utilization facilities decreases as capacity increases.

4. Discussion

In this study, we suggested a method to determine the economically feasible capacity of an RWH system based on a cost–benefit analysis. As the social and environmental benefits are ambiguous to evaluate and are not real economic benefits, this study set the benefits as the real profits: the governmental financial support for the installation of RWH, profits from replacing water usage with rainwater usage, and water utility billing relief. The costs were defined as installation and operation costs of the considered RWH systems.

First, the cost–benefit analysis was performed using the NPV and BCR; then, NPV was indicated as an appropriate method to analyze the economic feasibility of RWH systems. BCR indicates economics as a ratio of costs to benefits. BCR tends to derive a low-investment alternative because it presents the most efficient way to invest as the best alternative. Under the same conditions in Section 3.2, BCR and NPV present 285 ${\mathrm{m}}^3$ and 1105 ${\mathrm{m}}^3$ as the most economic capacity of the RWH system. When the capacity is 285 ${\mathrm{m}}^3$, temporal reliability is 16.7%, and net profit is KRW 366 million (USD 281,538). The 16.7% of reliability means that rainwater supply is available about 61 days a year, indicating that the operational efficiency is significantly unreasonable in the study area. In the case of NPV, the maximum net profit is derived as KRW 581 million (USD 446,923), which is a larger net profit than the case of BCR. Therefore, the BCR may be inappropriate for applying cost–benefit analysis for a large-scale RWH, because BCR is not reflecting the investment scale.

Second, we reviewed the effectiveness of Korean governmental financial support to encourage the installation of RWH via cost–benefit analysis. Incheon in South Korea provides financial support, such as subsidies for installation costs and water utility billing relief. As a result of comparing the maximum NPV according to the presence or absence of financial support, the maximum net profit increases by 52%. In addition, the marginal capacity, which is the maximum capacity of RWH to generate revenue, increases from 2770 ${\mathrm{m}}^3$ to 3385 ${\mathrm{m}}^3$ with financial support. That is, governmental financial support can contribute to the economic feasibility of RWH but also the installation and operation of a more effective RWH. The above results show as an example how much actual worth a support program presented only as a percentage can actually generate.

Finally, we present the results of the uncertainty modeling and sensitivity analysis for the inflation and discount rates that are uncertain parameters of NPV. In addition, as it takes time to determine the maximum NPV for the scenarios of inflation and discount rates, the simple optimization model using the PSO is presented to reduce the computation time in Section 2.4.2. As a result, as the present value of the future value increases, the volatility of the derived maximum NPV and capacity of RWH significantly increases. Giving high future value can lead to an increase in the capacity of the RWH corresponding to the maximum NPV, which may lead to concern about over-design of the RWH. Thus, in the case of cost–benefit analysis for RWH, caution is needed when determining the inflation and discount rates.

The appropriate discount rate for long-term environmental projects, such as projects to cope with climate change, and the appropriate rate for other countries, are presented in Table A1. Most developed countries recommend an appropriate discount rate of less than 3% for long-term projects. The conservative tendency to apply low discount rates to long-term projects reflects uncertainty about the future. In contrast, in the case of developing countries, a high discount rate of more than 10% is announced for economic growth and infrastructure improvement.

5. Conclusions

This study suggested a method to evaluate the economically feasible capacity of an RWH system using cost–benefit analysis. The cost–benefit analysis was performed using the NPV and BCR; then, an appropriate method was suggested to analyze the economic feasibility of a selected RWH system. Because the potential rainwater usage varies depending on the capacity of the RWH system in question, a simulation model was developed based on the reservoir mass balance equation, which could determine the yield of the considered RWH system, and the costs and benefits were defined. We defined the costs as the installation and operation costs of the considered RWH system. The benefits were the profits from replacing water usage with rainwater usage, the governmental financial support for the installation and operation costs of the RWH system, and water utility billing relief. South Korean policies regarding financial support for the installation and operation costs of RWH systems were also presented in this study, and the effects of those policies were reviewed using cost–benefit analysis. In addition, the key parameters to consider in cost–benefit analysis were identified via a sensitivity analysis.

As a result, the NPV, which could derive the threshold as the capacity increased, was determined to be appropriate for analyzing the economic feasibility of RWH systems. Because the BCR presents the economic feasibility of a project as a ratio, it was challenging to derive an absolute value for economic feasibility using this criterion. Thus, it was determined to be inappropriate to conduct cost–benefit analysis using BCR for relatively large-scale RWH systems. When comparing the maximum NPV for cases with and without help from governmental financial support programs, we found that the maximum NPV with financial support programs was nearly twice as large as that without. This study could be meaningful as an economic analysis case that considered the benefits for the users of RWH systems from the current financial support programs in South Korea. The results from the sensitivity analysis, in the case of cost–benefit analysis for RWH, showed that caution is needed when determining the discount and inflation rates because the future value may change significantly depending on the discount and inflation rates used.

In this study, the results were derived through the simulation of a designed RWH system in CN1-D, Incheon, South Korea. However, different results could also be derived, because RWH systems are designed in various ways depending on the purpose of their use, the installation location, and the support policies in the relevant country. Thus, the method proposed in this study needs to be verified through application to various meteorological characteristics.