Risk analysis of combined rainwater detention and pumping systems

Storm sewer systems (SSSs) are complex, with many hydraulic, mechanical and electrical components which may fail during natural extreme events, changing environmental conditions (including urban development), or simply due to poor maintenance. System complexity and management are important and still debated concepts within the framework of SSS risk analysis. A new probabilistic model for a conceptualized urban SSS, including a storage unit (SU) and a pumping station (PS), shows how single-component risk analysis can be extended to complex SSSs and demonstrates the combined effect of key design parameters (SU volume, detention time, prescribed outflow discharge) and management strategy on the overall SSS risk of failure. The risk of failure evaluated in a typical case study, demonstrates that economic restrictions leading to the loss of reliability of PS elements and the lack of redundant mechanical elements represent a major threat to SSSs and suggests a new risk-based definition of ’extended’ SU.


Introduction
As the population increases around urban areas and land use gives rise to changes in the disturbed water cycle in newly developed and developing areas, undesirable effects are produced on surface and subsurface water bodies (Salas andObeysekera 2014, Fletcher et al. 2008): increasing imperviousness may reduce water-table recharge whereas run-off volumes and peak flows increase. As a consequence, urban subcatchments and streams are more prone to flooding (Villarini et al. 2009, Vogel et al. 2011, Leopold 1968). Urban water systems dynamically co-evolve with society, enabling rapid growth and adaptation to environmental changes (Urich and Rauch 2014). Among structural technologies, best management practices (BMPs) help to restore pre-development run-off conditions. BMPs detain run-off volume and release it at a slower rate into downstream water drainage systems or water bodies, allowing for water quality control (Salas and Obeysekera 2014, Villarini et al. 2009, Hejazi and Markus 2012. As decentralized BMPs are currently implemented on a local scale, their ability to provide water quality and quantity control is limited by the amount of water managed in a "non-sustainable" way (Roy et al. 2008).
Development commonly reduces land availability in urban areas, so storage units (SUs) must be constructed below ground, and pumping stations (PSs) must ensure that rainfall volumes are delivered above ground for treatment, sent to an already existing body of water, or to the downward drainage system. The capability of combined SU and PS systems to deliver a service to customers (i.e. releasing at sufficiently slow rates volumes of water from SU during a storm event with no overflow, and improving water quality) typically depends on hydrology and good management. The latter is influenced by human decisions in response to changes in the economy, innovation technology, and non-stationary environmental conditions. Poor management may be a consequence of changing economic conditions and political actions leading to cuts in technical personnel and lowering of the priority for repairs in cases of failure of repairable mechanical and electrical parts.
A wide range of urban stormwater models is available for designing storm sewer systems (SSSs) and predicting the effects of low impact structural measures, including ponds, stormwater tanks and bio-retention devices, which are typical BMPs with SU. Their potential uses cluster around planning or preliminary design, when sustainability principles start to be converted to solutions (Elliott and Trowsdale 2007). Uncertainty is not systematically included in problem solving neither risk within the society is well understood (Hlavinek et al. 2009). The interplay of human and technical systems is rarely taken into account by conventional stormwater models, even though technical, socioeconomic and ecological issues need joint consideration and integration (Rauch et al. 2005). "There is need for better methods to address the complex interactions of urban infrastructure systems, physical environment, level of services and social factors" (Hlavinek et al. 2009).
Models of sewer pumping stations which assess the deterioration (Wirahadikusumah et al. 2001, Korving et al. 2006) and risk of failure of system components (Jin and Mukherjee 2010, Chughtai and Zayed 2008, Hahn et al. 2006, such as parallel pumping units Salandin 2014, Ursino et al. 1996), or others which examine the hydraulic functioning of the system (Bennis et al. 2003, Ermolin et al. 2002 rarely take into due account random flow rates resulting from urban rainfall drainage and detention. The probability approach (Adams and Papa 2000) provides analytical equations to estimate the risk of failure of components of the urban drainage system based on continuum rainfall records Adams 1999, Guo andAdams 1998). Although many studies have applied a probabilistic approach to BMP overflow risk analysis (Zhang and Guo 2014, Zhang and Guo 2013, Guo et al. 2012, Balistrocchi et al. 2009, Guo and Baetz 2007, Guo 2001, and to insufficient treatment risk analysis (Park et al. 2015, Park andRoesner 2012), failure-prone PS and SUs, which are parts of complex SSSs, have not yet been considered in the same context. A new probabilistic model was formulated for a conceptualized urban SSS. Risk analysis focuses on the risk of failure of both SU and PS, so that the complexity of the SSS involves the interplay of two failure-prone elements on a local scale. The simple model described here represents a first step toward (1) the definition of risk models for complex SSS systems, which is an important and still unresolved task in sustainable water management (Urich and Rauch 2014) and (2) the quantitative evaluation of socio-hydrological systems, which is a new topic, currently under debate (Montanari 2015, Sivapalan 2015.The effects of PS management policies are analyzed here in a range of cases based on literature data. Gathering data from a variety of sources, adapting socio-hydrological models to what is available, communicating and using model outcomes in policy applications are challenging and worth exploring aspects in water resources research and management (Sivapalan 2015, Sivapalan et al. 2012).

Conceptual model of a storm sewer system (SSS) and failure scenarios
In a conceptualized SSS, detention is achieved by a storage capacity (SU) and a pumping station (PS) ensuring flow continuity to the final destination, because gravity drainage is not possible (Figure 1). The risk of SSS failure is expressed as a function of the risk of failure of the two key elements -SU and PS. These are connected in series and are thus both "necessary", so that, if the incoming rainfall volume exceeds the storage capacity of SU or if the peak inflow is greater than the capacity of the PS, the SSS fails. Two set-ups are examined here: above-ground SU downstream of the PS Figure 1(a) and SU upstream of it (Figure 1(b).
The storage capacities of the urban catchment, drainage network and PS are conservatively neglected. Therefore, the combined SU and PS systems may fail in the case of an exceptional storm event in which: (1) its flow rate exceeds the PS discharge capacity (situation of above-ground SU); (2) its volume exceeds the SU volume; (3) two exceptionally violent storm events occur with a short inter-arrival time and the volume of the second event exceeds the available SU volume. In addition, the SU, which provides flow control and water quality control, may also fail if the rainfall volume leaves the SU before treatment is complete, i.e. after a residence time shorter that the design detention time.

Rainfall probability distribution
The analytical expression for the risk of SSS failure is obtained under the assumption that the rainfall depth (h), the duration of the rainfall event (τ ), and the length of the dry period in between two rainfall events (t) are three independent random variables with exponential probability density functions: f (h), f (τ ) and f (t) according to Adams and Papa (2000). (1) In equation (1) ζ is the inverse of the expected value of rainfall depth, in equation (2) λ is that of rainfall duration and in equation (3) ψ is that of the expected value of rainfall inter-arrival.
Characterizing the frequency distribution of rainfall is based on statistical analysis of rainfall records at gauge stations and on proper definition of inter-event time (IETD), which is required to isolate storm events from continuum time series (Adams and Papa 2000).

Pump and PS reliability
The reliability of a pump is defined as the probability that it will not fail during time interval [0, t]. The probability that it will fail at time t, provided that it was operating in [0, t], is α(t)·dt, where α(t) is the pump's failure rate. If the pump is not working at time t = 0, the probability that the component repairs are completed before time t is β(t) · dt, where β(t) is the pump repair rate (Mays and Cullinane 1986). In general, the rates α(t) and β(t) may vary during the lifetime of the PS.
However, when α(t) and β(t) are assumed to be timeindependent, mathematical tractability becomes much simpler, and the reliability a of each pump may be evaluated as the steady state solution of a Markov process with constant rates α and β (Henley and Kumamoto 1992), according to the following equation expression: Conversely, the risk of pump failure f is: The PS is modeled as a complex system with n identical components (pumps), which are connected in parallel, each pump having discharge capacity Q p and reliability a, given by equation (4). Backup capacity is provided by r additional pumps with capacity Q p and reliability a, which are redundant in standby. The probability that n + r − i of n + r pumps are not working at time t and thus, that the discharge capacity of the PS is i Q p may be expressed by a binomial distribution (Ursino and Salandin 2014), according to equation (6):

Risk of SU overflow and risk of insufficient water treatment
The components of a conceptualized SSS with above-ground SU Figure 1(a) involve the following: (i) the network transferring flow Q to (ii) a PS equipped with n + r pumps, each with discharge capacity Q p and reliability a, and (iii) the SU with a regulated outlet to downward drainage, further treatment or a water body. The SU's maximum detention volume and detention time are V s and T d . Stormwater is released from the SU at prescribed rate Q 0 after each storm event, either for as long as it contains rainwater, or a new rain event occurs. Even though, in reality, runoff detained in the SU can be conveniently released also during run-off inflow, here the resulting additional storage volume Q 0 τ is conservatively neglected, as well as the storage capacity of the network and the PS. PS inflow Q is evaluated under the following assumptions: (1) the rainfall-runoff transformation is stationary and deterministic, (2) the initial condition of the drainage catchment may be neglected and (3) Q may be estimated by the rational method: where S is the contributing catchment area, ϕ is the runoff coefficient, and h and τ are rainfall depth and duration.
Assuming that the SU is full at the end of the first of two consecutive rain events, the risk of overflow for a SSS with PS and above ground SU may be written as a function of four dimensionless groups: k ζ , k λ , k ψ,0 and k T ,a (Table 1).
Figure 1(b) shows the sequence of components of SSS with below ground SU as follows: (i) the network, which transfers runoff volume V = ϕ S h to (ii) the below-ground tank with storage volume V s and (iii) the PS, which draws water to downward drainage, treatment, or a water body at rate i Q p < n Q p ≤ Q 0 in the inter-arrival time t between two consecutive rain events.
The probability R b of an overflow may be written as a function of four dimensionless groups: k ζ , k λ , k T ,b and k ψ (Table 1).
Pollutants washed-off from urban catchment surfaces and attached to solid sediments, may be removed by sedimentation, provided that the residence time within the SU is sufficiently long (Guo 1996). The operating conditions of detention facilities are generally characterized by intermittent and variable rainfall, leading to variable inflows and storage levels. The effective detention time is the ratio between the rain volume detained in the SU and the release rate. Therefore, the effective detention time may be shorter than design detention tine T d . The risks of insufficient water treatment of SSS with above and below ground SU are: where dimensionless groups are define in Table 1.
Assuming that the SU is empty at the beginning of each rain event, the risks of overflow and insufficient detention time (Equations 8-10 and 11) become: A more detailed treatment of the risk models, including derivation is reported in Supplementary Material.

Risk of failure of SSS
Probability P(θ ) that the SSS does not perform its task in stationary conditions is obtained by combining the risks of overflow, too short detention times within the SU, and PS failure, as: where R(i) is the combined risk of overflow and too short detention time.
Equation (16) can estimate the risk of failure of the SSS whenever rain falls, as a function of the constant parameters:

T d a, n, r].
Equation (16) is a particular application of the "loading resistance interference" concept (Mays and Cullinane 1986) where the incoming rainfall volume is the load and the resistance is the combined system capacity to detain volume and release it to downward drainage, treatment or water body. The interaction of the load and resistance probability density functions resembles the interaction of two stochastic processes: rainfall and SSS performance.
The basic steps in the risk assessment by use of equation (16) are: (1) estimate the probability density function of rainfall depth duration and interevent time (statistics were taken here from Adams and Papa (2000)); (2) derive the distribution of inflow volumes by use of proper hydrological model (inflow probability distribution is derived here according to Ursino and Salandin (2014) by use of rational model (7)); (3) estimate the SU detention volume and PS capacity according to a design manual (e.g. CIRIA (2001)); (4) choose the system set-up (possible sequences of SU and PS are shown in Figure 1); (5) estimate the reliability of one pump (failure and repair rates were taken here from Korving et al. (2006)); (6) define SSS's failure as the risk of overflow, or insufficient water quality control, or a weighted average of the two; (7) apply risk interference concepts (equation (16)). Basic assumptions (Equations 1-6) allowed the analytical solution of both: the risk of overflow and the risk of insufficient water quality control (Equations 8-15). Provided a site-specific statistical characterization of the main random variables, other than the one used here, the risk assessment is possible by numerical integration of equation (16).
Equation (16) may be used to evaluate: (1) the annual risk of SSS failure (not shown here), where N is the average number of rainfall events in one year and 1 − P(θ ) N is the annual reliability of the SSS; (2) the risk of failure of the SSS over its design life M (not shown here): Equation (18) holds for unchanged climate, land use and managing conditions, although it may easily be generalized for non-stationary conditions (θ 1 , θ 2 , ...θ M ), where θ j is the set of climate and design parameters which accounts for the yearly stage of development and the economical and environmental scenarios at year j, in scenarios of changing climate, land use or management strategy (Read andVogel 2015, Salas andObeysekera 2014).
The risk of failure P(θ ) depends on climate (through ζ , λ ψ and IETD), the three hydraulic parameters Q, Q 0 and V s , the detention time T d , the reliability a, and the numbers n and r of the necessary and redundant pumps in the PS. The definition of design parameters of minimal models is complex, site-specific, and should preferably take into account uncertainty and nonstationary states of the urban catchment (not shown here). In the following, the real world complexity is condensed into dimensionless, reasonable effective parameters and Q, Q 0 and V s , per unit catchment area, are expressed as follows:

Application
Constructed ponds, which provide removal of total suspended solids up to 70%, have detention volumes normalized over catchment area V s S −1 = 2 ÷ 20 mm, correspondingly, C 1 = 0.4 ÷ 4 (Papa et al. 1997). Similarly, underground stormwater detention systems, designed to detain and treat, for instance, the first 5 ÷ 25 mm of rainfall have The inflow rate may be up to one order of magnitude higher than the outflow rate, in environments which undergo rapid changes in use, so that restoration or maintenance of pre-development conditions implies C 2 = 1 ÷ 10 C 3 .
Derivation of rain events from continuum records with a separation time equal to design detention time T d , may tend to overestimate detention volumes (Guo 1996), thus, IETD = 0.5 T d = 12 hours. Climatic parameters for Canada and the United states are ζ = 0.08 ÷ 0.3 mm −1 , λ = 0.06 ÷ 0.15 hr −1 and ψ = 0.008 ÷ 0.017 hr −1 (Adams and Papa 2000, Zhang and Guo 2014, Zhang and Guo 2013. In the hypothetical examples discussed in this section ζ = 0.2 mm −1 ; λ = 0.1 hr −1 ; ψ = 0.01 hr −1 . The analysis of 5 years failure records of two sewer PS in the Netherlands showed that: (1) the failure rate is erratic with interarrival times between 8 and 100 days; (2) clusters of dependent failures may occur; (3) the average repair time is between 1 hour and several days, emphasizing the importance of proper maintenance and management policies (Korving et al. 2006). Pump reliability may be therefore a = 0.99 ÷ 0.5. Figure (2) shows the combined risk of failure of SSS with above-ground (continuous line) and below-ground SU (dashed line) for various PS set-ups (different n and r) and reliability a of the pumps. In the case of above-ground tanks, the risk of failure of the SSS, P(θ ) (equation 16) is evaluated for R(i) = 0.5 · (R a + R T d ,a ) (Equations 8 and 10), whereas for below-ground SU it is R(i) = 0.5 · (R b + R T d ,b ) (Equations 9 and 11). In a real case, R(i) may be defined according to the design storm water management standard, whether it is pollutant removal, detention or both. Thus different weights may be given to the risk of overflow and to the risk of insufficient water quality control. P(θ ) increases with C 1 up to a certain maximum and then decreases. This is particularly evident in SSS with below ground SC. Partitioning of total PS capacity into a larger number of components n without redundancy does not substantially affect the risk of failure. Below-ground tanks are more reliable than above-ground ones over a restricted range of relatively small C 1 , and both are less prone to failure as the number r of redundant pumps increases. The setup influences the load density function, and thus the load-resistance interference, because below ground SU can store small rain event avoiding overflow even though the PS is on failure. The risk of SSS failure with back-up capacity (r > 0) is more sensitive to changes in a, and thus to changes in management strategy. Figure 3(a) shows the combined risk of overflow and insufficient detention for various storage volumes (C 1 ) and outflow rates (C 3 ).
Large outflow rates lessen the risk of overflow by reducing tank draw-down time and increase that of too short detention. Figure 3(a) shows that the risk of failure of SSS with aboveground SU decreases with increasing C 3 when C 1 is below a certain threshold, whereas it increases with increasing C 3 when C 1 is above that threshold. Extended SUs have a very high probability of not achieving the established water quality objective, and the low risk of overflow does not counterbalance the loss of reliability regarding water quality. The value of threshold C 1 corresponding to switching between 'small' and 'extended' SUs, in this case study is C 1 ≈ 2. Figure 3(b) shows the risk of failure P(θ ) of SSS with aboveground SU, evaluated for R(i) = 0.5·(R 1 a +R 2a +R Td,a ) (Equations 12 and 14), whereas for below-ground tanks it is: R(i) = 0.5 · (R b + R Td,b ) (Equations 13 and 15). Thus, Figure 3(b) shows the risk of SSS failure evaluated under the assumption that the tank is empty at the onset of each flow event. The risk of overflow is underestimated and the risk of insufficient detention time is overestimated. In the case study presented here, this model-ing assumption is conservative only for SSS with 'small' belowground SU and above-ground SU.

Discussion
Conventional stormwater models (Elliott and Trowsdale 2007) do not take into due consideration the interplay of human and technical systems, even though technical socio-economic and ecological issues need consideration and integration (Rauch et al. 2005). A new probability model links risk assessment to complexity, non-stationarity and socio-hydrology.
In previous risk models uncertainty is restricted to climate parameters (Guo 2001, Guo and Baetz 2007, Guo et al. 2012, Zhang and Guo 2013, Zhang and Guo 2014, or to stormwater effluent concentration and BMP removal rate (Park et al. 2015, Park andRoesner 2012). Risk models of PS components (Hahn et al. 2006, Jin and Mukherjee 2010, Ermolin et al. 2002 do not take into due account the stochastic nature of run-off flow, neither the hydraulic functioning of SU. Combining socio-economic aspects affecting the risk of failure of PS and technical issues concerning the BMP performance represents a first step toward the quantitative evaluation of sociohydrological systems, which is a new topic, currently under debate (Montanari 2015, Sivapalan 2015. This step is taken here within the framework of reliability interference theory. Systematic SSS's failure records could be used to validate risk models and give advice to the involved parties. Thus systematic data collection and circulation among meteorological station holders, infrastructure managers, socio and demographic observatories and engineers which are aware of uncertainty in problem solving should become part of standard operation (Hlavinek et al. 2009).
Further study of the two-way interaction between hydrology and water management may advance the comprehension of feedback and non stationary dynamics of the co-evolution of land and society. The need to evaluate positive and negative feedback between societal and environmental changes and determine the appropriate complexity for models of sociohydrological systems has been repeatedly emphasized in recent times (Montanari 2015, Montanari et al. 2013, Sivapalan et al. 2012.