# Applying the Theory of Reliability to the Assessment of Hazard, Risk and Safety in a Hydrologic System: A Case Study in the Upper Sola River Catchment, Poland

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

_{af}, given as the mean flow of the largest observed floods, then a time series analysis will show periods during which the hydrological system functions reliably (τ

_{i}), moments at which failure occurred (t

_{i}), periods during which failure endures (τ

_{wi}), and periods of system renewal (ϑ

_{i}) during which the system returns to full fitness after the period of failure (end of flooding) (Figure 1).

- $t$ is time,
- ${t}_{i}$ the moment failure occurs,
- $P(.)$ the probability of non-failure occurrence up to time t
_{i}, and - $\tau $ the duration of object function without failure i.

_{i}is as follows:

- $F\left(t\right)$ is a cumulative distribution function, otherwise known as the object failure function, and $R\left(t\right)+F\left(t\right)=1,\text{}R\left(t\right)=1-F\left(t\right)\text{}\mathrm{or}\text{}F\left(t\right)=1-R\left(t\right)$.

#### 2.1. The Reliability Structure of a Hydrologic System (Reliability Block Diagram)

#### 2.2. Mixed Structure of a Hydrologic System

- i is the element number, where i = 1, river valley retention; i = 2, levee; and i = 3, polder (Figure 2),
- ${F}_{i}\left(t\right)$ is the failure function of the i
^{th}element, and - ${R}_{i}\left(t\right)$ is the reliability function of the i
^{th}element.

#### 2.3. The Fault Tree Method (FTM) in the Reliability Analysis of Hydrological Systems

_{i}) of undesirable events A

_{i}found in FTM are dependent on the time (t) of failure occurrence (e.g., of the undesirable event) and are equal to:

- ${R}_{l}\left(t\right)$ is the reliability function of reliability of the l
^{th}element of the object, in which the l^{th}failure, ${F}_{l}\left(t\right)$ appeared at time t.

_{i}), the value of the reliability function R

_{l}(t) of the l

^{th}element of the hydrological system, in which failure F

_{l}(t) occurred as a result of undesirable events occurring at time t (Equation (9)), the reliability or failure of the entire hydrological system can be calculated.

#### 2.4. Definition of Risk

#### 2.5. Qualitative and Quantitative Assessment of Risk

#### 2.6. Relationship between Measures of Hydrological Risk and Measures of Reliability (Failure)—Threat and Losses

_{HR}, failure F (reliability R)—threat and losses T, can be expressed mathematically as:

- ${M}_{HR}\left(l,t\right)$ is a measure of risk of emergence of losses l in the hydrological system at time t of the system’s functioning,
- F(t) is the system failure function—probability of occurrence of event A at time t,
- $R\left(t\right)$ is the system reliability function—probability of non-occurrence of event A at time t,
- T(l) is the probability that the occurrence of an undesirable event A causes losses greater or equal to l, $T\left(l\right)=P\left[L\left(t\right)\ge l\setminus A\right]$,
- $P\left(A\right)$ is the probability of occurrence of the undesirable event, A, viewed as a measure of failure F(t) (or reliability R(t)) of the hydrological system,
- $P\left[L\left(t\right)\ge l\setminus A\right]$ is the probability of occurrence of losses greater or equal to l under the condition that undesirable event A occurred. It serves as the measure of threat of the occurrence of losses in the hydrological system resulting from its malfunctioning.

#### 2.7. Identification of Risk

- m is the number of secondary events sequences,
- n is the number of secondary events in the j
^{t}^{h}sequence, - ${p}^{\left(kj\right)}$ is the probability of occurrence of the j
^{th}sequence caused by the occurrence of the k^{th}undesirable event, - ${p}_{i}^{\left(kj\right)}$ is the probability of occurrence of the i
^{th}secondary event in the j^{th}sequence caused by the k^{th}undesirable event - $R{L}^{\left(kj\right)}$ is the risk level associated with the j
^{th}sequence of secondary events caused by occurrence of the k^{th}undesirable event.

^{(2)}of sequence no. 2 initiated by initial event IE, as well as probabilities of every secondary event, p

_{1}, p

_{2}and p

_{3}, would be estimated by experts or statistical methods (see Section 4).

## 3. Methodology

- i
- Identification of threat formation mechanisms, i.e., threats in the form of a surplus (flooding) of surface water in three environments: Canals of rivers, natural and artificial (man-made) reservoirs and periodically on the catchment area. In the example of the Sola River catchment at the Zywiec post (Section 4), the maximum flows from winter and summer seasons causing floods were identified as threat in Zywiec town and its surroundings.
- ii
- Determination of defense mechanisms appropriate for specific types of threats, i.e., taking into account technical and non-technical activities, what should be done? For instance, in the case of a huge flood, technical flood protection infrastructure can be installed for flood reduction. This include retention reservoirs with constant flood reserve, dry reservoirs and polders with locks and spillways as well as objects preventing flooding outside the intended area, i.e., levees, dry reservoirs and polders without locks, channels of relief and maintenance and adjustment of the riverbed capacity. Non-technical flood protection activities can include hydrological education about extreme events, information on the occurrence of risks of flood, appropriate land-use planning, insurance as well as legal and institutional systems.
- iii
- Identification of the reliability structure of a hydrological system, i.e., creating a reliability block diagram (see Figure 2) on the basis of existing objects in the studied system. The measures of the reliability analysis can be evaluated: Reliability function R(t) (Equations (1)–(8), (21) and (26)) and failure function F(t) (Equations (1)–(8)), function of failure intensity λ(t) (Equation (25)), function of cumulative intensity of failures Λ(t) (Equation (29)) and the expected value of time the system functions without failure ET (Equations (27) and (28)). The reliable structure of the hydrologic system can be described by the FTM as a probabilistic model of the system (see Figure 3 and Equations (9) and (10)).
- iv
- Qualitative risk assessment and prioritizing risk levels of identified threats, using ETM as a probabilistic model of risk occurrence in the hydrologic system (Section 2).
- v
- Quantitative risk assessment by evaluation probability of threat and consequences of their occurrence, using ETM (see Figure 4 and Equations (13)–(20), (43) and (44)).
- vi
- Evaluation of hydrologic risk measures of the entire system and its particular elements of flood protection infrastructure. The hydrologic risk measure M
_{HR}(Equations (11), (12), (22), (24) and (47)) and Safety Guarantee Indicator (SGI) (Equation (45)) and Flood Risk Indicator (FRI) (Equation (46)) should be estimated. - vii
- Evaluation of risk of losses in the hydrologic system using FTM and ETM probabilistic models (Equations (30)–(33) and (40)–(44)) as well as the Ranking Method (see Chapter 4.3. and Equations (34)–(39)) and Risk Matrix (see Chapter 4.4. Qualitative Method of Risk Assessment).
- viii
- Finally, the risk of hydrological extreme events can be managed.

## 4. Results and Discussion

#### 4.1. Place and Data for Case Study

^{2}of flysch—thick layers of sandstone partitioned by layers of slate. The river network is relatively dense. The highest point in the catchment is Pilsko (1557 m a.m.s.l.), and the lowest Zywiec (342 m a.m.s.l.), with an overall mean altitude of 683 m a.m.s.l. Land use in the Sola River catchment includes forest (55% by area), arable farming and grassland (44%), and urban areas (1%).

#### 4.2. General Measures of Reliability, Failure, Risk and Safety

- n(t) is the number of events that cause system failure, and
- N is the total number of events of the phenomenon affecting the system.

_{HR}, of maximal floods can be approximately given as:

- n(l, t) is the number of undesirable events that occurred at time t of system function and caused losses greater or equal to l,
- N is the number of all events occurring at time t.

_{b}on the system’s R(t) and F(t) and risk, ${M}_{HR}$, caused by maximal floods, the same calculation for Q

_{b}= Q

_{af}= Q

_{max50%}= 285 m

^{3}s

^{−1}was performed, for which the number of biggest undesirable events that may cause losses was equal to n(l,t) = 27, and R(t) = 0.53 and F(t) = 0.47 and ${M}_{HR}$ = 0.47. Obviously, with a reduction in the threshold Q

_{b}, the risk of losses as a result of extreme flood occurrence increases, and the Sola River catchment’s reliability of operation decreases. The adoption of a given threshold level, under which phenomena will be judged as undesirable events, has a significant impact on the assessment of the hydrological system’s failure, and consequently on its safety.

_{5}—fatalities as a result of undesirable event A. Should the levee fail or overflow, the n

_{l}

_{5}= 43 would be the number of potential deaths. The number n

_{A}of all occurrences of event A is equal to the number of flood appearances, where peak flow is greater than the design flood ${Q}_{d}={Q}_{max}^{1\%}=1243{\mathrm{m}}^{3}{\mathrm{s}}^{-1}$, for which the levee was designed and built. Thus far, over the period of 1956–2012, only one flood with ${Q}_{p}^{56-12}=1250{\mathrm{m}}^{3}{\mathrm{s}}^{-1}$ was greater than the design flood, Q

_{d}, then n

_{A}= 1. Accordingly, the probability that casualties, L, is greater than or equal to l in category l

_{5}is:

- n(t, t + Δt) is the number of events causing system failure within the time interval from t to t + Δt,
- Δt is the width of the time interval,
- N(t) is the number of events impacting the system at time t minus the number of events causing system failure within the time interval from t to t + Δt, i.e., the number of events impacting the system at the beginning of each period Δt [N(t = 0) = N],
- N is the total number of events affecting the system throughout the period of its operation.

^{−1}. Accordingly, the expected time of the system operating without failure for the Sola River watershed at Zywiec, is $ET=\frac{1}{\Lambda \left(t\right)}=2\mathrm{years}4\mathrm{months}$.

#### 4.3. Fault Tree and Event Tree Methods for Quantitative Estimation Threat and Risk

_{11}—river bed heavily developed, A

_{12}—bad river regulation, A

_{13}—too small a spacing of levee. The latter caused undesirable event A

_{1}—lack of river valley retention, as well as basic events A

_{21}—catastrophic flood, A

_{22}—bad technical condition of levee and A

_{23}—polder misuse. All these together were the cause of occurrence of undesirable event A

_{2}—insufficient security of protected land when exposed to flooding.

_{1}and A

_{2}(Figure 3), which together led to the occurrence of top event A (i.e., flooding in the town of Zywiec) call for the calculation of the probability of occurrence in the case of a serially structured system:

- $P\left({A}_{i}^{ss}\right)$ is the probability of event A
_{i}at the entrance to a logical sum that describes the serial structure of system in terms of reliability.

- $P\left({A}_{i}^{ps}\right)$ is the probability of event A
_{i}at the entrance to a logical product that describes the parallel structure of the system in terms of reliability.

_{11}, A

_{12}and A

_{13}as well as A

_{21}, A

_{22}and A

_{23}, were ranked by a group of five experts using the Ranking Method. Based on measured data and using statistical methods, the head of the team of five experts estimated the probability of events A

_{12}and A

_{21}as p(A

_{12}) = 0.03 and p(A

_{21}) = 0.001. These events were denoted B

_{1}and B

_{2}and used to calibrate the equation:

- a
_{0}and a_{1}are calibration coefficients, - $p\left({A}_{i,j}\right)$ is the probability of undesirable basic event A
_{ij}, where $p\left({A}_{12}\right)\equiv p\left({B}_{1}\right)\mathrm{and}p\left({A}_{21}\right)\equiv p\left({B}_{2}\right)$ (Table 3), - $\overline{pos}$is the mean position of the event in the ranking (Table 3).

_{0}and a

_{1}in this case, we know from Table 3 that $p\left({A}_{12}\right)\equiv p\left({B}_{1}\right)=0.03$, $p\left({A}_{21}\right)\equiv p\left({B}_{2}\right)=0.001$, ${\overline{pos}}_{{B}_{1}\equiv {A}_{12}}=1.6$, and$\text{}{\overline{pos}}_{{B}_{2}\equiv {A}_{21}}=5.0$. Accordingly, we can write:

_{11}, A

_{13}, A

_{22}and A

_{23}, respectively p(A

_{11}) = 0,011, p(A

_{13}) = 0.0027, p(A

_{22}) = 0.0012 and p(A

_{23}) = 0.005 as well as the known values of probabilities p(A

_{12}) = 0.03 and p(A

_{21}) = 0.001.

_{j}(t), can be estimated from the estimated probabilities p(A

_{i}), as can the failure function, F

_{j}(t):

#### 4.4. Qualitative Method of Risk Assessment

^{4}–10

^{5}Euros. One should keep in mind that when determining weights of scale of the probability category (Table 4) in the process of a quantitative assessment of probability (frequency) of undesirable event occurrence, the average return period T must be considered and associated with the probability of exceedance p, i.e., the probability of achieving or exceeding the value of p in each year the hydrological system functions.

## 5. Summary and General Conclusions

- (i)
- Floods of low probability, or extreme event scenarios; representing, as per [46], areas where the probability of flooding is low and is ≤0.2% (return period T = 500 years), or there is a non-zero probability of occurrence of extreme events in the area;
- (ii)
- floods with a medium probability (likely return period≥100 years); areas where the probability of floods is average and is 1% (T = 100 years);
- (iii)
- floods with a high probability, where appropriate; areas where the probability of flooding is high and is 10% (T = 10 years).

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Hydrograph with periods of reliable hydrological system operation (τ

_{i}), moments of failure occurrence (t

_{i}), duration of failure (τ

_{wi}) tied to the duration of system renewal (ϑ

_{i}). Q

_{bf}is the boundary flow of the river flood, often accepted as the minimal flow of annual maximum floods over the observation period, Q

_{af}—maximum flow not causing flood losses, often accepted as the mean flow of annual maximum floods for the observation period or Q

_{max,T=2 year return period}(Q

_{max,p=50%}).

**Figure 2.**Reliability structure (reliability block diagram) of the Sola River catchment’s hydrological system up to the Zywiec post. This structure reflects system function and element reliability and does not represent the actual location of system elements.

**Figure 3.**Fault tree for the structure of the hydrological system of the Sola River catchment up to the Zywiec post.

**Figure 4.**Example of event tree for part of Sola River basin system in Zywiec. (•—lack of disaster, ✕—disaster).

**Table 1.**Values of peak flow of Maximum Credible Flood, maximum flow over the monitoring period (1956–2012), mean and minimum flow of annual maximum floods, and maximum floods with a T-year return period in years for the Zywiec post on the Sola River.

Characteristic of Flood | Discharge (m^{3}s^{−1}) |
---|---|

Peak flow of Maximum Credible Flood (${Q}_{p}^{MCF}$) | 1833 |

Maximum flow over the observation period of 1956–2012 (${Q}_{p}^{56-12}$) * | 1250 |

Mean flow of annual maximum floods 1956–2012 (${Q}_{\overline{AMF}}$) | 354.5 |

Minimal flow of annual maximum floods 1956–2012 (${Q}_{AMF}{}_{min}$) | 92.6 |

${Q}_{max}^{T=2}={Q}_{max}^{50\%}$ | 285 |

${Q}_{max}^{T=10}={Q}_{max}^{10\%}$ | 679 |

${Q}_{max}^{T=20}={Q}_{max}^{5\%}$ | 850 |

${Q}_{max}^{T=100}={Q}_{max}^{1\%}$ | 1243 |

${Q}_{max}^{T=200}={Q}_{max}^{0.5\%}$ | 1410 |

${Q}_{max}^{T=500}={Q}_{max}^{0.2\%}$ | 1631 |

${Q}_{max}^{T=1000}={Q}_{max}^{0.1\%}$ | 1797 |

${Q}_{max}^{T=10000}={Q}_{max}^{0.01\%}$ | 2346 |

**Table 2.**Values of failure intensity, λ(t), and reliability function R(t) based on $\overline{\lambda}$ and λ(t) for consecutive decades of the observation period 1956–2012 for the Sola River catchment at Zywiec station.

Parameter | Span of Years | |||||
---|---|---|---|---|---|---|

1956–1965 | 1966–1975 | 1976–1985 | 1986–1995 | 1996–2005 | 2006–2012 | |

λ(t) based on Equation (25) | ||||||

i | 0 | 1 | 2 | 3 | 4 | 5 |

t + Δt | 0 + 10 | 10 + 10 | 20 + 10 | 30 + 10 | 40 + 10 | 50 + 7 |

n(t, t + Δt) | 4 | 4 | 4 | 2 | 5 | 2 |

N(t) * | 57 | 53 | 49 | 45 | 43 | 38 |

λ(t) | 7.0 × 10^{−3} | 7.5 × 10^{−3} | 8.2 × 10^{−3} | 4.4 × 10^{−3} | 11.6 × 10^{−3} | 7.5 × 10^{−3} |

$R\left(t\right)\text{}\mathrm{based}\text{}\mathrm{on}\text{}\overline{\lambda}$ | ||||||

$\overline{\lambda}$ | 7.7 × 10^{−3} | 7.7 × 10^{−3} | 7.7 × 10^{−3} | 7.7 × 10^{−3} | 7.7 × 10^{−3} | 7.7 × 10^{−3} |

R(t) ** | 0.93 | 0.86 | 0.79 | 0.73 | 0.68 | 0.64 |

$R\left(t\right)\text{}\mathrm{based}\text{}\mathrm{on}\text{}\lambda \left(t\right)$ | ||||||

R(t) | 0.932 | 0.928 | 0.921 | 0.957 | 0.890 | 0.949 |

Expert No. | Position (Rank) of Undesirable Event | |||||
---|---|---|---|---|---|---|

A_{11} | A_{13} | A_{22} | A_{23} | B_{1} = A_{12} | B_{2} = A_{21} | |

1 | 4 | 3 | 5 | 2 | 1 | 6 |

2 | 3 | 4 | 4 | 3 | 2 | 4 |

3 | 1 | 5 | 6 | 5 | 2 | 6 |

4 | 3 | 3 | 5 | 4 | 2 | 4 |

5 | 2 | 5 | 4 | 3 | 1 | 5 |

Sum of ranks | 13 | 20 | 24 | 17 | 8 | 25 |

Mean ranking (Σ ranks ÷ 5), $\overline{pos}$ | 2.6 | 4.0 | 4.8 | 3.4 | 1.6 | 5.0 |

$p\left({A}_{i,j}\right)$ | 0.011 | 0.0027 | 0.0012 | 0.005 | 0.03 | 0.001 |

Qualitative Evaluation of Probability | Quantitative Evaluation of Probability (Frequency of Event Occurrence) | Weight | |
---|---|---|---|

Frequent | F | T = more often than once a year | 5 |

Very Likely | VL | T = once a period from 1 year to 10 years, $p\in \left(1,0.1\right]$ | 4 |

Probable | P | T = once a period from 10 to 20 years, $p\in \left(0.1,0.05\right]$ | 3 |

Unlikely | U | T = once a period from 20 to 100 years,$p\in \left(0.05,0.01\right]$ | 2 |

Nearly Impossible | NI | T = once a period from 100 to 1000 years, $p<0.01$ | 1 |

Category of Losses and Damages | Consequences | Weight | |
---|---|---|---|

Disastrous | D | Collective losses including fatalities, very big economic losses above 10^{6} Euro | 5 |

High | H | Individual losses including fatalities or severe collective losses excluding fatalities, big economic losses in range 10^{5}–10^{6} Euro | 4 |

Significant | S | Heavy individual losses or light collective, economic losses in range 10^{4}–10^{5} Euro | 3 |

Small | SM | Light individual losses and no collective losses, minor economic losses in range 10^{3}–10^{4} Euro | 2 |

Negligible | N | Lack of casualties, economic losses below 1000 Euro | 1 |

Category of Probability | Category of Losses | ||||
---|---|---|---|---|---|

D = 5 | H = 4 | S = 3 | SM = 2 | N = 1 | |

Risk Levels | |||||

F = 5 | F∙D = 25, UR | F∙H = 20, UR | F∙S = 15, UTR | F∙SM = 10, CTR | F∙N = 5, CTR |

VL = 4 | VL∙D = 20, UR | VL∙H = 16, UTR | VL∙S = 12, CTR | VL∙SM = 8, CTR | VL∙N = 4, AR |

P = 3 | P∙D = 15, UTR | P∙H = 12, UTR | P∙S = 9, CTR | P∙SM = 6, CRT | P∙N = 3, AR |

U = 2 | U∙D = 10, CTR | U∙H = 8, CTR | U∙S = 6, CTR | U∙SM = 4, AR | U∙N = 2, AR |

NI = 1 | NI∙D = 5, CTR | NI∙H = 4, AR | NI∙S = 3, AR | NI∙SM = 2, AR | NI∙N = 1, AR |

Risk Levels | Scale of Risk Levels | Interpretation |
---|---|---|

UR | 20–25 | Inacceptable Risk |

UTR | 15–19 | Uncontrolled Tolerable Risk, tolerated only if risk reduction is difficult to achieve or costs are disproportionately high in relation to benefits, i.e., to potentially obtain an improvement of safety. |

CTR | 5–14 | Controlled Tolerable Risk, tolerated only when costs of its reduction are adequate for the established level of safety. |

AR | 1–4 | Acceptable Risk |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ozga-Zielinski, B.; Adamowski, J.; Ciupak, M.
Applying the Theory of Reliability to the Assessment of Hazard, Risk and Safety in a Hydrologic System: A Case Study in the Upper Sola River Catchment, Poland. *Water* **2018**, *10*, 723.
https://doi.org/10.3390/w10060723

**AMA Style**

Ozga-Zielinski B, Adamowski J, Ciupak M.
Applying the Theory of Reliability to the Assessment of Hazard, Risk and Safety in a Hydrologic System: A Case Study in the Upper Sola River Catchment, Poland. *Water*. 2018; 10(6):723.
https://doi.org/10.3390/w10060723

**Chicago/Turabian Style**

Ozga-Zielinski, Bogdan, Jan Adamowski, and Maurycy Ciupak.
2018. "Applying the Theory of Reliability to the Assessment of Hazard, Risk and Safety in a Hydrologic System: A Case Study in the Upper Sola River Catchment, Poland" *Water* 10, no. 6: 723.
https://doi.org/10.3390/w10060723